PREFACE - University of California, San...

PREFACE

i

ii PREFACE

Effects originating from light-matter coupling have stimulated the development ofoptics for the last three centuries. Nowadays, the limits ofclassical optics can be reachedin a number of solid state systems and quantum optics has become an important tool forunderstanding and interpreting modern optical experiments. Rapid progress of crystalgrowth technology in the XXth century allows the realization of crystal microstructureswhich have unusual and extremely interesting optical properties. This book addressesthe large variety of optical phenomena taking place in confined solid state structures:microcavities. Microcavities serve as building blocks for many opto-electronic devicessuch as light-emitting diodes and lasers. At the edge of research, the microcavity repre-sents a unique laboratory for quantum optics and photonics.The central object of studiesin this laboratory is theexciton-polariton: a half-light, half-matter quasiparticle exhibit-ing very specific properties and playing a key role in a numberof beautiful effectsincluding parametric scattering, Bose-Einstein condensation, superfluidity, superradi-ance, entanglement, etc. At present, hundreds of research groups in the world work onfabrication, optical spectroscopy, theory and applications of microcavities. The progressin this interdisciplinary field at the interface between optics and solid state physics isextremely rapid. We expect appearance of a new generation ofopto-electronic devicesbased on microcavities in 2010s.

Both rich fundamental physics of microcavities and their intriguing potential appli-cations are addressed in this book oriented to undergraduate and postgraduate studentsas well as to physicists and engineers. We describe the essential steps of developmentof physics of microcavities in their chronological order. We show how different types ofstructures combining optical and electronic confinement have come into play and wereused to realize first weak and later strong light-matter coupling regimes. We speak aboutphotonic crystals, microspheres, pillars and other types of artificial optical cavities withembedded semiconductor quantum wells, wires and dots. We present the most strikingexperimental findings of the recent two decades in the opticsof semiconductor quantumstructures.

The first Chapter of this book contains an overview of microcavities. We present thevariety of semiconductor, metallic and dielectric structures used to make microcavitiesof different dimensions and briefly present a few characteristic optical effects observedin microcavities.

The next two Chapters (2 and 3) are devoted to the fundamentalprinciples of opticsessential for understanding optical phenomena in microcavities. We provide overviewsof both classical and quantum theory of light, discuss the coherence of light, its polari-sation, statistics of photons and other quantum characteristics. The reader will find herethe basic principles of the transfer matrix technique allowing for easy understandingof linear optical properties of multilayer structures as well as the basics of the secondquantization method. We consider planar, cylindrical and spherical optical cavities, in-troduce the whispering gallery modes and Mie resonances.

In Chapters 4 and 5 we give the theoretical background for themost important light-matter coupling effects in microcavities considered from the point of view of classical(Chapter 4) and quantum (Chapter 5) optics. We formulate thesemi-classical non-localdielectric response theory and study the dispersion of exciton-polaritons in microcavi-

PREFACE iii

ties. As an important toy-model we consider a single excitonstate coupled to a singlelight mode and study many variations of it. We describe important nonlinear effectsknown in atomic cavities (such as the Mollow triplet).

In Chapter 6 we discuss the physics ofweak-coupling, when interaction of the light-field with the exciton acts as a perturbation on its state and energy. We discuss thePurcell effect which symbolises this regime and lasing as its most important application.We also describe nonlinear effects such as bistability.

Chapter 7 addresses the resonant optical effects in microcavities in thestrong cou-pling regime. We overview the most spectacular experimental discoveries in this areaand present the quasimode model of parametric amplificationof light. We also discussthe quantum properties of optical parametric oscillators based on microcavities.

Chapters 8 and 9 discuss the future of microcavities. Chapter 8 is devoted to theBose-Einstein condensation of exciton polaritons and polariton lasing. At the moment ofwriting, polariton lasers remain more a theoretical concept than commercial devices butwe believe that in a few years they will become a reality. Thus, for the first time wouldBose condensation be observed at room temperature and used for creation of a newgeneration of opto-electronic devices. The path toward this breathtaking perspectiveand most serious obstacles on this way which are not yet overcome, are tackled in thisChapter.

The subject of Chapter 9 is “spin-optronics”: a new subfield of solid state opticswhich emerged very recently due to the discoveries made in microcavities and otherquantum confined semiconductor structures. How to manipulate the polarisation of lighton a nanosecond and micrometre scale? What would be the polarisation properties ofpolariton lasers and which mechanisms govern spin-relaxation of exciton-polaritons?These questions are treated in this chapter.

The Glossary of Microcavities is addressed to a non-specialist who is searching forthe qualitative understanding of the physics of microcavities or to any reader who hasno time to go through the entire book but needs a simple and concise answer to oneof specific questions related to microcavities. In the Glossary a number of importantrelevant issues are treated without any equations on a simple and accessible level for thegeneral reader. We pay special attention to explanation of terms frequently used in thisfield of physics, for example, “exciton-polariton”, “Rabi-splitting”, “strong coupling”,“Bragg mirror”, “VCSEL”, “photonic crystal”, etc. . .

The book is intended as a working manual for advanced or graduate students andbeginning researchers in the field. It is written to a high standard of scientific and mathe-matical accuracy, but to allow an agreeable reading throughthe essential points unham-pered by details, many sophistications or difficulties, as well as side issues or extensions,have been relegated to footnotes. These would be most profitably considered in a secondreading.

Exercises are sprinkled throughout the text and are an important part of it. Theyshould be read as a minimum for otherwise the notions they introduce will be missingfor later development of the regular text. Starred exercises are straightforward or sys-tematic, those doubled starred are conceptually challenging or require involved compu-tations, those tripled starred are difficult and almost qualify as research problems. We

iv PREFACE

use the international system of units, while in numerical examples the energies will begiven in electron-Volts and the distances will be given in microns or nano-meters. Anextensive bibliography used throughout the text appears atthe end of the text in Harvardformat (identified by first author and date of publication).

Microcavities represent a young and rapidly developing field of physics. Our bookcover the state of the art in this field in the first half of 2006 observed from the prism ofpersonal experience of four authors who actively worked in physics of microcavities alarge part of their scientific lives. We wanted to give a personal touch to this book andwe do not claim to be objective, which at this stage of the fieldis very difficult. We shallbe very grateful to any feedback, comments and critical remarks from our Readers!

November 2006,The authors.

ACKNOWLEDGEMENTS

This book owes much to our collaboration with active researchers in the field. It isour pleasure to express our gratitude to Yuri Rubo, Ivan Shelykh, Kirill Kavokin, Pav-los Lagoudakis, Mikhail Glazov, Maurice Skolnick, David Whittaker, Daniele Sanvitto,Pavlos Savvidis, David Lidzey, Elena del Valle and Carlos Tejedor.

v

Alexey KavokinTo Sofia Kavokina,

Jeremy BaumbergTo Melisia Murry,

Guillaume MalpuechTo Anne Tournadre,

Fabrice LaussyTo my father,Raymond Laussy.

(1953–2006)

CONTENTS

1 Overview of Microcavities 11.1 Properties of microcavities 2

1.1.1 Q-factor and finesse 21.1.2 Intracavity field enhancement and field distribution 31.1.3 Tuneability and mode separation 31.1.4 Angular mode pattern 41.1.5 Low threshold lasing 41.1.6 Purcell factor and lifetimes 51.1.7 Strong vs. Weak coupling 5

1.2 Microcavity realizations 51.3 Planar microcavities 6

1.3.1 Metal microcavities 81.3.2 Dielectric Bragg mirrors 9

1.4 Spherical mirror microcavities 101.5 Pillar microcavities 121.6 Whispering gallery modes 14

1.6.1 Two-dimensional whispering galleries 151.6.2 Three-dimensional whispering galleries 17

1.7 Photonic crystal cavities 191.7.1 Random lasers 20

1.8 Plasmonic cavities 201.9 Microcavity lasers 211.10 Conclusion 21

2 Classical description of light 222.1 Free space 23

2.1.1 Light field dynamics in free space 232.2 Propagation in crystals 26

2.2.1 Plane waves in bulk crystals 262.2.2 Absorption of light 302.2.3 Kramers-Kronig relations 31

2.3 Coherence 312.3.1 Statistical properties of light 312.3.2 Spatial and temporal coherence 322.3.3 Wiener-Khinchin theorem 372.3.4 Hanbury Brown–Twiss effect 40

2.4 Polarisation-dependent optical effects 422.4.1 Birefringence 422.4.2 Magneto-optical effects 43

2.5 Propagation of light in multilayer planar structures 44

vii

viii CONTENTS

2.6 Photonic eigenmodes of planar systems 482.6.1 Photonic bands of 1D periodic structures 51

2.7 Planar microcavities 582.8 Stripes, pillars, and spheres: photonic wires and dots 64

2.8.1 Cylinders and pillar cavities: 652.8.2 Spheres: 68

2.9 Further Reading 72

3 Quantum description of light 733.1 Pictures of quantum mechanics 74

3.1.1 Historical background 743.1.2 Schrodinger picture 743.1.3 Antisymmetry of the wavefunction 833.1.4 Symmetry of the wavefunction 843.1.5 Heisenberg picture 863.1.6 Dirac (Interaction) Picture 91

3.2 Other formulations 933.2.1 Density matrix 933.2.2 Second quantization 953.2.3 Quantisation of the light field 96

3.3 Quantum states 973.3.1 Fock states 973.3.2 Coherent states 983.3.3 Glauber–Sudarshan representation 993.3.4 Thermal states 1003.3.5 Mixture states 1023.3.6 Quantum correlations of quantum fields 1033.3.7 Statistics of the field 1073.3.8 Polarisation 110

3.4 Outlook on quantum mechanics for microcavities 1123.5 Further reading 112

4 Semi-classical description of light-matter coupling 1144.1 Light-matter interaction 115

4.1.1 Classical limit 1154.1.2 Einstein coefficients 117

4.2 Optical transitions in semiconductors 1204.3 Excitons in semiconductors 124

4.3.1 Frenkel and Wannier-Mott excitons 1244.3.2 Excitons in confined systems 1284.3.3 Quantum wells 1284.3.4 Quantum wires and dots 132

4.4 Exciton-photon coupling 1344.4.1 Surface polaritons 1374.4.2 Exciton-photon coupling in quantum wells 139

CONTENTS ix

4.4.3 Exciton-photon coupling in quantum wires and dots 1434.4.4 Dispersion of polaritons in planar microcavities 1474.4.5 Motional narrowing of cavity polaritons 1574.4.6 Microcavities with quantum wires or dots 161

5 Quantum description of light-matter coupling in semiconductors 1655.1 Historical background 1665.2 Rabi dynamics 1665.3 Bloch equations 169

5.3.1 Full quantum picture 1725.3.2 Dressed bosons 175

5.4 Lindblad dissipation 1835.5 Jaynes–Cummings model 188

5.5.1 Excitons in quantum dots 1945.5.2 Gaussian toy model 197

5.6 Dicke model 2025.7 Excitons in semiconductors 203

5.7.1 Excitons as bosons 2055.8 Exciton-photon coupling 205

5.8.1 Dispersion of polaritons 2065.8.2 The polariton Hamiltonian 2085.8.3 Coupling in quantum dots 209

6 Weak-coupling microcavities 2116.1 Purcell effect 212

6.1.1 The physics of weak-coupling 2126.1.2 Spontaneous emission 2136.1.3 The case of QDs, 2D excitons and 2D electron-hole pairs 2156.1.4 Fermi’s golden rule 2166.1.5 Dynamics of the Purcell effect 2196.1.6 Case of QDs and QWs 2216.1.7 Experimental realisations 222

6.2 Lasers 2246.2.1 The physics of lasers 2256.2.2 Semiconductors in laser physics 2296.2.3 Vertical-Cavity Surface-Emitting Lasers 2326.2.4 Resonant cavity LEDs 2366.2.5 Quantum theory of the laser 237

6.3 Nonlinear optical properties of weak-coupling microcavities 2426.3.1 Bistability 2436.3.2 Phase matching 245

6.4 Conclusion 245

7 Strong coupling: resonant effects 2467.1 Optical properties background 247

x CONTENTS

7.1.1 Quantum well microcavities 2477.1.2 Variations on a theme 2497.1.3 Motional narrowing 2517.1.4 Polariton emission 251

7.2 Near-resonant-pumped optical nonlinearities 2537.2.1 Pulsed stimulated scattering 2537.2.2 Quasimode theory of parametric amplification 2587.2.3 Microcavity parametric oscillators 260

7.3 Resonant excitation case and parametric amplification 2637.3.1 Semi-classical description 2637.3.2 Stationary solution and threshold 2647.3.3 Theoretical approach: quantum model 2657.3.4 Three-level model 2667.3.5 Threshold 269

7.4 Two-beam experiment 2697.4.1 One beam experiment and spontaneous symmetry breaking 2697.4.2 Dressing of the dispersion induced by polariton condensates 2717.4.3 Bistable behavior 272

8 Strong coupling: polariton Bose condensation 2748.1 Introduction 2758.2 Basic ideas about Bose-Einstein condensation 275

8.2.1 Einstein proposal 2758.2.2 Experimental realization 2778.2.3 Modern definition of Bose-Einstein condensation 278

8.3 Specificities of excitons and polaritons 2798.3.1 Thermodynamic properties of cavity polaritons 2808.3.2 Interacting bosons and Bogoliubov model 2818.3.3 Polariton superfluidity 2848.3.4 Quasi-condensation and local effects 287

8.4 High-power microcavity emission 2898.5 Threshold-less polariton lasing 2928.6 Kinetics of formation of polariton condensates: semi-classical picture 297

8.6.1 Qualitative features 2978.6.2 The semi-classical Boltzmann equation 3008.6.3 Numerical solution of Boltzmann equations, practical aspects 3028.6.4 Effective scattering rates 3028.6.5 Numerical simulations 303

8.7 Kinetics of formation of polariton condensates: quantum picture in theBorn-Markov approximation 306

8.7.1 Density matrix dynamics of the ground state 3078.7.2 Discussion 3118.7.3 Coherence dynamics 312

CONTENTS xi

8.8 Kinetics of formation of polariton condensates: quantum picture beyondthe Born-Markov approximation 314

8.8.1 Two-oscillator toy theory 3148.8.2 Coherence of polariton laser emission 3248.8.3 Numerical simulations 3308.8.4 Order parameter and phase diffusion coefficient 331

8.9 Semiconductor luminescence equations 3348.10 Claims of excitons and polaritons Bose-Einstein condensation 3378.11 Further reading 337

9 Spin and polarisation 3399.1 Spin relaxation of electrons, holes and excitons in semiconductors 3409.2 Microcavities in the presence of magnetic field 3459.3 Resonant Faraday rotation 3469.4 Spin relaxation of exciton-polaritons in microcavities: experiment 3499.5 Spin relaxation of exciton-polaritons in microcavities: theory 3549.6 Optical spin Hall effect 3589.7 Optically induced Faraday rotation 3609.8 Interplay between spin and energy relaxation of exciton-polaritons 3629.9 Polarisation of Bose condensates and polaritons superfluidity 3669.10 Magnetic field effect and superfluidity 3689.11 Finite temperature case 3739.12 Spin dynamics in parametric oscillators 3759.13 Classical nonlinear optics consideration 3759.14 Polarized OPO: quantum model 3779.15 Conclusions 3799.16 Further readings 380

A Linear algebra 387

B Scattering rates of polaritons relaxation 390B.1 Polariton-phonon interaction 390

B.1.1 Interaction with longitudinal optical phonons 391B.1.2 Interaction with acoustic phonons 392

B.2 Polariton electron interaction 393B.3 Polariton-polariton interaction 394

B.3.1 Polariton decay 395B.4 Polariton-structural disorder interaction 396

C Derivation of the Landau criterion of superfluidity and Lan dau formula 397

D Landau quantisation and renormalisation of Rabi splitting 399

References 402

1

OVERVIEW OF MICROCAVITIES

In this chapter we provide an overview of microcavities. We present thevariety of semiconductor, metallic and dielectric structures used to makemicrocavities of different dimensions and briefly present afewcharacteristic optical effects observed in microcavities. Many importanteffects briefly mentioned in this chapter are commented in greater extentin the following chapters.

2 OVERVIEW OF MICROCAVITIES

A microcavity is an optical resonator close to, or below the dimension of the wave-length of light. Micron- and sub-micron sized resonators use two different schemesto confine light. In the first, reflection off a single-interface is used, for instance froma metallic surface, or from total internal reflection at the boundary between two di-electrics. The second scheme is to use microstructures periodically patterned on thescale of the resonant optical wavelength, for instance a planar multilayer Bragg reflectorwith high reflectivity, or a photonic crystal. Since confinement by reflection is requiredin all three spatial directions, combinations of these approaches can be used within thesame microcavity. In this chapter we will explore a number ofthe basic microcavity de-signs, and contrast their strengths and weaknesses. For practical purposes in this book,the discussion of microcavities will be limited to cavitiesin which all dimensions arebelow 100µm.

Fig. 1.1: (a) single interface reflection and (b) interference from multiple interfaces.

1.1 Properties of microcavitiesTo help survey microcavity designs it is helpful to motivatea comparison of differentoptical properties of a microcavity. We assume in this section a microcavity with totalpower reflectivityR, and round trip optical lengthL. The resonant optical modes withina microcavity have characteristic lineshapes, wavelengthspacings and other propertiestht control their use. A longitudinal resonant mode has an integral number of half wave-lengths that fit into the microcavity, while transverse modes have different spatial shape.However in a microcavity this traditional distinction can lose its precision as modes allexist on the same footing.

1.1.1 Q-factor and finesse

The quality-factor(or Q-factor) has the same role in an optical cavity as in anLCRelectrical circuit, in that it parametrises the frequency width of the resonant enhance-ment. It is simply defined as the ratio of a resonant cavity frequency,ωc, to the linewidth(FWHM) of the cavity mode,δωc:

Q =ωc

δωc=

λcδλc

(1.1)

Thefinesseof the cavity is defined as the ratio of free spectral range (the frequencyseparation between successive longitudinal cavity modes)to the linewidth (FWHM) ofa cavity mode (see Fig. 1.2):

PROPERTIES OF MICROCAVITIES 3

F =∆ωc

δωc=

π√R

1 −R(1.2)

TheQ-factor is a measure of the rate at which optical energy decays from within thecavity (from absorption, scattering or leakage through theimperfect mirrors) and whereQ−1 is the fraction of energy lost in a single round trip around the cavity. Equivalentlythe exponentially decaying photon number has a lifetime given byτ = Q/ωc.

Fig. 1.2: (a) longitudinal mode has integral wavelengths along the main cavity axis, giving (b) mode spectrum.

Because the mode frequency separation∆ωc = 2πcL is similar to the cavity mode

frequency in a wavelength-scale microcavity, the finesse and theQ-factor are not verydifferent. This is not the situation for a large cavity in which case theQ-factor becomesmuch greater than the finesse because of the long round trip propagation time. Insteadthe finesse parametrises theresolving poweror spectral resolutionof the cavity.

1.1.2 Intracavity field enhancement and field distribution

The on-resonance optical intensity enhancement is given by

Iintracavity

Iincident≃ 1

1 −R=

F

π√R

(1.3)

assuming the mirror losses dominate the finesse. In a travelling wave cavity this willbe uniformly distributed. However in a standing wave microcavity, this enhancementis found in the form of spatially-localised interefence peaks. Hence it is not alwayssimple to couple an emitter directly to this enhanced optical field. The enhanced opticalfield inside the microcavity can be usefully harnessed for enhanced nonlinear opticalinteractions.

1.1.3 Tuneability and mode separation

The separation of longitudinal modes in a microcavity,∆ωc, is inversely proportionalto the cavity length. However cavities other than confocal cavities have transverse op-tical modes at different frequencies, and these also scale similarly with cavity length.Hence microcavities have many fewer optical modes in each region of the spectrumthan macroscopic cavities. This can mean that specifically tuning the cavity mode to aparticular emission wavelength becomes more important than in large cavities.

Various techniques for spectral tuning of modes have been advanced, however noneis as yet ideal due to the difficulty of modifying the round trip phase byπ without intro-ducing extra loss. The simplest way is to scan the cavity length, however in cases where


this can be altered, it becomes more difficult to then maintain consistently a fixed lengthonce the desired tuning has been reached. For many other monolithic systems, tuning ofthe cavity modes is extremely difficult, and is most advancedfor semiconductor lasers(see section 1.5).

1.1.4 Angular mode pattern

Microcavities are typically small in all three spatial directions, with aspect ratios closerto unity than macroscopic cavities. As a result, the angularmode emission patterns,which are Fourier related to the cavity mode spatial distribution, tend to have a widerangular acceptance. This means that microcavities emit into a large solid angle on reso-nance.

On the other hand, emission from a microcavity is still beamed into particular di-rections. For instance, by embedding an LED active region within a planar microcavity,the light is emitted in a forward directed cone when the electroluminescence is resonantwith the cavity mode.

1.1.5 Low threshold lasing

There are two reasons why microcavities can have lower lasing thresholds: their reducednumber of optical modes, and their reduced gain volume.

In a microcavity, an embedded emitter has a reduced range of optical states intowhich it is likely to emit. In free space it can emit into any solid angle and frequency,but the microcavity acts to structure the optical density ofstates around the emitter. Thisis particularly a strong effect when the emitter has a large linewidth (eg. an electron-holepair in a semiconductor) as the spectral overlap with different cavity modes is reducedby reducing the size of the cavity. Also in a microcavity, theangular acceptance of anyparticular microcavity mode is much larger. Because the lasing threshold occurs at thepoint at which a spontaneously emitted photon returns to theemitter and stimulates thenext photon emission, reducing the number of cavity modes has the effect of reducingthe laser threshold because spontaneously emitted photonsare more likely to return tothe emitter. This effect is contained within aspontaneous emission coupling factor, β,which is defined as the fraction of the total spontaneous emission rate that is emitted intoa specific (laser) mode. This is typically below 10−5 in bulk lasers, but can be> 10%in microcavities. The laser threshold is given by

Pthr =~ω2

c

2Qβ(1.4)

A small optical cavity means that it contains a smaller volume which is pumpedelectrically or optically to provide gain. Because such systems have to be pumped sothat one of the energy levels is brought into inversion, the total energy needed to reachinversion scales with the volume of active material.

As a result of these two effects, microcavities have the smallest known thresholdsof any laser, having now reached the state of a single-photonintracavity field thresholdlevel—the first photon emitted turns the laser on.

MICROCAVITY REALIZATIONS 5

1.1.6 Purcell factor and lifetimes

Embedding an emitter inside a microcavity can lead to additional effects due to thechange in the optical density of states. When the emitter linewidth is smaller than that ofthe cavity mode (δλc), the emitter can be considered to couple to an optical continuumand the emission kinetics is given by Fermi’s golden rule. This describes the emissionlifetime, τ , modified from free space (τ0), in terms of the detuning between the emitter(λe) and cavity:

τ0τ

= FP2

3

|E(r)|2

|Emax|2δλ2

c

δλ2c + 4(λc − λe)2

+ f (1.5)

which is controlled by the Purcell Factor given by

FP =3

4π2

λ3c

n3

Q

V(1.6)

wheren is the refractive index of the cavity,V is the effective volume of the mode, andEmax is the maximum field intensity in the cavity.

The crucial ratioQ/V allows the emitter to emit much faster into the optical field(if both spectral and spatial overlaps are optimised). It also allows decay to suppressed,though competition with non-radiative recombination generally means that this is ac-companied by a decrease in emission efficiency. Typically the ratioQ/V is difficult toenhance arbitratily since smaller cavities often have restrictions in the maximumQ-factor that is possible. The theory of Purcell effect is presented in Chapter 6.

1.1.7 Strong vs. Weak coupling

If a resonant absorber is embedded inside a microcavity, than another new regime ofoptical physics can be reached when the absorption strengthis large and narrow-bandenough. If the total scattering rates of both the cavity photons and the excited absorberare less than the rate at which they couple with each other, new mixed light-modes calledpolaritonswill result. Spectrally tuning the absorber to the cavity resonance leads tomixing of the photon and absorber, resulting in new polariton states at higher and lowerenergies. This effect will be dealt with in detail in Chapters 7–8.

The condition for strong coupling is thus that the light-matter induced splitting be-tween the new polariton modes (known as the Rabi splitting,Ω) is greater than thelinewidths of either cavity photon (γc, controlled by the finesse) or the exciton (γx, con-trolled by the inhomegeneous broadening of the excitons in the sample). On resonance,the new polaritons are half-light, half-matter, with wavefunctions which are superposi-tions:

ψ =1√2ψX ± ψC (1.7)

whereψX , ψC are the exciton and photon wavefunctions, respectively.

1.2 Microcavity realizations

The most common microcavity is theplanar microcavityin which two flat mirrors arebrought into close proximity so that only a few wavelengths of light can fit in between


Fig. 1.3: Strong coupling of excitons inside a QW microcavity, with repulsion of initial cavity photon (C)and exciton (X) states producing upper (UP ) and lower (LP ) polaritons.

them. To confine light laterally within these layers, a curved mirror or lens can be in-corporated to focus the light, or they can be patterned into mesas.

An alternative approach for microcavities uses total internal reflection within a highrefractive index convex body, to producewhispering gallery modeswhich can existwithin spheres (3D modes) or disks (2D modes), or more complicated topological struc-tures.

Finally,photonic crystalsemploy periodic patterning in 2 or 3 dimensions to confinelight to a small volume surrounding a defect of the structure.

The key issues that should borne in mind when considering microcavities are:

• their optical losses or finesse• coupling to incident light• optical mode volume• fabrication complexity and tolerance• incorporation of active emitters, and• practicality of electrical contacting.

1.3 Planar microcavities

The well-known Fabry-Perot cavity comprised of two plane mirrors can perform effec-tively when the mirror separation,L, is only a few wavelengths of light. The resultinglongitudinal cavity modes are equally spaced in frequency,apart from shifts caused bythe variation with wavelength in the phase change on reflection on each of the mirrors.These Fabry-Perot modes have a characteristic dispersion in their frequency as the an-gle of incidence,θ, is increased. Essentially the condition for constructiveinterferenceafter one round trip enforces a condition on the wavevectork⊥ = k cos θ perpendicularto the mirror surfaces: at higher angles of incidence an additional wavevector parallel tothese surfaces means that the totalk = 2π/λ and hence the cavity frequency increases.

k⊥ × 2L = 2mπ (1.8)

hence

ω =mπc/L

√

n2 − sin2 θ(1.9)

PLANAR MICROCAVITIES 7

CharlesFabry (1867-1945) and AlfredPerot (1863-1925).

Fabry developped the theory of multibeam interferences at the heart of the Fabry-Perot interferome-ter. He published 15 articles derived from his invention with Perot in the course of 1896-1902, applying itwith great success to spectroscopy, metrology and astronomy. Alone or with others, he derived from theinterferometer a system of spectroscopic standards, demonstrated Doppler broadening in the emission lineof rare gas and, in 1913, evidenced the ozone layer in earth’supper atmosphere. When he was elected tochair at the French “Academie des sciences” in 1927, he gathered 51 votes whereas all other candidates cameout with only one, including Paul Langevin. An enthusiasticteacher and populariser of science, some of hislectures were so popular that the doors had to be closed for lack of space half an hour before the beginning.He is quoted as having said “My whole existence has been devoted to science and to teaching, and these twointense passions have brought me very great joy.”

Perot did not climb to the same fame as Fabry beside their joint naming of the interferometer, and the mostvalued source of information about his life is in fact the obituary written for him by Fabry, in “Alfred Perot”,Astrophys. J.64, 208 (1926). While his colleague and friend—apart from the theory—would also carry outmost of the measurements and calculations, Perot was mainly involved in the design and construction of theapparatus where he deployed great skills which brought to the system immediate fame. He also initiated theproject by consulting Fabry on a problem of spark dischargesof electrons passing between close metallicsurfaces. He later developed interest for experimental test of general relativity with some positive outcomesbut a final failure to evidence a gravitational redshit.

Further informations can be found in sources compiled by J. F. Mulligan for the centenary anniversary of theinterferometers in “Who were Fabry and Perot?”, Am. J. Phys. 66, 9, (1998).

wheren is the average refractive index of the microcavity.Planar microcavities illuminated with plane waves of infinite extent in the plane of

the mirrors do not have any additional transverse modes. Such a situation is practicallyunrealistic, and there is always some limit to the lateral extent either from the size of themirrors, the width of the illuminating beam, or aperturing effects inside the microcavity.The natural basis set for planar microcavities are waveletsof extent in both real and mo-mentum space. One useful variety are Airy modes, because these are anchored arounda particular point on the mirror (which we normally impose byilluminating or detect-ing at a particular position). However for most purposes plane waves in the transversedirection are used to describe the field distribution.

Microcavities in which the mirrors are not exactly parallelcause incident light toslowly “walk” towards the region of larger cavity length through multiple reflections,as illustrated on Fig. 1.4. The accelaration of the confined light in this direction can bedirectly tracked in time and space. Thus this lateral walk-off acts as an extra loss in the


cavity modes, reducing the finesse.

Fig. 1.4: Wedged microcavity, showing walk-off of incidentlight in multiple reflections.

It is helpful to consider approaches to making these devicesthrough comparisonbetween two sorts of reflectors: metals and distributed Bragg reflector (DBR) stacks:

1.3.1 Metal microcavities

The modes of a metal microcavity are limited by the fundamental material parametersof loss and reflectivity in metal films of varying thickness. For wavelengths further intothe infrared this situation improves and the finesse can be high, reachingQ-values of109 for superconducting cavities at microwave frequencies. However for microcavitiesaround the optical region of the spectrum, the modes haveQ < 500 (see Fig. 1.5).

100

80

60

40

20

0

Reflection/T

ransm

issio

n(%

)

200015001000500Wavelength (nm)

Au 25nm /SiO2 590nm / Au 25nm

expt theoryRT

Fig. 1.5: Reflection and Transmission of a planar microcavity consisting of a gold-coated 590nm glass spacer.

The boundary conditions for reflection of light at a metal imply that the optical fieldis nearly zero at the mirrors while penetration of light intothe metal mirrors is smallcompared to the wavelength. Note also that the phase change on reflection from themetal varies with wavelength depending on the dielectric constants (see Eq. 2.133).


The ultimateQ-factor for a metal cavity is set by the trade-off between thereal andimaginary parts of the dielectric constant,n = n + iκ, which control reflectivity andabsorption.

On the other hand in a metal microcavity the modal extent can be relatively small,with penetration of light into the barriers limited to the exponential decay of the electricfield in the metal,∝ exp(−δz) with δ = 2πκ/λ andz the distance along the normal tothe surface.

1.3.2 Dielectric Bragg mirrors

The situation is different when multilayers of many pairs ofalternating refractive indexare used to make cavity mirrors (see Section 2.6.1). The complete structure has an extracavity spacer in between the Bragg mirrors, and the whole canbe considered a 1Dphotonic crystal cavity with a central defect. A scanning electron micrograph of thecross section of a typical semiconductor DBR microcavity isshown in Fig. 1.6.

Fig. 1.6: Scanning electron micrograph of GaAs/AlGaAs DBR microcavity on a GaAs substrate, from Sav-vidis et al. (2000).

The finesse of this cavity is set by the reflectivity of each mirror which depends onthe number of pair repeats and the refractive index constrast between the two materialsused. The key condition is that the optical path in each of thelayers is a quarter of thedesired centre wavelength of reflection. In this case, the resonant field is maximum atthe dielectric interfaces and there is significant penetration of light into the surround-ing mirror stacks (Fig. 1.7c). The reflectivity has a centralflat maximum, which dropsoff in an oscillating fashion either side of the reflection- or stop-band. The spectralbandwidth of the mirror is set by the refractive index difference between the materials(Section 2.6.1).

The penetration into the mirrors limits the minimum modal length of the DBR mi-crocavity (see Eq. 2.124), so the cavity mode volume is larger than for metal Fabry-Perotmicrocavities. For instance the cavity mode and the first mode of the Bragg mirror side-bands have similar extents (Fig. 1.7 c, d).

The finesse of these microcavities is ultimately limited by the number of multilayersthat can be conformally deposited, whilst ensuring that roughness or surface crackingwhich creates scattering loss does not increase unduly. Electron-beam evaporation ofdielectrics in general produces the smoothest results for oxides, while MBE of semicon-ductors produces very high quality epitaxial single-crystal Bragg mirrors. Other tech-niques have also been investigated such as controlled etching of layer porosity in Si,


Fig. 1.7: (a,b) Reflection of a planar DBR microcavity consisting of a top & bottom mirror with 15 & 21repeats of GaAs/AlAs. In (a) a 240nm-thick bulk GaAs cavity is incorporated while in (b) 3 InGaAs QWsare incorporated in the same region. (c,d) field distributions corresponding to (a) at wavelengths of (c) 841nmand (d) 787nm.

polymer multilayers, and chiral liquid-crystalline phases. Finding suitable materials forBragg mirrors in the UV spectral region is in general more challenging, and the avail-able refractive contrast limits the bandwidth here. In practice, achieving transmissivityratios of 106 on and off the resonance wavelengths is achievable across wide spectralbandwidths.

Some of the earliest active planar microcavities were optically-pumped dyes flow-ing between two DBR mirrors. These demonstrated many of the features resulting fromlateral confinement (in this case by the localised optical pump beam) of the microcavitymodes in the weak coupling regime. By controlling the separation between the mirrors,the longitudinal mode can be tuned into resonance with the dye emission. The far fieldpattern shows how the lateral coherence length changes as the stimulated photon ampli-fication turns on. By optical pumping two neighbouring positions inside the dye-filledmicrocavity, the lateral coherence properties could be investigated as a function of theirseparation.

1.4 Spherical mirror microcavities

In order to fully control the photonic modes in the microcavity the light has to beconfined in the other two spatial directions. The way this is conventionally achievedin macroscopic cavities is to use mirrors with spherical curvature. This can also beachieved in microcavities, however new methods have to be utilised to produce spheri-cal optics with radius of curvature below 100µm.

Developing standard routes to spherical mirrors and polishing mirrors to ultra-lowroughness can give reflectivity in excess ofR > 0.9999984. These have been success-

SPHERICAL MIRROR MICROCAVITIES 11

fully used in 40µm long cavities for atomic physics experiments, providingQ-factorsexceeding 108. For such extremely narrow linewidths, active stabilisation of the cavitylengths is mandatory on the level of several pm using piezo-electric transducers, see thediscussion by Rempe et al. (1992).

Another route suggested has been to trap micron-sized air bubbles in cooling glassand then cut such frozen bubbles in two for subsequent coating. This has been suc-cessfully used by Cui et al. (2006) to create microcavities with finesse up to severalhundred.

A separate technique has been to develop templating for micron-scale mirrors. Inthis strategy, latex or glass spheres of a selected size are attached to a conducting sur-face. This is followed by electrochemical growth of reflective metals around them. Sub-sequent etching of the templating spheres leaves sphericalmicro-mirrors with smoothsurfaces. Such mirrors down to 100nm radius of curvature have been produced, for in-stance by Prakash et al. (2004).

a)

b)

c)

Fig. 1.8: Spherical gold mirrors (a,b) templating process,(c) SEM and optical micrograph, of 5µm diameter,5 µm radius of curvature mirrors, from Prakash et al. (2004).

For all these spherical mirror microcavities, the optical mode spectrum of the micro-cavity becomes fully discretised into transverse and longitudinal modes. In the paraxialapproximation these Laguerre-Gauss modes are given by:

ωnpq =c

L

[

n+ εn + (2p+ q) arctan

√

L

R− L

]

(1.10)

wheren, p, q are integers for the longitudinal, axial, and azimuthal mode indices,Ris the radius of curvature of one mirror (the other mirror is plane) and they are sepa-rated by a distanceL. The phase shifts from the mirror reflections areε(λ). Typicallyin macroscopic scale cavities the azimuthally-symmetric mode symmetry is broken by


slight imperfections in the mirror shape and the astigmaticalignement, and horizon-tal/vertical TEmn and TMmn modes result. However for microcavities, the dominantsplitting is due to the breakdown of the paraxial approximation, which assumes all raysare nearly parallel to the optic axis so thatsin θ ≃ θ. In microcavities, the cylindricalmode symmetry is maintained, while mode splittings are morepronounced for fieldswith different extents in the lateral direction (Fig. 1.9).

Fig. 1.9: (a) Spherical-planar microcavity,L=10µm, (b) Cavity mode transmission spectrum vs. radial dis-tance from centre of planar top mirror, (c) Optical near-field image of mode atω=1.932eV, from B. Pennington(2006).

Enhancing the finesse further requires deposition of DBR mirrors inside the spheri-cal micro-mirror. This becomes progressively more difficult as the radius of the spheri-cal mirror reduces, and the mode size shrinks laterally. Hence there is a minimum limitto the mode volume of spherical microcavities.

1.5 Pillar microcavities

Another way to confine the lateral extent of the photonic modes inside planar micro-cavities is to etch them into discrete mesas. Total internalreflection is used to confinethe light laterally, while the confinement vertically is dependent on reflection from DBRmirrors.

For semiconductors with high refractive indices, like GaAs, the lateral confinementis quite strong. One way to think of these microcavities is aswaveguides which have re-flectors at each end. Light propagating at any angle less than73 to the external wall sur-face is totally internally reflected. Once again the modes have discrete energies, whichare further separated in frequency as the pillar area is reduced. From the waveguidepoint of view, the modes can be labelled as TEMpq. A lowest order description of themodes in square pillars assumes the electric field vanishes at the lateral surfaces, andhence

ωnpq =c

n

√

k20 + k2

x + k2y (1.11)

wherek0 = 2πn/λ0 denotes the wave vector of the vertical cavity, and in the lateraldirectionskx,y = (mp,q + 1)π/D, for a square pillar of sideD with lateral photonquantum numbersmp,q ∈ N labelling the transverse modes.

Typically DBR planar microcavities are used as the basis of the pillar, and the etch-ing proceeds to just below the central defect spacer. In suchstructures, further splitting

PILLAR MICROCAVITIES 13

Fig. 1.10: Pillar microcavity from an etched planar DBR semiconductor microcavity (left), with emissionmode spectrum (right), from Gerard et al. (1996).

of degenerate modes is observed, due to the effects of pillarshape (square, elliptical,rectangular), strain (in non-centro-symmetric crystals like III-V semiconductors), andimperfections in the perimeter of the pillar.

The discrete modes of a hard apertured circular microcavitycorrespond to Airymodes,Al, in the angular dispersion,

Al(θ) =2Jl(kL cos θ)

kL cos θ(1.12)

whereJl is the first order Bessel function. The angle-dependent far-field coupling andenergies of these modes in larger microcavities falls on topof the in-plane dispersionof the planar microcavity showing the close connection between mode area and energysplitting. Depending on the geometry of the pillars, the modes may be mixed together,producing orthogonal TE/TM modes rather than axial symmetric modes as in the spher-ical microcavities.

A number of researchers have demonstrated that the growth ofthin semiconductornanowires (such as GaN or ZnO) can produce a similar microcavity effect, with thewaveguide modes confined at each end by the refractive index contrast at air and sub-strate interfaces. In general however the smaller the pillar area, the more difficult it isto couple light efficiently into or out of the microcavities.In addition such pillar mi-crocavities are inherently solid, and emitters have to be integrated into the structure.Typically for semiconductors, the diffusion length of carriers is microns, and thus oncethe pillar diameter is reduced below this, non-radiative relaxation on the surfaces of thepillar dominates the emission process reducing the emission intensities.

Typically, pillar DBR microcavities with diameters of 5µm showQ factors in excessof 104, and these can be straightforwardly measured by incorporating an emitting layersuch as InAs quantum dots in the centre of the cavity stack. TheseQ-values are limitedby imperfections in the DBR mirrors (mostly caused by the buildup of strain within themirror stacks). The other main loss from the micro-pillar isthe light which can leak out


the sides of the pillar. To improve the finesse of such microcavities, they can be coatedwith metal around their vertical sides.

The most common active pillar-type microcavities are vertical-cavity semiconductoremitting lasers (VCSELs), which form the basis of a huge and thriving industry due totheir ease of large-scale manufacturing and of in-situ testing of devices in wafer form.These VCSELs normally use a combination of pillar etching and further lateral controlof the electrical current injection by progressively oxidising an incorporated AlAs layerwhich thus forms an insulating AlOx annulus. The oxide also provides extra photonconfinement (as the refractive index is much lower than the semiconductor interior),improving the coupling of the optical mode with the electron-hole recombination gainprofile.

Tuning of VCSELs has been demonstrated using MEMs technologies, in which theupper mirror is suspended above the active cavity and lower DBR mirror, and can bemoved up and down to tune the main cavity mode wavelength. Other cavity tuning

Fig. 1.11: (a) VCSEL design incorporating oxide apertures,(b) MEMS VCSEL structure with cantileveredtop mirror, from Chang-Hasnain (2000).

schemes use temperature or current through the mirrors to modify the refractive indexof the cavity. Wide bandwidth tuning across hundreds of nanometres in the visible andnear infrared has not yet been achieved in integrated structures, though this would beextremely useful for many applications. More information on VCSELS can be found inChapter 6.

1.6 Whispering gallery modes

When light is incident at a planar interface from a high refractive indexn1 to a lowrefractive indexn2 medium, it can be completely reflected provided the angle of inci-dence exceeds the critical angle,θc = arcsin(n2/n1). This total internal reflection canbe used to form extremely efficient reflectors in a microcavity, not dependent on metalor multilayer properties. Because the light skims around the inside of such a high re-fractive index bowl, it resembles the whispering gallery acoustic modes first noticed byLord Rayleigh in St. Paul’s Cathedral in London.

However, in general it is not possible to make extremely small microcavities usingthis principle. The reason is that diffraction plays an increasing role at small scales, andany planar surfaces have to be connected by corners which actas leaks for the diffractinglight.

WHISPERING GALLERY MODES 15

Whispering gallery modes can be classified with integers corresponding to radial,n,and azimuthal,l, mode indices. High-Qwhispering gallery modes generally correspondto large numbers of bounces,l, (thus at glancing incidence to the walls) withn ∼ llocalizing the mode near the boundary walls (see Section 2.8.2).

The limit of these structures is a circle or in general a distorted or curved shape.In this regime of curved interfaces, the total internal reflection property is modified,and light can tunnel out beyond a critical distance to escape. A simple way to see thisis to map the curved interface into a straight interface by varying the refractive indexradially (in a well defined way, see Fig. 1.12). This leads to an equation for the radialoptical potential which for the perpendicular wavevector looks like tunneling through atriangular barrier. Solving this leads to the reflectivity per bounce,R:

ln(1 −R) = −4πρ

λh[

ln(h+√

h2 − 1) −√

1 − h−2]

(1.13)

whereh = n cosα, andα is the angle of incidence on the curved boundary.

Fig. 1.12: (a) Tunneling through total internal reflection at curved interface, (b) conformal mapping and (c)optical potential.

This shows the limits of the total internal reflection even for structures which wouldgive high reflectivity at a planar interface. For instance for micron-sized silica spheres,the reflectivity per round trip drops below 50% whenl < 5 bounces.

The variants of whispering gallery microcavities can be classified according to theirgeometry, whether the multiple total internal reflections lie in a plane (2D) or circulatealso in the orthogonal direction (3D).

1.6.1 Two-dimensional whispering galleries

For any sufficiently high refractive index convex shape, light can be totally internallyreflected around the boundary. Providingn > 1/ arcsin 30 = 2, triangular whisperinggallery modes are the lowest order number of bounces possible (Fig. 1.13).

In real structures, the third dimension remains important,and confinement in thisdirection is provided by a waveguide geometry, or DBR reflectors. Hence the geomet-rical form of 2D whispering galleries is generally a thin disk of high refractive indexmaterial on a low effective refractive index substrate. An extreme form of this geometry


Fig. 1.13: Microdisk triangle laser, with external coupling waveguide, from Lu et al. (2004).

is provided by the thumb-tack microcavity, in which a semiconductor disk is undercutand supported by a thin central pillar.

Fig. 1.14: Left: microdisk (lower ring) with upper electrical contact, from Frateschi et al. (1995) and right:microdisk intensity pattern inside disk.

The modes of a circular microcavity take the form of spherical Bessel functionsinside the disk (Section 2.8). As the microcavity becomes smaller, the photonic modeseparation increases, and can be estimated from the simple multi-bounce model basedon a ray treatment for constructive interference,kL = m2π, afterl bounces, hence

ωlm =m2πc

n2lR sin(π/l)≃ mc

nR(1.14)

with the disk radiusR and refractive indexn.As most of the field of the whispering gallery modes is in the outer part of the disk,

the inside of the disk can be removed to form microring resonators with very similarcharacteristics. The lack of emission from the central partwhen optically pumped re-duces the background spontaneous emission, thus reducing the laser threshold as longas scattering losses from the whispering gallery mode are not also increased.

Since the leakage from a circular microdisk is frustrated evanescent coupling toair, it produces a structure which is extremely hard to selectively couple to and from.Favoured versions have included waveguide in close proximity to provide selectiveevanescent coupling in particular directions (Fig. 1.15).However such coupling is ex-ponentially sensitive to the coupling gap between disk and waveguide, giving strict fab-rication tolerance on the nm scale.

WHISPERING GALLERY MODES 17

Fig. 1.15: Microring resonator with integrated waveguide coupling, from Xu et al. (2005).

To ameliorate this sensitivity, the basic shape of the micro-disk can be altered, forinstance to a stadium shape which has bow-tie modes in a ray picture.

Fig. 1.16: Bowtie resonator, from Nockel (2001).

Such shapes have classical chaotic ray orbits, and are related to a general class of“quantum billards” devices for ballistic electron transport. These have more sharplycurved sections, where most of the output coupling occurs, and hence the microcavityselectively emits in particular directions. A variety of other modified shapes have beenproduced to induce unidirectional, or spatially-stabilised output (Fig. 1.17).

Since 2D whispering gallery structures can be convenientlyproduced by conven-tional lithography processes, they are suited to dense integration applications. Howeverthe main issue is that of obtaining efficient outcoupling without destroying the highQ-factor, and introducing huge fabrication sensitivity. Byusing high refractive indexcontrast GaAs on AlOx disks,Q-factors> 104 have been observed in 2µm diameterwhispering gallery resonators. Nevertheless one of the remaining issues is the difficultyof tuning the wavelength of these cavity modes, and getting access to the internal elec-tric field.

1.6.2 Three-dimensional whispering galleries

Very similar behaviour to the 2D disks is observed for 3D structures. The difference isthat the confinement in the third dimension is now provided not by a thin waveguide,


Fig. 1.17: Cog microdisk laser, to pin the azimuthal standing wave field, from Fujita & Baba (2002).

but by additional total internal reflection.The simplest example is the spherical microcavity, which can be simply formed

by melting the end of a drawn optical fibre. The resulting sphere produced by surfacetension of the glass perches on top of the remaining fibre, andfor light resonating aroundthe equator forms an excellent high-Q cavity.

Fig. 1.18: Spherical glass resonator atop optical fibre, with lasing circular whispering gallery mode, fromVahala (2003).

Because of the increasing evanescent loss as the curvature increases, the highestQ (∼ 109) cavities are found in the largest spheres (> 100µm). For micron-sizedspheres,Q-factors drop to several 1000, and the mode spectral spacingincreases totens of nanometers. Typically, sets of azimuthal modes are seen (with the samel), with

PHOTONIC CRYSTAL CAVITIES 19

different out-of-plane mode indices,m, provided by the extra degree of freedom in 3D(see Section 2.8). These have also been spatially mapped using scanning near-field op-tical microscopy showing the way the orbits converge aroundan equatorial orbit.

As in the 2D micro-disks, deformations of the sphere provideanother control vari-able for the emission direction and mode symmetry. Investigations of elongated liquiddroplets filled with dye have shown the relation between unstable classical ray orbits anddiffraction. Once again, the most difficult problem is that of efficiently coupling lightinto and out of the whispering gallery. As before, smaller spheres show lowerQ-factorsdue to the curvature of the interface which permits light to evanescently escape.

1.7 Photonic crystal cavities

Photonic crystals arise from multiple photon scattering within periodic dielectrics, andalso exist in 2D and 3D versions. The ideal 3D photonic crystal microcavity would bea defect in a perfect 3D photonic lattice with high enough refractive index contrast thatthere is a bandgap at a particular wavelength in all directions. In principle this wouldprovide the highest optical intensity enhancement in any microcavity, however currentlyno fabrication route has yet demonstrated this. Currently,scattering determines the prop-erties of most 3D photonic crystals.

Instead, 2D photonic crystals etched in thin high refractive index membranes haveshown the greatest promise, with the vertical confinement coming from the interfacesof the membrane.

Fig. 1.19: (a) Photonic crystal resonator (5 hole defect) close to a photonic crystal waveguide coupler, (b)emission spectrum from a number of microcavities of different shapes, from Akahane et al. (2005).

These can showQ-factors exceeding105 while producing extremely small mode


volumes, which are advantageous for many applications. Themain issues for such mi-crocavities are the difficulty of their fabrication, the large surface area in proximity tothe active region (which produces non-radiative recombination centres and traps diffus-ing electron-hole pairs), and the difficulty in tuning theircavity wavelengths. Howeverthey currently offer the best intensity enhancement of any microcavity system.

1.7.1 Random lasers

Related to regular 3D photonic crystals are media with wavelength-scale strong scatter-ers which are randomly positioned. Such photonic structures support photonic modeswhich can be localised, with light bouncing around loops entirely inside the medium.By placing gain materials such as dyes inside these media, and optically pumping them,so-called “random lasing” can result. While the optical feedback is not engineered inspecific orientations in these devices, it is spontaneouslyformed by the collection ofrandom scatterers.

1.8 Plasmonic cavities

Recently a new class of microcavity has emerged which is based on plasmons localisedto small volumes close to metals. For noble metals particularly, a class of localised elec-tromagnetic modes exists at their interface with a dielectric, known as “surface-plasmonpolaritons”. If the metals are textured on or below the scaleof the optical wavelength,these plasmons can be localised in all three directions, producing 0D plasmonic modes.Four implementations are of note: flat metals, metallic voids, spherical metal spheres,and coupled metal spheres, in order of increasingly confinedoptical fields (Fig. 1.20).While plasmons bound to flat metal surfaces are free to move along the surface, theplasmons on nanostructures can be tightly localised.

Fig. 1.20: Plasmon localisation: on flat noble metals, metallic voids, metal spheres, and between metalspheres.

As in any microcavity, these confined modes can be coupled to other excitationssuch as excitons in semiconductors, and this has now become of interest as the plas-monic spatial extent can be significantly smaller than the wavelength of light, downto below 10nm. On the other hand, the presence of absorbtion from the metal due toplasmon-induced excitation of single electrons places strict limits of the utility of theseplasmonic modes. Strong coupling has recently been observed in such plamonic micro-cavities using organic semiconducotrs.

MICROCAVITY LASERS 21

1.9 Microcavity lasers

Microcavities can be used as laser resonators, providing the gain is large enough tomake up for the cavity losses. Because of the short round triplength, the conditionson the reflectivity of the cavity walls are severe. However these cavities are now thestructure of choice (in the form of VCSELs), due to their easeof integratable manufac-ture and their performance. Small cavity volumes are also advantageous for producinglow threshold lasers as the condition for inversion can be reached by pumping fewerelectronic states. On the other hand the total power produced from a microcavity is ingeneral restricted as eventually the high power density causes problems of thermal load-ing, extra electronic scattering, and saturation. One advantage of a microcavity laser isthe reduced number of optical modes into which spontaneous emission is directed, thusincreasing the probability of spontaneous emission in a particular mode and thus reduc-ing the lasing threshold. This is discussed in detail in Chapter 6. Polariton lasers areexpected to have a lower threshold than VCSELs as they do not require inversion ofpopulation (see Chapter 8). They represent one of the currently hottest subject in thephysics of semiconductors. Their characteristics are discussed in detail in the last twoChapters.

1.10 Conclusion

This survey of microcavities shows the wide range of possible designs and blend ofoptical physics, microfabrication and semiconductor engineering that is required to un-derstand them. These will be taken apart and then put back together in the successivechapters.

2

CLASSICAL DESCRIPTION OF LIGHT

In this Chapter we introduce the basic characteristics of light modes infree space and in different kinds of optically confined structuresincluding Bragg mirrors, planar microcavities, pillars and spheres. Wedescribe the powerful transfer matrix method which allows for solutionof Maxwell’s equations in multilayer structures. We discuss thepolarisation of light and mention different ways it is modified includingthe Faraday and Kerr effects, optical birefringence, dichroism, andoptical activity.

FREE SPACE 23

2.1 Free space

2.1.1 Light field dynamics in free space

We start from Maxwell’s (1865) equations in the vacuum (setting the charge densitydistributionρ and electric currentJ identically to zero):1

∇∇∇ ·E(r, t) = 0 , ∇∇∇× E(r, t) = − ∂tB(r, t) , (2.2a)

∇∇∇ ·B(r, t) = 0 , ∇∇∇× B(r, t) =1

c2∂tE(r, t) . (2.2b)

whereE andB are the electric and magnetic fields, respectively, both a function ofspatial vectorr and timet, andc is the vacuum speed of light. However the mathematicalstructure is much simpler in reciprocal space where fields are expressed as a function ofwavevector instead of position, helping to lay down a simpler mathematical structure.The link between the field as we introduced itE(r, t) and its weightEEE (k, t) in the planewave basiseik·r is assured by Fourier transforms:

E(r, t) =1

(2π)3/2

∫

EEE (k, t)eik·r dk , (2.3a)

EEE (k, t) =1

(2π)3/2

∫

E(r, t)e−ik·r dr . (2.3b)

James ClerkMaxwell (1831–1879) put together the knowl-edge describing the basic laws of electricity and magnetismto set up the consistent set of Equations (2.15) which formthe foundations of electromagnetism.

One of the giants of science, Einstein said of his work itwas the “most profound and the most fruitful that physicshas experienced since the time of Newton.”, Planck that“he achieved greatness unequalled” and Feynman that “themost significant event of the 19th century will be judgedas Maxwell’s discovery of the laws of electrodynamics”. Adeeply religious man, Maxwell composed prayers that werelater found in his notes. He once said that he was thanking“God’s grace helping me to get rid of myself, partially inscience, more completely in society.”

1Maxwell equations in presence of charge and currents are, inMKS units:

∇∇∇ · E(r, t) =1

ε0ρ(r, t) , ∇∇∇× E(r, t) = − ∂tB(r, t) , (2.1a)

∇∇∇ · B(r, t) = 0 , ∇∇∇× B(r, t) =1

c2∂tE(r, t) +

1

ε0c2J(r, t) . (2.1b)

24 CLASSICAL DESCRIPTION OF LIGHT

The properties of differential operators with respect to Fourier transformation turnMaxwell equations (2.2) into the set oflocal equations:

ik · EEE (k, t) = 0 , ik× EEE (k, t) = − ∂tBBB(k, t) , (2.4a)

ik ·BBB(k, t) = 0 , ik×BBB(k, t) =1

c2∂tEEE (k, t) . (2.4b)

with obvious notations: the cursive letter and its straightcounterpart are Fourier trans-form pairs related the one to the other like (2.3). The locality is a first appealing feature,as opposed to real space equations (2.2) where the value of a field at a point dependson other field values in an entire neighbourhood of this point. The picture clarifies fur-ther still by separating the transverse⊥ and longitudinal‖ components of the field,EEE (k, t) = EEE ⊥(k, t) + EEE ‖(k, t) (with likewise definitions for other fields), the longitu-dinal component at pointk being the projection on the unit vectorek ≡ k/k, i.e.,

EEE ‖(k, t) =(

ek · EEE (k, t))

ek (2.5)

(with likewise definitions for other fields) and the perpendicular component at pointkbeing the projection on the plane normal to the unit vectorek, i.e.,EEE ⊥(k, t) ≡ EEE (k, t)−EEE ‖(k, t).

Back in real space,E⊥(r, t) andE‖(r, t) are obtained respectively by Fourier trans-form ofEEE ‖(k, t) andEEE ⊥(k, t) as given above, which correspond to the divergence-freeand the curl-free components of the field.2 The field is transverse in vacuum (the mag-netic field always is) and these transverse components obey the set of coupled linearequations derived from (2.4):

ik × EEE ⊥(k, t) = −∂tBBB(k, t)ic2k ×BBB(k, t) = ∂tEEE ⊥(k, t)

(2.7)

This linear system is diagonalised by introducing the new mode amplitudesa andb as:3

2The benefits of decomposing the field into its transverse and longitudinal components are more com-pelling when charges are taken into account, as is the case in(2.1). Then the dynamics of the field arising fromthe interplay between its electric and magnetic components(the transverse part) and the dynamics created bythe sources (responsible for the static field if they are at rest) are consequently clearly separated. For instance,dotting (2.5) withk/k2, one gets from theEEE ‖(k, t) = −i(k)k/(ε0k2), which, by Fourier transformationof both sides, gives the electric field well known from electrostatic:

E‖(r, t) =1

4πε0

Z

ρ(r′, t)r − r′

|r − r′|3dr′ (2.6)

The fact that (2.6) is local in time is a mathematical artifice: only the whole fieldE(r, t) has a physicalsignificance and other instantaneous effects from the transverse field correct those of the longitudinal field.

3The diagonalisation of (2.7) can be made by evaluating the cross product ofk with both sides of firstline, yielding on l.h.s.ik × (k × EEE ⊥) = −ik2EEE ⊥ sinceEEE ⊥ is transverse, and on r.h.s.−∂t(k × BBB).Introducingω = ck, (2.7) then reads

iωEEE ⊥(k, t) = ∂t(cek ×BBB(k, t))iωcek ×BBB(k, t) = ∂tEEE⊥(k, t)

(2.8)

FREE SPACE 25

a(k, t) ≡ − i

2C (k)(EEE ⊥(k, t) − cek ×BBB(k, t)) , (2.10a)

b(k, t) ≡ − i

2C (k)(EEE ⊥(k, t) + cek ×BBB(k, t)) , (2.10b)

with C a real constant as yet undefined (written as such for later convenience). SinceEandB are real, relation (2.3) implies thatEEE (−k, t)∗ = EEE (k, t) (the same forBBB) whichallows us to keep only one variable, e.g.,a, and writea(−k, t)∗ instead ofb(k, t).Inverting, one obtains the expression of the physical fields(in reciprocal space) in termsof its fundamental modes:

EEE ⊥(k, t) = iC (k)(a(k, t) − a∗(−k, t)) (2.11a)

BBB(k, t) =iC (k)

c(ek × a(k, t) + ek × a∗(−k, t)) (2.11b)

This mathematical formulation has therefore replaced the electric and magneticfields by a set of complex variablesa(k, t) (which give the transverse components ofthe fields) and the phase-space variables of sources(ri, ∂tri/m). All quantities of in-terest which can be expressed in terms of the fields and the dynamical variables cannaturally be written with the new set of variables. The energy of the field relevant forour description of the vacuum comes with the Hamiltonian:

H⊥ = ε0

∫

C (k)2[a∗(k) · a(k) + a(−k) · a∗(−k)] dk (2.12)

As a function of this new variable for the field, (2.7) becomes:3

∂ta(k, t) = iωa(k, t)∂ta

∗(−k, t) = −iωa∗(−k, t)(2.13)

whereω = ck. This is the main result which asserts thatthe free electromagnetic fieldis equivalent to a set of harmonic oscillators.4 These oscillators—the complex-valuedvectorsa of (2.13)—result from the mathematical manipulations which we have de-tailed. The physical sense attached to such an oscillator israther meager in its classicalformulation, being at best described as a modal amplitude for the field and thereforechiefly as a mathematical concept. Still, this affords a straightforward canonical quanti-sation scheme. But this also provides an insightful interpretation fora.

Summing and subtracting both lines yields

∂t(EEE ⊥(k, t) ± cek ×BBB(k, t)) = ±iω(EEE ⊥(k, t) ± cek ×BBB(k, t)) (2.9)

which, when expressed in terms ofa(k, t) andb(k, t) = a(−k, t)∗, is the main result (2.13).4Inserting back (2.10) in Maxwell equations with source terms, Eqs. (2.1), the equations of motion ob-

tained following the same procedure become

∂ta(k, t) = −iωa(k, t) +i

2ε0C (k)JJJ ⊥(k, t) (2.14)

GenerallyJJJ ⊥(k, t) depends nonlocally ona. Interactions with sources therefore couple together the variousmodes of the fields which are otherwise independent.


2.2 Propagation in crystals

2.2.1 Plane waves in bulk crystals

We now describe propagation in dielectric or semiconductormaterials. Maxwell equa-tions get upgraded to the following closely related expressions:

∇∇∇ · D =ρ

ε0, (2.15a)

∇∇∇ · B = 0 , (2.15b)

∇∇∇× E = −1

c∂tB , (2.15c)

∇∇∇× B =1

ε0c2∂tJ +

1

c2∂tD . (2.15d)

whereρ is the free electric charge density5 andJ is the free current density.6 In thefollowing, we only describe dielectric or semiconductor materials whereρ = 0 andJ =0. Theelectric displacement fieldis defined as

D = ε0E + P = εεεE (2.16)

whereP is thedielectric polarisationvector andεεε is thedielectric constant.In the mostgeneral case,εεε is a tensor. By a proper choice of the system of coordinates itcan berepresented as a diagonal matrix:

εεε =

ε1 0 00 ε2 00 0 ε3

. (2.17)

If the diagonal elements of this matrix are not equal to each other, the crystal is opticallyanisotropic. The effect of optical birefringence specific for opticallyanisotropic mediais briefly discussed at the end of this section. Throughout this book we always assumeε1 = ε2 = ε3 = ε (unless it is explicitly indicated that we consider an anisotropic case).We largely operate with the quantityn =

√

ε/ε0 known as therefractive indexof themedium. In crystals having resonant optical transitions, Eqs. (2.15) have two types ofsolutions. One of them is given by the condition

∇ · E = 0 , (2.18)

It corresponds to the transverse waves having electric and magnetic field vectors per-pendicular to the wavevector. The transverse waves in mediaare analogous to the light-waves in vacuum. Another type of solution are the longitudinal waves. For them

ε = 0, ∇ · E 6= 0, H = 0, ∇× E = 0 . (2.19)

In these modes the electric field is parallel to the wavevector, and the magnetic field isequal to zero everywhere. They have no analogy in vacuum (which hasε > 0 at all

5Free electric charge are those that do not include dipole charges bound in the material.6Free current density are those that do not include polarisation or magnetization currents bound in the

material.

PROPAGATION IN CRYSTALS 27

Fig. 2.1: Elliptically polarized light. The curve shows thetrajectory followedby the electric field vector of the propagating wave togetherwith its projec-tions onx andy axes and on the plane normal to the direction of motionwhere the “elliptical” character of the polarisation becomes apparent.

frequencies) but play an important role in resonant dielectric media where the dielectricconstant can vanish at some frequencies. In any case, the solution of equations (2.15)can be represented as a linear combination of plane waves where the coordinate depen-dence of the electric field is given by

E(r, t) = E0 exp(

i(k · r− ωt))

. (2.20)

k is thewavevectorof light, its modulus obeys

k = nω

c, (2.21)

andE0 is the amplitude of the plane wave. Its vector character is responsible for the po-larisation of light. Note that the choice of the sign in the exponential factor of Eq. (2.20)is a matter of convention. Note also that the vector can change with time even for afreely propagating plane wave. The longitudinal waves are linearly polarized along thewavevector, by definition. For the transverse waves it is convenient to take the curlof both parts of Eq. (2.15c) and substitute the expression for ∇ × B from MaxwellEq. (2.15d). This yields

∇×∇× E =n2

c2∂tE (2.22)

therefore, from the identity∇ × ∇ × A = ∇(∇ · A) − ∇2A (for any fieldA) andsubstituting Eq. (2.20), one obtains the wave equation:

∇2E = −k2E (2.23)

The general form of the polarisation for transverse waves can be seen as follows.Consider a plane wave propagating in thez-direction. The vectorE0 can havex- andy-components in this case.


E =

(

ExEy

)

=

(

E0x cos(kz − ωt)E0y cos(kz − ωt+ δ)

)

(2.24)

with7 δ ∈ [0, 2π[. This vector is the real part of theJones vectorof the polarized lightintroduced in 1941 by R. Clark Jones for concisely describing the light polarisation.Since the phase constants can be picked arbitrarily for one component of the vector, weset it to 0 forEx. After an elementary transformation

EyEy0

= cos(kz − ωt+ δ) = cos(kz − ωt) cos δ − sin(kz − ωt) sin δ (2.25)

which yields

ExEx0

cos δ − EyEy0

= sin(kz − ωt) sin δ =

√

1 −(

ExEx0

)2

sin δ (2.26)

squaring, we finally obtain

(

ExEx0

)2

+

(

EyEy0

)2

− 2ExE0x

EyE0y

cos δ = sin2 δ (2.27)

which is the equation for an ellipse in the(Ex, Ey) coordinate system, inclined at anangleφ to thex axis given by

tan(2φ) =2E0xE0y cos δ

E20x − E2

0y

. (2.28)

The general polarisation is therefore elliptical. Ifδ = 0 or π, light is linearly polar-ized, if δ = π

2 or 3π2 andE0x = E0y, it is circularly polarized. We callright-circular

polarisationdenotedσ+ for the case whereδ = π2 , andleft-circular polarisation, de-

notedσ−, for the case whereδ = 3π2 .

The electric field vector of the circularly polarized light rotates around the wavevec-tor in the clockwise direction or anticlockwise directionsfor σ+ andσ− polarisation,respectively (if one looks in the direction of propagation of the wave).

In reality, light is usually composed of an ensemble of planewaves of the form (2.20)with their phases more or less randomly distributed. As a result light can be partiallypolarized or unpolarized. In this case its intensityI0 > (|E0x|2 + |E0y |2)/2 whilethe equality holds for fully polarized light. It is convenient to characterize the partiallypolarized light by so-calledStokes parametersproposed by the English physicist and

7The notation[a, b[ means all the values betweena includedandb excluded. It is a common notation incountries such as France or Russia. Another widely spread convention use parenthesis for exclusion, whichwould read[a, b) in our case. Both which are occasional source for confusion are recognised standard of theISO 31-11 that regulates mathematical signs and symbols foruse in physical sciences and technology.

PROPAGATION IN CRYSTALS 29

Henri Poincare (1854–1912) and George GabrielStokes(1819–1903).

Poincare was a French mathematician and philosopher. The number of significant contributions hemade in numerous fields is overwhelming. Among extraordinary achievements, he discovered the firstchaotic deterministic system and understood correctly ahead of his generation the implication of chaos.He formulated the “Poincare conjecture” one of the most difficult mathematical problems and one of themost important in topology only just recently solved by Grigori Perelman. The Poincare sphere arose inthe context of this conjecture. He was the first to unravel thecomplete mathematical structure of specialrelativity—christening Lorentz transformations which heshowed form a group—though he missed thephysical interpretation. He never acknowledged Einstein’s contribution, still referring to it by the end of hislife as the “mechanics of Lorentz”. He is also famous for the philosophical debates which opposed him tothe British philosopher Bertrand Russel. He liked to changethe problem he was working on frequently as hethought the subconcious would still work on the old problem as he would on the new one, a good match tohis famous saying “Thought is only a flash between two long nights, but this flash is everything.”

Sir George Gabriel Stokes was an Irish mathematician and physicist, also with exceptional productivity in awide arena of science. He is most renowned for his work on fluiddynamics (especially for the Navier-Stokesequation), mathematical physics (with the Stokes theorem)and optics (with his description of polarisation).In 1852 he published a paper on frequency changes of light in fluorescence, explaining the “Stokes shift.”His production is even more remarkable given that he kept unpublished many of his first rate discoveries,like Raman scattering in the aforementioned work, that LordKelvin begged him without success to bringto print, or the spectroscopic techniques (in the form of chemical identification by analysis of the emittedlight) which he merely taught to Lord Kelvin (then Sir William Thomson), predating Kirchoff by almost adecade. In a letter he humbly attributed the entire merit to the later, saying that some of his friends had beenover-zealous in his cause. He had a tumultuous mixture of professional and sentimental feelings (not aidedby the historical context in Cambridge) and it is reported his bride-to-be almost called off the wedding uponreceiving a 55 page letter of the duties he felt obliged to remind her.

mathematician George Gabriel Stokes in 1852. The Stokes parametersS0,1,2,3 are de-fined as a function of the total intensityI0, the intensity of horizontal (x-linear) po-larisationI1, the intensity of linear polarisation at a 45 angleI2 and the intensity ofleft-handed circularly polarized lightI3, and then defining

S0 = 2I0 , (2.29a)

S1 = 4I1 − 2I0 , (2.29b)

S2 = 4I2 − 2I0 , (2.29c)

S3 = 4I3 − 2I0 . (2.29d)

These are often normalized by dividing byS0. For the fully polarized light, they canalso be re-expressed in terms of the electric field as


S0 = E20x + E2

0y , (2.30a)

S1 = E20x − E2

0y , (2.30b)

S2 = 2E0xE0y cos δ , (2.30c)

S3 = 2E0xE0y sin δ . (2.30d)

In this form, it is clear that a relationship exists connecting these parameters:

S21 + S2

2 + S23 = S2

0 (2.31)

For partially polarized light this equality is not satisfied.The components of the Stokes vector have a direct analogy with the components

of the quantum-mechanical pseudospin introduced for a two-level system, as will beshown in detail in Chapter 9. Condition (2.31) defines a sphere in the(S1, S2, S3) set ofcoordinates, called thePoincare sphere(as originally proposed in 1892 by the Frenchmathematician Henri Poincare).

Partially polarized signals can be represented by augmenting the Poincare spherewith another sphere whose radius is the total signal power,I. The ratio of the radii ofthe spheres is the degree of polarisationp = Ip/I. The difference of the radii is theunpolarized powerIu = I − Ip. By normalizing the total powerI to unity, the innerPoincare sphere has radiusp and will shrink or grow in diameter as the degree of polar-isation changes. Modifications in the polarized part of the signal cause the polarisationstate to move on the surface of the Poincare sphere.

2.2.2 Absorption of light

Absorbing media are characterized by a complex refractive index with a positive imag-inary part in the convention adopted here, see Eq. (2.20):

n = n+ iκ (2.32)

Fig. 2.2: Three-dimensional representation of thePoincare sphere. The Stokes parameters consti-tute the cartesian coordinates of the polarisationstate, which is represented by a point on the sur-face of the sphere. The radius of the Poincaresphere corresponds to the total intensity of thepolarized part of the signal.

COHERENCE 31

An electromagnetic wave propagating in a given direction (say, z-direction) is evanes-cent in this case and can be represented as

E =

(

E0x cos(nωc z − ωt)E0y cos(nωc z − ωt+ δ)

)

exp(−κωcz) . (2.33)

The parameter

α = κω

c(2.34)

is known as the absorption coefficient.α−1 is a typical penetration depth of light in anabsorbing medium. It can be as short as a few tens of nanometers in some metals.

2.2.3 Kramers-Kronig relations

The German physicist Ralph Kronig and the Dutch physicist Hendrik Anthony Kramershave established a useful relation between the real and imaginary parts of the dielectricconstant, known as the “Kramers-Kronig relations”:

ℜε(ω) = ε0 +2

πP

∞∫

0

Ωℑε(Ω)Ω2 − ω2

dΩ (2.35a)

ℑε(ω) =2ω

πP

∞∫

0

ℜε(Ω) − ε0Ω2 − ω2

dΩ (2.35b)

where the above integrals are to be understood in the sens of Cauchy andP denotes theirCauchy principal value.8 In optics, especially nonlinear optics, these relations can beused to calculate the refractive index of a material by the measurement of its absorbance,which is more accessible experimentally. The link betweenn andα follows as:

n(ω) = 1 +c

πP∫ ∞

0

α(Ω)

Ω2 − ω2dΩ (2.36)

with c the speed of light.

2.3 Coherence

2.3.1 Statistical properties of light

When it is mature enough, a physical theory needs to be extended to a statistical de-scription, where lack of knowledge of the system is distributed into probabilities for thevarious “ideal” situations to arise. In the case of light theideal case would be typicallya sinusoidal wave of well defined frequencyω, amplitudeE0 and phaseϕ, propagatingalong some axisx with wavevectork = ω/c. Disregarding the polarisation degree of

8The Cauchy principal value of an integral which presents a certain type of singularity is obtained byexcluding the singularity from the integration by approaching it asymptotically from both sides (on the realaxis).


freedom (for which the argument extends in a straightforward and identical way) thesolution thus reads:

E(x, t) = E0ei(ωt−kx+ϕ) (2.37)

However such a perfect case is an ideal limit impossible to meet in practise. Forinstance, in the case where light is generated by an atom, obvious restrictions such as afinite time of emission (within the lifetime of the transition) already spoil the monochro-matic feature (light of a well defined frequency): the wave has finite extension in timeand therefore a spread in frequency. The uncertainty in timeemission results in uncer-tainty of the phase. Taking into account light from an ensemble of atoms brings furtherstatistics and more complications can be, and often must be,also added. Some sys-tems can bypass such shortcomings, as is the case of the laserwhere many atoms emitphotons made identical by the stimulated process with hope of overcoming the finitelifetime of a single atom and providing a continuously emitting medium, thereby reduc-ing the spread of frequency. Very small linewidths are indeed achieved in this way butwhatever source of light one considers, there is ultimatelyalways a complication whichrequires a statistical treatment. One universal cause calling for statistics is temperature.In the case of the laser, the complication arises from spontaneous emission, that is,the fact that some atoms of the collective ensemble decay independently of the others,bringing some noise in the system. We shall see, however, that which property of lightis affected can be subject to some control. This suggests that this requirement for statis-tics is ultimately linked to Heisenberg’s uncertainty principle, although the argumenttranslates almost verbatim to classical concepts, as illustrated in Fig. 2.3.

2.3.2 Spatial and temporal coherence

Coherencemeasures the amount of perturbation—or noise—in the wave, that we shallcall thesignalas the notion of coherence lies in information theory. Namely, coherencerefers to the ability of inferring the signal at remote locations (in our case, in space ortime) from its knowledge at a given point. In the upper case ofFig. 2.3, there is fullcoherence as the value at a given single point determines thestate of the field entirelyand exactly. In the second case, this is limited in the windowwhere the field is defined(where it is nonzero in this case). If the horizontal axis of left panel of Fig. 2.3 is time thecorresponding interval defines itscoherence time, if it is space, it defines itscoherencelength. From the symetry of time (t) and space (x) in Eq. (2.37), the two notions sharenot only similar definition and behaviour but are linked the one to the other by theexpressionτc = lc/c with τc andlc the coherence time and length, respectively, andcthe speed of light. A general formula for the coherence time for a wavetrain of spectralwidth ∆λ centered aboutλ is:

τc =λ2

c∆λ(2.38)

The coherence length of a laser can be as high as hundreds of kilometers. For a dis-charge lamp it reduces to a few millimetres. It is very small given the speed at whichlight travels but it is high enough for experimental measurements by the time of Huy-gens and Fresnel.

COHERENCE 33

0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=cos(ω0t)

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=e-γtcos(ω0t), γ=1

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=Σi cos(ωit), <ω>=ω0

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

Fig. 2.3: Statistical description of light. Even the ideal case of light in vacuum—which from Maxwell equa-tions arise as sinusoidal functions (upper case) of well defined frequency—need statistical description. Onesimple illustration is the finite lifetime of the emitter, resulting in a Lorentzian spread in frequency (centralcase). Another more realistic case displays superpositions of different wavetrains

This definition or notion of coherence gives a good vivid picture of the concept, butit is mainly rooted in classical physics. As such, it describes perfectly well coherenceof classical waves like sound or water waves. In a modern understanding of optics, thisdescribesfirst order coherence, still an important property of light, but definitely notencompassing the whole aspect of the problem.

The principle for measuring (first order) coherence followseasily from the intuitivedescription that we have given. The notion that the knowledge of a signal at some pointinforms about its values at other points is mathematically described bycorrelation, andin this case, since it refers to the signal itself, byautocorrelation. The signal itself be-


comes arandom variable, here the fieldE, which means that when prompted a value(when it is measured, for instance), the “variable” is sampled according to a probabil-ity distribution. Physically, when the experimental set-up performs a measurement, itdraws one possible realisation of the experiment. From our description of coherence,we want to know how much the knowledge of, e.g., the fieldE(t0) at a given timet0,tells us about the valuesE(t) at other timest (hereE is still a random variable). Away to quantify this is to take the productE(t0)E(t) and to average over many possi-ble realisations. We denote〈E(t0)E(t)〉 this average. If the field is coherent, that is, ifthe value att determines that att0, each product will bear a fix relationship that willsurvive the average. If however the two values are not related to each other, the productis random and averages to zero. In intermediate cases, the degree of correlation definesthe degree of coherence. To retain some mathematical properties, the product is takenhermitian,〈E∗(t0)E(t)〉, and is normalised to make it independent of the field’s abso-lute amplitude, as well as writingt0 − t = τ to emphasise the importance of “delay” in“confronting” the two values of the field, we arrive at thefirst order coherence degree:

g(1)(τ, t) =〈E∗(t)E(t+ τ)〉

〈|E(t)|2〉(2.39)

This is a complex number in general satisfying the followingproperties:

g(1)(0, t) = 1, g(1)(τ, t)∗ = g(1)(−τ, t) . (2.40)

Eq. (2.39) depends explicitly on time as can be the case if thesystem is not inequilibrium or in a steady state. Note that such a measure of coherence extends to otherdegrees of freedom of the fields.9 Often however one considers the time independentcase which in this context translates asstationary signals, i.e., systems which still varyin time “locally” but do not depend on time in an absolute way,as is the case in apulsed experiment for instance with clear difference before, during and after a pulse, asopposed to continuous pumping where on the average the system does not evolve.

The distinction in terms of the notions we have just presented is that for a stationaryprocess the probability distribution is time independent,but the experimental quantityremains the random variable which fluctuates according to its distribution. However itsmean and variance are also constant. An obvious example is when time variation ofthe fields are limited to fluctuations, though this is only a special case of a stationaryprocess. A periodic signal is also stationary. Theergodic theoremasserts that for suchprocesses time and ensemble averages are the same, so that inEq. (2.39),t is taken toassume values high enough so that steady state is achieved.10 What kind of average is

9A more general formula quantifying simultaneously coherence both in time and space reads:

g(1)(r1, t1; r2, t2) = 〈E∗(r1, t1)E(r2, t2)〉‹

q

〈|E(r1, t1)|2〉〈|E(r2, t2)|2〉 (2.41)

10One usually gets rid of the time dependence ofg(1) by assuming any given value, typically zero. Math-ematically this is made more rigorous for processes whose initial condition is not the steady state by takingthe limit of arbitrary hight and a more intimidating notation reads:

COHERENCE 35

effectively performed is a rather moot point. There is ensemble averaging in most casesbecause many emitters combine to provide light, all of them independently emittingdifferent “wavetrains” which correspond to an ensemble averaging. On the other hand,the coherence time of light is much too short to be detected directly by a detector whichperforms itself a time averaging. The importance of statistics and the relevance of cor-relations especially as defining coherence in optics has been realised in a large measurethanks to the guidance of Born and Wolf.

Max Born (1882–1970) and EmilWolf (b. 1922) emphasised the importance of coherence in optics by devel-oping its statistical theory. They co-author “Principles of Optics” (1959) and “Optical coherence and quantumoptics” (1995) which are two classic authoritative texts.

Born was awarded the 1954 Nobel prize (half-prize) for “his fundamental research in quantum mechanics,especially for his statistical interpretation of the wavefunction.” Educated as a mathematician (with Hilbert),he identified the indices of Heisenberg’s notations for transitions rates between orbitals as matrix elements.Systematising the approach with Pascual Jordan (then his student), they submitted a paper entitled “Zur Quan-tummechanik” (M. Born and P. Jordan, Zeitschrift fur Physik,34, 858, (1924)) bearing the first publishedmention of the term “quantum mechanics.”

Wolf is the physicist of optics par excellence, bringing many advances in statistical optics, coherence, diffrac-tion and the theory of direct scattering and inverse scattering. He discovered the “Wolf Effect” which is aredshift mechanism to be distinguished from the Doppler effect. As a signature of his eponymous father, theeffect follows from partial coherence effects. It was experimentally confirmed the year following its predictionin 1987.

The materialisation of the mathematical procedure outlineabove is made in the lab-oratory with an interferometer, such as the Michelson interferometer or Mach-Zehnderinterferometer. In these devices, the field is superimposedwith a delayed fraction ofitself and the time-averaged intensity of the light is collected at the output. Oscillationsin this intensity build upfringeswith visibility defined as

V =Imax − Imin

Imax + Imin(2.43)

with Imin/max the minimum, maximum intensity of the resulting interference pattern,respectively, and this equates with the modulus of coherence degree:

g(1)(τ) = limt→∞

〈E∗(t)E(t + τ)〉〈|E(t)|2〉

(2.42)


V = |g(1)(τ)| (2.44)

The visibilityV varies with the time delayτ and of course also with position if related tothe more general definition Eq. (2.41). It assumes values between zero, for an incoherentfield, and one, for a fully coherent one. Intermediate valuesdescribe partial first ordercoherence.

Albert AbrahamMichelson (1852—1931) received the first American Nobel prize for physics in 1907 “forhis optical precision instruments and the spectroscopic and metrological investigations carried out with theiraid”. He designed the Michelson interferometer (shown on right) which superposes the light field onto itselfwith a delay (imparted by the moving mirror), allowing to measure its first order coherence.

As a trivial example, the sine wave Eq. (2.37) gives by directapplication of thedefinition Eq. (2.39)

g(1)(τ) = exp(−iωτ) (2.45)

which corresponds to full first order coherence (as its modulus Eq. (2.44) is one). Incomparison, a field which results from two sources, each perfectly coherent (in the sensethat they are both a sinusoidal wave of the type of Eq. (2.37))and with a dephasingϕbetween them, give as the total scalar field:

E(x, t)/E0 = exp(

i(k1z − ω1t))

+ exp(

i(k2z − ω2t+ ϕ))

(2.46)

We assume they have common amplitudeE0. If ϕ is kept fixed in the averaging, thesame results as for the purely sinusoidal field apply. If, however,ϕ varies randomlybetween measurements, the result of exercise 2.1 is obtained.

Exercise 2.1 (∗) Show that the field given by Eq. (2.46) withϕ varying randomly yields:

V = |g(1)(τ)| = | cos(1

2(ω1 − ω2)τ)| . (2.47)

Another frequent expression forg(1) brings together in Eq. (2.50) (derived in exer-cise 2.2) the decoherence resulting from two typical atomicdephasing processes:colli-sion broadeningandDoppler broadening.

COHERENCE 37

Collision broadening is associated with the abrupt change of phase due to collisionsbetween emitters, which have probabilityγc per unit time to see their phase randomisedduring emission (associated with the chance of colliding with a neighbour).

Doppler broadening results from the Doppler effectwhich shifts the frequency ofemitters as a function of their velocity. From kinetic theory it can be shown that thedistribution of frequencies is normal

f(ω) =1√2π∆

exp

(

− (ω0 − ω)2

2∆2

)

, (2.48)

centered about the unshifted frequencyω0 and with root mean square:

∆ = ω0

√

kBT/(mc2) (2.49)

at temperatureT for emitters of massm. This effect is therefore important in a gas atnon vanishing temperatures.

Exercise 2.2 (∗∗) Model the light field of sources subject to collision and Dopplerbroadening and show that their first order coherence degree is given by:

g(1)(τ) = exp(−iω0τ − γc|τ | −1

2∆2τ2) (2.50)

Observe how in Eq. (2.50) the dephasing which ultimately loses completely the phaseinformation results in a decay of|g(1)| rather than in oscillations as was the case inEq. (2.47), as a result of the finite number of emitters contriving to dephase the system.This will become clear in the next Section which links first order coherence to theemitted spectra.

2.3.3 Wiener-Khinchin theorem

An important relationship was established by Wiener and Khinchin between thespectraldensityof a stochastic process and itsautocorrelation function. Namely, they form aFourier transform pair.

The spectral density is, physically, the decomposition of asignal (or field) into itscomponents of given frequencies. The relation between frequencyω and energyE,11

E = ~ω (2.51)

with ~ = h/(2π) the reduced Planck constant,12 provides the experimental way torecord such a spectra: photons of a given energy are counted over a given time to buildup a signal. The name of “power spectra” is also commonly used, since an intensity per

11The relation (2.51) is Planck’s hypothesis that energy is emitted by quanta, the quantization being pro-vided by the box of the black body. It was fully developed by Einstein for his explanation of the photoelectriceffect.

12The reduced Planck constant~ is a shortcut forh/2π whereh = 4.135 667 43(35) × 10−15 eV s isthe Planck constant proper. It is the quantum of action and ofangular momentum, or, following uncertaintyprinciple, the “size” of a cell in phase space.


NorbertWiener (1894–1964) and Aleksandr YakovlevichKhinchin (1894–1959)

Wiener was an American mathematician who pioneered an important branch of study of stochastic processeswith his generalized harmonic analysis. He is also creditedas the founder of “cybernetics”. His absent mind-edness and lively nature sparked many famous anecdotes. In his obituary, the Times reports how he “couldoffend publicly by snoring through a lecture and then ask an awkward question in the discussion”. In his essay“the quark and the jaguar”, Gell-Mann remembers how Wiener would hinder the circulation in the universityby sleeping in the stairs. The first tome of his autobiographyis titled “Ex-Prodigy: My Childhood and Youth”.Working with physicists, he remarked that “one of the chief duties of the mathematician in acting as an ad-viser to scientists is to discourage them from expecting toomuch from mathematics.”

Khinchin was one of the most prominent Russian mathematician in the field of probabilities, largely domi-nated at that time by the Soviet school. He unravelled the definition of stationary processes and developed theirtheoretical foundations. He published “Mathematical Principles of Statistical Mechanics” in 1943 which heextended in 1951 to the highly respected text “Mathematical foundations of quantum statistics”. He aimed at acomplete mathematical rigour in his results as characterises well Wiener-Khinchin theorem, since—applyingto signals without a Fourier transform—it is far more involved mathematically than physicists usually appre-ciate.

unit time is measured. We now discuss what physical quantities it relates to and how tomodel them mathematically.

Intuitively, given a time varying signalE(t), one can quantify in the spirit of Fourierhow much the harmonic componenteiωt exists in the signal by overlapping both andaveraging to get a number:

E(ω) =

∫ ∞

−∞E(t)eiωt dt . (2.52)

The “weight” amplitudeE is now itself a function ofω. Its modulus square for a givenfrequency is related to the “strength” of this harmonic in the signal, and thus wouldseem to serve as a good spectral density, and sometimes does:

S(ω) = |E(ω)|2 (2.53)

COHERENCE 39

However we already pointed out that an important class of physical signals are sta-tionary which—being globally time invariant—are not square integrable13 and conse-quently do not admit a Fourier transform, which dooms the above approach. Wienershowed in 1930 that, for a large class of functionsz (encompassing all the cases rele-vant for us), the integralΓ(τ) = limT→∞

12T

∫ T

−T z∗(t)z(t + τ) dτ exists and also its

Fourier transform:

S(ω) =1

2π

∫ ∞

−∞Γ(τ)eiωτ dτ . (2.54)

Γ(τ) as defined above (and by Wiener) does not refer to an ensemble as there is noaveraging. It required Khinchin’s inputs on stationary random processes (that we willnot detail) and the ergodic theorem to arrive at theWiener-Khinchin theoremsuch as itis known today, which is Eq. (2.54) together with the definition already discussedΓ(t−t′) = 〈E∗(t)E(t′)〉 for the autocorrelation function (reverting notation toE for thefunction which has beenz up to now). Hence:

S(ω) =1

2π

∫ ∞

−∞〈E∗(t)E(t+ τ)〉eiωτ dτ (2.55)

Normalising Eq. (2.55) shows that

σ(ω) =1

2π

∫ ∞

−∞g(1)(τ)eiωτ dτ (2.56)

is thelineshapeof the emission, that is, the normalised spectral shape.Applying formula (2.56) to Eq. (2.50) shows that:

• A pure coherent state (without dephasing) emits a delta-function spectrum: thereis no spread in frequency, as expected.

• Homogeneous broadening, associated to exponential decay ofg(1), results in aLorentzian lineshape:

σ(ω) =1

π

γc/2

(ω − ω0)2 + (γc/2)2(2.57)

• Inhomogeneous broadening, associated to Gaussian decay ofg(1), results in aGaussian lineshape:

σ(ω) =1

∆√

2πexp

(

− (ω − ω0)2

2∆2

)

. (2.58)

By convolution the general case forg(1) given by Eq. (2.50) combining homoge-neous and inhomogeneous broadenings yields the following expression of the lineshape(known as theVoigt lineshape):

σ(ω) =

∫ ∞

−∞

1

∆√

2πexp

(

− x2

2∆2

)

1

π

γc/2

(ω − x)2 + (γc/2)2dx . (2.59)

The Voigt lineshape interpolates between the two other cases, as shown on Fig. 2.4.

13A function f is square integrable ifR

|f |2 exists. It is a less stringent condition than integrabilitywhichis required for many properties relative to integration to be meaningful.


0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4 -3 -2 -1 0 1 2 3 4 5

ω

γc/2=0 ∆=1.6γc/2=1 ∆=1 γc/2=1.8 ∆=0

Fig. 2.4: Typical lineshapes (emission spectra) emitted bya source with two mechanisms of broadeningresulting in exponential and Gaussian decay of the coherence degreeg(1). The limiting cases recover therespective Lorentzian and Gaussian lineshapes (lower and upper lines). The general case in between is knownas theVoigt lineshape.

2.3.4 Hanbury Brown–Twiss effect

Up to now, we have investigated “coherence” as a concept of correlations in the fieldbut in fact, of correlations in theamplitudesof the field, as those have been the relativevalues ofE which were compared. Again, this corresponds experimentally to splittingthe beam in two and superimposing it onto itself with a delay,as schematised by theMichelson interferometer on page 36. The same approach extends to other quantitiesderiving from the field. The next step is to quantify correlations in theintensitiesratherthan amplitudes, as has been done by Hanbury Brown & Twiss (1956) who correlatedthe signals of two photomultiplier (PM) tubes collecting light from the star Sirius assketched on the next page.

Hanbury Brown and Twiss evidenced a positive simultaneous correlation betweenthe two signals, meaning that the detection of a signal on anyone of the PMs wasmatched by detection of a signal on the other PM more often than if the sources wereuncorrelated (if the PMs were aimed at two different stars, for instance). This is knownasbunching, as the detections appear to be grouped together. We shall see in the nextchapter that at the quantum level, this means that photons—the particles actually de-tected and amplified by the PMs—are lumped together in the light emitted by Sirius.This experiment which in the above case is made at two different locations can be madewith an extra delay in time. With a delay, one measures the likelihood that, given afirst photon is measured at timet, a second one is measured at timet + τ . The HBTsetup then consists of a 50/50 beam-splitter directing the light on two photomultipliers(typically avalanche photodiodes). The time elapsed between the detection of two con-secutive photons is measured and the numbern(τ) of photons pairs separated by a timeintervalτ is counted. This number gives theprobability of joint detectionP2(t, t + τ)

COHERENCE 41

Robert Hanbury Brown (1916–2002) invented—with Richard Twiss—the intensity interferometer. Thesetup, now used in various “HBT experiments”, is schematised by Hanbury Brown & Twiss (1958) for ap-plications in radio-astronomy to measure the radius of a star. Two stellar mirrorsA spatially separated by adistanced collect light and focus it on photomultipliersP . B are amplifiers andC operates the coincidence(after delayτ has been imparted on one of the line).M integrates the signal.

The concept of photon interferences was initially met with great opposition, also from the highest authoritiesof the time such as Richard Feynman. In his autobiography, “BOFFIN : A Personal Story of the Early Days ofRadar, Radio Astronomy and Quantum Optics”, Hanbury Brown remembers “As an engineer my educationin physics had stopped far short of the quantum theory. Perhaps just as well, otherwise like most physicists Iwould have come to the conclusion that the thing would not work[. . . ]In fact to a surprising number of peoplethe idea that the arrival of photons at two separated detectors can ever be correlated was not only hereticalbut patently absurd, and they told us so in no uncertain terms, in person, by letter, in print, and by publishingthe results of laboratory experiments, which claimed to show that we were wrong. . .”

at timest andt + τ . This probability can be linked to intensity correlations from pho-todetection theory as has been done by Mandel et al. (1964).14 They are proportional:

P2(t, t+ τ) = α2〈I(t)I(t+ τ)〉 (2.60)

whereI = |E|2 is the intensity, andα represents the quantum efficiency of the photo-electric detector.

Intensity correlations being the straightforward next order extension of amplitudecorrelations, Eq. (2.39), we introduce the notationg(2) and call “second order coherencedegree” the quantity

g(2)(t, τ) =〈E∗(t)E∗(t+ τ)E(t)E(t + τ)〉

〈E∗(t)E(t)〉2 . (2.61)

We also introduce the centered, normalised correlation function:

λ(τ) =〈∆I(t)∆I(t + τ)〉〈I(t)I(t + τ)〉 (2.62)

14The proportionality betweeng(2) and the joint detection probability is established in the full quantumcase in exercise 3.21 on page 105.


where∆I = I−〈I〉. Now Eq. (2.60) readsP2(t, t+τ) = α2〈I(t)〉〈I(t+τ)〉[1+λ(τ)]and from the first result of exercise 2.3, it follows that

P2(t, t) ≥ P2(t, t+ τ) . (2.63)

Exercise 2.3 (∗) Show thatλ(τ) (of Eq. (2.62)) satisfiesλ(τ) ≤ λ(0) and also thatlimt→∞ λ(t) = 0. Discuss the sign of this quantity.

We already noted that the detection process is at the quantumlevel, detecting singlephotons,15 whereas it was the field amplitude which was previously measured. For thisreason the HBT experiment is often regarded as the quantum optical measurement parexcellence, though it is is merely correlating intensitiesand has been used by HanburyBrown and Twiss in a classical context (more of which is investigated in the problemat the end of this chapter). However it is true that it also characterises light bearing aquantum character, as shall be explained in the next chapter.

2.4 Polarisation-dependent optical effects

We now give a brief account of the main optical effects dealing with the polarisation oflight. Most of them have been observed in microcavities. Some of them allow measure-ment of the most important intrinsic characteristics of microcavities.

2.4.1 Birefringence

Birefringence is the division of light into two components (anordinaryand anextraor-dinary ray), found in materials which have different indices of refraction in differentdirections (no andne for ordinary and extraordinary rays, respectively). Birefringenceis also known as double refraction.

The quantity referred to asbirefringenceis defined as

∆n = ne − no (2.64)

Crystals possessing birefringence include hexagonal (such as calcite), tetragonal,and trigonal crystal classes and are known as uniaxial. Orthorhombic, monoclinic, tri-clinic crystal exhibit three indices of refraction. They are known as biaxial. Birefringentprisms include the Nicol prism, Glan-Foucault prism, Glan-Thompson prism, and Wol-laston prism. They can be used to separate different incident polarisations.

Dichroismis the selective absorption of one component of the electricfield of a lightwave, resulting in polarisation.

Optical activityarises when polarized light is passed through a substance containingchiral molecules (or nonchiral molecules arranged asymmetrically), and the direction ofpolarisation can be changed. This phenomenon is also calledoptical rotation.

The Kerr effect discovered in 1875 by the English physicist John Kerr consists inthe development of birefringence when an isotropic transparent substance is placed in

15A single photon is detected in an avalanche photodiode by exciting a single electron-hole pair across thebandgap. The high electric field across the diode accelerates the electron which acquires sufficient energy toexcite further electron-hole pairs—the avalanche—and producing a sizable current pulse.

POLARISATION-DEPENDENT OPTICAL EFFECTS 43

MichaelFaraday (1791–1867) and an illustration of theFaraday effect, similar to optical activity as in bothcases the polarisation plane of light rotates as it propagates through the medium. The difference is that theFaraday rotation is independent of propagation direction and can be accumulated if light makes several round-trips through the medium, while the rotation caused by optical activity changes its sign for light propagatingin the opposite direction. Thus, after any round-trip inside an optically active medium the polarisation of lightremains the same.

Faraday was a self-taught genius who developed an interest for science by reading books he had to bind asa poor apprentice working in a bookshop. Through attendanceand hard work he gained access to experi-mental laboratories where he excelled and developed experimental setups of an unprecedented standard. Hisevidence that magnetism could affect rays of light proved a relationship between the two, a finding preparingthe great first unification in Physics of electricity and magnetism as two facets of electromagnetism. To thespectroscopist James Crookes, he once advised “Work. Finish. Publish”.

an electric fieldF. It is used in constructing Kerr cells, which function as variable waveplates with an extremely fast response time, and find use in high-speed camera shutters.Because the effect is quadratic with respect toF, it is sometimes known as thequadraticelectro-optical effect. The amount of birefringence (as characterized by the change inindex of refraction ) due to the Kerr effect can be parametrized by

∆n = λ0KF2 (2.65)

whereK is the Kerr constant (in cgs units of cm statv−2) andλ0 is the vacuum wave-length. The phase change∆φ introduced in a Kerr cell of thicknessd under an appliedvoltageV is given by

∆φ =2πKλV 2

d2(2.66)

2.4.2 Magneto-optical effects

2.4.2.1 Faraday effect The English physicist Michael Faraday experimentally dis-covered diamagnetism and observed what is now called theFaraday effectin 1845. Hedemonstrated that, given two rays of circularly polarized light, one with left-hand andthe other with right-hand polarisation, the one with the polarisation in the same directionas the electricity of the magnetizing current travels with greater velocity. That is whythe plane of linearly polarized light is rotated when a magnetic field is applied parallelto the propagation direction (see Fig. 2.5).

The empirical angle of rotation is given by

α = V Bd , (2.67)


Fig. 2.5: Orientation of electric and magnetic fields in TE- and TM-polarized incident on a planar boundary.

whereV is named the Verdet constant after the French physicist Emile Verdet (withunits of arc minutes cm−1 Gauss−1). The Faraday rotation angle is a measure of mag-netisation induced in the medium by a magnetic field. In the quantum mechanical de-scription, it occurs because imposition of a magnetic fieldB alters the energy levels ofatoms or electrons (Zeeman effect).

2.4.2.2 Magneto-optical Kerr effect This effect has very much in common with theFaraday effect. It also consists in rotation of the polarisation plane of light in the mediahaving a non-zero magnetization in the direction of light propagation. The differencebetween the two effects consists in the experimental configuration used to detect thepolarisation rotation. For the Faraday effect the polarisation of the transmitted light isanalysed, while for the magneto-optical Kerr effect the polarisation of reflected light iscompared with polarisation of the incident light. As a largepart of reflected signal comesfrom the surface reflection usually, the Kerr rotation is very sensitive to the surfacemagnetization.

Exercise 2.4 (∗∗) Consider a dielectric slab of thicknessd subjected to a magneticfieldB oriented normally to the surface of the slab. Lett+ andt− be amplitude trans-mission coefficients of the slab forσ+ andσ− polarized light, respectively, and|t+| =|t−|. Find the Verdet constant of the material.

2.5 Propagation of light in multilayer planar structures

In this Section we present thetransfer matrix methodwhich solves Maxwell equationsin multilayer dielectric structures. We consider the example of a periodical structure (aso-calledBragg mirror) and derive general equations for photonic eigenmodes in planarstructures. In the beginning we consider propagation of light normal to the layer planesdirection. Then we generalise to the oblique incidence case. We generalize the transfermatrix approach for TE and TM linear polarisations. By definition, TE-polarized (alsoreferred to ass-polarized) light has the electric field vector parallel to the layer planes,

PROPAGATION OF LIGHT IN MULTILAYER PLANAR STRUCTURES 45

TM-polarized light (also referred to asp-polarized) has the magnetic field vector parallelto the planes (see Fig. 2.5).

The behaviour of the electromagnetic field at the planar interface between two di-electric media with different refractive indices is dictated by Maxwell equations (2.15).They can be solved independently in the two media and then matched for the elec-tric and magnetic fields by the Maxwell boundary conditions at the interface. Theseconditions require continuity of the tangential components of both fields. They can bemicroscopically justified for any abrupt interface in the absence of free charges and freecurrents.

Consider a transverse light-wave propagating along thez-direction in a mediumcharacterised by a refractive indexn that is homogeneous in thex, y plane but possiblyz-dependent. The wave equation (2.23) in this case becomes for the field amplitude:

∂2zE = −k2

0n2E (2.68)

wherek0 is the wavevector of light in vacuum. The general form of the solution ofEq. (2.68) reads

E = A+ exp(ikz) +A− exp(−ikz) (2.69)

wherek = k0n, A+ andA− are coefficients. Using the Maxwell equation (2.15d) onecan easily obtain the general form of the magnetic field amplitude

B = nA+ exp(ikz) − nA− exp(−ikz) (2.70)

If we consider reflection of light incident from the left sideof the boundary(z = 0)between two semi-infinite media characterized by refractive indicesn1 (left) andn2

(right), the matching of the tangential components of electric and magnetic fields gives

A+1 +A−

1 = A+2 , (2.71a)

(A+1 −A−

1 )n1 = A+2 n2 (2.71b)

whereA+1 , A−

1 andA+2 are the amplitudes of incident, reflected and transmitted light,

respectively. One can easily obtain the amplitude reflection coefficient

r =A−

1

A+1

=n1 − n2

n1 + n2(2.72)

and the amplitude transmission coefficient

t =A+

2

A+1

=2n1

n1 + n2. (2.73)

Thereflectivity, which is the ratio of reflected to incident energy flux, is given by

R = |r|2 (2.74)

and thetransmittance, or ratio of transmitted to incident energy flux, is

T =n2

n1|t|2 . (2.75)

In the last formula, the factorn2/n1 comes from the ratio of light velocities in thetwo media.


In multilayer structures, direct application of Maxwell boundary conditions at eachinterface requires solving a substantial number of algebraic equations (two per inter-face). A convenient method allows the number of equations tobe solved to be strictlyminimised (four in the general case) and is thetransfer matrix methodwhich we nowbriefly describe.

Let us introduce the vector

ΦΦΦ(z) =

(

E(z)cB(z)

)

=

(

E(z)− ik0∂zE(z)

)

, (2.76)

whereE(z) andB(z) are the amplitudes of the electric and magnetic field of any light-wave propagating in thez direction in the structure. Note thatΦΦΦ(z) is continuous atany point in the structure due to Maxwell’s boundary conditions. In particular, it iscontinuous at all interfaces wheren changes abruptly.

By definition the transfer matrixTa across the layer of widtha is a matrix whichenforces

TaΦΦΦ|z=0 = ΦΦΦ|z=a (2.77)

It is easy to verify by substitution of the electric and magnetic amplitudes (2.69) and(2.70) into Eq. (2.77) that ifn is homogeneous across the layer,

Ta =

(

cos ka in sin ka

in sinka cos ka

)

. (2.78)

The transfer matrix across a structure composed ofm layers is found as

T =

m∏

i=1

Ti (2.79)

whereTi is the transfer matrix across theith layer. The order of multiplication in (2.79)is essential. The amplitude reflection and transmission coefficients (rs andts) of a struc-ture containingm layers and sandwiched between two semi-infinite media with refrac-tive indicesnleft andnright before and after the structure, respectively, can be foundfrom the relation between the fieldsΦΦΦ on either side of the structure:

T

(

1 + rsnleft(1 − rs)

)

=

(

tsnrightts

)

. (2.80)

One can easily obtain

rs =nrightt11 + nleftnrightt12 − t21 − nleftt22t21 − nleftt22 − nrightt11 + nleftnrightt12

, (2.81a)

ts = 2nleftt12t21 − t11t22

t21 − nleftt22 − nrightt11 + nleftnrightt12. (2.81b)

The intensities of reflected and transmitted light normalised by the intensity of theincident light are given by

R = |rs|2, T = |ts|2nright

nleft(2.82)

respectively.

PROPAGATION OF LIGHT IN MULTILAYER PLANAR STRUCTURES 47

ReneDescartes(1596–1650) and WillebrordSnellius (1580–1626).

Descartes is regarded as the father of modern mathematics and natural philosophy. His thinking instigatedthe revolution in western science which would break from eastern influence and mark the dominance ofEuropean thinking. His famous saying “je pense donc je suis” (I think therefore I am) in “Discourse onmethod” rank among the most influential philophical statements. His name became synonymous of “rational”and “meaningful”, an eponymous tribute which extent and glory was never equalled after him.

Snell was a Dutch astronomer and mathematician. He discovered the law of refraction that is now named afterhim (although in a few countries but especially in France this law is called Snell-Descartes). Arab astronomersapparently knew it long time before Snell from the work of IbnSahl (984). Snell also worked out a remarkablyaccurate value of the radius of the earth (for his time) and devised a new method to computeπ, the first suchimprovement since ancient time.

Reciprocally, the transfer matrix across a layer can be expressed via reflection andtransmission coefficients of this layer. If the reflection and transmission coefficients forlight incident from the right-hand side and left-hand side of the layer are the same,andnleft = nright = n (the symmetric case of, in particular, a quantum well embeddedin a cavity), the Maxwell boundary conditions for light incident from the left and rightsides of the structure yield:

T

(

1 + rsn(1 − rs)

)

=

(

tsnts

)

and T

(

ts−nts

)

=

(

1 + rs−n(1 − rs)

)

. (2.83)

This allows the matrixT to be expressed as

T =1

2ts

(

t2s − r2s + 1 − (1+rs)2−t2sn

−n((rs − 1)2 − t2s) t2s − r2s + 1

)

. (2.84)

For a quantum well, as will be shown in next section,ts = 1 + rs and Eq. (2.84)becomes

TQW =

(

1 0−2nrs/ts 1

)

. (2.85)

In the oblique incidence case, in the TE-polarisation, one can use the basis(Eτ (z), cBτ (z))

T ( T means transposition) whereEτ (z) andBτ (z) are the tangen-tial (in-plane) components of the electric and magnetic fields of the light wave. In this


case, the transfer matrix (2.78) keeps its form provided that the following substitutionsare made:

kz = k cosφ, n→ n cosφ (2.86)

whereφ is the propagation angle in the corresponding medium (φ = 0 at normal inci-dence).

In the TM-polarisation, following Born & Wolf (1970), we nowuse the basis(cBτ (z), Eτ (z))

T which still allows the transfer matrix (2.78) to be used, providedthat the following substitutions are done:

kz = k cosφ, n→ cosφ

n. (2.87)

Note that the transfer matrices across the interfaces are still identity matrices, andEq. (2.79) for the transfer matrix across the entire structure is valid.

In the formulas (2.81–2.83) for reflection and transmissioncoefficients, one shouldreplace, in the TE-polarisation

nleft → nleft cosφleft, nright → nright cosφright (2.88)

and in the TM-polarisation

nleft →cosφleft

nleft, nright →

cosφright

nright, (2.89)

whereφleft, φright are the propagation angles in the first and last media, respectively.The same transformations would be applied to the transfer matrices (2.84) and (2.85).Note that any two propagation anglesφi, φj in the layers with refractive indicesni, njare linked by theSnell-Descarteslaw:

ni sinφi = nj sinφj , (2.90)

which is also valid in the case of complex refractive indices, when the propagationangles formally become complex as well.

2.6 Photonic eigenmodes of planar systems

Consider a multilayer planar structure characterized by a transfer matrixT being aproduct of the transfer matrices across all the layers as given by Eq. (2.79). Photoniceigenmodes of the structure are the solutions of the Maxwellequations which decay out-side the structure and hence with the following boundary condition: no light is incidenton the structure neither from the left (z → −∞) nor from the right side (n → +∞).This means that the electric field of the eigen mode atz → ∞ can be represented as

E0 exp(ikx + iky + ikz) (2.91)

with ℜ(kz) ≥ 0 andℑ(kz) ≤ 0 while atz → ∞ the electric field can be represented inthe form of (2.91) withℜ(kz) ≤ 0 andℑ(kz) ≥ 0. In TE-polarisation, let us choose the

PHOTONIC EIGENMODES OF PLANAR SYSTEMS 49

system of coordinates in such a way that electric and magnetic field of the light modeare oriented as follows:

E =

0Ey0

, B =

Bx0Bz

. (2.92)

The transfer matrixTTE links the vectors(Ey , cBx)T at the left and right bound-aries of the structure, so that

TTE

(

Elefty

cBleftx

)

= A

(

Erighty

cBrightx

)

. (2.93)

whereA is a complex coefficient.Substitution of the electric field (2.69) into the first of Maxwell equations (2.15a)

yieldskzk0Ey = cBx , (2.94)

wherek0 = ω/c. This allows to rewrite Eq. (2.93) as

TTE

1kleftz

k0

= A

1krightz

k0

, (2.95)

wherekleftz andkright

z arez-components of the wavevector of light on the left and rightsides of the structure, respectively. By elimination ofA, Eq. (2.95) can be easily reducedto a single transcendental equation for the eigenmodes of the structure:

tTE11

krightz

k0+ tTE12

kleftz kright

z

k20

− tTE21 − kleftz

k0tTE22 = 0 , (2.96)

wheretij are the matrix elements of the transfer matrixTTE . Solutions of Eq. (2.96) arecomplex frequencies, in general. Only those having a positive real part and negative (orzero) imaginary part have a physical sense. The imaginary part of the eigenfrequency isinversely proportional to the lifetime of the eigenmode, i.e., a characteristic time spentby a photon going back and forth inside the structure before escaping from it to the con-tinuum of free light modes in the surrounding media. So-calledwave-guidedor guidedmodesare those which have an infinite lifetime (and consequently,zero imaginary partof the eigenfrequency), see Fig. 2.6.

The equation for eigenfrequencies of TM-polarized modes can be obtained in asimilar way. One can choose the system of coordinates in sucha way that electric andmagnetic field of the light mode are oriented as follows:

E =

Ex0Ez

, B =

0By0

. (2.97)


Fig. 2.6: Propagation of guided modes in planar structures (a) is based on the total internal reflection effect (b).

The transfer matrixT links the vectors(cBy,−Ex)T at the left and right boundariesof the structure, so that

TTM

(

Blefty

−Eleftx

)

=

(

Brighty

−Erightx

)

. (2.98)

The Maxwell equation (2.15a) yields in this case

− kzn2k0

By = Ex (2.99)

wheren is the refractive index of the media. This allows to rewrite Eq. (2.98) as

TTM

(

1kleft

z

n2leftk0

)

= A

(

1kright

z

n2rightk0

)

. (2.100)

wherenleft andnright arez-components of the wavevector of light on the left and rightsides of the structure, respectively. By elimination ofA, Eq. (2.100) can be easily re-duced to a single transcendental equation for the eigenmodes of the structure:

tTM11

krightz

n2rightk0

+ tTM12

kleftz kright

z

n2leftn

2rightk

20

− tTM21 − tTM

22

kleftz

n2leftk0

= 0 . (2.101)

At normal incidence, equations for the eigenmodes of light in TE- (Eq. (2.96)) andTM- polarisations (Eq. (2.101)), become formally identical. This is quite reasonable, asat normal incidence there is no difference between TE- and TM-polarisations. One canshow using transformations (2.87) and (2.88) that they bothreduce to

t11nright − t12nrightnleft − t21 + t22nleft = 0 (2.102)

with tij being elements of the transfer matrix at normal incidence defined by Eqs. (2.78–2.79). Comparing condition (2.102) and expressions (2.71b) and (2.81b) one can seethat reflection and transmission coefficientsrs andts become infinite at the complexeigenfrequencies of the system. This result also holds for oblique incidence. It is not


unphysical. We note that in optical measurements of reflectivity and transmission wealways detect signals at real frequencies and the conditionholds

|rs|2 +nright

nleft|ts|2 ≤ 1 . (2.103)

(the equality holds in case of no absorption and scattering).

2.6.1 Photonic bands of 1D periodic structures

Consider an infinite structure whose refractive index is homogeneous in thexy-planeand whose dependency on the coordinatez is a periodic function with periodd. Theshape of this function is not essential, and we shall only assume that a transfer matrixTd

across the period of the structure can be written as a productof a finite number of ma-trices of the form (2.72). Let an electromagnetic wave propagate along thez-direction.For this wave

TdΦ∣

∣

z=0= Φ

∣

∣

z=d, (2.104)

whereΦ(z) is defined by Eq. (2.76). According to Bloch theorem, it can berepresentedin the form:

Φ(z) = eiQz(

UE(z)UB(z)

)

(2.105)

whereUE,B(z) have the same periodicity as the structure andQ is a complex numberin the general case.

Note that the factoreiQz is the same for electric (E) and magnetic (B) fields in alight wave because in the normal incidence case they are linked by the relation

B(z) = − i

ck0

∂E(z)

∂z(2.106)

Substitution of Eq. (2.105) into Eq. (2.104) yields

TdΦ∣

∣

z=0= eiQdΦ

∣

∣

z=0, (2.107)

thus,eiQd is an eigenvalue of the matrixTd and therefore

det(Td − eiQd1) = 0 (2.108)

Solving Eq. (2.108), we use an important property of the matrix Td following fromEqs. (2.78—2.79):

det(Td) = 1 (2.109)

Thus we reduce Eq. (2.108) to

1 − (T11 + T22)eiQd + e2iQd = 0 , (2.110)

whereTij are the matrix elements ofTd. Multiplying each term bye−iQd we finallyobtain:


cos(Qd) =T11 + T22

2. (2.111)

The right-hand side of this equation is frequency-dependent. The frequency bandsfor which

∣

∣

∣

∣

T11 + T22

2

∣

∣

∣

∣

≤ 1 (2.112)

are allowed photonic bands. In these bandsQ is purely real and the light wave canpropagate freely without attenuation. On the contrary, thebands for which

∣

∣

∣

∣

T11 + T22

2

∣

∣

∣

∣

> 1 (2.113)

are usually calledstop-bandsor optical gaps(see Fig. 2.7). In these bandsQ has anonzero imaginary part that determines the decay of propagating light waves. All this iscompletely analogous to electronic bands in conventional crystals. Eqs. (2.111—2.113)are also valid in the oblique incidence case, while the form of the matrixTd is sensitiveto the angle of incidence and band boundaries shift as one changes the incidence angle.

Fig. 2.7: Experimental and theoretical transmittance of a periodic structure composed by Si3N4 and SiO2dielectric layers (a) compared to the calculated photonic dispersion for this structure (b), from Gerace et al.(2005). The stop-bands are shown in grey.

We remind that aBragg mirror is a periodic structure composed of pairs of layers ofdielectric or semiconductor materials (see Fig. 2.8) characterised by different refractive


indices (sayna andnb). The thicknesses of the layers (a andb, respectively) are chosenso that

naa = nbb =λ

4. (2.114)

Condition (2.114) is usually called theBragg interference conditiondue to its sim-ilarity to the positive interference condition for X-rays propagating in crystals pro-posed in 1913 by English physicists Sir William Henry Bragg and his son Sir WilliamLawrence Bragg. The Bragg mirrors are also frequently called distributed Bragg re-flectorsor DBRs. The wavelength of lightλ marks the centre of the stop-band of themirror. For the wavelengths inside the stop-band the reflectivity of the mirror is close tounity. In the following we assumena < nb (na is the refractive index of the first layerfrom the surface,nb is the refractive index of the next layer). We describe the opticalproperties of the mirror within its stop-band using the transfer matrix approach.

Fig. 2.8: Electronic microscopy image of the GaN/AlGaN Bragg mirror grown by E. Calleja’s group inMadrid and reported by Fernandez et al. (2001).

At normal incidence, the transfer matrices across the layers that compose the mirrorare:

Ta =

(

cos(kaa)ina

sin(kaa)

ina sin(kaa) cos(kaa)

)

, Tb =

(

cos(kbb)inb

sin(kbb)

inb sin(kbb) cos(kbb)

)

,

(2.115)whereka = ωna/c andkb = ωnb/c. The transfer matrixT across the period of themirror is their product:

T = TbTa . (2.116)

An infinite Bragg mirror represents the simplest one-dimensional photonic crystal.Its band structure is given by the equation (2.111). Its solutions with realQ form allowed


William Henry Bragg (1862–1942) and William LawrenceBragg (1890–1971).

Father and son, the two Braggs shared the 1915 Nobel Prize in Physics “for their services in theanalysis of crystal structure by means of X-rays.” When a young Bragg, aged 5, broke his arm by fallingoff his tricycle, he was radiographed byX-rays which his father had recently learned about from Rontgen’sexperiments. In 1912, aged 22 and a first year university student, Bragg discussed with his father of his ideasof diffraction by crystals which he would develop into Bragg’s law. His father developed the spectrometer.He became the youngest Nobel prize. He is also credited as having played an important role in his support ofidentifying the DNA double helix, as then head of the Cavendish laboratory.

photonic bands while solutions with complexQ having a nonzero imaginary part formphotonic gaps or stop-bands.

At the central frequency of the stop-band, given by

ω =2πc

λ(2.117)

the matrixT becomes:

T =

(−na

nb0

0 − nb

na

)

. (2.118)

Its eigenvalues are

exp[iQ(a+ b)] = −nanb

, exp[−iQ(a+ b)] = −nbna

. (2.119)

The reflection coefficient of a semi-infinite Bragg mirror atω = ω can be foundfrom the condition:

T

(

1 + rnleft(1 − r)

)

= −nanb

(

1 + rnleft(1 − r)

)

, (2.120)

which readily yieldsr = 1.In the vicinity of ω, one can derive a simple and useful expression for the reflection

coefficient, leaving in the matrixT only terms linear in

x = (ω − ω)λ

4c. (2.121)

The matrix is written in this approximation as:


T = −(

na

nbi(

1na

+ 1nb

)

x

i(na + nb)xnb

na

)

. (2.122)

Eq. (2.120) yields in this case

r =nleft

(

na

nb− nb

na

)

− i(na + nb)x

nleft

(

na

nb− nb

na

)

+ i(na + nb)x= exp

(

inanbλ

2nleft(nb − na)c(ω − ω)

)

= eiα(ω−ω) .

(2.123)The coefficient

LDBR =nanbλ

2(nb − na)= αn0c (2.124)

is frequently called theeffective lengthof a Bragg mirror. Note that it is close butnot exactly equal to the penetration lengthL of the light field into the mirror atω = ω.The lengthL can be easily obtained from the eigenvalues of the matrix (2.118):

L =a+ b

ln nb

na

. (2.125)

One can see from Eq. (2.123) that atω = ω the reflection coefficient of the Braggmirror is equal to 1, which means that the amplitudes of incident and reflected waveshave the same sign and absolute value at the surface of the mirror. That is why, themaximum (antinode) of the electric field of light is at the surface. We note that this isonly true if na < nb. In the opposite case, the amplitude reflection coefficient at thecentre of stop-band changes sign and the electric field has a node at the surface.

For a finite-size mirror, the reflection coefficient within the stop-band is differentfrom unity due to the non-zero transmission of light across the mirror. It can be foundfrom the matrix equation:

TN

(

1 + rn0(1 − r)

)

=

(

tnf t

)

, (2.126)

wherer andt are the amplitude reflection and transmission coefficients of the mirror,Nis the number of periods in the mirror andnf is the refractive index behind the mirror.At the centre of the stop-band:

r =

(

nb

na

)2N

− nf

n0

(

nb

na

)2N

+nf

n0

, t =

(

− nb

na

)N

(

nb

na

)2N

+nf

n0

. (2.127)

As follows from these formulas, the higher the contrast betweenna andnb, the betterthe reflectivity of the mirror.


A further important characteristic of a Bragg mirror is the width of its stop-band andthis can be found from Eq. (2.111). The boundaries of the firststop-band are given bythe condition:

T11 + T22

2= −1 . (2.128)

Substituting the matrix elements of the product of matrices(2.115) one easily obtains

cos2 Ω − 1

2

(

nbna

+nanb

)

sin2 Ω = −1 , (2.129)

whereΩ = kaa = kbb, therefore

cosΩ = ±nb − nanb + na

. (2.130)

This allows us to obtain the stop-band width (in frequency):

∆ =8c

λ

(

π

2− arccos

nb − nanb + na

)

≈ 8c

λ

nb − nanb + na

. (2.131)

The stop-band width increases with an increase in the contrast between the two refrac-tive indices. Fig. 2.9 shows the calculated reflectivity of Bragg mirrors made of differentsemiconductor and dielectric materials but all havingλ = 1550nm. One can see thatthe stop-band width can achieve a few hundreds of nanometersfor high contrast ofrefractive indicesna andnb.

Finally, under oblique incidence the optical thickness layers composing a Braggmirror change. The phase gained by light crossing a layer of thicknessa at an angleφais given by

θ =ω

cnaa cosφa , (2.132)

wherena is the refractive index of this layer. It is evident that the frequency whichfulfills the Bragg interference conditionθ = π/2 is higher for oblique angles than fora normal angle. This is why, at oblique angles, stop-bands ofany Bragg mirror shifttowards higher frequencies. More details on the phases of reflection coefficients of themirrors at oblique incidence can be found in the textbook by Kavokin & Malpuech(2003)

Metallic reflectivityis usually not so perfect as dielectric reflectivity. Metalsreflectlight because they have a large imginary component of the refractive index. Consideran interface between a dielectric having the real refractive indexn1 and a metal havingthe complex refractive indexn2 = n + iκ (we note that the absorption coefficient ofthe metal is proportional to the imaginary part of its refractive index:α = ωκ/c). Thereflection coefficient for light incident normally from the dielectric to the metal reads:

r =1 − n− iκ

1 + n+ iκ= exp

(

−2i arctanκ

1 + n

)

− 2n√

(1 + n)2 + κ2exp

(

−i arctanκ

1 + n

)

. (2.133)


Fig. 2.9: Reflectivity of different Bragg mirror structures. All stop-bands are centered at the same Braggwavelength of 1550nm. (from the teaching materials of the Institut fur Hochfrequenztechnik, Technical Uni-versity of Darmstand).

Clearly, the reflectivity of a metal increases with increaseof κ and decrease ofn.In the limit of n → 0, κ → ∞, the reflection coefficientr → −1 which means thatthe incident and reflected waves compensate each other and the electric field intensityis close to zero at the surface of metal.

In everyday life one uses metallic mirrors. A method of backing a plate of flat glasswith a thin sheet of reflecting metal came into widespread production in Venice duringthe 16th century; an amalgam of tin and mercury was the metal used. Thechemicalprocess of coating a glass surface with metallic silver was discovered by Justus vonLiebig in 1835, and this advance inaugurated the modern techniques of mirror making.Present-day mirrors are made by sputtering a thin layer of molten aluminum or silveronto a plate of glass in a vacuum. The metal used determines the reflection character-istics of the mirror; aluminum is cheapest and yields a reflectivity of around 88%-92%over the visible wavelength range. More expensive is silver, which has a reflectivity of95%-99% even into the far infrared, but suffers from decreasing reflectivity (< 90%) inthe blue and ultraviolet spectral regions. Most expensive is gold, which gives excellent(98%-99%) reflectivity throughout the infrared, but limited reflectivity below 550nmwavelength, resulting in the typical gold colour.

Exercise 2.5 (∗∗) Find the frequencies of the eigenmodes of an optical cavity composedby a homogeneous layer of widtha and refractive indexn sandwiched between twomirrors having the amplitude reflection coefficientsr.


Fig. 2.10: The real and imaginary part of the refractive index of gold, from Torok et al. (1998).

Exercise 2.6 (∗∗) If one of the layers in the infinite Bragg mirror has a different thick-ness from all other layers, it acts as a single defect or impurity in an ideal crystal.Localized photonic modes appear at such a defect. Find theireigenenergies.

2.7 Planar microcavities

In microcavities, the cavity layer can be considered as a “defect” layer within a regularBragg structure (see Exercise 2.6 in previous section). TheFabry-Perot confined modesof light appear within the cavity layer under condition (from Eq. (2.80)):

reikzLc = ±1 , (2.134)

whereLc is the cavity width andr is the reflection coefficient of the Bragg mirror.Alternatively, cavities with metallic mirrors can be used.In this case the reflection co-efficient r is given by Eq. (2.133). In this Section we only consider dielectric Braggmirrors characterized by the reflection coefficientr ≈ 1.

At normal incidence, for the ideal infinite Bragg-mirror (r = 1) a linear equationfor the frequencies of the eigenmodes can be written:

α(ωc − ω) + kzLc = jπ, j ∈ N . (2.135)

The difference between microcavities and conventional cavities is in the value ofLc.In case of microcavities, it is of the order of the wavelengthof visible light dividedby the refractive index of the cavity material (i.e. typically 0.2-0.4µm). The size ofconventional optical cavities is much larger. That is why the indexj of the eigenmodesof microcavities is low and the spacing between their frequencies is so large that usuallyeach stop-band contains only one microcavity mode. On the contrary, in conventionalcavities, the spacing between eigenmodes is small and many of them are present withineach stop-band. Usually, the microcavity width is designedto be an integer number of


times larger than one of the regular layers in Bragg mirrors,hencekzLc = jπ for ωc =ω. The electric field profile in the eigenmode of a typical planar microcavity is shownin Fig. 2.11.

Fig. 2.11: Refractive index profile and intensity of electric field of the eigen mode of a typical planar micro-cavity.

Transmission spectra of microcavities show peaks at the frequencies correspond-ing to the eigenmodes. Light is able to penetrate inside the cavity and be transmittedthrough it at these frequencies. This is an optical interference effect which can be alsounderstood as the resonant tunnelling of photons: the photon from outside excites theeigenmode of the cavity and then jumps out crossing the mirrors. Fig. 2.12 shows thetransmission spectrum of a HfO2/SiO2 microcavity compared with the reflectivity ofthe single Bragg mirror.

Fig. 2.12: Transmission spectrum of a HfO2/SiO2 microcavity and reflectivity of the single HfO2/SiO2

Bragg mirror containing 7 pairs of quarter-wave layers, from Song et al. (2004).

In the absence of absorption or scattering the reflectivityR is linked to the transmis-sionT by a simple relation:


R = 1 − T , (2.136)

so that the reflection spectra of dielectric microcavities exhibit dips identical to the peaksof transmission spectra.

One can see that the transmission peak corresponding to the cavity mode is broad-ened. Broadening is inevitably present because of the finitethickness of the Bragg mir-rors and resulting possibility for light to tunnel through the cavity even within the stop-bands of the mirrors. The quality factor of the cavityQ is defined as the ratio of thefrequency of the cavity mode to the full-width at half-maximum of the peak in trans-mission corresponding to this mode. The quality factor can be also defined as:16

Q = ℜωcU

dU/dt, (2.137)

whereU is the electromagnetic energy stored in the cavity,dU/dt is the rate of theenergy loss due to the tunnelling of light through the mirrors during a period of timedtandωc is the complex eigenfrequency of the cavity mode given by Eq.(2.134). Wenote that equation (2.135) is not exact in real structures having finite Bragg mirrors.In particular, it yields purely real solutions while true eigenfrequencies of the cavitymodes are complex. To determine the quality factor of cavityit is important to know theimaginary part ofωc as we show below.

The probability that a photon goes outside is proportional to the number of photonsinside the cavity, which yields an exponential dependence of the energyU of the cavitymode on time:

U(t) = U0e−ℜωct/Q (2.138)

τ = Q/ℜωc is the lifetime of the cavity mode. Having in mind the link between theenergy of an electromagnetic field and its complex amplitudeE(t), namelyU(t) ∝|E(t)|2, we obtain

E(t) = E0 exp(−ℜ(ωct/2Q)) exp(−iℜωct) , (2.139)

whereE0 is the coordinate dependent amplitude. Standard Fourier transformation givesus the frequency dependence of the field amplitude:

E(ω) =1√2π

∫ ∞

0

E0 exp(−ℜ(ωct/2Q)) exp(−iℜωct)dt , (2.140)

so that

U(ω) ∝ |E(ω)|2 ∝ 1

(ω −ℜωc)2 + (ℜωc/2Q)2. (2.141)

Expression (2.141) determines the transmission spectrum of the cavity. The resonanceshape has a full width at half-maximum equal toℜωc/Q which shows the equality

16We noteℜz andℑz the real part and imaginary part of the complex quantityz ∈ C, respectively.


of two definitions of the quality factor we have given. The denominator of expres-sion (2.141) vanishes if

ω = ℜωc + iℜωc/2Q . (2.142)

Having in mind that the complex eigenfrequency of the cavityis one which corre-sponds to the infinite transmission (see Section 2.6), we obtain from Eq. (2.142)

Q =ℜωc2ℑωc

, (2.143)

For a specific normal mode of the cavity this quantity is independent of the mode ampli-tude. The imaginary part of the eigenfrequency of the cavitymode can be easily foundfrom Eq. (2.134) as

ℑωc = − 1

αln |r| , (2.144)

where the absolute value of the reflection coefficient of the Bragg mirror with a finitenumber of layers is given by Eq. (2.127). In high quality microcavities, the quality factorcan achieve a few thousands.

We note, that the quality factor is different for different eigen modes of the samecavity, in general. In particular,Q → ∞ (if there is no absorption) for the guidedmodes which have purely real eigenfrequencies. From Eqs. (2.138, 2.143) follows thatthe lifetime of the cavity mode

τ =1

2ℑωc. (2.145)

It characterizes the average time spent by each photon inside the cavity before goingout by tunnelling through the mirrors. The lifetime of guided modes is infinite. Theoret-ically, light never goes out from the ideal infinite waveguide. In reality, each photonicmode of any structure has a finite lifetime. The photons escape from the eigenmodes dueto scattering by defects, interaction with the crystal lattice, absorption, etc. The finessof the cavity,F , (see Chapter 1), is linked to the lifetime of a cavity mode bythe rela-tion F = ∆ωcτ where∆ωc is the splitting between real parts of the eigenfrequenciesof the neighbouring cavity modes.

In the following we omit the prefixℜ while speaking about the real part of the cavityeigenfrequency and will simply note itωc for brevity. However, we shall remember thatit also has an imaginary part,ℑωc = γc.

The deviation of the cavity mode frequency from the center ofthe stop-band ofthe surrounding Bragg mirrorsω always takes place in realistic structures where thethicknesses of all layers slightly change across the sample. The detuning

∆ = ωc − ω (2.146)

is an important parameter, which governs the splitting between TE- and TM-polarizedcavity modes at oblique incidence.


At ∆ = 0 one can find the in-plane dispersion of the cavity Fabry-Perot mode asthe solution to exercise 2.5 of the previous Section:

ω ≈ cπj

ncLc+ck2xyLc

2ncπj, (2.147)

with nc being the cavity refractive index, which yields the effective mass of the photonicmode, from~ω(k) ≈ ~ωc + ~2k2/(2mph) where

mph =ncπj

cLc. (2.148)

To take into account the polarisation dependence of the dispersion of microcavitymodes one should take into account the angle-dependence of the reflection coefficientof a Bragg mirror. At oblique incidence, one can conveniently define the centre of thestop-band as a frequencyω for which the phase of the reflection coefficient of the mir-ror is zero. The transfer matrices are modified in the case of oblique incidence andare different for TE- and TM-polarisations, as described inthe previous Section (seeEqs. (2.86–2.87)). Condition (2.123) still holds at oblique incidence. It allows one toobtain the reflection coefficient of the Bragg mirror in the form

rTE,TM = rTE,TM exp(iαTE,TM(ω − ωTE,TM)) (2.149a)

= rTE,TM exp(

inccLTE,TM

DBR cos(φ0(ω − ωTE,TM))

(2.149b)

where for TE-polarisation:

rTE =

√

1 − 4nfn0

cosϕfcosϕ0

(

na cosϕanb cosϕb

)2N

, (2.150a)

ωTE =πc

2(a+ b)

na cosϕa + nb cosϕbnanb cosϕa cosϕb

, (2.150b)

LTEDBR =

2n2an

2b(a+ b) cos2 ϕa cos2 ϕbn2

0(n2b − n2

a) cos2 ϕ0, (2.150c)

whereN is the number of periods in the mirror,ϕ0 is the incidence angle,ϕa,b are thepropagation angles in layers with refractive indicesna, nb, respectively, andϕf is thepropagation angle in the material behind the mirror which has a refractive indexnf .They are linked by the Snell-Descartes law:

n0 sinϕ0 = na sinϕa = nb sinϕb = nf sinϕf (2.151)

In TM-polarisation:


rTM =

√

1 − 4nfn0

cosϕ0

cosϕf

(

na cosϕbnb cosϕa

)2N

, (2.152a)

ωTM =πc

2

na cosϕb + nb cosϕananb(a cos2 ϕa + b cos2 ϕb)

, (2.152b)

LTMDBR =

2n2an

2b(a cos2 ϕa + b cos2 ϕb)

n20(n

2b cos2 ϕa − n2

a cos2 ϕb), (2.152c)

One can see thatωTM increases faster thanωTE with an increase of the incidence angle.LDBR increases with the angle in TM-polarisation and decreases in TE-polarisation.Finally, r increases with angle in TE-polarisation and decreases in TM-polarisationif n0 = nf .

Substituting into the equation for the cavity eigenmodes (2.134) the renormalizedstop-band frequencies (2.139, 2.143) and effective lengths (2.140, 2.144), one can ob-tain as shown by Panzarini et al. (1999a), the angle-dependent TE-TM splitting of thecavity modes:

ωTE(ϕ0) − ωTM(ϕ0) ≈LcLDBR

(Lc + LDBR)22 cosϕeff sin2 ϕeff

1 − 2 sin2 ϕeff

∆ , (2.153)

whereϕeff ≈ arcsin n0

ncsinϕ0 andLDBR is given by (2.124). Obviously, the splitting

is zero atϕ0 = 0 as there is no difference between TE- and TM-modes at normalincidence. One can see that the sign of the TE-TM splitting isgiven by the sign ofthe detuning between the cavity mode frequency at normal incidence and the centerof the stop band. Changing the thickness of the cavity one cantune∆ and change theTE-TM splitting in large limits. Usually, the TE-TM splitting is much smaller than theshift (2.136) (note the relationkxy = (ω/c) sinϕ0).

Finally note that in addition to the Fabry-Perot cavity modes described above, themicrocavities possess rich spectra of guided modes. Their spectrum can be found fromEq. (2.96) (for TE-polarisation) and Eq. (2.101) (for TM-polarisation) atkxy > ω/c.

In summary to this Section, the finite transmittivity of the Bragg mirrors leads tothe broadening of the peaks in transmission and dips in reflection corresponding to thecavity modes. This broadening is related to the imaginary part of the eigenfrequencyof the modes and is characterized by a quality factor of the cavity Q. The dispersion ofconfined light modes in microcavities is parabolic to a good accuracy, while the parabolacan have a different curvature in TE- and TM-polarisations.The splitting of TE and TMcavity modes can have a positive or negative sign depending on the difference betweenthe position of the mode atkxy = 0 and the center of the stop-band of surroundingBragg mirrors.

Exercise 2.7 (∗∗) Find the quality factor of a GaAs microcavity (refractive indexnc =3.5, thicknessLc = 244nm) surrounded by AlAs/Ga0.8Al0.2As Bragg mirrors contain-ing 10 pairs of layers each (refractive indices of AlAs,na = 3.0, of Ga0.8Al0.2As,nb = 3.4, layer thicknessesa = 71nm,b = 63nm respectively).


2.8 Stripes, pillars, and spheres: photonic wires and dots

Progress in fabrication of so-calledphotonic structures, i.e., dielectric structures withintentionally modulated refractive indices, has made important the detailed understand-ing of the spectra, shape and polarisation of the confined light modes in these structures.In general, this is not an easy task as the variety of photonicstructures studied till nowis huge and the art to design them (“photonic engineering”) is developing rapidly.

The starting idea of the photonic crystal engineering formulated in 1986 by theAmerican physicist Eli Yablonovitch was to create the bandgap for light using the pe-riodic dielectric structures. The interference effects inplanar structures can induce for-mation of the stopbands or one-dimensional photonic gaps aswe have discussed inthe previous section in relation to the Bragg mirrors. More complex structures wherethe refractive index is modulated along three cartesian axes allow for creation of three-dimensional photonic gaps. Theoretically, photonic crystals represent ideal non-absorbingmirrors and can be efficiently used for the lossless guiding of light. They have a hugepotentiality for applications in future integrated photonic circuits as Fig. 2.13 shows.

Lord Rayleigh (1842–1919) discovered and interpreted cor-rectly what is now known asRayleigh scatteringand surfacewaves, now called solitons.

He is by far much more renown under his peerage than underhis real name, John Strutt. However, he acquired the title byhisthirties. Other exceptional achievements include co-discovery ofargon for which he was awarded the Nobel prize in 1904. Withthe Rayleigh scattering, he was the first to explain why the skyis blue (this was in 1871). A gifted experimentalist despitetougheconomy resulting in basic equipment, he pushed teaching oflaboratory courses to undergraduates with fervour. His interestsalso touched less mundane topics such as “insects and the colourof flowers”, “the irregular flight of a tennis ball”, “the soaringof birds”, “the sailing flight of the albatross” and of course, theproblem of the Whispering Gallery. In a presidential British As-sociation address, he said: “The work may be hard, and the disci-pline severe; but the interest never fails, and great is the privilegeof achievement.”

In reality, inevitable imperfections in photonic crystalslead to the losses because ofthe Rayleigh scattering of light. A detailed analysis of various photonic crystal systemscan be found in the textbooks by Yariv & Yeh (2002) and Joannopoulos et al. (1995).The description of photonic crystals is out of scope of this book. We shall mostly ad-dress the light-matter coupling in microcavities, i.e., cavities in the photonic structures.The photonic engineering is indeed a powerful tool for the control of light-matter cou-pling strength: the density of states of the photon modes governs efficiently the emis-sion of light by the media (which is referred to as thePurcell effect, addressed in de-tail in Chapter 6). In planar structures considered in the previous Section the photonicmodes have two degrees of freedom linked to the in-plane motion (Fig. 2.14a). Addi-tional confinement of light can be achieved in stripes where only the motion along thestripe axis remains free (Fig. 2.14b). Stripes as well as cylinders can be calledpho-

STRIPES, PILLARS, AND SPHERES: PHOTONIC WIRES AND DOTS 65

Fig. 2.13: Future concept optical integrated circuits withuse of photonic crystals, by Noda et al. (2000). Thecomplete three-dimensional photonic gap would allow lossless propagation of light in bent waveguides.

tonic wires. More radical enhancement of the photonic confinement can beachieved inpillar cavities (Fig. 2.14c). Here in-plane photonic confinement is not perfect, so thatthe leakage of light from the pillar is possible for a part of the eigenmodes, but thequality factor of the pillar can be high enough to strongly enhance the efficiency oflight-matter coupling with respect to the planar cavities.Finally, a realistic object al-lowing for a three-dimensional photonic confinement is a dielectric (or semiconductor)sphere (Fig. 2.14d). Both pillar cavities and spheres can bereferred to asphotonic dots.The light modes confined in such “dots” have a discrete spectrum and quite peculiarpolarisation properties. They can be coupled to the opticaltransitions inside the “dot”which is potentially interesting for observation of the Purcell effect and various non-linear optical effects. In the rest of this Section we give some basic formulae for thestructures having a cylindrical symmetry (cylinders and pillar cavities) and sphericalsymmetry (spheres).

2.8.1 Cylinders and pillar cavities:

Let us solve the wave equation (2.23) in cylindrical coordinates. The Laplacian operatorreads in this case:

∇2 =1

r

∂

∂r

(

r∂

∂r

)

+1

r2∂2

∂θ2+

∂2

∂z2, (2.154)

wherer, θ, z are cylindrical coordinates (see Fig. 2.15).Let us consider an infinite cylinder of radiusa and dielectric constantε. For sim-

plicity we only consider the modes with cylindrical symmetry, which means that electricand magnetic fields are independent ofθ. Solving the wave-equation with the cylindri-cal Laplacian (2.154) one can represent thez-components of the field amplitudes in thiscase as


Fig. 2.14: (a) Schematic representation of aplanar microcavity ; (b) aphotonic stripe as seen by electronicmicroscopy by Patrini et al. (2002); (c) apillar as seen by electronic microscopy by the group in Sheffield,where the pillars can be grown elliptically to split the polarisation states; (d) schematic representation of asphere (photonic dot).

Fz(ρ) = J0(γρ), ρ ≤ a , (2.155a)

Fz(ρ) = AK0(βρ), ρ ≥ a , (2.155b)

whereFz is either electric or magnetic field,J0 is the Bessel function,K0 is the modi-fied Bessel function,A is a constant which can be determined from Maxwell boundaryconditions (see Section 2.5), which require in our case conservation of thez- andθ-components of the fields,γ = ((ω2n2/c2) − k2

z)1/2, β = (k2

z − (ω2/c2))1/2 andkzis the wavevector of light along the axis of the cylinder. Other components of the fieldscan be found using the Maxwell equations (2.15c) and (2.16).Inside the cylinder:

Bρ =ikzγ2

∂Bz∂ρ

, (2.156a)

Bφ =in2kzcγ2

∂Ez∂ρ

, (2.156b)

Eφ = −Bρ , (2.156c)

Eρ =c

n2Bφ , (2.156d)

k0 = ω/c and outside the cylinder


Fig. 2.15: Cylindrical coordinates.

Bρ =−ikzβ2

∂Bz∂ρ

, (2.157a)

Bφ =−ikzβ2c

∂Ez∂ρ

, (2.157b)

Eφ = −Bρc , (2.157c)

Eρ = Bφc . (2.157d)

The triplets(Bz, Bρ, Eφ) (TE-mode) and(Ez, Eρ, Bφ) (TM-mode) are indepen-dent of each other. Let us find the spectrum of TE-modes. From Eqs. (2.156, 2.157) thefield components inside the cylinder can be expressed as:

Bz =1

cJ0(γρ), Bρ = − ikz

cγJ1(γρ), Eφ =

ikzγJ1(γρ) (2.158)

and outside the cylinder:

Bz =1

cAK0(βρ), Bρ = − ikzA

cβK1(βρ), Eφ =

ikzA

βK1(βρ) . (2.159)

Application of Maxwell boundary conditions atρ = a yields

− J1(γa)

γJ0(γa)=

K1(βa)

βK0(βa). (2.160)

This is a transcendental equation for the eigenfrequenciesof the cylinder which deter-mines the dispersion of both guided modes (real eigenfrequency, infinite lifetime andquality factor,(ω/c) < kz ≤ n(ω/c)) and Fabry-Perot modes (complexω having a


finite negative imaginary part, finite lifetime and quality factor,0 ≤ kz ≤ (ω/c)). In asimilar way one can obtain the spectrum of TM-modes:

−n2J1(γa)

γJ0(γa)=

K1(βa)

βK0(βa). (2.161)

In cylindrical waveguides, light can freely propagate along thez-axis. The electricand magnetic fields of the propagating modes can be found by multiplication of theamplitudes found above by an exponential factorexp

(

i(kzz−ωt))

. In pillar microcav-ities, light propagation is confined in thez-direction usually by the Bragg mirrors (seeFig. 2.14c for an electron microscopy image). Because of thephotonic confinementin the z-direction,kz takes now discrete values approximately given by Eq. (2.135),whereLc would be the distance between the two Bragg mirrors. The spectrum of eigen-frequencies of the pillar microcavity is also discrete which allows it to qualify as aphotonic dot. As in the infinite cylinder, the eigen modes of such a dot can be formallydivided in two categories: the “Fabry-Perot” modes having0 ≤ kz ≤ (ω/c) and the“guided” modes with(ω/c) ≤ kz. The profile of electric and magnetic fields in thez-direction can be obtained as for the planar cavities. If theBragg mirrors confiningthe pillar cavity are infinite, the “guided” modes have an infinite lifetime, while “Fabry-Perot” modes have a finite lifetime and finite quality factor in all cases. In realistic struc-tures, the lifetime of all the modes is finite while it can become very long in the caseof efficient photonic confinement. The quality factor recordvalues, obtained in toroidalmicroresonators, exceed108, as shown by Armani et al. (2003), which corresponds to alifetime of order of10−7s.

Pillar microcavities have attracted special attention very recently due to the exper-imental observation of the strong coupling of light with individual electron-hole statesin semiconductor quantum dots embedded inside the cavities. This is further discussedin Chapter 4.

2.8.2 Spheres:

To describe the light-modes in the dielectric or semiconductor microspheres it is conve-nient to rewrite the wave-equation (2.23) in spherical coordinates.

The Laplacian in spherical coordinates reads:

∇2 =1

r2∂

∂r

(

r2∂

∂r

)

+1

r2 sin2 φ

∂2

∂θ2+

1

r2 sinφ

∂

∂φ

(

sinφ∂

∂φ

)

, (2.162)

wherer, θ, φ are the radius, polar and azimuthal angle, respectively (see Fig. 2.16). Letus consider a sphere of radiusa and dielectric constantε surrounded by vacuum.

The solution for the amplitudes of electric and magnetic field inside the sphere canbe represented for a given mode as:

F inr

F inθ

F inφ

=

a1

a2

a3

jl(kinr)Pml (cos θ)eimφ , (2.163)


Fig. 2.16: A dielectric sphere and the path of light in the whispering gallery mode (left); localization of lightin a sphere due to multiple internal reflections (right)

whereFr, Fθ, Fφ are the components of either electric or magnetic field,jl(x) is thespherical Bessel function of the first kind,Pml (x) is the associated Legendre polyno-mial, l andm are angular and azimuthal mode numbers,a1, a2, a3 are coefficients(different for electric and magnetic fields, of course). An additional radial number ofthe moden allows the radial wavenumber inside the cavity to be linked to the cavityradiusa: kin ≈ (πn/a) with n ∈ N.

The fields outside the sphere are given by

F outr

F outθ

F outφ

=

b1b2b3

hl(koutr)Pml (cos θ)eimφ , (2.164)

wherehl(x) is the spherical Hankel function of the first kind,kout is the wavenum-ber outside the cavity,b1, b2, b3 are coefficients. The links between linear coefficientsfor electric and magnetic field are always given by the Maxwell equations (2.15c),(2.16). In general form they are rather complex. We address the interested reader tothe book by Chew (1995) containing a rigorous derivation of the spectra of the eigen-modes of dielectric spheres. Interestingly, there is no allowed optical modes having aspherical symmetry (i.e. having the angular numberl = 0). Such a mode would havea diverging magnetic field in the centre of the cavity which contradicts the Maxwellequation (2.15b). As in pillars, the sets of equations for TE- and TM- modes can bedecoupled. TE-modes in this case are defined as those havingEin

r = Eoutr = 0 and for

TM-modesBinr = Bout

r = 0.The spectrum of eigenmodes of the sphere is discrete. It can be obtained by matching

of the fields (2.163) and (2.164) by Maxwell boundary conditions requiring

F inθ = F out

θ , F inφ = F out

φ . (2.165)

Substitution of the functions (2.163, (2.164) into the conditions (2.165) gives theequations for eigenmodes. In TE-polarisation:


H ′l(kouta)Jl(kina) =

√εJ ′l (kina)Hl(kouta), (2.166)

and in TM-polarisation:√εH ′

l(kouta)Jl(kina) = J ′l (kina)Hl(kouta), (2.167)

whereJl(x) = xjl(x) andHl(x) = xhl(x), with ′ meaning derivative over the argu-ment of the function.

While an exact spectrum of the light modes in a sphere requires solution of thetranscendental equations (2.166, 2.167), for qualitativeunderstanding of localization oflight in the sphere the images of ray propagation and arguments of geometrical optics arevery helpful. Actually, light can be trapped by total internal reflection near the sphere’ssurface in a resonant so-calledwhispering gallerymode localized around the equator.

In the case of a dielectric sphere the whispering gallery modes are the eigenmodeshaving high numbersl andm (high usually means higher than 10 in this context).Fig. 2.16 shows schematically how the whispering gallery modes appear. Light is prop-agating along the surface of the sphere each time experiencing an almost total internalreflection (not exactly total because of the curvature of thesurface). The cyclic boundaryconditions determine the energy spectrum of such modes.

These modes have a huge (but finite) quality factor and a long (but finite) lifetime.They can be qualified as “quasi-waveguided” modes. Propagation of whispers in thedome of Saint Paul’s cathedral is assured by such “quasi-waveguided” acoustical wavessubject to cyclic boundary conditions.

Whispering gallery modes have been studied in micrometer-size liquid droplets andglass spheres from the early days of laser physics. Now very high quality spheres areobtained by melting a pure silica fiber in vacuum, as done by Collot et al. (1993). Thetransverse dimensions of the modes can be reduced down to a few microns, the sphere’sdiameter being about 100µm. The mode is strongly confined. Fig. 2.17 shows the calcu-lated field intensity in a TE-polarized whispering gallery mode in a silica microsphere.

Fig. 2.17: TE whispering gallery modes with mode numbersn = 1, m = l = 20 (from lectures notes byIkka Nitonen.)

The value of the angular numberl for such a mode is close to the number of wave-lengths of light on the optical length of the equator of the sphere. The valuel −m+ 1


is equal to the number of the field maxima in the polar direction (i.e. perpendicular tothe equatorial plane). The radial numbern is equal to the number of maxima in the di-rection along the radius of the sphere, and2l is the number of maxima in the azimuthalvariation of the field along the equator. The resonant wavelength is determined by thenumbersn andl. The modes with lower indicesl andm have a lower quality factor andshorter lifetime. They can be referred to asFabry-Perot modes. These modes are bettersuited for coupling to the material of the sphere than whispering gallery modes as theypenetrate deeper inside the sphere.

Eli Yablonovitch, produced the first artificialphotonic crystals, the engineered counterparts of such struc-tures as on the right from Yablonovitch (2001): a butterfly wing which—with its incomplete bandgap—produces iridescent colours. Yablonovitch (1987) authored the second most highly cited Physical ReviewLetter.

The German physicist Mie solved in 1908 the problem of scattering of a plane waveby a dielectric sphere and demonstrated the existence of resonances, now known asMieresonances, linked to the eigenmodes of the sphere including the whispering galleryand Fabry-Perot modes. Mie theory has allowed, in particular, to describe the scatteringof light of Sun by droplets of water in the clouds. It explainsthe colour of the sky andappearance of rainbows and glories. Fig. 2.8.2 shows the results of calculation of thecolour of sky in the presence of two rainbows performed within Mie theory assuming500 micron-size water drops randomly distributed in the atmosphere. The simulationresult is superimposed with a photograph to demonstrate theaccuracy of the model.

Exercise 2.8 (∗) The dome of Aya Sofya Mosque in Istambul has a radius of 31 meters.Find the wavelength of the whispering gallery mode of this dome having an angularnumberl = 31.

Exercise 2.9 (∗∗∗)The classical Hanbury Brown-Twiss effectIn this chapter we have focussed on the Hanbury Brown-Twiss effect in the temporal

domain and will again when we come back to it from the quantum point of view inthe next chapter. Historically the method of intensity interferometry of light arose as ameans of measuring the angular diameter of stars, and is related to spatial correlations.

Compute the intensity correlation in space (at zero time delay) given by expression

C = 〈E∗(r1)E∗(r2)E(r1)E(r2)〉 (2.168)


GustavMie (1869–1957). In background, a computer simulation (withMiePlot) for r = 500µm waterdrops, superimposed on a photograph of a primary and secondary rainbows.

Mie is known essentially for the solutions he provided to theproblem of light described by Maxwell equa-tions interacting with a spherical particle (commonly but incorrectly called “Mie theory”, when instead of a“theory” this is the analytical solution of the equation of an actual theory, namely, electromagnetism). Initiallya pure theorist, he indulged in experimental work toward theend of his career.

for incident fieldE impinging on two pointsr1 and r2 (spatially separated detectorson earth) and which is collected from the angular spread of the source seen from theearth as planewaves with wavevectorsk and k′ i.e., such thatE(r) = Ek exp(ik ·r) + Ek′ exp(ik′ · r) for both detectors. Compare with amplitude correlations suchas measured by a conventional optical interferometer and discuss how the apparentdiameter of the star can be deduced from varying|r1 − r2|. What is the advantage ofthe HBT setup over, e.g., the Michelson one?

2.9 Further Reading

Many excellent books on light propagation in various photonic structures are availablewhich will supplement usefully the content of this chapter.The interested reader willfind further details on the subject of this Chapter in Born & Wolf (1970) and Jack-son (1975) who give a general picture of classical optics, and Yariv & Yeh (2002) andJoannopoulos et al. (1995) textbooks, which are devoted to optical properties of di-electric structures including photonic crystals. More details on transfer matrix methodfor description of the optical properties of microcavitiescan be found in the textbookby Kavokin & Malpuech (2003). Rigorous derivation of the spectra of some photonicstructures including spheres is given in the monograph by Chew (1995).

3

QUANTUM DESCRIPTION OF LIGHT

In this chapter we present a selection of important issues, concepts andtools of quantum mechanics, which we investigate up to the level ofdetails required for the rest of the exposition, disregarding at the sametime other elementary and basic topics which have less relevance tomicrocavities. In the next chapter we will also need to quantise thematerial excitation as well, but for now we limit to light which allows usto lay down the general formalism for two special cases—the harmonicoscillator and the two-level system.

73

74 QUANTUM DESCRIPTION OF LIGHT

3.1 Pictures of quantum mechanics

3.1.1 Historical background

Historically Quantum Mechanics assumed two seemingly different formulations: one byHeisenberg, calledmatrix mechanics, to which we come back in Section 3.1.5, and an-other shortly to follow by Schrodinger, based onwavefunctions. Although highly com-petitive at the start, the two “theories” now framed in modern mathematical notationsdisplay clearly their interconnection and unity.17 As the twopicturesare useful both forphysical intuition and practical use, we study both of them.We start with Schrodingerpicture which offers the best support for the postulates of Quantum Mechanics in theinterpretation of the so-calledCopenhagenschool, which is nowadays the commonlyagreed set of working rules to deal with practical issues, although as a worldview thisinterpretation is now largely discarded.18

3.1.2 Schrodinger picture

In the Schrodinger picture, one starts with theSchrodinger equation

i~∂

∂t|ψ〉 = H |ψ〉 (3.1)

written here with Dirac’s (1930) notation ofbra andkets, an elegant convention cap-turing the essentials of the mathematical structures, as isdiscussed below.H is thequantum Hamiltonianof the system to be specified for each case under considerationand~ is the reduced Planck’s constant.

The first postulate of quantum mechanics: the quantum state

The postulates which govern quantum mechanics, essentially laid down by von Neu-mann (1932), provide the recipe to use the formalism and relate it to experiments:

I — A quantum system is described by a vector—called thestateof the system—in aseparable, complex Hilbert spaceH. This vector, in Dirac’s notations, is noted|ψ〉whereψ is the set of variables needed to fully describe the system, but the notationused symbolically affords powerful abstract manipulations.19

17Quantum Theory brought about many interesting developments in the history of science for all thecontroversies among its founding fathers and their personal views which often were the occasion for greatdrama. Beyond the famous opposition between Bohr and Einstein, there were also even animus feelings be-tween Schrodinger and Heisenberg, and hearted oppositionamidst political tensions between Heisenberg andBohr who worked together on the Copenhagen interpretation.A theatrical unravelling on the birth of Quan-tum Mechanics based on recently released documents providethe impetus for the recent play of M. Frayn,Copenhagen.

18In the field ofinterpretation of Quantum Mechanics, although there is no consensus, the modern trendfavours the theory ofdecoherenceand Everett interpretations of consistent realities (or parallel universes). Weshall briefly touch upon some of these aspects which intersect with the physics of microcavities, but otherwisewill remain oblivious and stick to the conventional Copenhagen interpretation. For further studies, cf., e.g.,Quantum Theory and Measurement, J. A. Wheeler and W. Zurek (Princeton Series in Physics), 1984.

19The main advantage of Dirac’s notation is the considerable simplification it brings when handling thedualspace ofH. Whereas a ket is a vector of some given nature, a bra is actually a linear application defined

PICTURES OF QUANTUM MECHANICS 75

Paul Dirac (1902–1984), WernerHeisenberg (1901–1976) and ErwinSchrodinger (1887–1961) in theStockholm train station, 1933, before the Nobel prize ceremony to award the 1932 prize to Heisenberg forthecreation of quantum mechanicsand the shared 1933 prize to Schrodinger and Dirac forthe discovery of newproductive forms of atomic theory. The delay in awarding the 1932 Nobel prize was due to the defiance of theNobel committee towards quantum mechanics.

In this chapter, the quantum system of ultimate interest is light, for which two ab-stract systems will eventually prove sufficient to describeit fully: a two-level systemwith associated vector spaceH2 = α |0〉 + β |1〉 , α, β ∈ C (which will describethe polarisation state of light) and an harmonic oscillator, which in stark contrast tothe simple spaceH2 requires a functional space of square modulus integrable functionsHa = |ψ〉 , |〈ψ|ψ〉|2 < ∞ (and which will describe the oscillations of the normalmode of the light field, cf. section 2.1.1). These two specificcases will allow us to illus-trate in very different cases the mechanism of the theory. Wewill describe the two-levelsystem in terms of a spin and the harmonic oscillator in termsof a mechanical oscilla-tor, allowing us to recourse to widely used language and intuition. When it is time toreturn to these notions to what we initially planned them for—the quantum description

on the ket space. With a few practise, one can almost forget entirely this underlying mathematical structure.Such shortcuts motivated Bourbaki mathematician Jean Dieudonne to state “It would appear that today’sphysicists are only at ease in the vague, the obscure, and thecontradictory”. An interesting discussion ofthese conflicting approaches is given by Mermin in “What’s Wrong with This Elegance?” of March 2000issue of Physics Today. We will, of course, make ample use of such simplifications. Such “rules of thumbs”are as follows: the ket|ψ〉 goes to〈ψ|, coefficients are conjugated,α |ψ〉 → 〈ψ|α∗ and operators aretranspose-conjugated, always written in reverse order, sothat ΩΛ |ψ〉 → 〈ψ|Λ†Ω†. So for instance, the“dual” of Schrodinger equation reads

−i~ ∂∂t

〈ψ| = 〈ψ|H (3.2)

sinceH is hermitian.


David Hilbert (1862–1943) and Johnvon Neumann(1903–1957), two pure mathematicians as emblems ofthe inherent abstract nature of Quantum theory. The quantumstate is best described asa vector in a Hilbertspace, as has been axiomatised by von Neumann.

Hilbert’s interest in physics started in 1912 and became hismain preoccupation. It provided impetus bothto quantum mechanics and relativity. He invited Einstein togive lectures on general relativity in Gottingenat which occasion some believe he derived the Einstein field equations. He put forth 23 problems at theInternational Congress of Mathematicians in Paris in 1900 setting the edge of mathematical knowledge at thenew century. The 6th one is “Axiomatize all of physics.” It is, as yet, unresolved.

Von Neumann was an extraordinary prodigy. At six, he could mentally divide two eight-digit numbers. He wasfamous for memorising pages on sight and, as a child, he entertained guests by reciting the phone-book. Besideaxiomatisation of quantum physics which he connected to theHilbert spaces—thereby solving the 6th problemin this particular case—he made crushing contributions—when he did not create the field—to functionalanalysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics, statistics,game theory and complexity theory. Many place him as among the greatest genius. Fellow mathematicanStanislaw Ulam’s biography “Adventures of a Mathematician” is largely a tribute to his mentor with manyanecdotes of this peculiar character, famous for his hazardous driving, taste for parties and hypnotization bywomen.

of light—we shall stick to the common practise of keeping thevocabulary of spin andanalogies of classical mechanics, so these asides are not completely out of purpose. Tolater link with the statistical interpretation, we furtherdemand that

〈ψ|ψ〉 = 1 (3.3)

Exercise 3.1 (∗) Show that the normalisation condition, Eq. (3.3), remains satisfied atall times through the dynamics of Schrodinger equation (3.1).

We have noted|0〉 and|1〉 two basis vectors ofH2. Mathematically it is convenientto relate them to the canonical basis, i.e.,

|0〉 =

(

1

0

)

and |1〉 =

(

0

1

)

. (3.4)

Physically, we could choose to represent the first state withright circular polarisation ofthe light mode,|〉 = |0〉, and the second with left circular polarisation, and|〉 = |1〉.We will refer to such states as spin-up and spin-down, respectively (for a true spin1

2


particle we might prefer the depiction|↑〉 and|↓〉). The first postulate says that the mostgeneral state for a two-level system is|ψ〉 = (α, β)T /

√

|α|2 + |β|2 with α, β ∈ C or,if the normalisation has been properly ensured

|ψ〉 = α |0〉 + β |1〉 (3.5)

With the description in terms of Jones vectors, such a general state describes an arbitrarypolarisation. Basis (3.4) is not always the most convenient. In generic terms, anotherimportant basis reads

|+〉 =|0〉 + |1〉√

2, |−〉 =

|0〉 − |1〉√2

(3.6)

Here we have chosen the conventional notations for generic two-level systems, todayreferred to asqubitsfor their fundamental role inquantum computation, but of courseit transposes immediately with states of polarisations.

Exercise 3.2 (∗) Based on the definitions of the various possible polarisation states,express the states of linear horizontal and vertical polarisation|↔〉 , |l〉 and lineardiagonal polarisation|ւր〉 , |ցտ〉 as a function of states|〉 and|〉. Typical exampleswould be the right and left circular polarisation of light, given by, respectively:

|〉 =|↔〉 + i |l〉√

2, |〉 =

|↔〉 − i |l〉√2

(3.7)

Obtain all other relations between any two bases. Write state (3.5) in each of thesebases. How are states of elliptical polarisation described?

Exercise 3.3 (∗) Two bases are said to be conjugate when any vector of the first onehas equal projection on all vectors of the second. Study the conjugate character ofbases encountered so far.

The inner product is the vector scalar product, i.e.,

if |ψ〉 =

(

α

β

)

and |φ〉 =(γ

δ

)

, 〈ψ|φ〉 = (α∗, β∗)(γ

δ

)

= α∗γ + β∗δ . (3.8)

This illustrates the richness of Dirac’s notation as the bravector〈ψ| is now tentativelyassociated to(α, β)∗ and the inner product reads as a product of bra and ket vectors,forming a “braket” (whence the names of each vector in isolation). Being of finite di-mension,H2 is trivially complete and separable.

A choice of basis forHa first demands a choice of a space where to project thestates of the system. An oscillator could be characterised in real space by its position asit oscillates, or in momentum space by its velocity. A classical oscillator would requirespecification of both of these at a particular time to be fullyspecified. In quantum me-chanics, as the dynamics is ruled by a first-order differential equation equation (3.1), thestate is fully characterised by one only of these pieces of informations. Later we shallsee that the simultaneous specification of both is in fact impossible.


If the state is projected in real space, a basis could consistof all the states|x〉 describ-ing an oscillator located atx on a 1D axis (without loss of generality). The first postulatein this case asserts that the most general state for an oscillator is|ψ〉 =

∫

ψ(x) |x〉 dxNow the linear superposition require an integral as there isa continuous varying set ofbasis states.

The second and third postulates: observables and measurements

II — A physicalobservableis described by an hermitian operatorΩ onH. The possiblevalues obtained from an observable are its eigenvalues. If the eigenstates ofΩ arefound to be|ωi〉 with associated eigenvaluesωi, the resultωi0 is obtainedfor a system in state|ψ〉 with probability |〈ωi0 |ψ〉|2.

III — After measurement of an observableΩ which has returned the valueωi0 , the statehascollapsedto |ωi0〉, so that repeating the observation yieldsωi0 this time withcertainty.

An observableis a property of the state that one can determine through an appropri-ate measurement on the system, or vividly “something which can be observed”. Suchaccuracy in defining basic notions has been made compulsory after the counter-intuitiveimplications of quantum mechanics, of which we shall see a few in the following.

For a quantum system which has variables with classical counterparts, as is the casewith a quantum oscillator for which a position and momentum can still be measured,Bohr formulated the prototype of the second postulate in what came to be known asthe correspondence principle, which asserts that the classical variablesx (for position,here in 1D) andp (momentum, also in 1D) are upgraded in quantum theory to hermitianoperatorsX andP defined, in the position basis, as

〈x|X |y〉 = xδ(x − y) and 〈x|P |y〉 = −i~δ′(x− y) (3.9)

whereδ′ is the derivative of the delta function.Any dynamical variable function of these variables extendsto the quantum realm

in this way, so for instance the kinetic energy12mv

2 is written 12p

2/m and its quan-tum counterpart reads12P

2/m. The classical Hamiltonian20 also extends in this wayto a quantum Hamiltonian, which appears in Schrodinger equation (3.1). Therefore,one quantises an harmonic (mechanical) oscillator of massm and force constantκin phase-space of position-momentum(x, p) starting from its (classical) HamiltonianHc = p2/(2m) + κx2/2 to read quantum-mechanically

H =1

2ω(X2 + P 2) (3.10)

through correspondence

20The Hamiltonian in classical mechanics is an analytic function which describes the state of a mechanicalsystem in terms of its phase-space variables, typically position and momentum variables. It is a reformulationof Newton mechanics which is more suited to shift to quantum mechanics. For most practical use, the Hamil-tonian of a system can be understood as the energy of the system written in terms of specified coordinates.For more detailed discussions, see for instanceClassical Dynamics of Particles and Systems, S. T. Thorntonand J. B. Marion, Brooks Cole (2003).


x→ (κm)−1/4X and p→ (κm)1/4P (3.11)

with ω ≡√

κ/m, where along with quantisation of scalars(x, p) we have scaled theoperators in term of a dimensionless variableω (as we eventually wish to describe os-cillations of light modes, without reminiscence of any mechanical embryos)

For a quantum system which has no mechanical counterpart, asis the case withspin,21 the definition of mathematical observables to describe experimentally measur-able properties of a system, is the result of guesswork to adjust with experimental facts.This needs not concern us however as this procedure as been carried out long time agofor all properties which we will need to describe quantum mechanically in a microcav-ity.

The second postulate states that inH2, such an observable is described by a2 × 2hermitian matrix. Any such matrix can be decomposed as a linear superposition, overC,of the identity1 = (1 0

0 1) and thePauli matrices:

σx =

(

0 11 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 00 −1

)

(3.12)

written here, as will always be the case unless specified otherwise, in the canonicalbasis (3.4). If we consider for instance the observableSz = ~

2σz, which provides aphysical dimension to the result obtained, the second postulate asserts that ifSz is mea-sured on a system in the state (3.5), the possible outputs are±~/2 (the eigenvalues ofSzwhich are obtained straightforwardly as the operator is diagonal) and+~/2 is obtainedwith probability |α|2 while −~/2 is with probability1 − |α|2. Postulate III states thatafter the measurement,|ψ〉, previously in the superposition (3.5), has collapsed to oneof the eigenstates|0〉 or |1〉, depending on which eigenvalue has been obtained.

So when a photon is absorbed, it transfers to the detecting material an angular mo-mentum of±~, depending on which state of circular polarisation it is detected in. Theoutcome is deterministic if the photon was in one eigenstate|〉 or |〉. But accordingto postulate II and exercise 3.2 which obtains the possible linear polarisations as super-position of circular ones, then a linearly polarised photonstill impinges one quantum ofangular momentum, but now with a given probability. It is only when a beam made upof many photons is considered that distinguishing featuresof linear polarisation (likezero average angular momentum) appear. Still all photons ina pure linearly polarisedbeams are identical.

Statistical interpretation

Quantum mechanics, according to postulate II, is a probabilistic theory: the outcome ofa given experiment is in general unknown, the theory can onlyaccount for the statisti-cal spread of repeated measurements. In this context, relevant quantities to compute areaverage and spread about this average, that is, the value obtained when an experiment isrepeated on different systems all in the same quantum state.There should be an ensem-ble of systems as once a measurement has been made on one of them, it has collapsed

21Although we contend that spin also describes polarisation,which we have seen is a property of classicallight as well; this point will be clarified in section 3.2.3.


on an eigenvalue of the observable, so the next measurement should not be made on thesame physical system but on another system in the same initial quantum state.

Sinceωi0 is obtained with probability|〈ψ|ωi0 〉|2 whenΩ is measured on|ψ〉, theaverage value of this observable, written〈Ω〉, is the weighting of all possible outcomes(that is,Ω eigenvalues) and so is

∑

i ωi|〈ψ|ωi〉|2, so that through bra and ket algebra

〈Ω〉 =∑

i

ωi〈ψ|ωi〉〈ωi|ψ〉 (3.13a)

= 〈ψ|(

∑

i

ωi |ωi〉〈ωi|)

|ψ〉 (3.13b)

= 〈ψ|Ω |ψ〉 (3.13c)

since (3.13b) isΩ spelled out in its eigenstates basis, cf. Eq. (A.3) in appendix. Notethat 〈Ω〉 does not specify on which quantum state the average has been taken, whichis usually clear from the context. In case where the specification is important, Dirac’snotation once again provide a most convenient alternative (3.13c).

The dynamics of such an average follows from Schrodinger equation as:22

∂

∂t〈Ω〉 =

i

~〈[H,Ω]〉 . (3.15)

Uncertainty principle

Coming back to a single experiment on a two-level system, assume that the value+~/2has been obtained as the result of measuringSz on state (3.5), so that the system has nowcollapsed to state|0〉 according to the third postulate. If another measurement ofSz isperformed, the same value+~/2 will be obtained with probability one, and−~/2 withprobability |〈1|0〉|2 = 0. So the result is deterministic in this case. But what if the ob-servable associated to, say,Sx = (~/2)σx is now measured? One can check that states(3.6) are the eigenstates ofSx with eigenvalue±~/2 so that the system will collapse onone of them as the result of the measurement, with probability 1/2 (cf. exercise 3.2).If, after this, one returns toSz , the result has been randomised completely and the firstmeasurement will yield any possible value±~/2 with probability1/2. This is a man-ifestation of theuncertainty principlewhich arises from the second and third postulatefrom non-commuting operators.

22Spelling out the derivation of the equation of motion for a quantum average:

∂

∂t〈Ω〉 =

∂

∂t〈ψ|Ω |ψ〉 (3.14a)

=“ ∂

∂t〈ψ|

”

Ω |ψ〉 + 〈ψ|Ω“ ∂

∂t|ψ〉

”

(3.14b)

=i

~〈ψ|HΩ − ΩH |ψ〉 (3.14c)

which, contracted, yields the result. In (3.14b) we took advantage of the linearity of the differential on Hilbertspaces and their dual reminding that in this caseΩ is time-independent. In (3.14c) we substituted Schrodingerequation in the ket (3.1) and bra (3.2) spaces.


Generally, it is a necessary and sufficient condition for twooperatorsΩ andΛ toshare a common basis of eigenstates on the one hand and to commute, [Ω,Λ] = 0,on the other hand. The uncertainty principle in its most general form reads as a lowerbound for the spread in the distributions of two observables:

Var(Ω)Var(Λ) ≥(

i

2〈[Ω,Λ]〉

)2

, (3.16)

where Var(Ω) = (Ω − 〈Ω〉)2.

Exercise 3.4 (∗∗) Prove Eq. (3.16) with Schwarz inequality applied to vectors(Ω −〈Ω〉) |ψ〉 and(Λ − 〈Λ〉) |ψ〉.

From (3.16) it is apparent that the commutation relations between operators are animportant ingredient of quantum mechanics. One can check that

[X,P ] = i~ (3.17)

by computing−i~(x∂x − ∂xx) on a generic test function after applying the correspon-dence principle backward to get back to scalars from the operators. This is, when appliedto the general formula, the origin for the most famous form ofthe uncertainty relation:

∆x∆p ≥ ~

2(3.18)

Composite systems and symmetry

IV — The Hilbert space of a composite system is the Hilbert space tensor product ofthe state spaces associated with the component systems.

This postulate extends in the expected way the rules of Quantum Mechanics froma one-dimensional case to many: the Hilbert space of the entire system dimensionalityscales with the number of degrees of freedom to be described quantum mechanically23

and observables which also inherit this tensor product structure. The additional variablecan pertain to the same particle, e.g., bei) another spatial dimension orii) a property ofan altogether different character like spin, oriii) can be the same variable but for anotherparticle. Starting with|ψ1D〉 the state of a particle inH1 a single-particle Hilbert space,these three cases would lead to, respectively,i) |ψ2D〉 to be projected on〈x| ⊗ 〈y| togive the function of two variablesψ2D(x, y), ii) |ψ1D〉 ⊗ σ andiii) |ψ1D〉 ⊗ |ψ′

1D〉.The general case of an observable beingΩ1⊗Ω2, it is customary to drop the explicit

tensor sign and abbreviate it into a product,Ω1Ω2 |ψ1〉 |ψ2〉 or even and as commonly,simply |ψ1, ψ2〉 or |ψ1ψ2〉 for the state.

Composite systems however do not simply transport the quantum “weirdness” of thesingle particle case to the higher dimensional one. They bring one conceptual difficulty

23Time is an example of a variable which remains classical in non-relativistic quantum mechanics, i.e.,which is not a quantum observable and therefore is not associated to an Hilbert space. This is to be contrastedwith special relativity where on by contrast time is shown tocarry equivalent features with space variables.


of their own rooted in correlations and known asentanglement, which is at the heart ofquantum information.

The counter-intuitive physics of entangled systems is bestexemplified followingBohm & Aharonov (1957) who consider the singlet spin state oftwo particles:

|Ψ〉 =1√2

(|↑↓〉 − |↓↑〉) (3.19)

We repeat that the notation|↑↓〉 is a shorthand for|↑〉 ⊗ |↓〉 and that first ket (in thiscase|↑〉) refers to one of the particles and second ket to the other particle. These twoparticles are separated though remaining in the state (3.19) (the spatial wavefunctionpart of the system has not been written, which would change toreflect this spatial sepa-ration; the spin state can remain the same for separated particles). When the separationis significantly large, the spin of the “first” (or “left”) particle is measured. As a result,the wavefunction collapses on the measured eigenstate. This, however, has the effect ofalso collapsing simultaneously the state of the other particle. Such an experiment ex-hibits nonlocal quantum correlations, i.e., correlations with no classical counterpart ina sense that we now discuss in greater detail.

There is first the obvious correlation of the measurement which says that if the leftbranch has measured, say, spin-up, then the other is assuredto measure spin-down. Thisis the correlation part just as it applies in a classical sense, and which ensures total spinconservation. However these correlations are non-classical because they also hold inall other bases although in these cases the wavefunction does not specify the outcome.Therefore if one measures the first qubit value in the basis|±〉 of (3.6) (with same eigen-values+1 and−1) and finds, say, spin-up again, then the other bit is also−1 in the newbasis. Although the wavefunction does not specify the values of all components, the cor-relations always match. These correlations are finallynonlocalbecause this agreementholds even for any separation with the bits possibly measured simultaneously. This hasbeen confirmed experimentally by Weihs et al. (1998).

V — The wavefunction changes or retains its sign upon permutation of two identicalparticles.

This important postulate24 can be motivated by the insightful quantum mechanicalproperty ofindistinguishable identical particles, which asserts that two particles of thesame species bear no absolute or independent role to the wavefunction which describesthem both. To make this explicit, consider the wavefunctionwritten as a function of thegeneralised coordinatesqi for the ith particle out ofN , that is,ψ(q1, · · · ,qN ). Thesystem would remain the same if thejth andkth particles were to be interchanged,provided that they are identical, i.e., refer to particles of the same species which cannotbe distinguished experimentally. The quantum state would therefore also remain thesame, i.e.,

24It is little appreciated that the indistinguishable characters of the quanta is a postulate, motivated byexperimental evidence, but which in principle can be violated without undermining quantum mechanics.Messiah & Greenberg (1964) have emphasised this point for elementary particles.


ψ(q1, · · · ,qj , · · · ,qk, · · · ,qN ) = αψ(q1, · · · ,qk, · · · ,qj , · · · ,qN ) (3.20)

with the phase factorα = eiθ which does not change the observables (and thereforeyields the same quantum state). Doing this twice yieldsα2 = 1, i.e.,α = ±1. Thisshows how postulate V derives from invariance of the quantumstate upon interchangeof identical particles.

Pauli (1940) has shown in the context of relativistic quantum field theory how thisproperty of the wavefunction relates to the spin of the particles thus described and istherefore also an intrinsic property always and consistently satisfied. Particles with inte-ger spin are calledbosons(after Bose) and those of half-integer spin are calledfermions(after Fermi). It is shown that wavefunctions for bosons keep the same sign as particlesare interchanged while those for fermions change sign.25

This has considerable physical consequences of strikinglydifferent characters de-pending on the sign, although the mathematical structure assumes a simple unifyingform:

ψ(qσ(1), · · · ,qσ(N)) = ζξψ(q1, · · · ,qN ) (3.21)

with ζ = 1 for Bosons andζ = −1 for Fermions.

whereσ ∈ S is a permutation26 of [1, N ] andξ the signature ofσ.

3.1.3 Antisymmetry of the wavefunction

In the case of fermions, (3.21) leads toPauli blockingor Pauli exclusionwhich as-serts that two fermions cannot occupy the same quantum state. This is clear from (3.20)with α = −1 since in this case the probability (amplitude) is zero when theqj = qkwhich makes it impossible to find the system with two particles in the same projec-tions |qi〉. Pauli blocking can also be stated with overall quantum states in which casethe entire wavefunction vanishes. If theith particle out ofN is in state|φi〉, the wave-function of the whole system reads as a determinant since this is the mathematical ex-pression to associate sign swapping to function (column) orcoordinates (row) permuta-tions:27

ψ(q1, · · · ,qN ) ∝ det1≤i,j≤N

(

φi(qj))

(3.22)

The constant of normalisation of (3.22) depends on the orthogonality of the|φi〉. Itdiverges as some of the|φi〉 overlap to unity, as a manifestation of Pauli exclusion.

25In addition to possible (small) deviations from Bose and Fermi statistics for elementary particles, whichwould be of a fundamental character, there are also deviations which result from cooperative or compositeeffects, or of reduced dimensionalities. Such emerging statistics are typical of solid-state physics and examplesare provided in next chapter withexcitons.

26A permutationσ of the set of integers[1, N ] = 1, 2, . . . , N is a one-to-one function from[1, N ]unto itself, e.g.,σ(1) = 2, σ(2) = 1 andσ(3) = 3 is a permutation of[1, 3]. There areN ! permutationsof [1, N ]. The set of all permutations is writtenS. The signatureξ of a permutation, also known as itsparity, is±1 according to whether an even or odd number of pairwise swappings is required to bring thesequence(1, . . . , N ) into (σ(1), . . . , σ(N)).

27With only two particles to simplify notations, in respective states|φi〉, i = 1, 2, the total Fermion

wavefunction readsψ(q1,q2) ∝ φ1(q1)φ2(q2) − φ1(q2)φ2(q1) =˛

˛

˛

φ1(q1) φ2(q2)φ1(q2) φ2(q1)

˛

˛

˛.


Exercise 3.5 ∗ Show that in the case where theN single particle states are orthogonal,〈φi|φj〉 = δi,j , the constant of normalisation of the symmetrized or antisymmetrizedwavefunction is1/

√N !.

3.1.4 Symmetry of the wavefunction

If the sign remains the same in (3.21), there is no cancellingand the probability doesnot vanish for any given superpositions. Rather the opposite tendency holds that theprobability is enhanced for two different particles to be found in the same quantumstate. The accurate and general formulation states thatthe probability thatN bosonsbe found in the same quantum state isN ! times the probability that distinguishableparticles be found in the same state28.

A more familiar statement is that ifN particles are in the same state, the probabilityfor the (N + 1)th to be found also in this same state isN + 1 times this probabilityfor distinguishable particles. It is proved with conditional probabilities, if “A” is thestatement “the(N +1)th particle is in some given state|ϕ〉” while “B” is the statement“the N th other bosons already are in state|ϕ〉, thus the probability (with respect todistinguishable particles) that “A” is realised given than “B” is P(A ∩ B)/P(B), thatis (N + 1)!/N !. ThisN + 1 coefficient characterisesbosonic stimulation.29

28The proof is instructive and goes as follows: let|Ψ〉 the wavefunction of the state be developed on abasis|φi〉 of H⊗N , first assuming distinguishable particles which do not require the symmetry postulate:

|Ψ〉C =X

i1,...,iN

αi1,...,iN |φi1〉 · · ·˛

˛φiN¸

(3.23)

(we sub-scripted the state with C for “classical”), then symmetrizing the state to ensure Bosonic indistin-guishability (3.21):

|Ψ〉B =1√N !

X

i1,...,iN

X

σ∈S

αi1,...,iN˛

˛φσ(i1)

¸

⊗ · · · ⊗˛

˛φσ(iN )

¸

(3.24)

The probability amplitude that all distinguishable particles be in the same quantum state, say|φ1〉, is

〈φ⊗N1 |Ψ〉C = α1,··· ,1 (3.25)

while for indistinguishable particles,

〈φ⊗N1 |Ψ〉B =N !√N !α1,··· ,1 . (3.26)

The ratio of these probabilities isN !. Note especially that it is independent ofα (which should be nonzero,meaning that one cannot find all particles in the same state ifthey do not all have a projection in this state).This ratio is therefore independent of any linear combination of theα and thereby of any state of the system,thus showing that the probability to find all particles in thesame state if they are indistinguishable bosonsisN ! the probability for distinguishable particles.

29As all results involving conditional probabilities—and inthis case further complicated by the quantuminterpretation—the bosonic stimulation is more subtle than it appears. However, considered in first order ofperturbation theory, it becomes an exact and useful conceptin the form of renormalisation scattering rates ofemission or in rate equations, as shall be seen in greater details in later chapters.


Solving Schrodinger equation

Now that the postulates and interpretation of quantum mechanics have been laid down,we can carry on with solving the equations.

Exercise 3.6 (∗) Reduce by separation of variables Schrodinger equation for a time-independent Hamiltonian to Schrodinger’s time-independent equation

H |φ〉 = E |φ〉 (3.27)

with |φ〉 now time-independent.

When the Hamiltonian is time-independent, a formal solution is obtained as

|ψ(t)〉 = e−iHt/~ |ψ(0)〉 (3.28)

All the postulates of quantum mechanics apply on|ψ(t)〉 which now just happensto change with time.

Exercise 3.7 (∗∗) Solve the Schrodinger equation (3.1) for the quadratic potential (3.10).Technically this requires finding the eigenstates and eigenenergies ofH .30

Exercise 3.8 (∗∗) Using the results of the previous exercise, study the time-dynamicsof the initial conditions〈x|ψ1〉 ∝ exp(−(x − x0)

2/L2) for various (relevant) sets ofparameters (x0, L).

Liouville-von Neumann equation

One can work with all operators by promoting the quantum state |ψ〉 to an operator

ρ = |ψ〉〈ψ| (3.30)

which is the projector of state|ψ〉, and is known as thedensity operator. Its main interestarises when statistical physics is added to quantum mechanics as we shall discuss insection 3.2.1. For now it is just another representation forthe state whose main effect31

is the introduction of new formulas to carry out computations, for instance the averageof an observableΩ in state (3.30) is now obtained through the formula:32

30The eigenstates|φn〉 read of Schrodinger equation for the harmonic potential are:

〈x|φn〉 =1√2nn!

“mω

π~

”1/4exp,

„

−mωx2

2~

«

Hn

„r

mω

~x

«

. (3.29)

The associated energy spectrum isEn = (n+ 1/2)~ω.31The density operator also makes clear that the overall phaseof a ket state is immaterial, as|ψ〉 =

α |0〉 + β |1〉 andeiφ |ψ〉 both have the same density operator independent ofφ.32The trace of an operatorΩ in some Hilbert space is defined as the sum of its diagonal elements (in

any base), i.e.,Tr(Ω) =P

i 〈φi|Ω |φi〉, and thus is a number (a real number if the operator is hermitian).The partial trace, say overH2, of an operatorΩ acting on a tensor Hilbert spaceH1 ⊗ H2, is the traceover diagonal elements ofH2 in any of its basis, leaving unaffected the projection ofΩ on H1. That is,decomposingΩ asΩ =

P

i

P

j ωijΩi ⊗ Ωj

TrH2Ω =

X

i

“

X

j

ωijX

k

〈φk|Ωj |φk〉”

Ωi =X

i

ωiΩi (3.31)

which is an operator onH1, with ωi the term in parenthesis. The generalisation to higher dimensions as wellas to any other selection of which spaces to trace out is obvious.


〈Ω〉 = Tr(ρA) (3.32)

as〈Ω〉 = 〈ψ|Ω |ψ〉 =∑

i 〈ψ|Ω |φi〉〈φi|ψ〉 by inserting the closure of identity (A.4),now 〈ψ|Ω |φi〉 and 〈φi|ψ〉 are two complex numbers, so they commute and〈Ω〉 =∑

i〈φi|ψ〉〈ψ|Ω |φi〉 =∑

i 〈φi| ρΩ |φi〉 by definition ofρ, which is the result.

Exercise 3.9 ∗ Derive from Schrodinger equation for|ψ〉 the equation for|ψ〉〈ψ|:

i~∂

∂tρ = [H, ρ] (3.33)

We shall refer to Schrodinger Eq. (3.33) cast with the density operator asLiouville–von Neumman equation(which holds for a more general density operator than (3.30),as will be seen in section 3.2.1).

3.1.5 Heisenberg picture

At this stage, we have presented the essential facts of quantum theory required for thequantum description of light which we shall soon undertake,of course omitting a lot ofmaterial not immediately or crucially needed for that purpose.

We now devote further considerations to alternate formulations along with inclu-sions of other physics, like statistical physics or thermodynamics, because of their im-portance to microcavity physics.

The formal integration of Schrodinger equation, (3.28), shows that the time evolu-tion of the state is a rotation in Hilbert space. Taking advantage of this fact, one can use abasis of rotating states and transfer the dynamics from the states to the operators, whichwere previously fixed, i.e., time-independent. Therefore,considering the time-varyingaverage33 〈Ω〉(t) which is a quantity physically measurable that should not depend onwhich formalism is used, and starting from the definition we have given (in Schrodingerpicture)

〈Ω〉(t) = 〈ψ(t)|Ω |ψ(t)〉 (3.34a)

= 〈ψ(0)| eiHt/~Ωe−iHt/~ |ψ(0)〉 (3.34b)

= 〈ψ(0)| Ω(t) |ψ(0)〉 (3.34c)

we arrive to a time-varying operatorΩ(t)

Ω = eiHt/~Ωe−iHt/~ (3.35)

which acts on time-independent state (frozen to their initial condition |ψ(0)〉). Thisformulation of quantum mechanics where operators carry thetime-dynamics and thestates are fixed, is called theHeisenberg picture. We used the average as an intermediatebetween the two pictures but there is an equation of motion for the operators directly,aptly calledHeisenberg equation, which reads

i~∂

∂tΩ(t) = [Ω(t), H ] (3.36)

33Now the time dependence is shown explicitly everywhere and is accurately attributed to which quantityis time dependent, e.g.,〈Ω(t)〉 is very different from〈Ω〉(t).


Exercise 3.10(∗) Derive Heisenberg equation (3.36) from Schrodinger equation (3.1)and also Schrodinger equation form Heisenberg equation, thereby proving the completeequivalence of the two formulations. Note that in the commutator of Eq. (3.36),Ω isthe time dependent Heisenberg operator whileH is the time-independent SchrodingerHamiltonian. In the course of your demonstration, show thatfor algebraic computationpurposes, Heisenberg equation can be read as:

i~∂

∂tΩ(t) = [Ω, H ] (3.37)

where tilde means the operator has been transformed according to (3.35), i.e.,[Ω, H ] =eiHt/~[Ω, H ]e−iHt/~. Therefore the algebra carried out in the commutator is timein-dependent throughout.34

Keeping in mind the functional analysis results of the quantum harmonic oscillator,we now consider the problem from an algebraic point of view, therefore closer in spiritto the Heisenberg approach (this solution is due to Dirac). It prefigures the analysis weshall make on the more complicated system of the light field insection (2.1.1).

Let us introduce theladder operators

a =1√2~

(X + iP ), a† =1√2~

(X − iP ) (3.38)

where the others quantities have been defined on page 78. It follows straightforwardlyfrom (3.17) and the above definition that

[a, a†] = 1 (3.39)

and also that the Hamiltonian (3.10) reads

34From Schrodinger equation to Heisenberg equation and back; we compute:

∂

∂tΩ =

∂

∂t

“

eiHt/~Ωe−iHt/~

”

=

„

∂

∂teiHt/~

«

Ωe−iHt/~ + eiHt/~

„

∂

∂tΩ

«

e−iHt/~ + eiHt/~Ω

„

∂

∂te−iHt/~

«

=

„

∂

∂teiHt/~

«

Ωe−iHt/~ + eiHt/~Ω

„

∂

∂te−iHt/~

«

sinceΩ is time independent

=

„

iH

~eiHt/~

«

Ωe−iHt/~ + eiHt/~Ω

„

− iH~

«

e−iHt/~

=i

~HΩ − i

~ΩH

sinceH commutes withe−iHt/~

=i

~[H, Ω] .


H = ~ω(

a†a+1

2

)

(3.40)

so that the eigenvalue problem now consists in finding|φ〉 such that

a†a |φ〉 = φ |φ〉 (3.41)

To do so, dot (3.41) with〈φ| (these states assumed normalised) to getφ = 〈φ| a†a |φ〉 =‖a |φ〉 ‖2 which shows that the eigenvaluesφ are positive. Using the algebraic rela-tions35

[a, a†a] = a , [a†, a†a] = −a† , (3.42)

we obtain the equality(a†a)a = a(a†a− 1) which leads to

(a†a)a |φ〉 = a(a†a− 1) |φ〉 = a(φ− 1) |φ〉 = (φ− 1)a |φ〉 (3.43)

the two ends showing thata |φ〉 is an eigenstate ofa†a, with eigenvalueφ − 1, at thepossible exception ifa |φ〉 = 0 (the null vector, since it cannot be an eigenstate). Asfor a |φ〉, dotting it with its conjugate yields

‖a |φ〉 ‖2 = (〈φ| a†)(a |φ〉) = 〈φ| a†a |φ〉 = α (3.44)

Iterating (3.43)n times shows thatan |φ〉 is also an eigenvector ofa†a but this timeswith eignvalueφ − n, which unless it becomes zero for some value ofn (interruptingthe iteration), will ultimately become negative in contradiction to what precedes. There-fore, the zero vector must be hit exactly, i.e., there existsn ∈ N∗ such thatan |φ〉 = 0butan−1 |φ〉 6= 0. Consider the normalised eigenstate|φ− n〉 = an |φ〉 /‖an |φ〉 ‖ witheigenvalueφ − n. Applying a on this state yields zero by definition ofn, while (3.44)gives its normed squared asφ− n. Equating the two provides the structure of the solu-tions,φ = n, therefore the eigenstates ofa†a are states|n〉 wheren ∈ N. The “bottom”state|0〉 satisfies

a |0〉 = 0 (3.45)

Note that|0〉 and0 are two different entities. The latter is the mathematical zero, and inthis case the zero vector of the Hilbert space, while the former is more of a notation fora complicated mathematical object, in this case a Gaussian function once projected in acoordinate space.

Exercise 3.11(∗) Repeat the previous analysis to prove the counterpart propertiesfor a†, i.e.,a† |φ〉 is an eigenvector ofa†a and‖a† |φ〉 ‖ =

√φ+ 1.

35Commutation relations between operators derived from the initial definition range from straightforwardto intractable with little difference in their shape. (3.42) is easily obtained from (3.39), for example by bruteexpansion:[a, a†a] = aa†a − a†aa = (aa† − a†a)a = a.


As opposed toa which eventually reaches the bottom rung|0〉 past which no otherstates can be obtained,a† can increase indefinitely the eigenstate label. These operatorsare quickly checked to satisfy:

a |n〉 =√n |n− 1〉 , (3.46a)

a† |n〉 =√n+ 1 |n+ 1〉 (3.46b)

and also, applying them in succession:

a†a |n〉 = n |n〉 . (3.47)

Higher order formulas are readily obtained, like Eq. (3.48)below, useful for rapidevaluation of quantities of interest as illustrated in exercise (3.12):

a†iaja†k |n〉 =(n+ k)!

(n+ k − j)!

√

(n+ i+ k − j)!

n!|n+ i+ k − j〉 (3.48)

with conditionj ≤ n+ k, the result being zero otherwise.

Exercise 3.12(∗) Derive Eq. (3.48) and use it to evaluate the following statesarising inconnection with boson-boson interactions:36 a2a†

2 |n〉, a†2a2 |n〉, 〈n| a2a†2, 〈n| a†2a2.

Also evaluate the application of the following operator on〈n| and|n〉, arising in masterequation approach to lasers:a†aa, aa†a, aaa†, aa†a†, a†aa†, a†a†a.

36Applying repeatedly formulas (3.46), written here for convenience with their counterparts in the dualspace,

a |n〉 =√n |n− 1〉 , 〈n| a = 〈n+ 1|

√n+ 1 , (3.49a)

a† |n〉 =√n+ 1 |n+ 1〉 , 〈n| a† = 〈n− 1|

√n . (3.49b)

one obtains straightforwardly

ai |n〉 =

s

n!

(n− i)!|n− i〉 , a†i |n〉 =

r

(n+ i)!

n!|n+ i〉 (3.50)

and therefore, still iterating:

(for i ≤ n+ j) aia†j |n〉 =

(n+ j)!√n!

p

(n+ j − i)!|n+ j − i〉 , (3.51a)

(for i ≤ n) a†jai |n〉 =

√n!

p

(n+ j − i)!

(n− i)!|n+ j − i〉 . (3.51b)

The result is zero if the condition on the left is not satisfied. Formula (3.48), repeated here for convenience,obtains in the same way:

a†iaja†k |n〉 =(n+ k)!

(n+ k − j)!

r

(n+ i+ k − j)!

n!|n+ i+ k − j〉 (3.48)

with again conditionj ≤ n+ k, the result being zero otherwise.Also note the useful relation obtaining|n〉 from the vacuum, following from (3.51b):

|n〉 =a†n

√n!

|0〉 . (3.52)


The Fock states are handy for calculations, and offer ade factocanonical basis, i.e.,

|0〉 =

10000...

, |1〉 =

01000...

, |2〉 =

00100...

, . . . |n〉 =

0...010...

, . . .

with 1 as the(n+ 1)th element of the vector. In this basis, the annihilation and creationoperators read:

a =

0 1

0√

2

0√

3. . .

. . .0

√n

. . .. . .

, a† =

01 0√

2 0√3 0

.. .. . .√

n+ 1 0. . .

. . .

where thenth column is being shown for the general case. In both cases, thediagonalis zero, and, for the annihilation operator, the line above the diagonal is nonzero, while,for the creation operator, the line below the diagonal is nonzero. This depends on theconvention. All the algebra can be computed with this representation and expressionsfor operators obtained in this way. The number operator, forinstance, is diagonal withvalues 0, 1, 2, . . .

If we return to the Hamiltonian (3.40), the energy spectrum is seen to consist ofequally spaced energy levels with energy difference~ω:

En =(

n+1

2

)

~ω (3.53)

The state|0〉 of lowest energy is theground stateof the system and is, by defini-tion, devoid of excitation. It is called thevacuum state(or a vacuum state in case ofdegeneracy). However its energy is nonzero since〈0|H |0〉 = ~ω/2. This term is as-sociated to quantum fluctuations which arise from the uncertainty principle, of whichwe shall see more later in a true field-theoretical setting. One more important point isthat this structure of equally spaced levels makes compelling the interpretation of|n〉as a state withn quantaof excitations, each of energy~ω, superimposed to the vacuumfluctuations which cannot be got rid of.

The effects of the various operators just defined on the state|n〉 with n excitationsentitle them to be calledannihilation, cf. (3.46a), andcreation operator, cf. (3.46b).


Their net result is indeed to raise or lower the number of excitations37. The annihilationoperating on the vacuum state, (3.45), results in cancelingthe term, which disappearsfrom the process being computed.

The Heisenberg picture has many advantages, including computational or algebraicsimplicity as we shall later appreciate. One further quality is the aid to physical intuitionafforded by this formalism as these are the observables which vary in time, in analogywith classical physics.

3.1.6 Dirac (Interaction) Picture

The Dirac picture, also known as theinteraction picture, is an intermediate case be-tween the Schrodinger picture (with fixed operators and time-varying states) and theHeisenberg picture (with time-varying operators and fixed states) where both operatorsand states are time-dependent. Which fraction of the dynamics is attributed to each ofthem depends on the decomposition of the Hamiltonian.

First with the case of a total HamiltonianH without time dependence, which isseparated as

H = H0 +HI (3.54)

whereH0 is typically a “simple” part of the dynamics, i.e., which canbe solved exactlyand whose solutions will define the states of the “free” particle, or a “dominant” partwhich will result in a fast dynamics, and the remainingHI , typically complicated orof a lesser magnitude, which will be interpreted as an interaction terms between theparticles defined byH0.

The underlying principle of interaction picture is to concentrate on the complica-tionHI in the Hamiltonian by embeddingH0 dynamics into the operator by defining:

Ω = eiH0t/~Ωe−iH0t/~ (3.55)

which equation of motion is reminiscent of the Heisenberg equation (3.36) but withsome important redefinitions of the quantities involved:

i~∂

∂tΩ(t) = [Ω(t), HI ] (3.56)

where tilde means the operator has been transformed to Heisenberg picture accordingto (3.55). As opposed to the Heisenberg equation, this transformation applies to whatremains of the Hamiltonian as well,HI = eiH0t/~HIe

−iH0t/~, which however canremain time independent under this transformation leadingto HI = HI . In all case,HI and notH—like in Heisenberg or Schrodinger equations—appears in the commu-tator. The similarities are strong enough however for Eq. (3.56) to be calledHeisenbergEquation in Interaction Picture.

Exercise 3.13(∗) Derive Eq. (3.56).

37A more accurate terminology would call these operators “ladder operators” in the case of a genericquantum harmonic oscillator, comprising araising and loweringoperator, and to reserve the terms foranni-hilation andcreationin a field-theoretic context.


If the computation is difficult, one can compute the commutation first, as[Ω, H] =

[Ω, H ].An efficient approach to solve a general problem is to separate it in a “dominant”

part which can be solved exactly or approximated easily and consider the minor partsleft out initially as perturbations. What is meant by dominant depends on the problemat hand, but a typical case would be the time-independent part of a general Hamilto-nianH(t), which we could separate as

H(t) = H0 +H1(t) (3.57)

whereH0 is time-independent, resulting in a uniform time rotation in Hilbert spacee−iH0t/~ |ψ0〉, cf. (3.28).

Heisenberg and Dirac pictures often require a lot of algebra, which means that com-mutators have to be computed by expansion and simplificationthrough gathering andcancelling with relations such as 3.39. To allow such an evaluation when faced with ageneral commutator of the type[

∏ni=1Ai,

∏mj=1 Bj ] whereAi, Bj are some operators,

the most general expansion is a sum over all combinations of commutators[Ai, Bj ]with other operators factored outside of the commutator. Their relative position is ofcourse important in the most general case where their relative commutation rules areunknown. Their placement is made as follows for the commutator [Ai, Bj]: all opera-torsA1 · · ·Ai−1 placed beforeAi and all operatorsB1 · · ·Bj−1 placed beforeBj areplacedbeforethe commutator[Ai, Bj ], in this order, and all operatorsAi+1 · · ·An andall operatorsBj+1 · · ·Bm are placedafter the commutator, in the opposite order ofAandB. This is illustrated below, for the case where the operatorswhich remain in thecommutator areAi andBj :

[(A1 · · ·Ai−1)Ai(Ai+1 · · ·An), (B1 · · ·Bj−1)Bj(Bj+1 · · ·Bm)]

(A1 · · ·Ai−1)(B1 · · ·Bj−1)[Ai, Bj ](Bj+1 · · ·Bm)(Ai+1 · · ·An)

Application of this rule on the six arbitrary operatorsA, · · · , F written as followsyields:

[ABC,DEF ] =[A,D]EFBC +D[A,E]FBC +DE[A,F ]BC (3.58)

+A[B,D]EFC +AD[B,E]FC +ADE[B,F ]C (3.59)

+AB[C,D]EF +ABD[C,E]F +ABDE[C,F ] (3.60)

There are in this case nine terms as there are3 × 3 combinations for commutators,with operators distributing as illustrated. Such evaluations usually simplify extensively

OTHER FORMULATIONS 93

when they are carried over a family of bosons operatorsai, obeying the following alge-bras:

[ai, aj ] = 0, [ai, a†j] = δij . (3.61)

For instance here a useful relation obtained in this way is:

a†a†aa = a†a(a†a− 1) (3.62)

Exercise 3.14(∗) Given three Bose operatorsa1, a2 anda3 obeying commutation re-lations (3.61), evaluate the following expressions often encountered in the Heisenbergand interaction pictures:[a1, a

†1a1], [a1, a

†21 a

21], [a1, a

†1a1a

†2a2], [a1, a

†n1 am1 ]

3.2 Other formulations

3.2.1 Density matrix

The formulation of the theory so far—in whatever representation—describe so-calledpure states, which for now we can regard as a synonym forket states, that is, states forwhich there exists a “wavefunction”|Ψ〉. A first reason why there would be no suchstate is if one would attempt to describe only part of a composite system. Namely, ifthe system of our interestS is in contact with another systemR (the notations are for“system” and “reservoir”), the (pure or ket) state to describe the whole system is, ingeneral

|ΨSR〉 =∑

i

∑

j

cij |φi〉 |ϕj〉 (3.63)

where the|φi〉 are basis states forS and |ϕj〉 are basis states forR. Note that in thesense that each state has its associated Hilbert space and set of observables, it is definedindependently of the other. If thecij are such that (3.63) can be written as a (tensor)product

|ΨSR〉 =(

∑

i

cSi |φi〉)(

∑

j

cRj |ϕj〉)

(3.64)

(that is, if cij = cSi cRj ), then|ΨSR〉 can be likewise decomposed as|ΦS〉 |ΦR〉 and the

systemS considered in isolation with its quantum state|ΨS〉.If, however, such a decomposition (3.64) is not possible, inwhich case the systemsS

andR are said to beentangled, then it is not possible to consider any one of themindependently, at least exactly. The density operator is the approximated state whicharises when the total wavefunction is averaged over unwanted degrees of freedom ofthe total system.

Exercise 3.15(∗∗) Show that the state which best describesS in isolation is

ρS = TrR(|ΨSR〉〈ΨSR|) (3.65)

where the partial trace32 is taken over the Hilbert space ofR.


Such an operatorρS—the so-calledreduced density matrix—which cannot be de-scribed by a ket state, is called amixed state—as opposed to thepure state—because ofthe “averaging” which mixes up quantum states.

Another reason to recourse to a density operator is when statistical mechanics isincorporated into Quantum Mechanics, i.e., when some indeterminacy is injected in thesystem as the result of knowledge not available only in practise (in direct analogy to theclassical statistical theory and direct opposition to the quantum indeterminacy which isintrinsic to the system). For instance the state of a radiation mode which can take uppossible energiesEi in thermal equilibrium at temperatureT requires such a statisti-cal description, as some excitations are randomly poured into or conversely removedfrom the system by thermal kicks issued by the reservoir. In accordance with thermo-dynamics, such a state would be described as being in the (pure) quantum state|Ei〉of energyEi with probabilitypi = e−Ei/kbT /Z with Z =

∑

i e−Ei/kbT the partition

function.The average〈Ω〉 given by (3.32) with density matrix

ρ =∑

i

pi |Ei〉〈Ei| (3.66)

yields〈Ω〉 =∑

i pi 〈Ei|Ω |Ei〉 from which it is seen that the average is now the quan-tum average (3.13c) weighted over the classical probabilitiespi, so(3.66) describes asystem where a quantum state is realised with probabilitypi.

It is important, though sometimes subtle, to distinguish between the quantum and aclassical indeterminacy. For instance the density matrix

1

2

(

1 11 1

)

=1√2(|〉 + |〉)(〈| + 〈|) 1√

2(3.67)

which describes a pure state of linear polarisation (cf. exercise 3.2), is physically differ-ent from the mixed state

1

2

(

1 00 1

)

=

|〉 with probability1/2|〉 with probability1/2

(3.68)

which describe a single photon which has either circular polarisation with probabil-ity 1/2 all along. This is different from (3.67) where the circular polarisation becomesleft or right only as a result of a measurement in this basis and is the rest of the timeotherwise undetermined.

It therefore appears that off-diagonal elements of the density matrix are linked withthe pure or mixed character of the state. Indeed as we shall have many occasions toappreciate in this text, these elements relate toquantum coherenceand their contributionto the equations of motion discriminates between classicaland quantum dynamics.

Mathematical properties

The density matrix is hermitian and its trace is unity in agreement with its statisticalinterpretation. The trace of its square is also one in case ofa pure state since in this

OTHER FORMULATIONS 95

caseρ = ρ2 as there exists|ψ〉 such thatρ = |ψ〉〈ψ| whenceρ2 = |ψ〉〈ψ|ψ〉〈ψ| = ρaccording to (3.3). But this is not the case for a mixed state and this provides a usefulcriterion for identification of pure states:

Exercise 3.16∗ Show that for a mixed state,

Tr(ρ2) < 1 . (3.69)

The minimum for the trace ofρ2—to indicate how “mixed” a state is—depends on thesize of the Hilbert space. Show that for a space of sizen it is given by1/n.

A density matrix is also positive38 and reciprocally, any positive operator whose traceequals 1 is eligible to be a density matrix.

3.2.2 Second quantization

We have presented in the previous sections the basic concepts of quantum physics fromits “mechanical” point of view where the object of quantization is a mechanical attributeof the particle, like its motion or its spin. When the physical object is a field—as is thecase with light—quantum mechanics is upgraded to the statusof aquantum field theory.There are various possible theoretical formulations but for the needs of this book weshall be content with a simple and vivid picture, known assecond quantization. Oneconceptual benefit of this reformulation of the theory is thevaluable concept it affordsof a particle as an excitation of the field, in the terms we are about to present. In theformulation given so far, we have already used repeatedly the term “particle” to describethe object to which to attach the wavefunction or one of its attribute (like the spin). Onedifficulty however arises when the number of particles is notconserved, as is the casein a statistical theory in the grand-canonical ensemble, orif particles are unstable andcan decay into other particles (calling for the necessity toremove and add particles inthe quantum system). Also particles are generally considered in a collection, so that thesymmetry requirements are to be taken into account.

Second quantization starts with theoccupation number formalismwhich providesan elegant and concise solution to all these desiderata. LetH1 be a single-particle basis,i.e., a set of states|φi〉 which are orthonormal and—assuming a discrete basis—suchthat any possible quantum state|ψ〉 can be written as|ψ〉 =

∑

i αi |φi〉 for a suitablechoice ofαi (which is also unique by orthogonality of the basis). By the fourth postulate,the wavefunction for a collection of particles with|Ψ〉 =

⊗Ni=1 |φi〉

⊗ni needs to besymmetrised. Consider a state|Ψ〉 of the system withni particles in state|φi〉 andwhich are not entangled. Without any symmetry requirement,such a state would readwhereN is the number of different quantum states (for a total of

∑Ni=1 ni particles in

the system). For fermions allni = 1.Second quantisation sheds much light on the dilemma of particle-number duality

and the so-calledcomplementarity39 which is naively perceived as a photon behavingsometimes like a particle, sometimes like a wave.

38A positive operatorM is such that for all state|φ〉, 〈φ|M |φ〉 ≥ 0.39It is interesting to remind Anderson who comments in Nature,437, 625, (2005) that


We have introduced with Eq. (3.38) the so-called ladder operatorsa anda† whichannihilate and create, respectively, excitations of the quantum harmonic oscillator. Be-cause the energy levels are equally spaced, cf. Eq. (3.53), we can understand such astate, say thenth one with energy(n + 1/2)~ω, as consisting ofn excitations eachwith energy~ω, with a remainder of~ω/2 in energy whenn = 0 which is thereforeidentified as energy of the vacuum.

3.2.3 Quantisation of the light field

The quantisation of the electromagnetic spectrum is known as quantum electrodynam-ics, (QED). For the spectral window of light which corresponds to high frequencies butstill moderate quanta energies, QED can be framed in a suitable way to make the mostof the various energy and time scales, which is known asquantum opticsand is the topicof this section.

Quantization of the field is made in the vacuum, leaving the complicated problemof the interaction with matter to the next chapter which willbring many different tech-niques to tackle this issue in various approximations. We therefore proceed from theclassical equations obtained in Section 2.1.1. We quantizethe field following Dirac(1927) and Fermi (1932) by canonical quantization of the variablesa related to the fieldthrough Eq. (2.10a) and which is known from Eq. (2.13) to undergo harmonic oscil-lation. To settle notations, we regard free space as a large cubic box of sizeL, withboundary conditions for the electromagnetic field of running waves40 ei(ωkt−k·r), asopposed to standing waves. A wavevector of these solutions is defined by:

k =

(

2πnxL

,2πnyL

,2πnzL

)

(3.70)

wherenx,y,z ∈ N andL is taken as high as necessary for the sought precision. The fieldamplitude in this box is

Ek =

√

~ωk

2ǫ0V(3.71)

Completing the modal expansion with quantised operators, the expression of thefield E (now also an operator) reads:

Niels Bohr’s ‘complementarity principle’—that there are two incompatible butequally correct ways of looking at things—was merely a way ofusing his pres-tige to promulgate a dubious philosophical view that would keep physicists work-ing with the wonderful apparatus of quantum theory. Albert Einstein comes offa little better because he at least saw that what Bohr had to say was philosophi-cally nonsense. But Einstein’s greatest mistake was that heassumed that Bohr wasright—that there is no alternative to complementarity and therefore that quantummechanics must be wrong.

40Boundary conditions of a running wave are such that once the wavefront reaches the end of the freespace within the box and touches the border, it goes through and reappear instantaneously on the other sideof the box. In this sense this boundary condition is not really physically relevant, and should be viewedas a mathematical trick to model the infinite universe in a more intuitive way, with labelsL to track ourwavevectors.

QUANTUM STATES 97

Vladimir Fock (1898–1974) gave his name to the Fock space and Fock stateafter building the Hilbert space for Dirac’s theory of Radiation, Zs. f. Phys.,49 (1928) 339.

Many other results and methods due to him such as the Fock proper timemethod, the Hartree-Fock method, the Fock symmetry of the hydrogen atomand still others make him one of the most popular names in quantum fieldtheories.

E =∑

k

Ekakei(k·r−ωt) + h.c. (3.72)

The vector nature ofa is associated to polarisation. We split these two aspects—field amplitude and polarisation—to deal with them separately which allows us to applythe results already presented regarding the formal quantumsystems of the harmonicoscillator and the two-level system. The vector variablea being transverse, we choosean orthogonal 2D basis in the transversek plane:

a(k, t) = a↑(k, t)e↑(k) + a↓(k, t)e↓(k) (3.73)

where(e↑(k), e↓(k),k) form an orthogonal basis of unit vectors. When need ariseswe will use the full vector expression, but without limitation we now focus on oneprojection only, saya↓, which for brevity we shall notea, thereby coming back to theelementary one-dimensional quantum mechanics which was our starting point.

3.3 Quantum states

We now discuss some commonly encountered quantum states in the light of the formal-ism of second quantization.

3.3.1 Fock states

An important and intuitive building block is theFock statewhich is the canonical basisstate of the Fock space. It is therefore the state with a definite number of particles (in thecase of the electromagnetic field, a definite number of photons) and is, in this respect, aphysical state as well as a mathematical pillar of second quantization. The vacuum|0〉and the single particle|1〉 are two typical, important and “relatively” easy states toprepare in the laboratory.41 The arbitrary case|n〉 with n ∈ N becomes increasinglydifficult with large values ofn. Small values have however been indeed reported in the

41To prepare a single photon state it suffices to dispose a thin absorbing media in front of a light withthickness increasing until an avalanche photodiode registers separate detections. Each detection correspondsto a quantum. This is assured to work provided that one is disposed to wait the necessary time. To provide asingle photon sourceon demandis an altogether more difficult problem, subject to active research because ofits application in quantum cryptography (see the Problem atthe end of this chapter).


literature, and cavities are precious tools to this end, as demonstrated for instance byBertet et al. (2002).

From the mathematical point of view, Fock states are useful for many computationalneeds. When quantum electrodynamical calculations are conducted pertubatively, theFock state plays a major role as each successive order of the approximation describesprocesses which increase or decrease the number of photons.A two particle interaction,such as the Coulomb interaction, reads in second quantization:

V =1

2

∑

k1,k2k3,k4

〈k3k4|V |k1k2〉 a†k3a†k4

ak2ak1 (3.74)

To such a process one can attach aFeynman diagramwhich displays graphicallyone term of the Born expansion of an interaction, as shown in Fig. 3.1.

|k1 + q〉 Vq = 4πe2/q2

|k1〉 |k2〉

|k2 − q〉

Fig. 3.1: A Feynman interaction for Coulomb interaction whereby two particles of momentak1 andk2 inthe initial state scatter to final statesk3 andk4 by exchanging momentumq.

3.3.2 Coherent states

Thecoherent statewas initially introduced by Schrodinger as the quantum state of theharmonic oscillator which minimises the uncertainty relation (3.16) for the observ-ablesP andX . From the derivation of the inequality of the generalised Heisenberguncertainty relation of exercice 3.4, we know that (3.16) isSbthe chwarz inequality indisguise applied on vectors

(Ω − 〈Ω〉) |α〉 and (Λ − 〈Λ〉) |α〉 , (3.75)

with nowΩ = P andΛ = X , and with uncertainty assumed to be equally distributed42

in both positionP and impulsionX . The inequality is optimised to its minimum whenthese vectors are aligned, which yields:

42When the uncertainty of a coherent state is not equally shared, i.e.,∆X 6= ∆P , although the prod-uct remains~/2, the state is said to besqueezedin the variable whose root mean square goes below~/2.In the dynamical picture provided by exercise (3.8), squeezed states correspond to the ground state of theharmonic oscillator, i.e., a Gaussian wavefunction, whosewidth is mismatched with the harmonic potentialcharacteristic length, so that as the state bounces back andforth in the trap, its wavepacket spreads or narrowsperiodically (it is said to “breathe”).

QUANTUM STATES 99

(P − 〈P 〉) |α〉 = i(X − 〈X〉) |α〉 (3.76)

or, written back in terms ofa, a†:

a |α〉 =1√2~

(〈X〉 + i〈P 〉) |α〉 (3.77)

so that thecoherent statewhich minimises the uncertainty relation appears as the eigen-state of the annihilation operator (we come back later to themore general case wherethe uncertainty is not equally distributed).

It was expected thatα could be complex sincea is not hermitian. From (3.77) itis seen that it actually spans the whole complex space. The phase associated with thiscomplex number (in the sense of a polar angle in the complex plane) maps to the phys-ical notion of phase since it arises as a complex relationsipbetween the phase spacevariables. The physical meaning of the (unsqueezed42) coherent state is that of the mostclassical state allowed by quantum physics, since it has thelowest uncertainty allowablein its conjuguate variables, and can be located in the complex plane as the position inthe phase space of the quantum oscillator, with position on real axis and momentumon imaginary axis. The Hamiltonian being time independent,the evolution of|α〉 isobtained straightforwardly from the propagator:

|α(t)〉 = e−iω(a†a+1/2)(t−t0) |α(t0)〉= e−iω(t−t0)/2

∣

∣

∣e−iω(t−t0)α(t0)⟩ (3.78)

so that the free propagation of the coherent state is rotation in Argand space, or harmonicoscillations in real space. Definitely it is a state of well defined phase.

In terms of Fock states, it reads:

|α〉 = exp(−|α|2/2)

∞∑

n=0

αn√n!

|n〉 (3.79)

with α ∈ C.Coherent states are normalised though not orthogonal:43

〈β|α〉 = exp(

− 1

2(|α|2 + |β2| − 2αβ∗)

)

(3.80)

with indeed〈α|α〉 = 1 but〈β|α〉 6= δ(α− β), being smaller the farther apartα, β in C.

3.3.3 Glauber–Sudarshan representation

Sudarshan (1963) promoted coherent states as a basis for decomposition of the densitymatrixρ, replacing it by a scalar functionP function of the complex argumentα of thecoherent state|α〉:

43The non-orthogonality of coherent states close in phase-space was one compelling support in favourof Glauber’s (1963) argument that the light emitted by the newly discovered maser called for a quantumexplanation over the classical model advocated by Mandel & Wolf (1961).


Roy Glauber (b. 1925) advocated the importance of quantum theory in op-tics and its implications for the notion ofcoherence. As such he is widelyregarded as the father of quantum optics and was awarded the 2005 Nobelprize in physics (half of the prize) “for his contribution to the quantum the-ory of optical coherence”.

Before being awarded the Nobel prize, Glauber was an active and long timesupporter of its parody—the “Ig Nobel” prize—where he traditionally sweptoff the stage the paper planes sent on it by the audience. He made a testimo-nial for the 1998 Ig Nobel Physics prize awarded for “unique interpretationof quantum physics as it applies to life, liberty, and the pursuit of economichappiness”.

ρ(t) ≡∫

P (α, α∗, t) |α〉〈α| d2α (3.81)

which allows one to carry out quantum computations with tools of functional analysisrather than operator algebra. Sudarshan incorrectly deduced that this proved the equiv-alence of the quantum formulation and classical one. Glauber, who also had in his firstpublication the insight of this decomposition, showed him wrong.44

We shall see many applications and usage of the Glauber–Sudarshan (or simplyGlauber) representation later on. Now we proceed to give itsexpression for some statesof interests.45 For the coherent state, it is straightforward by identification to obtain forrepresentation of|α0〉 that

P (α, α∗) = δ(α− α0) (3.82)

3.3.4 Thermal states

The incoherent superposition of many uncorrelated sourcesgenerates chaotic or a so-called thermal state, as is the case of the light emitted a lightbulb where each atomemits independently of its neighbour. The convolution ruleand thecentral limit theoremcombine to provide theP function of a such a state:a chaotic state without phase noramplitude correlations has a Glauber distribution which isGaussian in the complexplane:

44Upon award of the Nobel prize in 2005 to Glauber, Sudarshan sparked a controversy, questioning thisdecision and writing to the Nobel committee and the Times (who did not publish his input) where he claimedpriority on the representation. An extract of the letter of Sudarshan to the Nobel committee reads “[. . . ]Whilethe distinction of introducing coherent states as basic entities to describe optical fields certainly goes toGlauber, the possibility of using them to describe ‘all’ optical fields (of all intensities) through the diagonalrepresentation is certainly due to Sudarshan. Thus there isno need to ‘extract’ the classical limit [as statedin the Nobel citation]. Sudarshan’s work is not merely a mathematical formalism. It is the basic theory un-derlying all optical fields. All the quantum features are brought out in his diagonal representation[. . . ]”. Heconcludes “Give unto Glauber only what is his.”

45TheP representation for Fock states is highly singular, involving generalised functions of a much highercomplexity than for the coherent state, which somehow restrain its applicability. This can be linked to the non-classical character of such a field.

QUANTUM STATES 101

P (α, α∗) =1

πnexp(−|α|2/n) (3.83)

wheren = 〈a†a〉 is the average number of particle in the mode. From this expressionthe Fock state representation can be obtained:

Exercise 3.17(∗) Show that the density matrix of the thermal state in the Fock statesbasis is:

ρ =∞∑

n=0

nn

(n+ 1)n+1|n〉〈n| (3.84)

Observe that the density matrix Eq. (3.84) is diagonal (all the terms|n〉〈m|with n 6=m are zero). This means that there is no quantum coherence and the superposition isclassical. This is the fully mixed state (cf. Eq. (3.69)), where the uncertainty is entirelystatistical. The result can be alternatively obtained directly from the statistical argu-ment: in the canonical ensemble, the density matrix for a system with HamiltonianHat temperatureT is given by:46

ρ =exp(−H/kBT )

Tr(exp(−H/kBT ))(3.85)

which can be evaluated exactly whenH is the Hamiltonian for an harmonic oscillator,given by Eq. (3.40):

Exercise 3.18(∗) Show that Eq. (3.85) evaluates to

ρ =

[

1 − exp(

− ~ω

kBT

)

] ∞∑

n=0

exp(

− n~ω

kBT

)

|n〉〈n| (3.86)

for the harmonic oscillator Hamiltonian. Deduce the following relation between theaverage occupancy of the mode and temperature:

n = 〈a†a〉 =1

exp(

− ~ω

kBT

)

− 1

(3.87)

The important formula Eq. (3.87)—the thermal distributionof bosons at equilibrium(here for a single mode)—is named theBose–Einstein distribution.47

46This follows from the fundamental postulate for the canonical ensemble, which states that if a sys-tem is in equilibrium at temperatureT , the probability that it be found with energyEn is (1/Q) exp

`

−En/(kBT )

´

whereQ =P

n exp`

− En/(kBT )´

(known as thepartition function) andEn is the en-ergy of any of the states in which the system can be found. Therefore, taking|i〉 as the state with energyEiand Ω an operator, from the definition of quantum average that we have given, cf. Eq. (3.32), it followsthat〈Ω〉 = (1/Q)

P

|i〉〈i|Ω |i〉 exp`

− Ei/(kBT )´

.47The general formula for the Bose-Einstein distribution, with all modes (and possible degeneracy such

as that imparted by polarisation) and with a chemical potential is readily obtained along the same lines asexercise 3.18 and will be a central ingredient of the physicsof Chapter 8.


3.3.5 Mixture states

An important property ofP functions is that the superposition of uncorrelated fieldseach described by itsP function amounts to a field whose ownP function is obtainedby the convolution of that of the constituting fields. So for instance if we superpose twonon-correlated fields described respectively by functionsP1 andP2, the total field isgiven byP = P1 ∗ P2, or:

P (α, α∗) =

∫

P1(β, β∗)P2(α − β, α∗ − β∗)dβdβ∗ (3.88)

This has application for an important class of states, whichinterpolate between thecoherent state Eq. (3.82) and the thermal state Eq. (3.83), being rare in practise that astate is realistically completely coherent.48 When there is a large number of particles,the physically relevant case is that of an essentially coherent state, with, say,nc particles,to which is superimposed a fraction of a thermal state, withnt particles. Such a state isobtained in the optical field by interfering ideal coherent radiation with that emitted by ablack body. In the resulting field, we will conveniently refer to such particles as coherentand incoherent respectively, though of course once the two fields are merged, a particledoes not belong any longer to a part of this decomposition butis indistinguishable fromany other. This is just a vivid picture to describe a collective state which has some phaseand amplitude spreading. Onlync + nt = 〈n〉 is well defined.

TheP state which results from the convolution of a Gaussian centered aboutαcoh ∈C and a delta function centered aboutαth is a Gaussian centered aboutαcoh − αth, sothat theP state of the mixed state is a Gaussian centered aboutα0, as depicted inFig. 3.2. Its analytical expression reads:

Pm(α, α∗) =1

πnte−|α−nce

iϕ|2/nt . (3.89)

whereϕ is the mean phase of the state, inherited from the phase of thecoherent state.We conclude by illustrating the power of theP function as a mathematical tool by

providing a few techniques which we shall use later on. With proper notation, Eq. (3.77)reads

a |α〉 = α |α〉 (3.90)

which states that the coherent state is an eigenstate for theannihilation operator (recip-rocal equation is〈α| a† = 〈α|α∗). Such properties make the evaluation of many statesstraightforward, translating the operator as its eigenvalue. It also results in many sim-plifications of the mathematical analysis of equations, as operator algebra gets mappedto complex calculus. However during this translation from operators toc-numbers, theneed for the following can (and does) arise:

a† |α〉 (3.91)

48From photodetection theory one can show that the inefficiency of the detector results in a broadeningof the counting statistics of a coherent state of the kind of the mixed states for the density matrix of a Bosesingle-mode.

QUANTUM STATES 103

Fig. 3.2: Schematics representation ofP representations for a thermal state (Gaussian), a coherentstate (δfunction) and a superposition, or mixture, of the two, in which case the Gaussian gets displaced onto theδlocation.

(or the reciprocal〈α| a). This is made technically easy with a few tricks that can bedevised from the Fock representation of the coherent state,Eq. (3.79), as in this case:

Exercise 3.19(∗) By going back and forth to the Fock and coherent basis, derivethefollowing rule of thumb:

a† |α〉 =

(

∂

∂α+α∗

2

)

|α〉 . (3.92)

and following the same principle, compute the exhaustive set of possible combinationswhich arise in the conversion of the master equation with operators into a so-calledFokker-Planck equation forc numbers functions:

a |α〉〈α| a† , a†a |α〉〈α| , |α〉〈α| a†a , (3.93a)

|α〉〈α| aa† , a† |α〉〈α| a . (3.93b)

The master equation for a matrix can in this sense be directlyrewritten as a Fokker-Planck equation which is an equation of diffusion and drift for a probability distribution.A lot of insights into the quantum picture can be gained through this approach as wewill see when investigating the quantum interaction of light with matter in chapter 5.

3.3.6 Quantum correlations of quantum fields

We emphasised in section 2.3 how a realistic description of the optical field needs to takeinto account its statistical character, at the classical level as has been seen in the previouschapter but also at the quantum level with additional features of a specific quantumnature. The importance of statistics in optics has been realised by Mandel & Wolf (1995)but they missed the importance of quantum mechanics and it was for Glauber (1963)to formalise into the theory of Quantum Coherence. In all cases it is the fluctuationsof light which underly the notion of optical coherence. Since there are always at leastthe quantum fluctuations of the field, a fundamental definition of coherence is requiredat the quantum level. Glauber provided such a definition by emphasising the role of


correlations. One of his main contributions in this respectwas to separate the notion ofoptical coherence from that of monochromaticity, and the notion of interference fromthat of phase.

A generic property of the light field can depend on arbitrary high-order correlationfunctions. On the other hand a statistical classical model—that for all relevant detailswould be modelled after a stationary Gaussian stochastic process—has all its informa-tion contained already in its frequency spectrum (first order in the correlators). The firstsuch physical property to require a higher order correlation thang(1) was the bunchingof photons counted in a Hanbury Brown–Twiss experiment which motivated the workof Glauber. At a time when the Hanbury Brown–Twiss effect washighly controversial,Purcell (1956) stood in its favour and pointed out the quantum character of the effectfrom the Bose statistics, even predicting antibunching forfermions. In the abstract ofhis paper published in Nature, he comments:

“Brannen and Ferguson (preceding abstract) have suggestedthat the correlationbetween photons in coherent beams observed by Brown and T. (cf. above), if true,would require a revision of quantum mechanics. It is shown that this correlationis to be expected from quantum mech. considerations and is due to a clumpingof photons. If a similar experiment were performed with electrons a neg. cross-correlation would be expected.”

His semi-classical insight is based on quantum statistics of Bosons along the linesor arguments used in section 3.1.4 on the symmetric states ofBose particles. Let us con-sider two particles,a andb which can be detected by any one of two detectors,1 and2.In the case where particles are classical (distinguishable), the probabilityPc of detectingone particle on each detector is the sum of the probabilitiesof all possibilities, namelydetectinga in 1 andb in 2 or vice versa, with respective probabilities|〈a|1〉〈b|2〉|2and|〈a|2〉〈b|1〉|2 so that

Pc = 2|ab|2 . (3.94)

If particles are bosons, however, there is quantum interference of their trajectoriesat detection, and the amplitudes sum, rather than the probabilities (which is one tenetof how quantum mechanics extends classical mechanics). Theprobability remains themodulus square of the total amplitude, so that:

PQ = |〈a|1〉〈b|2〉 + 〈a|2〉〈b|1〉|2 (3.95)

= |2ab|2 = 2Pc (3.96)

which shows how the probability of joint detection increases for thermal bosons as com-pared to classical particles. As the latter display no correlation, this increase translates asa bunching of particles, that is, a tendency to cluster and arrive together at the detector.

Exercise 3.20(∗∗) Spell out the procedure outlined above for plane wave modes forthe photons (with|a〉, |b〉 dotting with momentum〈k| and position〈r|) and by usingannihilation operators to model the detection. Follow the routes of the particles and howthey interfere destructively. Recover the result of Problem 2.9 obtained in a classicalpicture.

QUANTUM STATES 105

We now systematise this idea to more general quantum states,a necessity followingfrom an insight by Glauber (1963), who inaugurated a series of publications to accountfor the Hanbury Brown-Twiss correlations in a fully quantumpicture. His approachcompeted with the classical one by Mandel et al. (1964) whichput a strong empha-sis on statistics but essentially excluding quantum effects, or at any rate excluding theconcept of nonclassical fields that we have touched upon in the previous section. Thesemiclassical picture, Glauber observes, based on a statistical approach requires solelythe knowledge of the power spectrum (2.56), whereas the fullquantum picture allowsstates which need to track up to arbitrary high number of correlators.

First Glauber derives the quantum analog of Eq. (2.60) with quantum fields andoperators and obtains Eq. (3.97).

Exercise 3.21(∗∗) The photodetection can be modelled by the ionisation of atoms ofthe active medium of the detector, these being at positionsr1 andr2, respectively. Call-ing wi the constant transition probability for an atom excited by the beam show (sum-ming over final electron states) that the probability of coincidence detectionw(t1, t2)at t1 andt2 is given by:

w(t1, t2) = w1w2

Tr(

ρE(−)(r1, t1)E(−)(r2, t2)E

(+)(r1, t1)E(+)(r2, t2)

)

Tr(

ρE(−)(r1, t1)E(+)(r1, t1))

Tr(

ρE(−)(r2, t2)E(+)(r2, t2))

(3.97)

A more direct route is to agree on some canonical quantisation of the classical no-tions of coherence developed in section 2.3, indeed still largely valid in the quantumregime. This require only the fieldE—which was ac-number quantity in the previouschapter—to now be promoted to its operator form Eq. (3.72). If we consider a singlemode (notinga the associated annihilation operator), we find the expressions ofg(n) forquantum fields withn = 1, 2 given by:49

g(1)(τ, t) =

⟨

a†(t)a(t+ τ)⟩

〈a†(t)a(t)〉 , (3.98a)

g(2)(τ, t) =〈a†(t)a†(t+ τ)a(t+ τ)a(t)〉

〈a†(t)a(t)〉2 . (3.98b)

which matches with Eq. (3.97).Once they are evaluated on the quantum state of a given system, the correlators

Eq. (3.98) becomec-number functions which can be processed in the same way asbefore, e.g., theg(1) fed into Eq. (2.56) provides the emitted spectra of the system, as inthe classical case, but this time taking into account the quantum dynamics of the systemand quantum fluctuations of the state.

49Higher order correlations, withn ≥ 3, are defined in the same way but they find little practical valueeven in fields where these quantities have been mastered for along time. Theoretically, however, Glaubercrowns the definition of coherence by stating that a field isall order coherentif g(n)(t, t + τ) = 1 forall n ∈ N and allτ ∈ R. That is, the correlations all factorise into products of single operator averages. Thisis the case for a monochromatic wave, for instance, or in terms of states, for a coherent state|α〉.


Fig. 3.3: Bunching (front) and antibunching (back) oflight as observed by Hennrich et al. (2005) by varyingthe average number of atoms in a cavity. At zero delay,g(2) has a peak when light arrives in lumps and a dropwhen photons avoids each other. The curves are sym-metric in time and tend asymptotically (though slowly)towards 1 for long delays.

For a single mode, the second order correlatorg(2) is often more interesting, espe-cially in the quantum regime. At zero time delay, it becomes

g(2)(0) =〈(a†)2a2〉〈a†a〉2 (3.99)

Observe indeed how this quantity is sensitive to the quantumstate considered:

Exercise 3.22(∗) Show that the following values ofg(2)(0) are obtained for the asso-ciated quantum states:

g(2)(0) State

1 − 1

nFock state|n〉

1 Coherent state|α〉 (cf. Eq. (3.79))

2 Thermal state (cf. Eq. (3.86))

The most noteworthy result of exercise (3.22) isg(2)(0) for the Fock state which iszero forn = 1 and is smaller than1 for all n ∈ N. This contradicts Eq. (2.63), if oneremembers that the two-time coincidence probabilityP2 is proportional tog(2)(τ) andthatg(2)(∞) = 1, which translates as the constrain:

g(2)(0) ≥ 1 (from classical model) (3.100)

The reason is because there is no classical state that can describe correctly a Fockstate, which thus poses itself as a purequantum fieldwithout any classical analog. Onthe other hand, coherent states and thermal states and all the mixed states interpolat-ing between them do have such classical counterparts and theeffects they display (likebunching or no-correlations) can be explained classicallyor semi-classically. Butanti-bunching, typically—that is the decrease of probability of a second detection once one

QUANTUM STATES 107

has been registered—cannot be reproduced by any classical model. How quantizationof the field bears on this problem can be illustrated as follows: If instead of the light of astar or the sun, a single photon is fed to the HBT setup, maximum anti-correlation resultfrom annihilation of a quantum, which in this case is an abrupt annihilation of the wholesignal. Indeed the branch which detects the photon destroysit and the probability to de-tect it on the other branch becomes zero (whereas if collecting from a star, detecting aphoton means increase of the chance that another one will arrive on the other branch).

How to say whether the “quantum character” of a field is relevant can be decidedsimply from the GlauberP representation: all states whoseP representation has a sin-gularity stronger than a Dirac delta function, is a quantum state without any classi-cal counterpart. Indeed one can show that theP distribution for the Fock state|n〉is a 2n partial derivative of the delta function.50 For other non-classical states, it canbe thatP becomes negative. Physically, wheneverP cannot be interpreted as a well-behaved probability distribution, the field is quantum. Otherwise a classical descriptionwith statistical weight of plane waves weighted by theP function will provide an ade-quate substitute. The first report of such a field without a good classical counterpart—byKimble et al. (1977) who observed photon antibunching in resonance fluorescence—still ranks among the few unambiguous direct evidence of quantization of the opticalfield.

Typical experimental results forg(2)(τ) are shown on Fig. 3.3 where light emit-ted from an atomic ensemble coupled to a single cavity mode offers the nice featureof changing from bunching to antibunching with the number ofatoms, as reported byHennrich et al. (2005). There is a peak at zero delay in the bunching case (in front ofthe plot) and a dip in the antibunching case (in back). Observe however that despitethe dip,g(2)(0) is still higher than 1 and Eq. 3.100 is not violated in this case, butg(2)(0) ≥ g(2)(τ) is and this also is a signature of a quantum field. This is one reasonwhy it is interesting to knowg(2)(τ) also at nonzero values, although quantum charac-teristics are usually more marked at zero delay and a decay towards uncorrelated valuesof the field are of course eventually obtained at long time (the timescale of this decay isanother good reason to measure or compute at nonzero delay).

3.3.7 Statistics of the field

The zero-delay second order coherence degree,g(2)(0), presents however strong exper-imental and theoretical assets, mainly because it is a single time parameter (althoughit is still g(2)(t, 0)). It embeds a lot of information in a single quantity, especially it isable to characterise the nonclassical character of the field, it has the strongest signaturescompared to other delays which suffer decoherence (and therefore decorrelation) andtheoretically it can be computed easily in the simple quantum-mechanical pictures wehave presented earlier in this chapter.

50Explicitly, the expression reads

PFock(α, α∗) =exp(|α|2)

n!∂2nαn,α∗nδ(α) . (3.101)


Fig. 3.4: Computer generated time series for detection of photons for:i) antibunched light such as emitted by resonance fluorescence of an atom (withg(2) < 1),ii) coherent light such as emitted by a laser or other coherent source (g(2) = 1),iii) bunched light such as emitted by a candle, the sun or any suchincoherent source (g(2) > 1) .

The quantity whichg(2)(0) captures in the most general case is the so-called “count-ing statistics” of the field, already discussed, and of whichwe give a graphical represen-tation in terms of a time series of detection events for threelimiting cases on Fig. 3.4,namely a Fock state (anti-bunching), coherent state (no correlation) and thermal state(bunching). Observe how the upper sequence—associated to Fock states and thereforeto a field of a highly quantum nature—the chain of detections looks like a stream of wellspaced events reaching the target one by one, as if emitted bya “photon gun”. Such kindof emitters are highly prized for quantum information processing.51 If an additional de-gree of freedom like polarisation can be controlled for eachemission, it is possible withsuch a device to setup a completely secure (unbreakable) cryptographic system, as isstudied in the problem at the end of this chapter.

The counting statistics is itself strongly linked to thestatistics of the state, whichis the probability of a given state to be found with a given occupancy numbern (to befound in the state|n〉). These probabilities are obtained from the diagonal elements ofthe density matrix in the basis of Fock states. Therefore, this statistics is defined as

p(n) = 〈n| ρ |n〉 (3.102)

From the definition Eq. (3.98b) and the Boson algebra, it is straightforward to obtainthe formula which linksg(2) to p(n) in the Fock state basis:

g(2)(0) =

∞∑

n=0

n(n− 1)p(n)

(

∞∑

n=0

np(n))2

(3.103)

with ρ the density matrix. It is easy to compute the statistics of most of the states con-sidered so far. The results are compiled in Table 3.1.

Other states present more obscure distributions. This is the case of the mixture state,for instance, which despite being merely a Gaussian with an offset, (cf. Eq. (3.89)), ismore involved mathematically.

51Antibunched states are more easily obtained with fermions,for instance they are easily formed by elec-trons passing through a large resistance due to Coulomb interaction.

QUANTUM STATES 109

Fock state Coherent state|α〉 Thermal state|n0〉 cf. Eq. (3.79) cf. Eq. (3.84)

Distributionp(n) δn,n0 e−|α|2 |α|2nn!

1

n+ 1

(

n

n+ 1

)n

〈a†a〉 =∑

n np(n) n0 |α|2 n

〈(a†a)2〉 − 〈a†a〉2 0 |α|2 n2 + n

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

Pro

babi

lity

n

<n>=5

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20

n

<n>=15

10

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20

n

<n>=15

10

Table 3.1 Probability distribution and moments of basic quantum states.i) First row is the probability to haven excitationsii) Second row is the mean number of excitationsiii) Third row is the varianceiv) Fourth row is the profile for averages〈a†a〉 = 1, 5 and10.

Exercise 3.23(∗∗) Consider the state which is the mixture of a thermal state with in-tensitynt and a coherent field with intensitync, whichP representation is given byEq. (3.89). Show that the distribution of this state is:

pm(n) = exp

(

− nχ

1 + n(1 − χ)

)

(n(1 − χ))n

(1 + n(1 − χ))n+1Ln

(

− χ

(1 − χ)(1 + n(1 − χ))

)

(3.104)whereLn is thenth Laguerre polynomial52 andχ is the coherent ratio (percentage of“coherent particles”):

χ =nc

nc + nt(3.105)

Distribution Eq. (3.104) of mixture states is plotted on Fig. (3.5) for values ofχranging from0 to 100% by step of10%, also with the two limiting cases of the purecoherent state (χ = 1) and the pure thermal state (χ = 0). The mathematical expressionis heavy but as a function it is well behaved and brings no problem for a numericaltreatment. For analytical efforts, as this distribution flattens very quickly with small

52The Laguerre polynomials are a canonical basis of solutionsfor the differential equationxy′′ +

(1 − x)y′ + ny = 0. They can be written asLn(x) = ex

n!dn

dxn

`

e−xxn´

and computed bycarrying out the derivation. First Laguerre polynomials are for n ≥ 0, Ln(x) = 1 (constant),−x + 1, 1

2(x2 − 4x + 2), . . . See Weisstein, Eric W. “Laguerre Polynomial.” From MathWorld at

http://mathworld.wolfram.com/LaguerrePolynomial.html.


incoherent fractions, it can profitably be replaced by a Gaussian which is much betterapproximation than to consider pure coherent states, as seen on Fig. 3.5. The first twomoments ofPm can be retained exactly, namely the meann and the variance computedas

σ2 = n+ n2t + 2ncnt (3.106)

which moreover linksχ andg(2) through:

g(2)(0) = 2 − χ2 . (3.107)

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

p(n)

n

Coherent fraction=0%10%20%30%40%50%60%70%80%90%

100%

Fig. 3.5: Statistics of mixtures of thermal and coherent states in proportions ranging from 0 (thermal state) to100 per cent (coherent state) of coherence, for an average number of particlen = 100. At 90% of coherence,the deviation is already very significant. The pure coherentstate which has been truncated extends four timeshigher than is displayed, see Fig. 3.6 where this state is fully displayed. On the other hand, the curves with10% of coherence and the thermal state are practically superimposed.

3.3.8 Polarisation

Elliptical polarisation

The polarisation state of a photon and of a beam of photons is described by directtransposition of the Jones vector into a quantum superposition of basis states. This canbe described using second-quantized formalism as well, as is done now in the mostgeneral case of elliptical polarisation.

A photon with circular polarisation degree given byP ≡ cos2 θ − sin2 θ is thecoherent superposition of a spin-up photon with probability cos2 θ and of a spin-downphoton with probabilitysin2 θ, therefore, its quantum state can be created from thevacuum|0, 0〉 (zero spin-up and zero spin-down photon) by application of the followingoperator:

|1, θ, φ〉 ≡ (cos θa†↑ + eiφ sin θa†↓) |0, 0〉 . (3.108)

QUANTUM STATES 111

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 20 40 60 80 100 120 140 160

p(n)

n

Coherent fraction=90%95%99%

99.9%100%

0

0.005

0.01

0 100 200

Fraction=90%Gaussian fit

Fig. 3.6: Statistics of mixtures with high fractions of coherence, in respective proportions of 90% (also dis-played on Fig. 3.5), 95%, 99%, 99.9% and 100% (coherent state) of coherence. For high values of coherence,the exact expression (3.104) can be approximated by a simpleGaussian, as illustrated in inset. The approxi-mation becomes much better to higher ratio of coherence.

where the angleφ is the in-plane orientation of the axis of the polarisation ellipse. Thisdefinesa†θ,φ the creation operator for an elliptically polarized photonas

a†θ,φ ≡ cos θa†↑ + eiφ sin θa†↓. (3.109)

The superposition ofn such correlated photons is obtained by recursive application ofthe creation operator:

|n, θ, φ〉 = a†nθ,φ |0〉 =1√n!

(cos θa†↑ + eiφ sin θa†↓)n |0, 0〉 , (3.110)

which we have normalized (here|0〉 is the vacuum in the space of elliptically polarizedstates).

Pseudospin formalism

To deal with the polarisation of a radiation mode, it can be convenient to introduce anew operator defined as

S = a↑a†↓ (3.111)

It is the ladder operatorSx + iSy of operatorsSx ≡ ℜa↑a†↓, Sy ≡ ℑa↑a†↓ andSz ≡a†↑a↑ − a†↓a↓ which follow a spin-12 algebra. For this reasonS is called thepseudospin.It is a powerful representation for two-level systems. Its in-plane components character-ize correlations that exist between spin-up and spin-down states. Intensities of linearlypolarized components of the emitted light are linked to the pseudospin as follows:

I↔ =n0

2+ 〈Sx〉, Il =

n0

2− 〈Sx〉 , (3.112)


wheren0 = 〈a†↑a↑ + a†↓a↓〉 is the total number of particles, and the degree of linearpolarisation,ρl, follows as

ρl =2|〈S〉|n0

, (3.113)

Exercise 3.24(∗∗∗) The BB84 protocol of single photon quantum cryptography.One desirable goal of cryptography is the secure communication of a key, which

is a binary digit list of arbitrary length which two partiesA andB should be able tocommunicate at will. The exact information carried by the key is not important as longas it is known by the two parties and them only.

Assume that the two parties use a quantum channel over which they can send singlephotons which reachB in the polarisation state they have been encoded byA. Theyalso use a classical channel over which they can communicateany information theyneed, given however that this information is thus also made available to potential code-breakers.

By using the concept of conjugate bases, cf. exercise (3.3) on page 77, design aprocess by whichA is able to generate and communicate toB a key which they theyboth know for sure has not been observed by any third party.

3.4 Outlook on quantum mechanics for microcavities

Quantum physics at large is a significant active and important area of research today. Itis at the outset of numerous series of topics which diverge from each other as they getmore specialised. One of these routes leads to microcavities.

What makes this topic specially attractive is the depth and extent that it affords, aprecious and rather uncommon quality in today’s research, where specialisation reducethe physics to its most intricate details. This point can be illustrated from the photogra-phy below taken at the occasion of the fifth Solvay conference. The first edition in 1911was also the first international conference (the series is still continued to this date, it isheld every three years) on the topic ofRadiation and the Quanta. Einstein attended asthe youngest participant (at 25 years old). We have put in bold the names of the sci-entists whose work is central to the physics which makes the topic of this book, andwithout whose knowledge, one cannot pursue useful research. As one can clearly see,microcavities physics essentially brings again to the forethe fundamental physics of thefathers of modern science.

3.5 Further reading

This chapter has reviewed some of the basic aspects of quantum mechanics and theirrelevance to the optical field to cover our needs for the more specific treatment in whatfollows and in a form and context such as they would typicallybe encountered in thephysics of microcavities. But dealing with such general issues, this chapter is also themost remote from the object of study of this book and therefore requires much supple-mentary reading to appreciate the subject in some depth. General quantum mechanicscan be obtained from countless sources, e.g., Merzbacher (1998). For quantum fieldtheory, attention should be directed towards texts writtenwith statistical physics or

FURTHER READING 113

A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder,E. Schrodinger, E. Verschaffelt,W. Pauli,W. Heisenberg, R.H. Fowler,L. Brillouin , (upper row);P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers,P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, (middle); I. Langmuir, M. Planck,Mme. Curie,H.A. Lorentz , A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson and O.W. Richardson(lower row) posing for the 1927 Solvay conference.

condensed matter in mind, as there is little to be gained fromthe more popular rela-tivistic formulation which has different concerns. Renowned classics are the textbooksby Abrikosov et al. (1963), Negele & Orland (1998) and Fetter& Walecka (2003). Fora special attention paid to the light field while still maintaining a strong field-theoreticapproach, one can refer to the textbook by Cohen-Tannoudji et al. (2001). For quan-tum optics proper, Mandel & Wolf (1995) provide the best reference, accompanied by avery large literature with such textbooks as those by Loudon(2000) or Scully & Zubairy(2002) among the most significant.

4

SEMI-CLASSICAL DESCRIPTION OF LIGHT-MATTERCOUPLING

In this chapter we consider light coupling to elementary semiconductorcrystal excitations—excitons—and discuss the optical properties ofmixed light-matter quasi-particles named excitons-polaritons, whichplay a decisive role in optical spectra of microcavities. Ourconsiderations are based on the classical Maxwell equations coupled tothe material relation accounting for the quantum properiesof excitons.

LIGHT-MATTER INTERACTION 115

4.1 Light-matter interaction

4.1.1 Classical limit

Although this chapter refers to a “semi-classical” treatment of light-matter interaction,there naturally exists a “pure classical treatment” where matter is put on an equal “clas-sical” footing with light (cf. chapter 2). This descriptionis moreover a very powerfulone and which will lay down an important concept: theoscillator strength. It is how-ever simple enough to serve as an introduction to the so-called semi-classical treatment,where parts of the quantum concepts are involved in a rather vague and intuitive wayinto the material excitation. This helpful and short description in full classical terms iswhat we address now.

Lorentz proposed a fully classical picture of light-matterinteractions where light ismodelled by Maxwell equations and the atom by a mechanical system of two masses—the nucleus and an electron—bound together by a spring.ω0 is therefore the naturalfrequency of oscillation. This purely mechanical oscillator will carry along many con-cepts into the quantum picture. The spring is set into motionwhen light irradiates theatom.

Hendrik A. Lorentz (1853-1928), who received the 1902 Nobel Prizein physics (with Zeeman) for his work on electromagnetic radiation,provided an insightful classical description of light-matter interactions.

His doctoral thesis in 1875 developed Maxwell’s theory of 1865 toexplain reflection and refraction of light. He is also noted for the LorentzTransformationof space and time dilations and contractions which wouldbe at the heart of Einstein’s special theory of relativity.

The gist of the physical implications of this assumption is retained in the simplestcase where the atom is fixed and the electron a distancex(t) away, moving under theinfluence of an applied electric fieldE(t), with equation of motion:

m0x+m02γx+m0ω20x = −eE(t) (4.1)

wherem0 is the mass of the electron,−e its charge andm0ω20 the harmonic potential

binding the electron (the spring). Excludingγ, this is merely Newton’s equationF =m0a of a dipole whose accelerationa is driven by a forceF . The loss term arises in thismodel from the fact that an oscillating dipole radiates energy.53 With this understanding,

53That a moving charge radiates energy is one of the problems with the classical picture of atoms wherethe electron is thought of as orbiting (hence, moving around) the nucleus, therefore doomed to spiral intoit as it loses energy. It cannot stay still neither because ofthe gravitational attraction of the nucleus. Bohrpostulated that the electron cannot move smootly towards its nucleus because of interferences along certainorbits, giving birth to the original quantum theory.

116 SEMI-CLASSICAL DESCRIPTION OF LIGHT-MATTER COUPLING

its expression can be related to fundamental constants by computing the rate of energyloss of a dipole:

Exercise 4.1 (∗∗) Show thatγ = e2ω20/(3m0c

3) with c the speed of light.

In our case where the oscillator models an atom, it is a good approximation to modelthe exciting field as an harmonic function of time—the vacuumsolution of Maxwellequations—withE(t) = E0 cos(ωt). One can solve Eq. (4.1) and obtain the steadystate of the system, where the electron also oscillates harmonically with the frequencyof the external force but with a different amplitude and a different phase:

x(t → ∞) = A cos(ωt− φ) (4.2)

One can see the intrinsic simplicity of the Lorentz oscillator. We are going to seenow its considerable richness. When we say “different amplitude and phase” of oscil-lation, we mean essentially the change in the response as a function of the excitingfrequencyω (see Fig. 4.1):

A(ω) =−eE0

m0

1√

(ω2 − ω20)

2 + (2γω)2(4.3a)

φ(ω) = arctan( 2γω

ω20 − ω2

)

(4.3b)

The external frequency that maximizes the amplitudeA for a given set of param-etersω0 and γ is the resonant frequency. We find it by taking the derivativeof theamplitude function and setting it to zero:

ωres =√

ω20 − 2γ2 . (4.4)

At this frequency the field is transferring energy most efficiently to the atom’s elec-tron. In the inset of Fig. 4.1) one can see how the resonant frequencyωres(γ) varieswith the damping. There is only a genuine resonance for caseswith γ < ω0/

√2, other-

wise the system is overdamped and does not oscillate. On the other hand, ifγ = 0 theresonant frequency is simply the natural frequency of the system. In the limiting casewhereγ ≪ ω0/

√2 (for frequencies close toω0), the energy transfer distribution is a

Lorentzian:γ

(ω − ω0)2 + (γ)2(4.5)

A more general method to solve Eq. (4.1), is to find the “output” or “answer”x(t)of the system to the “input” fieldE(t) = exp(iωt) through itstransfer functionH:

x(t) = H(ω)E(t) . (4.6)

The linearity of Eq. (4.1) allows one to computeH directly (by direct substitution)as


0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

Osc

illat

ion

ampl

itude

(a.

u.)

ω

γ = 0.1, 0.3, 0.7, 10

ω0 =1

0

0.5

1

0 0.2 0.4 0.6

γ

ωRes

0

0.5

1

0 0.2 0.4 0.6

γ

ωRes

Fig. 4.1: The amplitude of the oscillations of the electron,driven by a sinusoidal external field of frequencyω,shows a resonace whenω = ωres. In the inset plot we can see how the resonant frequencyωres varies withthe damping coefficientγ, becomingω0 for the undamped case. The natural frequency of the system isset toone.

H(ω) =e/m0

ω2 − ω02 − 2iωγ(4.7)

Then, the time evolution of the oscillator for an arbitrary excitationE(t)—other thanexp(iωt) for which Eq. (4.6) applies—can be obtained directly as:

x(t) =

∫ ∞

−∞h(t− τ)E(τ) dτ (4.8)

whereh is the Fourier transform ofH. The imaginary part ofH is calledabsorptionsignal as it provides similar information as the amplitudeA we have analysed in ourparticular case.

4.1.2 Einstein coefficients

A fundamental problem of light-matter interaction fallingout of the scope of classicalphysics as noted by the beginning of the XXth century is that of blackbody radiation.54

So fundamental was this problem that any mathematical trickto derive the solution wasproviding a physical insight into a worldview going much beyond that afforded by thethen existing models, which would flourish into quantum mechanics. The attempt byPlanck culminated in his law for blackbody radiation:

54A black bodyis an object which radiate energy originating from intrinsic emission by the object and isnot the result of reflection or transmission from external radiation. Therefore all these radiations from outsidethe object and which impinge on it are absorbed by it (and later re-radiated but only as a result of how theblack body stores energy to maintain its thermal equilibrium). This is an ideal limiting case (since all objectsreflect or transmit light to some extent) to investigate the thermodynamic of the electromagnetic field.


I(ν) =8πhν3

c31

ehν/(kBT ) − 1(4.9)

whereI is the spectral energy density (with dimension of joule per cubic meter persecond−1), h is Planck’s constant,ν the frequency,c the speed of light,kB Boltzmann’sconstant andT the temperature. Its main merit is the perfect accord with experimentaldata, previously afforded only when separating short and long wavelengths. Expres-sion (4.9) gave impetus to the concept of the photon.55 The attempt by Bose led to thenew statistics of Bosons and in 1916, the attempt of Einsteinhimself—already a key ac-tor in the two previous approaches—led to the fundamental processes of semiclassicallight-matter interaction. These are the topics of this section.

Einstein proposed the existence of three fundamental processes in the interaction oflight with matter, that is—in the view of the time—in the interaction of a photon withan atom. These are:

1. Absorption2. Spontaneous emission3. Stimulated emission

Fig. 4.2: The three fundamental processes of light-matter interaction in the semi-classical paradigm:Absorp-tion, Spontaneous emissionandStimulated emission.

The first two processes present no difficulty once the postulate of quantised energylevels is accepted: a photon of energyEf −Ei can be absorbed by an atom thus excitingit from energy levelEi to energy levelEf (energy is conserved and—this is the nov-elty arising from the semi-classical/old quantum theory—exchanged in discrete amountdefined by the atom structure).

Spontaneous emission is the reverse process where energy isreleased by the atomand a photon created which carries it away. It is calledspontaneousbecause an excitedatom will decay by itself into a lower energy level (until it reaches the ground state,where it stays until it gets excited again). The average timefor this transition is, likethe possible energies of the atom, a property of the atom.56 It will later be for quantummechanics to calculate it; at this stage this is a given constantτsp.

55Einstein interpreted Planck’s hypothesis in terms of “quanta” of light—one reason for which he is cred-ited as a founding father of Quantum Mechanics—which he usedfor his explanation of the photoelectriceffect. The name “photon” itself was coined in by the chemistGilbert Lewis as a support of a theory whichwas soon refuted and abandoned.

56Spontaneous emission is not anintrinsic property of the atom as it can be modified by changing theoptical environment of the atom. This important point is discussed in detail in Chap. 6.


To account for the Planck distribution, however, Einstein requires the introductionof the third process,stimulated emission.57 Because of this process, an excited atom—in addition to the decay by spontaneous emission (first process)—can also decay ifit is interacting with a photon, by emitting a clone of this photon. This is not veryintuitive but we now show that if it is accepted, the Blackbody radiation spectrum isreadily derived by mere rate equations and detailed balancearguments. Each of theseprocesses are associated to a probability of occurrence: the probability per unit time ofspontaneous emission is calledA, that is, if the density (number per unit volume) ofatoms in excited (resp. ground) state isn2 (resp.n1), there is a transfer from excited toground state populations following

dn1

dt= An2 . (4.10a)

Eq. (4.10a) implies thatA = τ−1sp .

Absorption is ruled by Einstein coefficientB12 which gives the probability per unittime and per unit energy density of the radiation field that the atom initially in state 1absorb a photon and jump to state 2 causing a change in the number density of atoms inthe ground state of:

dn2

dt= B12n1I(ν) (4.10b)

with I(ν) the spectral intensity of the radiation field at the frequency of radiationν =(E2 − E1)/h. This equation merely spells out the definition ofB given above, withI(ν) quantifying the number of photons available to excite the atom.

Stimulated emission, as said above, is induced by a photon bringing an atom in theexcited state to its ground state. It is therefore the same process as Eq. (4.10b) onlyreversed, with an excited atom as the starting state and finishing with one more groundstate atom. Calling the corresponding probability of this eventB21

dn1

dt= B21n2I(ν) . (4.10c)

At equilibrium all these three processes concur to establish the steady state condi-tionsdni/dt = 0. In this regime the average change in the populations of ground andexcited states is zero, being balanced by the losses and gains. The principle of “detailedbalance” postulates that such an equilibrium in population exchanged is reached bypairwise compensations of the typedn1/dt = −dn2/dt which leads to:

n2A− n1B12I(ν) + n2B21I(ν) = 0 (4.11)

which provides the energy density as a function of other parameters as

I(ν) =A

(n1/n2)B12 −B21(4.12)

We are investigating the energy distribution of the radiation field. That of the atoms(which form a classical system like a gas or a solid and nothing like a Bose condensate

57Historically one was speaking of “induced” rather than “stimulated” emission, but the new term whichis now prevailing is more common and vivid for other similar manifestations, such as “stimulated scattering”.


where their statistics could play a role) was well known fromthe earlier work of Boltz-mann and Maxwell. The kinetic theory of gas that they developed gives the populationsof atomic states after the ratio of their energy to the temperature:

ni = gi exp(

− EikBT

)

(4.13)

(with gi the degeneracy of the state), so that finally:

I(ν) =A

(g1/g2) exp(hν/kBT )B12 −B21(4.14)

Identifying this result with Eq. (4.9) provides the following expressions for Einsteincoefficients:

A =~ω3

π2c3B21 , (4.15a)

g1B12 = g2B21 . (4.15b)

4.2 Optical transitions in semiconductors

An arena of choice for microcavity physics is that of semiconductor physics. A semi-conductor is a solid whose electrical conductivity has behaviour and magnitudes in be-tween metals and insulators. This comes from the energy levels of such systems whichform bandsseparated bygapsof forbidden energies (or states). Consequently, semicon-ductors afford a great control of electronic excitations. We note here the most essentialfeatures of the structure of optical transitions in semiconductors.58

Fermi level Energy gap width Conductivity (Ω−1m−1)

metals Inside the bandany Up to6.3 × 107 (silver)semiconductors Inside the gap < 4eV Varies in large limitsdielectric Inside the gap 4eV Can be as low as10−10

Table 4.1 Classification of solids.

The discrete electronic levels of individual atoms formbandsin crystals where thou-sands of atoms are assembled in a periodic structure. There are also gaps between theallowed bands where no electronic states exist in an ideal infinite crystal. Those crystalswhich have a Fermi level59 inside one of the allowed bands aremetals, while the crystalshaving a Fermi level inside the gap aresemiconductorsor dielectrics. The difference be-tween semiconductors and dielectrics is quantitative: thematerials where the band gap

58Much more information on this subject can be found in CharlesKittel, Introduction to Solid StatePhysics (Wiley: New York, 1996) and Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt:Orlando, 1976).

59The Fermi energy is the energy below which, at zero temperature, all the electronic states are occupiedand above which all the states are empty. The Fermi level is the set of states with Fermi energy.

OPTICAL TRANSITIONS IN SEMICONDUCTORS 121

containing the Fermi level is narrower than about 4eV are usually called semiconduc-tors, the materials with wider band gaps are dielectrics. Inthis Chapter we consider onlysemiconductor crystals.

The eigenfunctions of electrons inside the bands have a formof so-calledBlochwaves. The concept of the Bloch waves was developed by the Swiss physicist FelixBloch in 1928 (see on page 170), to describe the conduction ofelectrons in crystallinesolids. TheBloch theoremstates that a wavefunction of an electronic eigenstate in aninfinite periodic crystal potentialV (r) can be written in form (see Section 2.6.1):

Ψk,n = Uk,n(r)eik·r, (4.16)

whereUk,n (calledBloch amplitude) has the same periodicity as the crystal poten-tial, k is a so-calledpseudo-wave vectorof an electron (further we shall omit “pseudo”while speaking about this quantity), andn is the index of the band.

Substitution of the wave-function (4.16) into the Schrodinger equation for an elec-tron propagating in crystal

− ~2

2m0∇2Ψk,n + V (r)Ψk,n = Ek,nΨk,n, (4.17)

with m0 being the free electron mass, one obtains an equation for theBloch amplitude:

− ~2

2m0∆Uk,n + V (r)Uk,n +

(~2k2

2m0+

~

m0(k · r)

)

= Ek,nUk,n (4.18)

wherep = ~

i∇. Consideration of the operators in the parentheses as a perturbationconstitutes the method of thek · p perturbation theorywhich readily enables solvingthe shape of the electronic dispersion in the vicinity ofk = 0 points of all bands andwhich appears to be strongly different from the free electron dispersion in vacuum.Approximating:

Ek,n ≈ E0,n +~2k2

2mn∗(4.19)

is called theeffective mass approximationwith mn∗ being the electron effective massin n-th band:60

1

mn∗=

1

m0+

2

m20

∑

l 6=n

|〈U0,l|p|U0,n〉|2E0,l − E0,n

. (4.20)

The frequencies and polarisation of the optical transitions in direct gap semiconduc-tors are governed by the energies and dispersion of the two bands closest to the Fermilevel61, referred to as the conduction band (first above the Fermi level) and the valenceband (first below the Fermi level; often several close bands are important).

Semiconductors can be divided into those with direct band gaps and indirect bandgaps. In indirect gap semiconductors (like Si and Ge) the electron and hole occupying

60In general, the effective mass is a tensor. It reduces to a scalar in crystals having a cubic symmetry.61In semiconductors, the Fermi level is situated in the gap. The width of this gap Eg governs the optical

absorption edge.


Fig. 4.3: Zinc-blende (left) and wurtzite (right) crystal lattices.

lowest energy states in conduction and valence bands cannotdirectly recombine emit-ting a photon due to the wavevector conservation requirement. While a weak emissionof light by these semiconductors due to phonon-assisted transitions is possible, they canhardly be used for fabrication of light-emitting devices and studies of light-matter cou-pling effects in microcavities. In the following, we shall only consider the direct gapsemiconductor materials like GaAs, CdTe, GaN, ZnO etc (see Fig. 4.4). Most of themhave either a zinc-blende or a wurtzite crystal lattice62 (see Fig. 4.3). In zinc-blendesemiconductors, the valence band splits into three sub-bands referred to as the heavy-hole, light-hole and spin-off bands (see Figure 4.5). Atk = 0 the heavy and light holebands are degenerate in bulk crystals, while this degeneracy can be lifted by strain orexternal fields. In the wurzite semiconductors the valence band is split into three non-degenerate subbands referred to as A, B, and C bands.

Dispersion of the light and heavy holes in zinc-blende semiconductors can be con-veniently described by theLuttinger Hamiltonian63

H =~2

2m0[(

γ1 +5

2γ2

)

∆ − 2γ3(∇J)2 + 2(γ3 − γ2)(

J2x

∂2

∂x2+ J2

y

∂2

∂y2+ J2

z

∂2

∂z2

)

],

(4.21)whereJα = ± 1

2 ,± 32 , α = x, y, z and,γ1, γ2, γ3 are Luttinger band parameters depen-

dent on the material.In the bulk zinc-blende samples, atk = 0 the degenerate light and heavy hole states

mean that the probability of an allowed optical transition from a heavy hole to an elec-tron state is three times higher than the probability of a transition from a light hole state(see Fig. 4.6). That is why illumination of a semiconductor crystal by circularly polar-ized light leads to preferential creation of electrons witha given spin projection. Thiseffect referred to asoptical orientationwill be discussed in more detail in Chapter 9.

Optical absorption spectra in semiconductors are governedby the density of elec-tronic states in the valence and conduction bands,g(E) = ∂n

∂E , wheren is the numberof quantum states per unit area. In bulk crystals, inside thebands the density of states

62A cubic phase is somewhat more exotic. It is found for GaN, forexample.63Proposed in the famous paper by J. M. Luttinger and W. Kohn, Physical Review 97, 869 (1955).

OPTICAL TRANSITIONS IN SEMICONDUCTORS 123

Fig. 4.4: Energy gaps and lattice constants in some direct-bandgap semiconductors and their alloys, fromVurgaftman et al. (2001).

Fig. 4.5: Schematic band structure of a zinc-blende semiconductor (a) with a conduction band (on the top),degenerated heavy and light hole bands (in the middle) and the spin-off band (at the bottom) and a wurtzitesemiconductor (b) with A, B and C valence subbands.

behaves as√E, which results in the corresponding shape of the interband absorption

spectra. Besides this, at low temperatures the absorption spectra of semiconductors ex-hibit sharp peaks below the edge of interband absorption (i.e. at frequenciesω < Eg/~,whereEg is the band-gap energy). These peaks manifest the resonant light matter cou-pling in semiconductors. They are caused by the excitonic transitions which will remainthe focus of our attention throughout this book.


Fig. 4.6: Polarisation of the interband optical transitions in zinc-blende semiconductor crystals. Solid, dashedand dotted lines showσ+, σ− and linearly polarized transitions, respectively.

4.3 Excitons in semiconductors

4.3.1 Frenkel and Wannier-Mott excitons

In the late 1920s narrow photoemission lines were observed in the spectra of organicmolecular crystals by Kronenberger & Pringsheim (1926) andI. Obreimov and W. deHaas. These data were interpreted by the Russian theorist Frenkel (1931) who intro-duced the concept of excitation waves in crystals and later coined the termexciton(Frenkel 1936).

By definition, the exciton is a Coulomb-correlatedelectron-hole pair. Frenkel treatedthe crystal potential as a perturbation to the Coulomb interaction between an electronand a hole belonging to the same crystal cell. This scenario is most appropriate in or-ganic molecular crystals. The binding energy of Frenkel excitons (i.e. the energy of itsionisation to a non-correlated electron hole pair) is typically of the order of 100-300meV. Frenkel excitons have been searched for and observed inalkali kalides by Ap-ker & Taft (1950). At present they are widely studied in organic materials where theydominate the optical absorption and emission spectra.

At the end of the 1930s, Wannier Wannier (1937) and Mott (1938) developed theconcept of excitons in semiconductor crystals where the rate of electron and hole hop-ping between different crystal cells much exceeds the strength of their Coulomb cou-pling with each other. Unlike Frenkel excitons, Wannier-Mott excitons have a typicalsize of the order of tens of lattices constants and a relatively small binding energy (typ-ically, a few meV).

Besides Frenkel and Wannier-Mott excitons, there are few other types of excitons.Thecharge transfer excitonsare spatially separated Coulomb-bound electron hole pairshaving a spatial extension of the order of the crystal lattice constant. The lowest en-ergy charge transfer exciton usually extends over two nearest neighbour moleculesin a molecular crystal and creates a so-called donor-acceptor complex. See, for in-stance Silinish (1980) for a more thorough treatment.

EXCITONS IN SEMICONDUCTORS 125

Yakov Il’ich Frenkel (1894–1952), Sir Nevill. FrancisMott (1905–1996) and GregoryWannier (1911–1983) gave their name to the two main categories of excitons.

Frenkel was a versatile physicist who made his main contributions in solid state physics. He wrotethe first paper devoted to the quantum theory of metals. He is now remembered for the exciton bearinghis name and theFrenkel defect. A very prolific writer, Kapitza reportedly once told him “You would be agenius if you published ten times less than you do” (his most noted work is the textbook “Kinetic Theoryof Liquids”). A good overview of his life and work is given by S. L. Lopatnikov and A. H.-D. Cheng. in J.Engrg. Mech.,131, 875, (2005).

Mott, the Nobel prize-winner for Physics in 1977 for “fundamental theoretical investigations of theelectronic structure of magnetic and disordered systems”,is most renowned for his mechanism explainingwhy material predicted to be conductors by band theory are inreality (so-called Mott-) insulators and fordescribing the transition of substances from metallic to nonmetallic states (Mott transition).

Wannier authored a series of important papers on the properties of crystals. His main achievement isa complete set of orthogonal functions, known as “Wannier functions”, which provide an alternativerepresentation of localized orbitals to the usual Bloch orbitals. According to his graduate student D. R.Hofstadter who pays him tribute in Phys. Rep.,110, 237 (1984), “it is not what [he] would want to be knownfor primarily. He was so involved in so many areas of physics,and his breadth was so refreshing, comparedto the narrow range of most physicists today, that I think he would wish to be remembered for that breadthand for his style, a style that stressed beauty and purity andfundamentality”.

Another example of a few-particle exciton complex in a quantum confined semi-conductor system is the so-calledanyon excitonappearing in the regime of the quantumHall effect as described by Rashba & Portnoi (1993). The energy of anyon excitonictransitions lies in the far infrared range thus these quasiparticles cannot be coupled tolight in optical cavities.

Recently, Agranovich et al. (1997) have proposed a concept of hybrid Frenkel-Wannier Mott exciton which can be formed in mixed organic-inorganic structures. Suchexcitons would combine a huge binding energy and relativelylarge size. Extensive in-formation on the Frenkel or hybrid excitons and their coupling with light in organic mi-crocavities can be found in the recent volume edited by Agranovich & Bassani (2003).

In the present Section we only discuss the Wannier-Mott excitons in semiconduc-tor structures. Such excitons can be conveniently described within the effective massapproximation which allows to neglect the periodic crystalpotential and describe elec-trons and holes as free particles having a parabolic dispersion and characterized by


Fig. 4.7: Wannier-Mott exciton is the solid state analogy ofa hydrogen atom, while they have very differentsizes and binding energies. Unlike atoms, the excitons havea finite lifetime.

effective masses dependent on the crystal material. Usually, the effective masses of car-riers are lighter than the free electron mass in vacuumm0. For example, in GaAs theelectron effective mass isme = 0.067m0, the heavy-hole mass ismhh = 0.45m0.

Consider an electron-hole pair bound by the Coulomb interaction in a crystal havinga dielectric constantε. The wavefunction of relative electron-hole motionf(r) can befound from the Schrodinger equation analogous to one describing the electron state in ahydrogen atom:

− ~2

2µ∇2f(r) − e2

4πε0rf(r) = Ef(r) (4.22)

with µ = memh/(me +mh) the reduced mass,r =√

x2 + y2 + z2 the distance be-tween electron and hole. The solutions of Eq. (4.22) are wellknown as they correspondto the states of the hydrogen atom with the following renormalizations:64

m0 → µ, e2 → e2/ε (4.23)

For example, the wavefunction of the 1s state of exciton reads:

f1s =1

√

πa3B

e−r/aB (4.24)

64In the hydrogen atom problem the reduced mass is equal in goodapproximation to the electron massm0

because of the very big mass of the nucleus.


with the Bohr radiusaB given as:

aB =~2ε0

4πµe2. (4.25)

The binding energy of the ground exciton state is

EB =(4π)2µe4

2~2ε2ε20=

(4π)2~2

2µa2Bε

20

. (4.26)

Given the difference between the reduced massµ and the free electron mass, andtaking into account the dielectric constant in the denominator, one can estimate thatthe exciton binding energy is about three orders of magnitude less than Rydberg con-stant. Table 4.2 shows the binding energies and Bohr radii for Wannier-Mott excitons indifferent semiconductor materials.

Semiconductor crystalEg (eV) me/m0 EB (eV) aB (A)

PbTe∗ 0.17 0.024/0.26 0.01 17000InSb 0.237 0.014 0.5 860Cd0.3Hg0.7Te 0.257 0.022 0.7 640∗∗

Ge 0.89 0.038 1.4 360GaAs 1.519 0.066 4.1 150InP 1.423 0.078 5.0 140CdTe 1.606 0.089 10.6 80ZnSe 2.82 0.13 20.4 60GaN∗∗∗ 3.51 0.13 22.7 40Cu2O 2.172 0.96 97.2 38∗∗∗∗

SnO2 3.596 0.33 32.3 86∗∗∗∗

Table 4.2 Strongly anisotropic conduction and valence bands, directtransitions farfrom the center of the Brillouin zone.∗ Strongly anisotropic conduction and valence bands, directtransitions far from thecenter of the Brillouin zone.∗∗ In the presence of magnetic field of 5T.∗∗∗ An exciton in hexagonal GaN.∗∗∗∗ The ground state corresponds to an optically forbidden transition, data givenfor n = 2 state.

The exciton excited states form a number of hydrogen-like series. Observation ofsuch a series of excitonic transitions in the photoluminescence spectra of Cu2O in 1951was the first experimental evidence for Wannier-Mott excitons (see Fig. 4.3.1). Thisdiscovery was made by the Russian spectroscopist Evgeniy Gross who worked in thesame institution—the Ioffe Physico-Technical institute in Leningrad—as Ya.I. Frenkelat that epoque.


Evgenii FedorovichGross (1897–1972) andthe experimental discovery of the exciton: theHydrogen-like “yellow” series in emission ofCu2O observed by Gross et al. (1956), withits numerical fit. Beside discovering experimen-tally the exciton, he is also noted for pio-neering in 1930 the experimental observationof Rayleigh scattering fine structure due toBrillouin-Mandelstam light scattering on acous-tic waves. A short account of his scientificachievements can be found in “Evgeni Fe-dorovich Gross”, B. P. Zakharchenya and A. A.Kaplyanski, Sov. Phys. Uspekhi,11, 141 (1968).

4.3.2 Excitons in confined systems

Since the beginning of the 1980s, progress in the growth technology of semiconduc-tor heterostructures encouraged study of Wannier-Mott excitons in confined systemsincluding quantum wells, quantum wires and quantum dots. The main idea behind de-velopment of heterostructures was to create artificially potential wells and barriers forelectrons and holes combining different semiconductor materials. The shape of the po-tential in conduction and valence bands is determined in these structures by positionsof the corresponding band edges in the materials used as wellas by the geometry of thestructure. Theband engineeringin semiconductor structures by means of high-precisiongrowth methods has allowed the creation of a number of electronic and opto-electronicdevices including transistors, diodes and lasers. It has also permitted discovery of im-portant fundamental effects including the integer and fractional quantum Hall effects,Coulomb blockade, light-induced ferromagnetism etc.

The large size of Wannier-Mott excitons makes them stronglysensitive to nanometer-scale variations of the band edge positions which can be easily obtained in modern semi-conductor nanostructures. The energy spectrum and wavefunctions of quantum confinedexcitons can be strongly different from those of bulk excitons. Here we consider bymeans of an approximate but efficient variational method theexcitons in quantum wells,wires and dots (see Figure 4.8). We will use the effective mass approximation. Whenwe refer to wavefunctions we always mean the envelope functions, neglecting the Blochamplitudes of electrons and holes. Note that in these examples we neglect the complex-ity of the valence band structure and consequent anisotropyof the hole effective masswhich sometimes strongly affect the excitonic spectrum in real semiconductor systems.More information on excitons in confined systems can be foundin the books by Bastard(1988), Ivchenko & Pikus (1997) and Ivchenko (2005).

4.3.3 Quantum wells

The Schrodinger equation for an exciton in a quantum well (QW) reads:


Fig. 4.8: Reduction of the dimensionality of a semiconductor system from 3D to 0D from a bulk semicon-ductor to a quantum dot. The electronic density of statesg(E) = dN/dE—with dN the number of electronquantum states within the energy intervaldE—changes drastically between systems of different dimension-alities as is shown schematically on the figure. This variation of the density of states is very important forlight emitting semiconductor devices.

− ~2

2me∇2e −

~2

2mh∇2h + Ve(ze) + Vh(zh) −

e2

4πε0ε|re − rh|Ψ = EΨ (4.27)

with Ve,h(ze,h) the confining potential for electron, hole on thez-axis which is thegrowth axis of the structure. Solving exactly Eq. (4.27) is not an easy task. We approachthe problem variationally over a class of trial functions having the form:

Ψ(re, rh) = F (R)f(ρρρ)Ue(ze)Uh(zh) (4.28)

where

R =mere +mhrh

me +mh(4.29)

is the exciton center of mass coordinate and

ρρρ = ρρρe − ρρρh (4.30)

is the in-plane radius-vector of electron and hole relative motion,r = (ρρρ, z). Fourcomponents of the trial function (4.28) describe the exciton center of mass motion, therelative electron-hole motion in the plane of the QW, and electron and hole motionnormal to the plane direction. The factorization of the exciton wave function makessense when the QW width is less than or comparable to the exciton Bohr diameter in thebulk semiconductor. In this case, electron and hole are quantized independently of eachother. On the other hand, in larger QWs, one can assume that the exciton is confined as awhole particle and keeps the internal structure of a 3D hydrogen atom. Here and further


we shall consider narrow QWs where Eq. (4.28) represents a good approximation. Thefour terms which compose the exciton wave-function are normalized to unity:

∫

|Ue(ze)|2dze = 1,

∫ ∞

0

|f(ρ)|22πρdρ = 1 ,

∫

|Uh(zh)|2dzh = 1,

∫ ∞

0

|F (R)|22πRdR = 1 . (4.31)

After substitution of the trial function (4.28) and integration overR, Eq. (4.27)becomes:

− ~2

2me

∂2

∂z2e

− ~2

2mh

∂2

∂z2h

− 1

ρ

∂

∂ρ

(

~2

2µρ∂

∂ρ

)

+ Ve(ze) + Vh(zh)

− e2

ε√

ρ2 + (ze − zh)2− P 2

exc

2(me +mh)− E

f(ρ)Ue(ze)Uh(zh) = 0

(4.32)

wherePexc is the excitonic momentum,P = 0 for the ground state. Eq. (4.32) can betransformed into a system of three coupled differential equations, each defining one ofthe components of our trial function. The equation forf(ρ) is obtained by multiplicationof both parts of Eq. (4.32) byU∗

e (ze)U∗h(zh) and integrating overze andzh. This yields:

−1

ρ

∂

∂ρ

(

~2

2µρ∂

∂ρ

)

− e2

ε

∫∫ |Ue(ze)|2|Uh(zh)|2√

ρ2 + (ze − zh)2dzedzh

f(ρ) = −EQWB f(ρ)

(4.33)whereEQWB is the exciton binding energy. The electron and hole confinement energiesEe andEh, and wavefunctionsUe,h(ze,h), can be obtained by multiplying Eq. (4.32)by f∗(ρ)U∗

h,e(zh,e) and integrating overze,h andρ:

− ~2

2me,h∇2e,h + Ve,h −

e2

ε

∫∫ |f(ρ)|2|Uh,e(zh,e)|2√

ρ2 + (ze − zh)22πρdρdzh,e

Ue,h(ze,h)

=Ee,hUe,h(ze,h)(4.34)

In the ideal 2D case,|Ue,h(ze,h)|2 = δ(ze,h) and equation (4.33) transforms into

− ~2

2µ

1

ρ

∂

∂ρ

(

ρ∂

∂ρ

)

− e2

ερ

f(ρ) = E2DB f(ρ) (4.35)

which is an exactly solvable 2D hydrogen atom problem. For the ground state:

f1S(ρ) =

√

2

π

1

a2DB

exp(−ρ/a2DB ) (4.36)

with


a2DB =

aB

2(4.37)

andaB the Bohr radius of the three dimensional exciton given by Eq.(4.25). The bindingenergy of the two-dimensional exciton exceeds by a factor of4 the bulk exciton bindingenergy:

E2DB = 4EB (4.38)

For realistic QWs, Eqs. (4.32—4.33) still can be decoupled if the Coulomb term in (4.33)is neglected. This allows to find the functions|Ue,h(ze,h)| independently from eachother as well asf(ρ). Solving Eq. (4.32) with a trial function

f(ρ) =

√

2

π

1

aexp(−ρ/a) (4.39)

wherea is a variational parameter, one can express the binding energy as:

EQWB (a) = − ~2

2µa2+

e2

4πε0ε

∫∫∫ |f(ρ)|2|Ue(ze)|2|Uh(zh)|2√

ρ2 + (ze − zh)22πρdρdzedzh (4.40)

Maximization ofEQWB (a) finally yields the exciton binding energy in a QW, whichranges fromEB toE2D

B and depends on the QW width and barrier heights for electronsand holes. The binding energy increases if the exciton confinement strengthens. Thatis why the dependence of the binding energy on the QW width is non-monotonic: forwide wells the confinement increases with the decrease of theQW width, while forultranarrow wells the tendency is inverted due to tunnelingof electron and hole wavefunctions into the barriers (Fig. 4.9).

Fig. 4.9: Exciton binding energy as a function of the QW width(schema). The insets show the QW potentialand wavefunctions of electron (convex shape) and hole (concave shape) for different QW widths.


4.3.4 Quantum wires and dots

Variational calculation of the ground exciton state energyand wavefunction in quantumwires or dots can be done using the same method of separation of variables and decou-pling of equations as for QWs. There exist a number of important peculiarities of wiresand dots with respect to wells, however.

For a wire, the Schrodinger equation for the wavefunction of electron-hole relativemotionf(z), with z the axis of the wire, reads:

− ~2

2µ∇2z −

e2

4πε0ε

∫∫ |Ue(ρe)|2|Uh(ρh)|2√

z2 + (ρe − ρh)2dρedρh

f(z) = −EQWWB f(z) (4.41)

with the kinetic energy for relative motion being along the axis of the wire.Ue,h(ρe,h)is the electron, hole wavefunction in the plane normal to thewire axis andEQWW

B is theexciton binding energy in the wire. Despite visible similarity to Eq. (4.33) for electron-hole relative motion wavefunction in a QW, Eq. (4.41) has a different spectrum anddifferent eigenfunctions. As a quantum particle in 1D Coulomb potential has no groundstate with a finite energy, the exciton binding energy in a quantum wire is drastically de-pendent on spreading of the functionsUe,h(ρe,h) and can, theoretically, have any valuebetweenEB and infinity. The trial function cannot be exponential (as itwould have adiscontinuous first derivative atz = 0 in this case). The Gaussian function is a betterchoice in this case. Usually, realistic quantum wires do nothave a cylindrical symmetry(most popular are “T-shape” and “V-shape” wires, see Fig. 4.10), which makes com-putation ofUe,h(ρe,h) a separate, difficult task. Moreover, the realistic wires have afinite extension in thez-direction which is comparable with the exciton Bohr-diameterin many cases. Even if the wire is designed to be much longer than the exciton dimen-sion, inevitable potential fluctuations inz-direction lead to the exciton localization. Thismakes realistic wires similar to elongated quantum dots (QDs).

An exciton is fully confined in a QD, and if this confinement is strong enough itswave function can be represented as a product of electron andhole wave-functions:

Ψ = Ue(re)Uh(rh) (4.42)

where the single-particle wavefunctionsUe,h(re,h) are given by coupled Schrodingerequations:

− ~2

2me,h∇2e,h + Ve,h −

e2

4πε0ε

∫ |Uh,e(rh,e|2|re − rh|

Ue,h(re,h) = Ee,hUe,h(re,h)

(4.43)with Ve,h is the QD potential for an electron, hole. In this case, the exciton bindingenergy is defined as

EQDB = E0e + E0

h − Ee − Eh (4.44)

whereE0e andE0

h are energies of non-interacting electron and hole, respectively, i.e.,the eigenenergies of the Hamiltonian (4.43) without the Coulomb term.


Fig. 4.10: Cross-sections of V-shape (a) and T-shape (b) quantum wires, from Di Carlo et al. (1998).

Fig. 4.11: Transmission electron microscopy image from Widmann et al. (1997) of the self-assembled QDsof GaN grown on AlN

In small QDs Coulomb interaction can be considered as a perturbation to the quan-tum confinement potential for electrons and holes. The exciton binding energy can beestimated using perturbation theory as

EB ≈ e2

4πε0ε

∫∫ |Ue(re)Uh(rh)|2|re − rh|

dredrh (4.45)

As in the wire, the exciton binding energy in the dot is strongly dependent on thespatial extension of the electron and hole wavefunctions and can range from the bulkexciton binding energy to infinity, theoretically. In realistic wires and dots, the binding


Solomon I.Pekar (1917–1985), John J.Hopfield (b. 1933) and Vladimir M.Agranovich (b. 1929).

Pekar was the leading theorist of the Physical Institute in Kiev. He created the theory of adiabatic polaronsin 1946 (the term polaron comes from him), and with his work onadditional light wave—stating that underexcitation by monochromatic light, the dielectric constant’s spatial dispersion near an excitonic resonancegives rise to an additional propagating polariton wave—he originated the theory of exciton-polaritons.

Hopfield, after important contributions to Physics, turnedhis interest to biology, where he made his mostsignificant contribution to science with his associative neural network now know as theHopfield network. Atthe time of writing, he is professor in the department of Molecular Biology at Princeton University and thePresident of the American Physical Society.

Agranovich is the head of the Theoretical Department of the Institute of Spectroscopy of the Russian Academyof Sciences. He made seminal contributions to the theory of excitons especially in organic crystals and is oneof the founders of the theory of polaritons. A proficient author, he fathered among other important work,“Crystal Optics with Spatial Dispersion, and Excitons” with Nobel laureate Ginsburg, a “Theory of Excitons”and recently the monograph “Electronic Excitations in Organic Based Nanostructures”. He currently holdsthe special position of “Pioneer of Nano-Science” at the University of Texas at Dallas.

energy rarely exceeds 4EB, however. At present, small QDs are mostly fabricated by so-called Stransky-Krastanov method of molecular beam epitaxy and have either pyramidalor ellipsoidal shape (see Fig. 4.11). In large quantum dots (“large” meaning “of a sizeexceeding the exciton Bohr diameter”) excitons are confinedas whole particles andtheir binding energy is equal to the bulk exciton binding energy. Good examples oflarge quantum dots are spherical microcrystals which may serve also as photonic dots.

Exercise 4.2 (∗) Find the binding energies of the first excited states of 2D and1D exci-tons.

4.4 Exciton-photon coupling

It has been clear since the very beginning of experimental studies of excitons that theeasiest way to create these quasiparticles is by optical excitation. In the mid-1950s,theorists understood that coupling to light strongly influences the physical propertiesof excitons and their energy spectrum. The Ukrainian physicist Pekar (1957) was thefirst to describe these changes of the exciton energy spectrum due to coupling to lightin terms ofadditional wavesappearing in the crystal. Almost simultaneously, the termpolariton appeared in the works of Agranovich (1957) (Russia) and Hopfield (1958)

EXCITON-PHOTON COUPLING 135

(U.S.A.) devoted to the description of photon-exciton coupling.65

We begin the description of exciton-polaritons from equations proposed by Hopfield(1958), adapting his notations to our exposition.

The first equation reads:

εBc2

∂2

∂t2E(r, t) + ∇×∇× E(r, t) = −4π

c2∂2

∂t2P(r, t) (4.46)

which directly follows from the wave-equation (2.22) and material relation Eq. (2.16).εB is the normalised background dielectric constant66 which does not contain the exci-tonic contribution. The link between the polarisation and the electric field is given bythe second Hopfield equation:

[

∂2

∂t2+ 2γ

∂

∂t+ ω2

0 − ω0

Mx∇2

]

4πP(r, t) = εBω2pE(r, t) (4.47)

whereωp is the so-calledpolariton Rabi frequency, ω0 is the exciton transition energydependent on the difference of energies of an electron and a hole composing the excitonand on the exciton binding energy, andMx = me +mh is the exciton translation mass.

Eq. (4.47) is derived assuming linear optical response of the system and consider-ing each exciton as an harmonic oscillator having its eigenfrequency corresponding tothe energy of the excitonic transition, with damping causedby exciton interaction withacoustic phonons. The polarisation created by excitons is taken to be proportional tothe amplitude of the harmonic oscillator, which constitutes the so-calleddipole approx-imation. A derivation in full details appears in the textbook of Haug& Koch (1990). Adouble Fourier transform of Eq. (4.47) yields

P(ω,k) = εBE(ω,k) +εBω

2pE(ω,k)

ω20 − ω2 − 2iωγ + ω0k2/Mx

(4.48)

In the vicinity of the resonant frequency, one can express the normalised dielectric func-tion from Eq. (4.48) as

ε(ω, k) = εB +εBωLT

ω0 − ω + k2/(2Mx) − iγ(4.49)

whereωLT is so-called longitudinal-transverse splitting. Forγ = 0 andMx → ∞, itis equal to the splitting between the frequencies at which the dielectric constant goesto infinity (ωT ) and to zero (ωL = ωT + ωLT). This splitting is a direct measure of

65Thepolaritonsin their most generally accepted terminology refer to mixedlight-matter states in crystals.They do not necessarily imply excitons, but can also be formed, typically, by optical phonons. In this textwe use the term “polariton” to meanexciton-polaritonsonly. The reader can find a recent starting point onphonon-polaritons in, e.g., the text of Stoyanov et al. (2002).

66εB is normalised so that it is equal to unity in vacuum.


the coupling strength between the exciton and light, and is proportional to the excitonoscillator strengthf :

f =(4πε0)

2√εBπ

m0c

e2ωLT . (4.50)

For the ground exciton state in GaAs,ωLT = 0.08meV, while in wide-band-gapmaterials (GaN, ZnO) it is an order of magnitude larger.

The dependence of the tensor of the dielectric susceptibility of a crystal on thewavevector of light is referred to in crystal optics asspatial dispersion. Eq. (4.49) de-scribes the spatial dispersion induced by excitons in semiconductors.

The relation between frequency and wavevector of the transverse polariton modesis given by

k2 = ε(ω, k)ω2

c2, (4.51)

which is a biquadratic equation with solutions:

k21,2 = −M

~

(

ω0 − ω − iγ − εB2M

ω2

c2

)

± M

~

√

(

ω0 − ω − iγ +εB2M

ω2

c2

)2

+2

MεBωLT

ω2

c2. (4.52)

For the longitudinal modes, the conditionε(ω, k) = 0 yields

k2L =

2Mx

~(ω − ω0 − ωLT + iγ) . (4.53)

One can see that at each frequency two transverse and one longitudinal polaritonmode with different wavevectors can propagate in the same direction. The appearance ofadditional light modes in crystals at the exciton resonancefrequency as a result of spatialdispersion was theoretically predicted by Pekar in 1957 andconfirmed experimentallyby Kiselev et al. (1973). Description of additional light modes in finite size crystalslabs requires Additional Boundary Conditions (ABC) whichhave to be imposed onthe dielectric polarisation in spatially dispersive media. Pekar proposed the followingform for the ABC:

P = 0 (4.54)

at the surface of the crystal. The condition comes from the physical argument that theexcitons which are responsible for appearance of the polarisationP exist only inside thecrystal. Physically, the Pekar conditions assume that the exciton wavefunction is con-fined within the crystal slab, which acts on the exciton as a potential well with infinitelyhigh barriers. The choice (4.54) of ABC is not the only possible one. In a number ofworks the concept of a so-called “dead-layer” is used, assuming that the exciton centreof mass cannot approach the interface closer than some critical length of the order ofthe exciton Bohr radius. In general, Neumann-type conditions on the polarisation and itsderivative may be formulated. Though Pekar ABC have proven to provide a good agree-ment between theoretical and experimental spectra of exciton-polaritons, the debate onthe exact form of ABC still continues.


Here we use the Pekar ABC to obtain the eigenfrequencies of exciton-polaritonmodes in a crystal slab of thicknessL. Assuming zero in-plane wave vector of all lightmodes (normal incidence geometry), one can readily obtain from Eq. (4.54) the condi-tions on wavevectorskj , (j = 1, 2,L):

kj =Nπ

L, N ∈ N (4.55)

Fig. 4.12 shows the dispersion of transverse (solid lines) and longitudinal (dashedline) polariton modes in GaAs calculated with Eqs. (4.52) and (4.53), respectively. Thevertical dotted lines show those values of the wavevectors which satisfy the condi-tion (4.55) for a given value ofL. Crossing points of the dotted lines and the dispersioncurves yield the discrete spectrum of eigenfrequencies of exciton-polaritons in the thinfilm. These frequencies correspond to resonances in reflection or transmission spectra.The splitting between neighboring eigenfrequencies increases with decrease of the ex-citon mass and the thicknessL. The fit of optical spectra of exciton-polaritons in thinfilms allows one to obtain the exciton mass with a good accuracy.

Fig. 4.12: Dispersion of the transverse (solid) and longitudinal (dashed) exciton-polariton modes in GaAs ascalculated by Vladimirova et al. (1996). Vertical dotted lines show positions of quantum confined polaritonmodes in a 1148A thick film of GaAs.

4.4.1 Surface polaritons

Surface polaritons result from exciton coupling with lightmodes having a componentof wavevector in the plane of the surface,kx > ω/c, i.e., outside of the light cone (weassume that TM-polarized light propagates along thex-axis, in the plane, so that themagnetic field vector of the light-wave is parallel to they-axis also in the plane). In thiscase, the light wave propagating along the surface of the crystal decays in vacuum. If weconsider the right surface, the electric field vector of sucha mode in vacuum behaves as:


E+ = E+0 exp

(

ikxx−√

k2x −

ω2

c2z

)

(4.56)

(z-axis is normal to the surface). The electric field vector in the crystal reads:

E− = E−0 exp

(

ikxx+

√

k2x −

ω2

c2ε(ω)z

)

(4.57)

whereε(ω) is the same as in Eq. (4.49). Dependent on the sign and value ofε(ω), theelectric field may decay or not inside the crystal. From Eqs. (4.56) and (4.57), one easilyobtains the ratio of thex- andz-components of the field:

E+z

E+x

=kx

i√

k2x − ω2

c2

,E−z

E−x

= − kx

i√

k2x − εω

2

c2

. (4.58)

The dispersion of surface polariton modes can be obtained from the Maxwell boundaryconditions, which require:

E−x = E+

x , ε(ω)E−z = E+

z . (4.59)

The second condition comes from the continuity of the magnetic field at the surface.From (4.58) and (4.59) we obtain:

ε(ω)

√

k2x −

ω2

c2= −

√

k2x − ε(ω)

ω2

c2, (4.60)

thus

ω = ckx

√

1 + ε(ω)

ε(ω), ε < 0 . (4.61)

To analyse the dispersion equation (4.61), let us consider the limitγ → 0 andMx →∞. ε(ω) is a real function schematically displayed in Fig. 4.13. Eq.(4.61) yields the realexciton-polariton eigenfrequencies ifε(ω) < −1. One can see that this condition canonly be satisfied within the frequency range

ω0 ≤ ω < ω0 + ωLT − δ , (4.62)

whereδ can be found from the condition

ε(ω0 + ωLT − δ) = −1 . (4.63)

The dispersion of surface polaritons can now be easily understood: it starts atω =ω0, kx = ω/c and goes toω → ω0 + ωLT − δ, kx → ∞ as shown on Fig. 4.14.

Exercise 4.3 (∗∗) A short pulse of light centred at the exciton resonance frequency isreflected from a semi-infinite semiconductor crystal. Calculate the time dependence ofthe intensity of reflected light neglecting the spatial dispersion of exciton-polaritons.

Exercise 4.4 (∗∗∗) A short pulse of light centred at the exciton resonance frequency istransmitted through a semiconductor crystal slab of thicknessd. Calculate the time de-pendence of the intensity of transmitted light neglecting the spatial dispersion of excitonpolaritons.


Fig. 4.13: Dielectric constant in the vicinity of the exciton resonance (scheme). The broadening and specialdispersion are neglected.

Fig. 4.14: Schematic dispersions of the surface polariton (solid), the transverse bulk polaritons (dashed)and the longitudinal bulk polariton (dotted). Spatial dispersion and broadening of the exciton resonance areneglected.

4.4.2 Exciton-photon coupling in quantum wells

In quantum confined structures the second Hopfield equation (4.47) cannot be directlyused as the exciton is no more a free moving particle and its wavevector in the con-finement direction is not defined. On the other hand, the theory of spatial dispersion inoptical media can still be applied to describe the dielectric response of quantum struc-tures containing excitons, where the exciton wavefunctionplays the role of a correlationfunction. Indeed, once an exciton is created, the dielectric polarisation changes in allpoints where its wavefunction spreads.

The first theoretical description of exciton-polaritons in2D structures was givenby Agranovich & Dubovskii (1966). In this Section we will follow the so-called non-local dielectric response theory, developed in the beginning of the 1990s by Andreaniet al. (1991) and Ivchenko (1992) to describe the optical response of excitons in QWs.


Fig. 4.15: Transmission intensity through a slab of Cu2O of thicknessd = 0.91mm as calculated by Panzarini& Andreani (1997).

We consider the simplest case of reflection or transmission of light through a QW inthe vicinity of the exciton resonance at normal incidence. We neglect the differencebetween background dielectric constants of the well and barrier materials (which isusually small) and only take into account the exciton-induced resonant reflection.

The non-local dielectric response theory is based on the assumption that the exciton-induced dielectric polarisation can be written in the form:

Pexc(z) =

∫ ∞

−∞χ(z, z′)E(z′)dz′ , (4.64)

whereχ(z, z′) = χ(ω)Φ(z)Φ(z′) (4.65)

with

χ(ω) =Q

ω0 − ω − iγ, Q = εBωLTπa

3B . (4.66)

Here,Φ(z) = Ψexc(R = 0, ρ = 0, ze = zh = z) is the exciton wavefunction (4.28)taken with equal electron and hole coordinates,ω is the frequency of the incident light,γ is the homogeneousbroadening of the exciton resonance, same as in the Hopfieldequations,ωLT andaB are the longitudinal-transverse splitting and Bohr radiusof exci-ton in the bulk material. Once the polarisation (4.64) is introduced, Eq. (4.46) becomesan integro-differential equation and can be solved exactlyusing the Green’s functionmethod. In this method the solution of Eq. (4.46) is represented in the form

E(z) = E0 exp(ikz) + k20

∫

Pexc(z′)G(z − z′)dz′ (4.67)

whereE0 is the amplitude of the incident light,k0 = ω/c and the Green’s functionGsatisfies the equation:


(

∂2

∂z2+ k2

)

G(z) = −δ(z), k =√εBk0 . (4.68)

Bearing in mind that∫∞−∞ f(z′)δ(z − z′)dz = f(z′), one easily checks thatG is

given by:

G(z) =i exp(ik|z|)

2k. (4.69)

Eq. (4.67) can be solved with respect toE(z). In order to do it, multiply the left andright parts of Eq. (4.67) byΦ(z) and integrate overz. This yields:

∫

EΦ(z)dz = E0

∫

Φ(z) exp(ikz)dz+

+ k20χ

∫∫

Φ(z)Φ(z′)G(z − z′)dzdz′∫

EΦ(z′′)dz′′ (4.70)

which means that∫

EΦ(z)dz =E0

∫

Φ(z) exp(ikz)dz

1 − k20χ∫∫

Φ(z)Φ(z′)G(z − z′)dzdz′. (4.71)

We now return to Eq. (4.67) and substitute Eq. (4.71) into itsright-hand side:

E =E0 exp(ikz) + k20 χ

∫

Φ(z′)G(z − z′)dz′∫

E(z′′)Φ(z′′)dz′′

=E0

[

eikz +k20χ∫

Φ(z′)G(z − z′)dz′∫

eikzΦ(z′′)dz′′

1 − k20 χ∫∫

Φ(z)Φ(z′)G(z − z′)dzdz′

]

(4.72)

Using Eq. (4.69) we finally obtain

E(z) = E0eikz +

ik02√εBQE0

∫

Φ(z′′)eikz′′

dz′′∫

Φ(z′)eik|z−z′|dz′

ω0 − ω − iγ −Q ik02√εB

∫∫

eik|z′−z′′|Φ(z′)Φ(z′′)dz′dz′′(4.73)

The amplitude reflection (r) and transmission (t) coefficients of the QW can then beobtained as

r =E(z) − E0(z)e

ikz

E0(z)e−ikz

∣

∣

∣

∣

z→∞and t =

E(z)

E0eikz

∣

∣

∣

∣

z→∞. (4.74)

If we consider a ground exciton state in a QW,Φ(z) is an even function and the integralson the right-hand side of Eq. (4.73) can be easily simplified.

In the casez → ∞,∫

Φ(z′)G(z − z′)dz′ =i

2k

∫

Φ(z′)eik(z−z′)dz′ =

ieikz

2k

∫

cos(kz′)Φ(z′)dz′

(4.75)and in the casez → −∞,


Fig. 4.16: Schema of multiple reflections of light within a crystal slab.

∫

Φ(z′)G(z − z′)dz′ =i

2k

∫

Φ(z′)e−ik(z−z′)dz′ =

ie−ikz

2k

∫

cos(kz′)Φ(z′)dz′

(4.76)

∫∫

G(z − z′)Φ(z)Φ(z′)dzdz′ =i

2k

[∫

Φ(z) cos(kz)dz

]2

− 1

2k

∫∫

Φ(z)Φ(z′) sin(k|z − z′|)dzdz′ . (4.77)

This allows us to obtain the reflection and transmission coefficients of the QW in asimple and elegant form:

r(ω) =iΓ0

ω0 − ω − i(Γ0 + γ), (4.78a)

t(ω) = 1 + r(ω) , (4.78b)

where

Γ0 =Qk0

2√εB

[∫

Φ(z) cos(kz)dz

]2

(4.79)

is an important characteristic further referred to as theexciton radiative broadening, and

ω0 = ω0 +Qk0

2√εB

∫∫

Φ(z)Φ(z′) sin(k|z − z′|)dzdz′ (4.80)

is the renormalisation of the exciton resonance frequency due to the polariton effect.The radiative broadeningΓ0 is connected to the exciton radiative lifetimeτ by the

relation:

τ =1

2Γ0(4.81)

which follows from the time-dependence of the intensity of light reflected by a QWexcited by an infinitely short pulse of light:


Fig. 4.17: Typical frequency- (a) and time-resolved (b) reflection spectra of a structure containing quantumwells (schema). In frequency-resolved reflectivity, the exciton resonance usually induces a characteristic mod-ulation on the top of strong background reflectivity dependent on the refractive index of the barrier materials.

R(t) =

∣

∣

∣

∣

1

2π

∫ ∞

−∞r(ω)e−iωtdω

∣

∣

∣

∣

2

= Γ20e

−2Γ0t . (4.82)

A finite exciton radiative lifetime is a peculiarity of confined semiconductor sys-tems. In an infinite bulk crystal, an exciton-polariton can freely propagate in any di-rection and its lifetime is limited only by non-radiative processes such as scatteringwith acoustic phonons. On the contrary, in a QW the exciton-polariton can disappear bygiving its energy to a photon which escapes the QW plane. The polariton effect (some-times referred to as theretardation effect) consists, in this case, in the possibility for theemitted photon to be reabsorbed once again by the same exciton. The chain of virtualemission-absorption processes leads to a finite value ofτ and is also responsible forrenormalisation of the exciton frequency (4.80). This renormalisation does not exceeda fewµeV in realistic QWs, although it becomes more important in quantum dots. Itdoes not play an essential role in microcavities, and we shall neglect it hereafter. Theradiative lifetimeτ is about 10ps in typical GaAs-based QWs. Although it is extremelyhard to observe free excitons in the photoluminescence as often PL is governed mainlyby excitons localised at imperfections of a QW, a lifetime of10±4ps for a free excitonhas been measured experimentally by Deveaud et al. (1991) ina record-quality (for thattime) 100A-thick GaAs/AlGaAs QW (see Fig. 4.18).

4.4.3 Exciton-photon coupling in quantum wires and dots

Quantum wires and dots scatter light. In the vicinity of the exciton resonance, this scat-tering has a resonant character and polariton effects take place. If a wave of light isincident on an array of quantum wires or dots, the interference of waves scattered bydifferent individual wires or dots results in enhanced signals in reflection and trans-mission directions. A regular array of identical wires or dots would diffract light incertain directions similarly to a crystal lattice diffracting X-rays. In realistic semicon-ductor structures, the inevitable potential disorder leads to random fluctuations of the


Fig. 4.18: Time-resolved photoluminescence spectra of a GaAs/AlGaAs QW measured by Deveaud et al.(1991) at different detunings between the excitation energy and the exciton resonance. These data allow anexciton radiative lifetime of about 10ps to be extracted.

exciton resonance frequency from wire to wire and from dot todot. This leads to theinhomogeneous broadeningof exciton resonances in reflection or transmission spectra.The stronger the inhomogeneous broadening, the larger the ratio of scattered to reflectedlight intensity.

Polariton effects in systems of quantum wires and dots manifest themselves in fi-nite exciton radiative lifetime, appearance of collectiveexciton-polariton states due tooptical coupling of different wires or dots and appearance of a specific polarisation-dependent fine-structure of exciton resonances. All these phenomena can be conve-niently described within the nonlocal dielectric responsetheory in a similar way to thatdone for quantum wells.

The dielectric polarisation induced by an exciton in a planearray ofN quantumwires or dots reads

Pexc =

N∑

n=1

TnΦn(r − Rn)

∫

E(r′)Φ(r′ − Rn)dr′ (4.83)

with

Tn =εBωLTπa

3B

ωn − ω − iγ, (4.84)

wherek0 is the wavevector of the incident light in a vacuum,ωLT is the longitudinal-transverse splitting in bulk,εB is the background dielectric constant,aB is the excitonBohr radius in bulk,γ is the homogeneous broadening andΦn(r) is the exciton wave-function in then-th wire or dot taken with equal electron and hole coordinater. Solv-ing the wave equation using the Green’s function method, onerepresents the electricfield as:


E(ω, r) = E0 exp(ik · r)

+ k20

N∑

n=1

Tn

∫

Φn(r′ − Rn)G0(r − r′)dr′

∫

E(r′′)Φn(r′′ − Rn)dr′′ , (4.85)

wherek = k0√εB is the wavevector of the incident light in the medium. In the case of

infinite quantum wires oriented along they-axis

G0(x, y, z) =i

4H

(1)0 (kxx+ kzz) (4.86)

whereH(1)0 is the Hankel function.G0 is a Green’s function which satisfies the equation

−(

∂2

∂x2+

∂2

∂y2+ k2

)

G0 = δ(x)δ(z) . (4.87)

In the case of QDs, Eq. (4.85) is the Green’s function for a zero-dimensional system.For realistic systems, the “quantum dot” model is more relevant, as the excitons inquantum wires are inevitably localized in they-direction due to potential fluctuations.This makes the wires similar to elongated dots. In the following we therefore consideran array of QDs.

The integral equation (4.85) can be treated analytically, see for instance the dis-cussion by Parascandolo & Savona (2005), but the calculation becomes heavy if thenumber of dots is large. A compact analytical expression forthe electric field can beobtained if we assume that the wavefunctions of excitons in all QDs are identical andthatE(ω, r+Rn) ≈ eir·RnE(ω, r). This would be true for a regular grating of identicalQDs. In all other cases, this is a more or less accurate approximation depending on thedegree of disorder in the system. Multiplying Eq. (4.85) byΦ(r) and integrating overrwe obtain in this case:

E(ω, r) = E0 exp(ik · r)

+ k20

N∑

n=1

Tneik·Rn

∫

Φ(r′ − Rn)G0(r − r′)dr′∫

E0eik·rΦ(r)dr

1 −∑Nn=1 Θn

(4.88)

whereΘn = k20Tne

ik·Rn∫∫

G0(r − r′)Φ(r′ − Rn)Φ(r)dr′dr.The Fourier transform of the electric field (4.88) yields itsdirectional dependence,

which can be represented in the form:

Ed(ω,ks) = E0δk,ks +E0

N∑

m=1

iΓQD0

ωm − ω − iγ

1

1 −∑Nn=1 Θn

exp(iks ·Rm) , (4.89)

where

ΓQD0 =

1

6ωLTk

30a

3B

[∫

cos(k · r)Φ(r)dr

]2

(4.90)

andδk,ks = 1 iff k = ks and is zero otherwise.


Fig. 4.19: Exciton radiative decay rate as a function of the radius of a spherical quantum dot R (schema).

The quantityΓQD0 is the exciton radiative decay rate in a single QD. This is an impor-

tant characteristic of light-matter coupling in QDs. It changes non-monotonically withthe dot size as Fig. 4.19 shows. If the size of the dotR is much smaller than the wave-length of light at the exciton resonance frequencyλ0, one can neglect the cosine in theright part of Eq. (4.90). In this case the radiative damping rate is directly proportional tothe volume occupied by the exciton wavefunction. This dependence has been obtainedtheoretically for the first time by Rashba & Gurgenishvili (1962) for impurity-boundexcitons in semiconductors. The minimum ofΓQD

0 corresponds to the strongest excitonconfinement atR ≈ aB. At very smallR the exciton is less confined due to penetrationof the electron and hole wavefunctions into the barriers. AtlargerR the exciton is con-fined as a whole particle inside the dot, and the volume occupied by its wavefunctionincreases proportionally to the size of the dot. When the size of the dot approaches thewavelength of light, the cosine in the right part of Eq. (4.90) can no more be neglected,as was shown by Gil & Kavokin (2002).ΓQD

0 has its maximum aboutR ≈ λ0/(π√εB)

and than decreases.Equation (4.89) describes all kinds of coherent optical experiments (reflection, trans-

mission and Rayleigh scattering) within the same semiclassical formalism and takesproperly into account the polariton effect. In the specularreflection direction, one canneglect the small portion of scattered light. Substitutingsummation by integration inEq. (4.89) we obtain the reflection and transmission coefficients of an array of QDs,rQD andtQD, as:

rQD =β

1 − βand tQD =

1

1 − β(4.91)

where

β = iΓr

∫

f(ν)

ν − ω − iγdν (4.92a)

Γr =k

2d2ωLTπa

3B

∫

[Φ(r) cos(k · r)dr]2 (4.92b)


whered is the average distance between the dots,f(ν) is the distribution function ofexciton resonance frequencies in the dots (

∫∞−∞ f(ν)dν = 1). The ratio of reflected

and average scattered intensities can be estimated from Eqs. (4.89–4.90), assuming thatthe light waves scattered by QDs lying within aΓr vicinity interfere positively in thereflection direction:

IrIs

≈ N

(

Γr∆

)2

, (4.93)

whereN is given by the number of QDs within the spot of light which illuminates thesample. In real systems,Γr ≫ ΓQD

0 , which means that optically coupled QDs emit lightmuch faster than single QDs. The radiative lifetime of an exciton in a single QD variesin large limits as a function of the QD size, typically in the range10−10—10−8s.

Exercise 4.5 (∗∗) Find the reflection coefficient of a system of two identical quantumwells parallel to each other and separated by a distanced.

4.4.4 Dispersion of polaritons in planar microcavities

Fig. 4.20: An electron microscopy image of a GaAs microcavity with GaAlAs/AlAs Bragg mirrors and thecalculated profile of electric field of the cavity mode.

4.4.4.1 Bulk microcavities: In Chapter 2, we considered the eigenmodes of planarmicrocavities in the absence of exciton-photon coupling. Imagine that the cavity isnow made from a material having an excitonic transition at the frequency close to theeigenfrequency of the photonic mode of the cavity. Neglecting the spatial dispersion ofexciton-polaritons in the cavity layer, one can solve the eigenvalue problem using thetransfer matrix technique as we did in Chapter 2. Imposing the boundary condition ofno light incident from left and right sides on the cavity, onecan readily obtain the matrixequation for the eigenfrequencies:


(

cos(kLc)i√ε(ω)

sin(kLc)

i√

ε(ω) sin(kLc) cos(kLc)

)

(

1 + rB√εB(rB − 1)

)

= A

(

1 + rB√εB(rB − 1)

)

(4.94)whereLc is the cavity width and

ε(ω) ≈ εB +εBωLT

ω0 − ω − iγ(4.95)

as follows from Eq. (4.49) if the wavevector dependent term in the denominator is ne-glected. EliminatingA we obtain

rBeikLc = ±1 . (4.96)

Near the frequencyω, which is the centre of the stop-band of the mirrors,rB can beapproximated by

rB =√R exp

[

incLDBR

c(ω − ω)

]

(4.97)

whereLDBR is the effective length of the mirror (see Section 2.5). Assumingω0 = ω,ωLT ≪ ω0, Γ ≪ ω0 and1 − R ≪ 1 one can reduce Eq. (4.96) to the two-coupledoscillator problem

(ω0 − ω − iγ)(ωc − ω − iγ) = V 2 (4.98)

whereωc − iγc is the complex eigenfrequency of the cavity mode in the absence ofexciton-photon coupling, and

V =

√

2ω0ωLTd

LDBR + d(4.99)

Eq. (4.98) has two complex solutions:

ω1,2 =ω0 + ωc

2− i

2(γ + γc) (4.100a)

±√

(

ω0 − ωc2

)2

+ V 2 −(

γ − γc2

)2

+i

2(ω0 − ωc)(γc − γ) . (4.100b)

The parameterV has the sense of the coupling strength between the cavity photonmode and the exciton.

If ω0 = ωc, the splitting between the two values is given by√

4V 2 − (γ − γc)2.If

V >

∣

∣

∣

∣

γ − γc2

∣

∣

∣

∣

(4.101)

the anticrossing takes place between the exciton and photonmodes, which is charac-teristic of thestrong-coupling regime. In this regime, two distinct exciton-polaritonbranches manifest themselves as two optical resonances in the reflection or transmissionspectra. The splitting between these two resonances is referred to as thevacuum-field


Rabi splitting. It reaches 4-15 meV in existing GaAs-based microcavities,up to 30 meVin CdTe-based microcavities, and is found to be as large as 50meV in GaN cavities.

If

V <

∣

∣

∣

∣

γ − γc2

∣

∣

∣

∣

(4.102)

the weak-coupling regime holds, which is characterised by crossing of the exciton andphoton modes and an increase of the exciton decay rate at the resonance point. Thisregime is typically used in vertical cavity surface emitting lasers (VCSELs).

Fig. 4.21: Real parts of the eigenfrequencies of the exciton-polariton modes in the weak-coupling regime(left) and strong coupling regime (right).

The spatial dispersion of exciton-polaritons leads to appearance of additional reso-nances in reflection and transmission and additional eigenmodes of the microcavity. Inorder to take into account the spatial dispersion one shoulduse the expression (4.49)for the dielectric function, which links it not only with thefrequency but also with thewavevector. The wavevector is no longer a unitary function of frequency, so that thesimple transfer matrix method described in Chapter 2 and used to obtain Eq. (4.98)fails. In order to calculate the optical spectra of semiconductor films containing excitonresonances we apply the generalised transfer matrix method. As was discussed in Sec-tion 4.4, the usual Maxwell boundary conditions are not sufficient in this case. We usePekar’s additional boundary conditions:

P |z=±Lc/2 = 0 , (4.103)

whereP is the exciton-induced dielectric polarisation, and corresponds to the bound-aries of the cavity. Eq. (4.103) yields a series of exciton-polariton eigenmodes (seeFig. 4.12). Their energies depend onLc and the exciton mass. The modes can be char-acterized by a quantum numberN equal to the number of nodes inF (R) for the cor-responding state plus one. These are polariton modes uncoupled to the cavity photonmode. To introduce the coupling one should substitute the amplitudes of all existingmodes of electromagnetic field into the Maxwell boundary conditions. For simplicity,


we consider only the normal incidence geometry (in-plane wave vector equal to 0) andonly the transverse light modes. In order to write down the transfer matrix taking into ac-count spatial dispersion of exciton-polaritons, one should describe propagation of fourwaves with amplitudesE±

1 , E±2 at the beginning and the end of the layer, respectively,

where “+” denotes a wave propagating in the positive direction and “−” denotes a wavepropagating in the negative direction.

The matrixQ connectingE±1 andE±

2 so that

(

E+2

E−2

)

= Q

(

E+1

E−1

)

(4.104)

can be written as

Q = −ε1ε2

(

λ+2 λ2

λ2 λ+2

)−1(λ+

1 λ1

λ1 λ+1

)

, (4.105)

with εj = c2k2j /ω

2 − εB and

λj = exp

(

ikjd

2

)

, λ+j = exp

(

−ikjd

2

)

, (4.106)

kj , (j = 1, 2) are wavevectors of the transverse polariton modes givenby Eq. (4.52).The transfer matrix across the cavity layer, i.e., the matrix that connects the in-planecomponents of the electric and magnetic fields at the boundaries, can be written as:

T =

[(

λ+1 λ1

n1λ+1 −n1λ1

)

+

(

λ+2 λ2

n2λ+2 −n2λ2

)

Q

]

×

×[(

λ1 λ+1

n1λ1 −n1λ+1

)

+

(

λ2 λ+2

n2λ2 −n2λ+2

)

Q

]−1

, (4.107)

wherenj = ckj/ω. Once its elements are known, the eigenmodes of the cavity can befound using the standard procedure described in Section 2.5from the equation:

T

(

1 + rB√εB(rB − 1)

)

= A

(

1 + rB√εB(1 − rB)

)

(4.108)

Generalization of this equation into the oblique incidencecase requires substantialmodifications of the transfer matrix (4.107). In the case of TM-polarized light, the lon-gitudinal polariton modes can be excited, and their amplitudes should be substitutedas well into the Maxwell boundary conditions. We address theinterested reader to thepaper by Vladimirova et al. (1996) and show here the dispersion curves of exciton-polaritons in a model bulk GaAs microcavity calculated in that work (Fig. 4.22). Re-markably, only the modes with evenN are coupled to light: this is because the excitonwavefunction should be of the same parity with the cavity mode to be coupled.

One can see that the light mode of the cavity goes through different polariton res-onances in the layer of GaAs forming a series of anticrossings. The splitting (Rabi


Fig. 4.22: Calculated exciton-polariton eigenenergies asfunctions of the incidence angle of light in TE(triangles) and TM (solid line) polarisations.

splitting) at the lowest of them (N = 2) is given with a good accuracy by2V , whereVis defined by Eq. (4.99).

The Rabi splitting for higher exciton states(ΩN ) is related to the splitting for thelowest state approximately as the ratio of squared overlap integrals of the excitonicpolarisation and electric field of the light mode:

ΩNΩ2

=

(

PNP2

)2

, N = 2, 4, 6 . . . (4.109)

where

PN ≈∫ π/2

−π/2sin(Nx) sin xdx =

2

N2 − 1(4.110)

This result easily follows from the quantum model as the matrix element of light-excitoncoupling is proportional toPN . in real bulk microcavities, the absorption of free e-hpairs aboveEg produces a large imaginary component toεB and washes out the higheranticrossings.

4.4.4.2 Microcavities containing quantum wells:Microcavities containing quantumwells (QWs) are most commonly used in practise. They allow for observation of thestrong coupling of a single exciton resonance and the cavitymode. The use of multipleQWs also helps to enhance the Rabi splitting and make the cavity polaritons more sta-ble. In order to obtain the dispersion equation of the polaritons in structures containingQWs it is convenient to use the transfer matrices written in the basis of amplitudes ofelectromagnetic waves propagating in positive and negative directions along the axis ofthe cavity.

Let us represent the electric field at the pointz of the structure as:

E(z) = E+(z) + E−(z) (4.111)


whereE±(z) is the complex amplitude of a light wave propagating in the positive/negativedirection. One can define the transfer matrixMa by its property:

Ma

(

E+(0)E−(0)

)

=

(

E+(a)E−(a)

)

. (4.112)

Consider a few particular cases.If the refractive indexn is constant across the layera, the transfer matrix has the

simple form:

Ma =

(

eika 00 e−ika

)

. (4.113)

The transfer matrix across the interface between a medium with refractive indexn1 anda medium with refractive indexn2 is

Ma =1

2n2

(

n1 + n2 n2 − n1

n2 − n1 n1 + n2

)

. (4.114)

It can be obtained using the condition (4.112) applied to thelight waves incident fromthe left side and right side of the interface, keeping in mindthe well-known expressionsfor the reflection and transmission coefficients of interfaces:

r =n1 − n2

n1 + n2, t =

2n1

n1 + n2. (4.115)

A transfer matrix across a structure containingm layers has the form

M =

2m+1∏

j=1

M2m+2−j (4.116)

wherej = 1, . . . , 2m + 1, labels all the layers and interfaces of the structure from itsleft to right side. The amplitude reflection and transmission coefficients (rs andts) ofa structure containingm layers and sandwiched between two semi-infinite media withrefractive indicesnleft, nright before and after the structure, respectively, can be foundfrom the relation

M

(

1rs

)

=

(

ts0

)

(4.117)

as

rs =m21

m11and ts =

1

m11. (4.118)

Note that here the refractive indicesnleft, nright do not appear explicitly as they arecontained in the transfer matrices across both surfaces of the structure.

If the reflection and transmission coefficients for light incident from the right-handside and left-hand side of the layer are the same andnleft = nright = n, Maxwell’s


boundary conditions for light incident from the left and right sides of the structure yield,in addition to Eq. (4.117), the following equation:

M

(

0ts

)

=

(

rs1

)

. (4.119)

In this case, the transfer matrix across a symmetric object (a QW embedded in thecavity, for example) can be written as:

M =1

ts

(

t2s − r2s rs−rs 1

)

. (4.120)

Consider a symmetric microcavity with a single QW embedded in the centre. In thebasis of amplitudes of light waves propagating in positive and negative directions alongthez-axis, the transfer matrix across the QW has the form

TQW =1

t

(

t2 − r2 r−r 1

)

(4.121)

wherer andt are the angle- and polarisation-dependent amplitude reflection and trans-mission coefficients of the QW derived previously. The transfer matrix across the cavityfrom one Bragg mirror to the other one is the product:

Tc =

(

eikLc/2 0

0 e−ikLc/2

)

1

t

(

t2 − r2 r−r 1

)(

eikLc/2 0

0 e−ikLc/2

)

, (4.122)

whereLc is the cavity width. The matrix elements read:

T c11 =t2 − r2

teikLc , T c12 =

r

t, (4.123)

T c21 = −rt, T c22 =

1

te−ikLc . (4.124)

To find the eigenfrequencies of the exciton-polariton modesof the microcavity, oneshould search for non-trivial solutions of Maxwell’s equations under the requirement ofno light incident on the cavity from outside. This yields

Tc

(

rB1

)

= A

(

1rB

)

, (4.125)

whererB is the angle-dependent reflection coefficient of the Bragg mirrors for lightincident from inside the cavity, introduced in Section 2.6.

Eliminating the coefficientA from Eq. (4.125), one obtains the following equationfor polariton eigenmodes:

T c21rB + T c22rBT c12rB + T c11rB

= rB (4.126)

This is already a dispersion equation because the coefficients of the transfer matrixandrB are dependent on the in-plane wavevector of light. Substituting the coefficients


(4.123) into Eq. (4.126), one can represent the dispersion equation in the followingform:

(

rB(2r + 1)eikLc − 1) (

rBeikLc + 1

)

= 0 (4.127)

Here we have used the relationt − r = 1, cf. Eq. (4.78b). Solutions of Eq. (4.127),coming from zeros of the second bracket on the left-hand side, coincide with pureoddoptical modes of the cavity. These modes have a node at the centre of the cavity wherethe QW is situated. Therefore, they are not coupled with the ground exciton state havinganevenwavefunction. The first bracket on the left-hand side of Eq. (4.127) contains thereflection coefficient of the QW, which is dependent on excitonic parameters. The zerosof this bracket describe the eigenstates of exciton-polaritons resulting from coupling ofeven optical modes with the exciton ground state. From now onwe shall consider onlythese states, neglecting excited exciton states that may becoupled to odd cavity modes.

For the even modes and normal incidence, if we take

rB = r exp(iα(ω − ωc)) ≈ r(1 + iLDBRncn

(ω − ωc)) , (4.128)

wherer is close to1 (see Section 2.5) and assumeeikLc ≈ 1 + i(ω − ωc)/cncLc, weobtain, using the explicit form for the reflection coefficient r:

r(1+ i(LDBR +Lc)ncc

(ω−ωc))(ω0−ω− i(γ−Γ0)) = ω0−ω− i(γ+Γ0) , (4.129)

which finally yields, after trivial transformations:

(ω0 − ω − iγ)(ωc − ω − iγc) = V 2 , (4.130)

where

γc =1 − r

rnc

c (LDBR + Lc)(4.131)

V 2 =1 + r

r

Γ0c

nc(LDBR + Lc)(4.132)

Here, quadratic terms in(ωc − ω) have been omitted. In all further calculations weassume for simplicityω0 = ω0. Eq. (4.130) is an equation for eigenstates of a sys-tem of two coupled harmonic oscillators, namely, the exciton resonance and the cavitymode. In this form Eq. (4.130) was published for the first timeby Savona et al. (1995)while its general form (4.126) was later obtained by Kavokin& Kaliteevski (1995). Itssolutions have the form (4.100). The weak-to-strong coupling threshold is defined inthe same way as for the bulk cavities (cf. Eqs. (4.101–4.102)). Note that all the abovetheory neglects the effect of disorder on the exciton resonance. Taking into account theinevitable inhomogeneous broadening of the exciton resonance and Rayleigh scatteringof exciton-polaritons, one should also modify the criterion for weak-to-strong couplingthreshold. The detuning of bare photon and exciton modes in amicrocavity is an im-portant parameter which strongly affects the shape of polariton dispersion curves to thestrong coupling regime as Figure 4.23 shows.


Fig. 4.23: Energies of exciton-polaritons at (a) negative,(b) zero and (c) positive detuning between the barephoton and exciton modes. Solid lines show the in-plane dispersion of exciton-polariton modes. Dashed linesshow the dispersion of the uncoupled exciton and photon modes. The calculations have been made withparameters typical of a GaN microcavity.

4.4.4.3 Oblique incidence case:Eq. (4.130) can be easily generalized for the obliqueincidence case by introduction of the dependence of the cavity and exciton eigenmodefrequenciesωc andω0 on the in-plane wavevectorkxy:

ωc =k2xy

2mph, ω0 =

k2xy

2Mexc(4.133)

whereMexc is the sum of the electron and hole effective masses in the QW planeandmph is the photon effective mass. In an idealλ-microcavity, the normal-to-planecomponent of the wavevector of the eigenmode is given bykz = 2π/Lc. The energy ofthe mode is

ωc =c

nc

√

k2xy + k2

z ≈ c

nckz

(

1 +k2xy

2k2z

)

=2πc

ncLc+

k2xy

2mph(4.134)

and thusmph = hnc/(cLc). This mass is extremely light in comparison to the exci-ton mass as it usually amounts to10−5—10−4m0, wherem0 is the free exciton mass.Note also that the in-plane wavevectorkxy is related to the angle of incidence of lightilluminating the structure,ϕ, by the relation:

kxy =ω

csinϕ . (4.135)


By measuring the angle dependence of the resonances in the reflection or trans-mission spectra of microcavities, one can restore the true dispersion curves of exciton-polaritons.

Fig. 4.24: Theoretical (lines) and experimental (squares)values of longitudinal-transverse splitting of the up-per (UP) and lower (LP) polariton branches in a GaAs-based microcavity with embedded QWs, by Panzariniet al. (1999b).

The coupling constantV is renormalised in the case of oblique incidence and itbecomes polarisation-dependent. In TE-polarisation,Γs0 = Γ0/ cosϕc whereϕc is thepropagation angle within the cavity. In TM-polarisation,Γp0 = Γ0 cosϕc. The effec-tive length of the Bragg mirrors,LDBR, is also angle- and polarisation-dependent: itdecreases with angle in TE-polarisation but increases in TM-polarisation. Also, the co-efficientr slightly depends on the angle (see Eqs. (2.150–2.152)).Allthese factors makethe coupling constant increase with angle in TE-polarisation, while in TM-polarisationthe opposite tendency occurs. This is why, in TM-polarisation at some critical anglethe strong coupling regime can be lost. Note also that in TM- and TE-polarisations theeigenfrequencies of pure cavity modesωTE,TM

c are split (see Eq. (2.153)) Fig. 4.24by Panzarini et al. (1999a) shows the longitudinal-transverse splitting of cavity polari-ton modes calculated and experimentally measured in a GaAs microcavity containingInGaAs QWs.

The longitudinal-transverse splitting becomes essentialat large in-plane wavevec-tors, and it has an impact on the spin-relaxation of exciton-polaritons as we shall dis-cuss in detail in Chapter 9. The splitting is dependent on twomain factors: the TE-TMsplitting of the bare cavity mode and the detuning between cavity and exciton modes.Therefore, it can be efficiently controlled and tuned withina wide range by changingthe cavity width.


4.4.5 Motional narrowing of cavity polaritons

Motional narrowing is the narrowing of a distribution function of a quantum particlepropagating in a disordered medium due to averaging of the disorder potential overthe size of the wavefunction of a particle. In other words, a quantum particle, which isnever localised at a given point in space but always occupiessome nonzero volume, hasa potential energy that is the average of the potential within this volume. This is why, ina random fluctuation potential, the energy distribution function of a particle is alwaysnarrower than the potential distribution function (see Fig. 4.25).

Fig. 4.25: Due to their finite De Broglie wavelength the quantum particles see the averaged potential. Theaveraging is done on the scale of the particle’s wavefunction. That is why the distribution function of quan-tum particles localized within some fluctuation potential is usually narrower than the potential distributionfunction.

Motional narrowing of exciton-polaritons in microcavities was the subject of sci-entific polemics at the end of the 1990s. The debate was initiated by an experimentalfinding of the Sheffield University group reported by Whittaker et al. (1996). Measuringthe sum of full widths at half minimum (FWHM) for two exciton-polariton resonancesin reflection spectra of microcavities as a function of incidence angle, the experimental-ists found a minimum of this function at the anticrossing of exciton and cavity modes(see Fig. 4.26). This result contradicts what one could expect from a simple model oftwo coupled oscillators. Actually, if the dispersion of microcavity polaritons is given byEq. (4.130) then the sum of the imaginary parts of the two solutions of this equation isalways−(γ + γc), independently of the detuningδ = ω0 − ωc. This follows from amore general property of any system of coupled harmonic oscillators to keep constantthe sum of eigenfrequencies, independently of the couplingstrength.

Clearly, Eq. (4.130) is no longer valid if, instead of a single free exciton transition,one has an infinite number of resonances distributed in energy. This is what happens inrealistic QWs, where in-plane potential fluctuations caused by the QW width and alloyfluctuations induce the so-calledinhomogeneous broadeningof an exciton resonance.An idea has been proposed that exciton-polaritons, having alighter effective mass thanbare excitons, are less sensitive to the disorder potential, which is a manifestation oftheir motional narrowing. Thus, the inhomogeneous broadening of exciton-polaritonmodes is less than that of a pure exciton state, which is a consequence of the polariton


Fig. 4.26: Left: linewidth of lower (solid) and upper (open)polaritons in an InGaAs microcavity as a functionof exciton fraction (or detuning), demonstrating linewidth narrowing on resonance, reported by Whittakeret al. (1996). Right: full width at half minimum of two polariton resonances in reflectivity spectra of a GaAs-based microcavity with an embedded QW, from Whittaker et al.(1996), as a function of the detuningδ =ω0 −ωc (circles correspond to the upper branch, squares correspond to the lower branch) in comparison witha theoretical calculation accounting for (solid lines) or neglecting (dashed lines) asymmetry in the excitonfrequency distribution from Kavokin (1997).

motional narrowing effect. At the anticrossing point this effect is especially strong, sinceat this point both upper and lower polaritons are half-excitons half-photons.

Further analyses by Whittaker et al. (1996) and Ell et al. (1998) have shown, how-ever, that experimentally observed narrowing of polaritonlines at the anticrossing pointis indeed caused by exciton inhomogeneous broadening, but not by the motional nar-rowing effect. On the other hand, the motional narrowing maymanifest itself in resonantRayleigh scattering or even photoluminescence.

In order to understand the inhomogeneous broadening effecton the widths of polari-ton resonances in microcavities, let us first consider its influence on the optical spectraof QWs.

We suppose that the in plane wavevector of any exciton interacting with the inci-dent light is the same as the in-plane wavevector of lightq, while the frequency ofexciton resonanceω0 is distributed with some function. This is a particular caseof amicroscopic model considering all exciton states as quantum-dot like and assuming nowavevector conservation. It is well adapted for description of reflection or transmission,i.e., experiments that conserve the in-plane component of the wavevector of light. Notethat most scattered light does not contribute to reflection and transmission spectra, whilea small part of the scattered light can re-obtain the initialvalue ofq after a second, third,etc. scattering act. The main reason why motional narrowinghas almost no influence onreflection spectra is that it is an effect which originates from the finite in-plane size ofthe exciton-polariton wave-function, or, in other words, implies scattering of exciton-polaritons in the plane of the structure. The impact of scattering is, however, negligiblysmall in reflection and transmission experimental geometries.


As in Section 4.4.2, we shall operate with the dielectric susceptibility Eq. (4.65),while also taking into account that the exciton resonance frequency is distributed withsome functionf(ω − ω0). We assume that in this case the dielectric susceptibility of aQW can be written in the form

χ(ω) =

∫

χ(ω − ν)f(ν − ω0) dν . (4.136)

If f is a Gaussian function, this integral can be computed analytically:

χ(ω) =1√π∆

∫

χ(ω − ν) exp

[

−(

ν − ω0

∆

)]

dν =i√πΘ

∆exp(−z2)erfc(−iz) ,

(4.137)with

χ(ω − ω0) =ε∞ωLTπa

3Bω

20/c

2

ω0 − ω − iγ, Θ = εBωLTπω

20a

3B/c

2 (4.138a)

and z =ω − ω0 + iγ

∆(4.138b)

where erfc is the complementary error function, and∆ is a width parameter of theGaussian distribution that describes exciton inhomogeneous broadening. We assume∆,γ > 0 and consider a normal incidence case for simplicity. Substituting the susceptibil-ity (4.136) into Eq. (4.64) and carrying out the same transformations as in Section 4.4.2,we obtain finally the amplitude reflection and transmission coefficients of a QW in theform:

r =iαχ

1 − iαχt = 1 + r (4.139)

whereα = Γ0/Θ andΓ0 is the radiative damping rate of the exciton in the case of noinhomogeneous broadening. This yields, using Eq. (4.137):

r = −√πΓ0 exp(−z2)erfc(−iz)

∆ +√π(Γ0 + i(ω0 − ω0)) exp(−z2)erfc(−iz) , t = 1 + r . (4.140)

We shall neglect renormalisation of the exciton resonance frequency due to the polaritoneffect, since it is much less thanΓ0, and also assumeω0 − ω0 ≈ 0. In the limit of smallinhomogeneous broadening, the complementary error function becomes:

lim|z|→∞

exp(−z2)erfc(−iz) =i√πz

(4.141)

which allows one to reduce Eq. (4.140) to the “homogeneous” formula (4.78) of An-dreani et al. (1998).

In realistic narrow QWs, the distribution of the exciton resonance frequency mayhave a more complex non-Gaussian distribution. Quite oftenit is asymmetric due to theso-called excitonic motional narrowing effect (to be distinguished from the motional


narrowing of exciton-polaritons). This effect comes from the blue-shift of the lower-energy wing of the excitonic distribution due to the lateralquantum confinement oflocalised excitons.

Exciton inhomogeneous broadening of any shape necessarilymodifies the equa-tion for exciton-polariton eigenmodes in a microcavity. Eq. (4.130) obtained for themodel of two coupled oscillators is no longer valid, and one should use the generalformula (4.128) instead. For the coupled exciton and photonmodes it reduces to:

rB(2r + 1)eikLc = 1 (4.142)

Using the same representation for the reflection coefficientof the Bragg mirror asin (4.129), and Eq. (4.139) forr with χ given by Eq. (4.102), we obtain after transfor-mations analogous to those in Section 4.4.2:

ωc − ω − iγc = V 2

∫ ∞

−∞

f(ν) − ω0

ν − ω − iγdν . (4.143)

In the limit of a small inhomogeneous broadening, it reducesto Eq. (4.130). As forEq. (4.143), it has between zero and two complex solutions depending on the shape ofthe distribution functionf(ν − ω0). In the general case, the sum of the imaginary partsof its eigenfrequencies varies as a function of detuning, which is not the case for thepure two coupled oscillator problem. Fig. 4.27 shows the dependencies of the FWHMof two polariton resonances on detuningδ = ω0 − ωc, in calculated reflection spectraof a GaAs-based microcavity with an embedded QW, in comparison with experimentaldata. One can see that the broadenings of the two polariton modes coincide near thezero-detuning point. At this point, both in calculation andexperiment, the sum of thetwo FWHM has a pronounced minimum.

This minimum is a specific feature of an inhomogeneously broadened exciton statecoupled to the cavity mode. It can be interpreted in the following way. The coupling tolight has a different strength for excitons from the centre and from the tails of an inho-mogeneous distribution. As the density of states of excitons has a maximum atω = ω0,these excitons have the highest radiative recombination rate and the strongest couplingto light. Now, the strong-coupling regime holds only for excitons situated in the vicin-ity of ω0, while the tails remain in the weak-coupling regime. The central part of theexcitonic distribution is of course less broadened than theentire distribution. Therefore,the two polariton modes that arise due to its coupling with the cavity photon are alsonarrower than one would expect for the case of all excitons equally coupled to light.At zero detuning, both polariton modes are far enough from the bare exciton energy, sothat the tails of the exciton resonance give no contributionto the FWHM of polaritonresonances. On the contrary, for strong negative or positive detunings, one of two polari-ton states almost coincide in energy with a bare exciton state, so that the line-shape ofthe corresponding spectral resonance is necessarily affected by the tails of the excitonicdistribution.

Note that this interpretation does not involve any motionalnarrowing. On the otherhand, specific effects of motional narrowing play a role in resonant Rayleigh scatter-ing of the cavity polaritons, and may also provide narrowingof the photoluminescence


Fig. 4.27: showing schematically the energy distributionsfunctions of bare excitons, bare photons andexciton-polaritons in microcavities. Sum of the broadenings of two polariton modes at the anticrossing pointis less than the sum of exciton and photon mode broadenings because of the preferential photon coupling withthe exciton states close to center of the inhomogeneous distribution of excitons.

lines. Only wavevector conserving optical spectroscopiessuch as reflection and trans-mission are not sensitive to the motional narrowing.

Another remark concerns the asymmetric behaviour of the broadenings of the twopolariton peaks in Fig. 4.26. This comes from the asymmetry of the excitonic distri-bution. It is a manifestation of theexcitonmotional narrowing that we have describedabove. Actually, the lower polariton branch is the result ofmixing between the photonmode and the lower part of the excitonic distribution, whichis sharper than the upperpart. This is why the lower branch has a narrower line-width at the anticrossing condi-tion.

4.4.6 Microcavities with quantum wires or dots

Planar microcavities with embedded quantum wires or quantum dots can exhibit theweak or strong exciton-photon coupling regime similarly tobulk cavities or microcav-ities with embedded quantum wells. An essential differencecomes from the enhancedresonant scattering of light by excitons in these structures. If the scattering is strong,exciton states in quantum dots or wires are coupled to the whole ensemble of the cavityphoton modes with different in-plane wavevectors. In this case, the polariton eigenstatecan hardly be represented as a linear combination of plane waves, and calculation ofthe spectrum of exciton-polaritons becomes a non-trivial task. However, for the cavitieswith embedded arrays of quantum wires or dots, scattering oflight can be neglected ifthe spacing between neighbouring wires (dots) is less than the wavelength of light andthe variation of the exciton resonance frequency form wire to wire (from dot to dot)is less than the Rabi splitting. In this case, one can still use Eq. (4.127) for polaritoneigenfrequencies, replacing the QW reflection coefficientr by the reflection coefficientof the array of wires or dots (4.91).


Fig. 4.28: Schematic diagram by Kaliteevski et al. (2001) ofa spherical microcavity with an embeddedquantum dot. The central core of radiusrQD is surrounded by a spherical Bragg reflector constructed fromalternating layers with refractive indicesna andnb.

Microcavities with a three-dimensional photonic confinement can exhibit strongcoupling between their confined photon modes and an exciton resonance in a single QD.Experimental evidence for strong coupling of a single QD exciton to a cavity mode hasbeen reported by Reithmaier et al. (2004), Yoshie et al. (2004) and Peter et al. (2005)(see Section 6.1). Here we consider the simplest model of a spherical microcavity ofradiusrQD with a spherical QD embedded in its centre (Fig. 4.29).

The eigenfrequencies of exciton-polariton modes can be found by the transfer matrixmethod generalized for spherical waves. Using the Green’s function technique to resolvethe Maxwell equations for a spherical wave incident on the QDin a similar way as weused above to describe scattering of a plane wave by a QD, one can obtain the reflectioncoefficient

rQD = 1 +2iΓQD

sp

ω0 − ω − i(γ + ΓQDsp )

, (4.144)

as described in more detail by Kaliteevski et al. (2001), with

ΓQDsp =

2

3πk4V 2

0 ωLTa3B (4.145)

V0 =∫

Φ(r)j0(kr)dr andk = ncω/c, (j0 is the zero order spherical Bessel functionandΦ the exciton wavefunction in the QD taken with equal electronand hole coor-dinates and assumed to have a spherical symmetry). The term 1in the right part ofEq. (4.144) comes from the fact that for a spherical wave incident on a center, the trans-mission contributes to reflectivity. The generalized transfer matrix method yields an


equation for the eigenfrequencies of the cavity polaritons, being a spherical analogue ofEq. (4.127):

h(2)1 (kR0) = rBrQDh

(1)1 (kR0) , (4.146)

where the functionsh are related to the spherical Hankel functionsH by h(1,2)l (x) =

√

π/2xH(1,2)l+1/2(x) andrB is the reflection coefficient of the Bragg mirror for the spher-

ical wave incident from inside the cavity.

Fig. 4.29: Schematic distribution by Kaliteevski et al. (2001) of the magnitude of the electric field intensity(gray scale) for thel = 1, m = 1 TM mode in a spherical microcavity with an embedded QD. The arrowsshow directions of the electric field vector of the mode

Eq. (4.146) yields the dispersion of all existing cavity modes including those cou-pled to the QD exciton. Each mode can be characterized by an orbital quantum numberland magnetic numberm and also TE or TM polarisation (corresponding to the modeswhere electric or magnetic field have no radial component, respectively). A detailedclassification of spherical polariton modes have been givenby Ajiki et al. (2002). Notethat there exist no allowed optical mode having a perfect spherical symmetry (l = 0) andthe photon mode withl = 1 has the lowest allowed energy (see Fig. 4.29). In particularcases, Eq. (4.146) can be reduced to the problem of two coupled harmonic oscillatorsfamiliar in planar microcavities. Assuming zero broadening of both exciton and photonmodes and approximatingrBR by

rBR ≈ exp(iβω − ωB

ωB) (4.147)

with β = πnanb/nc(nb − na), one can rewrite Eq. (4.146) as

βω − ωbωb

− 2ΓQDsp

ω − ωex+ 2kR0 = 2π(N + 1) , N ∈ N . (4.148)


In particular, if the uncoupled cavity mode has a frequency equal toωB, the eigen-frequencies of exciton-polariton modes are given by

ω1,2 = ωb ±√

2ΓQDsp ωb

β + 2ncωb

c R0. (4.149)

If the splitting exceeds the line-broadening of exciton andphoton modes, the strongcoupling regime can be observed.

Exercise 4.6 (∗∗∗) Find the frequencies of exciton-polariton eigenmodes in a planarmicrocavity with an embedded periodical grating of infinitequantum wires (QWWs).Consider diffraction effects to the first order.

5

QUANTUM DESCRIPTION OF LIGHT-MATTER COUPLING INSEMICONDUCTORS

In this chapter we study with the tools developped in Chapter3 the basicmodels which are the foundations of light-matter interaction. We startwith Rabi dynamics, then consider the optical Bloch equations whichadd phenomenologically the lifetime of the populations. Asdecay andpumping are often important, we cover the Lindblad form, a correct,simple and powerful way to describe various dissipation mechanisms.Then we go to a full quantum picture, quantising also the optical field.We first investigate the simpler coupling of bosons and then culminatewith the Jaynes–Cummings model and its solution to the quantuminteraction of a two-level system with a cavity mode. Finally weinvestigate a broader family of models where the material excitationcreation operator differs from the ideal limits of a Bose anda Fermifield.

166 QUANTUM DESCRIPTION OF LIGHT-MATTER COUPLING

5.1 Historical background

Many important concepts of light-matter interaction and especially their terminologyis rooted in the physics of nuclear magnetic resonance (NMR), where the effects werefirst observed and the models first developed. During World War II there was a stronginterest for radars and researchers investigated the radio-frequency range of the elec-tromagnetic field with great scrutiny, especially the meansof creation and the efficientdetection of such waves. Purcell, then at the MassachusettsInstitute of Technology,noted that magnetic nuclei—such as1H, 13C or 31P—could absorb radiowaves whenplaced in a magnetic field of specific strength. The perturbation of this state allows ac-curate measurements which are the basis of NMR and derived techniques. As the effectwas understood to be linked to the intrinsic magnetic properties of the nucleus, the dy-namics of its spin under the action of strong electromagnetic fields was studied. Thephysics of spin-radiowave interaction shares many similarities to that of atom-light in-teraction: the two levels of the spin-up/spin-down configurations become the groundand excited state of an atomic resonance, and the electromagnetic field merely changesin frequency, so that many results and concepts obtained in the former context reappearin light-matter interaction. We start our investigation ofthe physics of light-matter in-teraction with Rabi’s approach to the problem of nuclear induction. He had the simplestmodel, minimally quantized. His model cannot be further simplified without reducingto the Lorentz oscillator. He neglected lifetime, pumping,decoherence, and he got ridof all “complications”, even those he could have taken into account easily. For this rea-son, he unravelled the most important features of the problem and had his name pinnedin the physics of light-matter coupling. All subsequent approaches rely on his simpleresult, as we shall see in this chapter.

5.2 Rabi dynamics

Rabi investigated the coupling of a quantum two-level system driven by a sinusoidalwave, modelling a classical optical field interacting with aspin. We shall use in thefollowing the terminology of atoms. In the approximations of Rabi, we write the atomicwavefunction as:

|ψ(t)〉 = Cg(t) |g〉 + Ce(t) |e〉 (5.1)

which dynamics is entirely contained in the two complex coefficientsCg andCe, andthe Hamiltonian as:

H = Eg |g〉〈g| + Ee |e〉〈e| (5.2a)

−(

Vge |g〉〈e| + Veg |g〉〈e|)

E(t) (5.2b)

Throughout, subscripts “g” and “e” refers to “ground” and “excited” (level, energy. . . )Veg is the dipole moment of the transition (discussed more later). Only the atom istreated quantum-mechanically so the light energy does not enter into the Hamiltonian,it only acts as a time dependent interaction through thec-functionE. In the dipoleapproximation, the electric field is:

E(t) = E0 cosω0t (5.3)

RABI DYNAMICS 167

where the amplitudeE0 and the frequencyω0 are constant. To emphasise the temporaldynamics, we introduce the frequency associated to the levels energies:

Eg = ~ωg , Ee = ~ωe . (5.4)

Exercise 5.1 (∗) Show that for the system (5.1–5.4), Schrodinger’s equation (3.1) be-comes:

Ce = −iωeCe + iΩRe−iφ cos(ω0t)Cg

Cg = −iωgCg + iΩReiφ cos(ω0t)Ce

(5.5a)

with φ the complex phase ofVeg = |Veg|eiφ and in terms of the Rabi frequency

ΩR =Veg|E0|

~. (5.6)

The quantity given by formula (5.6) is an important parameter for the quantum dy-namics of a two level system. It is often referred to as an energy,~ΩR (theRabi energy),and more often still as twice this quantity under the name ofRabi splitting. We have al-ready encountered this in the semiclassical treatment of Section 4 in Eq. (4.47) as aterm quantifying the magnitude of the splitting (hence the name) of two resonances inthe polarisation. This was obtained without reference to quantum dynamics, as this isa result which in some cases can also be derived from a classical perspective, but theorigin of this popular term67 is in the quantum treatment, Eqs. (5.5a).

In the interaction picture (or in the language more suited tothis problem, in “rotatingframes”), the interesting dynamics is in the slow evolutiondue to the couplingΩR notin the rapid and trivial one imparted by the optical frequency, so that we redefine

cg = Cgeiωgt and ce = Cee

iωet (5.7)

which lead to:

ce = iΩR

2e−iφ[e−i(ω0−ωg)t + ei(ω0+ωg)t]cg

cg = iΩR

2eiφ[e−i(ω0−ωe)t + ei(ω0+ωe)t]ce

(5.8a)

where we have written the cosine as(eiω0t + e−iω0t)/2 to deal with complex expo-nentials only. Despite their simple appearance, these equations are not easy to solveexactly because of the many terms which appear by combinations of the two terms onthe right hand sides. They become straightforward however if a single term is retained.Now let us remind ourselves that the purpose of Eqs. (5.7) is precisely to separate theslow dynamics ofc coefficients from the rapid oscillation at ratesω = ωe − ωg, so thaton a small time interval over which Eqs. (5.8a) are integrated, the temporal evolution

67Numerous other accommodations of the adjective apply, suchas “Rabi flop” or “Rabi oscillation” todenominate the dynamics of Fig. (5.5a).


of exp(

i(ω0 + ωe/g)t)

ce/g is essentially given by the exponential which is oscillat-ing so quickly that it averages to zero even over a small interval wherec changes bya little amount. This is called therotating wave approximation(RWA) because of theway frequencies contrive to stabilise or cancel the time evolution depending whetherthey oscillate with or against the frame rotating with the field. In the RWA the solutionfollows quickly:

cg(t) =(

a1eiΩt/2 + a2e

−iΩt/2)ei∆t/2 (5.9a)

ce(t) =(

b1eiΩt/2 + b2e

−iΩt/2)e−i∆t/2 (5.9b)

with constants of integrations computed from initial conditions:

Exercise 5.2 (∗) Show that in solving Eqs. (5.8a), the following is obtained:

a1 =1

2Ω

[

(Ω − ∆)ce(0) + ΩRe−iφcg(0)

]

, (5.10a)

a2 =1

2Ω

[

(Ω + ∆)ce(0) − ΩRe−iφcg(0)

]

, (5.10b)

a3 =1

2Ω

[

(Ω + ∆)cg(0) + ΩReiφce(0)

]

, (5.10c)

a4 =1

2Ω

[

(Ω − ∆)cg(0) − ΩReiφce(0)

]

(5.10d)

and more importantly, in terms of newly introduced quantities:

The detuning: ∆ = ω − ω0 , (5.11)

The generalised Rabi frequency: Ω =√

Ω2R + ∆2 . (5.12)

From Eqs. (5.9–5.12), we can finally complete the description of the wavefunctionEq. (5.1):

ce(t) =

(

ce(0)

[

cos(Ωt/2) − i∆

Ωsin(Ωt/2)

]

+ iΩR

Ωe−iφcg(0) sin(Ωt/2)

)

ei∆t/2 ,

cg(t) =

(

cg(0)

[

cos(Ωt/2) + i∆

Ωsin(Ωt/2)

]

+ iΩR

Ωeiφce(0) sin(Ωt/2)

)

e−i∆t/2 .

(5.13a)

From the interpretation of the wavefunction in quantum mechanics,|cg|2 and|ce|2are probabilities to find the atom in its ground or excited state, respectively. If the atomis initially in its ground state (att = 0), the probability to find it in its excited state attime t is, from Eqs. (5.13):

Pe(t) = |ce|2 =ΩR

√

Ω2R + ∆2

sin2

√

Ω2R + ∆2

2t (5.14)

The Rabi frequency, which at resonance is proportional to the amplitude of the lightfield,E0, and to the matrix elementVeg, becomes renormalised with detuning. However

BLOCH EQUATIONS 169

0 5 10 15 20WRt

-3

-2

-1

0

1

2

3

DW

R

2 4 6 8 10WRt

0.2

0.4

0.6

0.8

1PeHtL

DWR=1

D=0

DWR=10

Fig. 5.1: Rabi oscillations in the probabilityPe (grey-scale coding, light colour corresponds to high proba-bility) as function of time and detuning (in units of∆/ΩR andΩRt). Departing from resonance increases thetransition rate but spoils its efficiency.

the transition efficiency gets spoiled, as seen in Fig. (5.1). The probability oscillates intime, so that continuously exciting a system is not the best way to excite it. Once theexcitation is created, further continuous excitation now works towards bringing backthe system to its ground state. Note that no mechanism of decay or relaxation has beenincluded in the simplest of the pictures and that if the external excitation is shut off,the system stays where it is forever. In effect, this tendency of an external excitationto induce the atom to de-excite is thestimulated emissionforeseen by Einstein andintroduced in Section 3, for which we have just provided a microscopic derivation.

5.3 Bloch equations

The model developed by Rabi contains the key elements of the dynamics, associated tothe Rabi frequencyΩR. However it is not realistic in many respects, if only because itlacks any form of dissipation. Losing excitation from the system or coupling it to someexterior reservoir will result in dephasing of the state,68 which is clearly not the casein the previous model where the quantum state is a wavefunction and therefore a purestate. Dissipation will induce a loss of quantum coherence in the sense of losing thequantum correlations beetween states which are embedded inthe off-diagonal elementsof the density matrix in this basis. A way to doctor this limitation in the above approachis therefore to rewrite the equation with a density matrix instead of with a wavefunctionand add phenomenological decay terms which reduce elementsof the density matrix.First, rewriting the equation in terms of

ρ = |ψ〉〈ψ| (5.15)

gives, in the simple case of Eq. (5.1):

68The interaction of the system with reservoirs which dephaseit imply that even in the case where theinitial state of the system is well known (pure state), it evolves with time into a mixture of states where onlyprobabilistic information remains.


EdwardPurcell (1912–1997) and FelixBloch (1905-1983), the 1952 Nobel prize-winners in physics for“ their development of new methods for nuclear magnetic precision measurements”.

Purcell worked during World War II at the MIT Radiation Laboratory on the development of the mi-crowave radar under the supervision of Rabi. Back in Harvard, he discovered nuclear magnetic resonance(NMR) with Pound and Torrey, published in Phys. Rev,69, 37 (1946). Other similar and importantcontributions include spin-echo relaxation and negative spin temperature. In another field but still revolvingabout the radio spectrum, he made with Ewen the first detection of the famous “21cm line”, due to thehyperfine splitting of hydrogen. This seeded the field of radioastronomy and allowed breakthrough in thestudy of galactic structure. Most importantly for the cavity community, he gave his name to the enhancementor inhibition of dipole radiation by boundary conditions (see next chapter).

Bloch’s education—like that of Purcell—was initially thatof an engineer, but he soon turned tophysics which gave him the opportunity to study with Schrodinger, Weyl and Debye, among others. WhenSchrodinger left Zurich, Bloch worked with Heisenberg instead and promptly after with Pauli, Bohr, Fermiand Kramers as parts of fellowships he earned. He formulatedin 1928 in his doctoral dissertation the theorembearing his name (that we presented on page 121) to describe the conduction of electrons in crystallinesolids. He met with Purcell in 1945 at a meeting of the American Physical Society where they realised theunity of their work on nuclear resonance. They agreed to share its experimental investigations, in crystals forPurcell’s group, in liquids for Bloch.

ρ =

(

|Cg(t)|2 Cg(t)Ce(t)∗

Cg(t)∗Ce(t)

∗ |Ce(t)|2)

(5.16)

and subsequently, its time equation of motion:

Exercise 5.3 (∗) Derive the following dynamics from Eq. (3.33) applied to Eqs. (5.15)and (5.1):

ρgg =i

~

(

VegEρeg − c.c.)

,

ρee = − i

~

(

V ∗egEρge − c.c.

)

,

ρge = −iω0ρeg −i

~VegE(ρee − ρgg

)

,

ρeg = iω0ρge +i

~V ∗

egE(ρee − ρgg

)

.

(5.17)

This is, up to know, strictly equivalent to Eqs. (5.5a) as allelements of the densitymatrix can be reconstructed from the knowledge of the two amplitudesCg/e. Also ob-serve the symmetries among the matrix elements, namely,ρgg = −ρee andρge = ρ∗eg.

BLOCH EQUATIONS 171

The latter follows from hermiticity of the density matrix. The former is an expressionof the conservation of probabilities: if the atom gets excited, the ground state gets de-populated. Because these matrix elements are real numbers,the complex conjugation isnot necessary (in the equations it is corrected by the off-diagonal element).

Lifetime and dephasing is now added phenomenologically to Eq. (5.17), by intro-duction of two characteristic times, known asT1 andT2,69 They account for lifetime(decay in population) and dephasing (decay in phase), respectively. Two typical phys-ical mechanisms responsible for these terms are spontaneous emission and atom-atomcollision, respectively. The spontaneous decay rate also dephases the system so that atypical expression forT2 could be:

1

T2=

1

2T1+

1

T ∗2

, (5.18)

whereT ∗2 is the mean time between atomic collisions which reset the phases of the

emitters. A microscopic derivation of these quantities is possible but of course moreinvolved mathematically. In the former case it requires quantization of the optical field(which here is still ac-functionE) and in the later introduction of a bath of other carriersand a statistical treatment. On the other hand, the phenomenological treatment is clearenough (we shall see more about their microscopic origin in what follows). Eqs. (5.17)read:

ρee = − i

~

(

V ∗egEρge − c.c.

)

− ρee

T1,

ρge = −iω0ρeg −i

~VegE(ρee − ρgg

)

− ρge

T2.

(5.19)

where we have also limited to independent terms only and performed the rotating waveapproximation.70

Eqs. (5.19) are called theoptical Bloch equations, after Bloch who derived them fora spin in an oscillatory electric field in a form suitable to bemapped onto the Blochsphere (or Poincare sphere). They do not admit analytical solution in the general casebut limiting cases of practical interest exist and are presented now.

Exercise 5.4 (∗∗) Consider trial solutions of the formρij(t) = ρij(0) exp(λt). Showthat at zero detuning and forVeg ≥ (1/2T1), the linearisation thus afforded admitssolutions:

λ1 = 0, λ2 = − 1

T1, λ3 = −3

2

1

T1+ iλ and λ4 = −3

2

1

T1− iλ (5.20)

where:

λ =√

|Veg|2 − 1/(2T1)2 . (5.21)

Consider now the caseVeg ≪ 1/T1 where the same method yields:

69In NMR, T1 andT2 are called the longitudinal and transverse relaxation times, respectively, becausethey cause the decay of orthogonal projections of the magnetic spin along its precession axis.

70See the discussion on page 168.


λ1 = 0, λ2 = − 2

T1, λ3 = − 1

T1+ i(ω0 − ω) and λ4 = − 1

T1− i(ω0 − ω) .

(5.22)

5.3.1 Full quantum picture

We now turn to a complete quantum treatment where the opticalfield gets quantizedas well. Successful descriptions of light–matter interactions in full quantum mechani-cal terms is of course a complicated problem. It culminates in the theory of quantum-electrodynamics (QED), which is to date the most succesful,accurate and inspiringphysical theory, among other points having served as a blueprint for a vast family ofsister theories—collectively known as the “standard model”—to describe the force ex-perimentally observed in nature (QED being the quantization of the electromagneticforce). Only gravity is resisting the axiomatisation and interpretation of QED.

Back to our concern of light-matter interaction, one starting point71 is to acknowl-edge that matter—being essentially neutral—couples with light through the electric fieldcomponent arising fromfluctuationsin the electric charges (since the total charge can-cels on the average). In the multipolar expansion of the field, the first nonzero fluctua-tions are thedipolar fluctuations for most cases of interest, including the simplest andgeneric case of an atom made up from a positive nucleus and an orbiting negative elec-tron. This case naturally also depicts the exciton, cf. Fig.4.7 on page 126. The potentialenergyU of a classical dipole72 d in an electric fieldE is easily derived from Newton’slaw and electrostatic energy, to yield

U = −d · E (5.23)

We have already quantized the fieldE, cf. Eq. (3.72). We now consider the atomdipolar momentd. We will simplify the above interaction by restraining to only twoatomic levels and a single mode of the electromagnetic field.This is a bold approxi-mation which leaves out a lot of the physics described by Eq. (5.23) that, even for thesimplest atom—the electron-proton pair bound as Hydrogen—describes the couplingof two infinite sets of energy levels. The considerable simplification that we shall makeproves however to be sound especially for microcavities whose merit is to filter outmodes of the electromagnetic field by imposing boundary conditions. For the atom, itsuffices to select two levels whose energy difference matches the energy singled out bythe optical mode to make all other transitions negligible incomparison. In the absenceof a cavity this can be approached by using laser light which mimicks a single modethanks to small energy fluctuations.

71Another, even more popular quantization scheme, is that of the HamiltonianH = V (r) + (p −qA(r, t))2/2m of the vector potentialA. This Hamiltonian or that derived from Eq. (5.23), are essentiallyequivalent and are exactly so at resonance. The so-calledAp Hamiltonian (of this footnote) is better adaptedto delocalised systems like electrons and hole in a bandstructure. Localised systems, on the contrary, likeatoms or quantum dots, find a better starting point with the dipole Hamiltonian.

72A dipoleis most simply visualised as the limiting case of two point-like particles of opposite charges+qand−q, and vanishing separationδ, beingd = qδud whereud is the unit vector pointing from the plus to mi-nus charge. Physical dipoles are two actual opposite charges separated by some distance small in comparisonto the distance at which the dipole is observed.

BLOCH EQUATIONS 173

We remind our notations for the two atomic modes|e〉 and |g〉 with energiesEe

andEg for “excited” and “ground”, respectively. At this stage andwith such nomencla-ture we have virtually already completely abstracted the “real” or “physical” atom away.Let us keep in mind, though, that|g〉 can equally well be the lower of two excited statesof the physical atom. In the spirit of Dirac,|g〉 is a simple notation for a potentiallycomplicated wavefunction. To illustrate the point we shallconsider the|Ψ100〉 state ofthe hydrogenoid atom as the ground|g〉 state and|Ψ211〉 as the excited|e〉 state. Theexpressions for these states follow from Schrodinger equation with Coulomb potentialas:

Ψ100(r) =1

(2a2B)3/2

exp(

− r

2aB

)

, (5.24a)

Ψ211(r) =

√

1

64πa3B

(

r

aB

)

exp(

− r/(2aB))

sin θeφ , (5.24b)

with r = (r, θ, φ) in Eq. (5.24b).

Exercise 5.5 (∗∗) Show that the dipole element operator73 for states (5.24) is:

d = |e〉〈g|deg + |g〉〈e| (deg)∗ , (5.25)

where:

deg = e

∫

Ψ∗211(r)rΨ100(r) dr , (5.26a)

= −27

35eaB(x − iy) . (5.26b)

For the following it is therefore mathematically advantageous to consider the com-plicated “object”d as a mere two by two matrix on a Hilbert space spanned by:

|e〉 =

(

10

)

, 〈e| =(

1, 0)

,

|g〉 =

(

01

)

, 〈g| =(

0, 1)

.

In this space,d can be expressed as function ofσ = (σx + iσy)/2 andσ† (seeEq. (3.12) on page 79) to become:

d = degx(σ + σ†) (5.27)

where, in this case, the orientation of the dipole has been arbitrarily taken along thexaxis so as to cancel the imaginary part (orientation along the latter would result in a

73A note of caution on notations: in Eq. (5.25),d is an operator whiledeg is a cartesian vector. Someauthors use a hat to denote explicitly an operator, i.e., they would write d = |e〉〈g|deg + h.c. As a matterof fact, d is also a vector, as is seen in Eq. (5.26b), whose components are operators. We do not follow thispractise because it is often clear enough from the context what is the mathematical nature of a variable, and itis helpful to retain the scope of an equation in both classical and quantum domains.


minus sign in Eq. (5.27) and a complex matrix element). So finally, the light-matterinteraction reads~g(σ† + σ)(a − a†) where:

~g =

√

~ω0

2ǫ0Vdeg (5.28)

Eq. (5.28) is a “typical” result for the cases we have been considering. It specificallydepends on the system at hand and how it is modelled. In this case we even have leftthe dipole element in its general form. If considering the complete, multimode lightfield, Eq. (3.72) should be used instead, as will be the case inthe next Chapter wherethe continuum of modes is needed to compute the radiative lifetime of a dipole in thevacuum.

With the free energy of the optical mode and the atom added, the “generic” completeHamiltonian is:

H = ~ωCa†a+

1

2~ωXσz + ~g(σ† + σ)(a− a†) . (5.29)

This system is the basis for what follows. To start with, we investigate quantumcoupling in simpler cases than Eq. (5.29) which despite its apparent simplicity is highlynon trivial. We shall in all cases reduce in this chapter the problem to single modes. Thiscould be the case when large splitting of energies separate higher states from those con-sidered, typically one cavity mode near resonance with an excitonic transition. Moreelaborate multimode couplings are treated in subsequent chapters. For now, however,neglecting off-resonant terms74 like a†b†, the Hamiltonian reduces in this approxima-tion to the following form:

H = ~ωaa†a+ ~ωbb

†b+ ~g(ab† + a†b) . (5.30)

For further clarity we also simplified notations:~ωa and~ωb are thefree energiesofthe modes, with second quantization operatorsa andb, we keep the notation∆ for thedetuning in the eigenfrequencies:

∆ = ωa − ωb (5.31)

and~g is for the coupling strength. One mode,a, which describes light, will be in allcases a pure Bose operator with commutation relation Eq. (3.39).

The operatorb which describes the material excitation, on the other hand,dependson the model of the matter-field, and could range—in the extreme—from another Boseoperator following the same commutations relations asa, to a fermion operator with an-ticommutation relations. It can also be, more generally, a more complicated expressionwhich follows from the rich structure of the particles involved, e.g., Eq. (5.147) for theexciton in the approximations outlined there.

These two limiting cases—coupling a single cavity mode to a Bosonic or Fermionicfields—are also of tremendous importance and many experimental configurations referto them in some approximations. Their simple mathematical forms allow exact solutionsto be obtained and therefore many insights to be gained.

74Neglecting terms which do not conserve energy likea†b† or ab—since they simultaneously create orannihilate two excitations, respectively—is the second quantized version of the rotating wave approximation.

BLOCH EQUATIONS 175

5.3.2 Dressed bosons

The most elementary problem of the kind (5.30) is the case where b is also a Boseoperator, namely

[b, b†] = 1 (5.32)

(cf. Eq. (3.39)). This is the elementary problem of couplingof two linear oscillators(hence the problem is refered to aslinear and this linearity will transpire in all whichfollows). We investigate the solutions in the various pictures of Quantum Mechanicsintroduced in Chapter 3.

The analysis of Eq. (5.30) can be made directly in the basis|i, j〉 with i excitationsin the matter field andj in the photonic field,i, j ∈ N. Those are calledbare statesincontrast to thedressed statesthat we consider hereafter. The value of this approach isthat the excitation, loss and dephasing processes generally pertain to the bare particles.For instance matter excitations are usually created by an external source (pumping) andlight excitations can be lost by transmission through the cavity mirror. This physics isbest expressed in the bare states basis.

As the system is linear, the integration is straightforwardin the bare states basis. Inthe Heisenberg picture:

Exercise 5.6 (∗) Show that the time evolution of the operatorsa and b under the dy-namics of Hamiltonian (5.30) is given by:

a(t) = exp(

− iωt)

(

a(0)[

cosGt

2− i

∆

Gsin

Gt

2

]

− 2ib(0)g

Gsin

Gt

2

)

(5.33a)

b(t) = exp(

− iωt)

(

− 2ia(0)g

Gsin

Gt

2+ b(0)

[

cosGt

2+ i

∆

Gsin

Gt

2

]

)

(5.33b)

whereω = (ωa + ωb)/2 andG =√

4g2 + ∆2.

Note that the commutation relation Eq. (3.39) remains well behaved at all timesthanks to the intermingling ofa andb operators. This can be illustrated on the case∆ =0 where the expressions 5.33a simplify considerably to:

a(t) = e−iωt[a(0) cos(gt) − ib(0) sin(gt)] , (5.34a)

b(t) = e−iωt[b(0) cos(gt) − ia(0) sin(gt)] . (5.34b)

in which case one gets

[a(t), a†(t)] = [a(0), a†(0)] cos2(gt) + [b(0), b†(0)] sin2(gt) = 1 . (5.35)

We shall see in the Schrodinger representation how this result manifests in complicatedcorrelations between the two states. This result should also make clear that equal-timecommutations hold for Bose operators. Observe indeed how[a(0), a†(t)] oscillates be-tween 0 and 1 with time.

Observables can be obtained directly in the Heisenberg picture from the solutionsEqs. (5.33a). For instance, the population is obtained in the exercise below.


Exercise 5.7 (∗) Show that the population operators for the coupling of two oscillatorswith initial condition|Ψ〉 = α |1, 0〉+ β |0, 1〉 are given by:

(a†a)(t) = |α|2 cos2Gt

2+

|∆α+ 2βg|2G2

sin2 Gt

2−ℑ(αβ∗)

4g

Gcos

Gt

2sin

Gt

2, (5.36)

and use this result to recover Rabi oscillations of Fig. 5.1.

On the other hand, Eq. (5.30) assumes a straightforward expression in the basis ofso-calleddressed stateswhich diagonalises the Hamiltonian as we now show explic-itly going back to the Schrodinger picture. The most general substitution that can beattempted is:

p = αa+ βb (5.37a)

q = γa+ δb (5.37b)

with α, β, γ, δ ∈ C. If we require thatp andq remain boson operators, i.e.,

[p, p†] = [q, q†] = 1 (5.38)

implying|α|2 + |β|2 = |γ|2 + |δ|2 = 1 (5.39)

as well as[p, q] = [p, q†] = 0 (5.40)

The second one impliesαγ∗ + βδ∗ = 0 (5.41)

while the first is automatically satisfied.Relations (5.37) reversed read

a =δp− βq

αδ − βγ(5.42a)

b =−γp+ αq

αδ − βγ(5.42b)

Their substitution in Eq. (5.30) yields

(αδ − βγ)2H = p†p~ω(|δ|2 + |γ|2) − ~gℑ(δ∗γ) (5.43a)

+ q†q~ω(|β|2 + |α|2) − ~gℑ(β∗α) (5.43b)

+ p†q~ω(−βδ∗ − γ∗α) + ~g(αδ∗ + βγ∗) + h.c. (5.43c)

We require that line (5.43c) be zero, which reduces to

αδ∗ + βγ∗ = 0 (5.44)

Fitting the above conditions, we are led toα = cos θ, β = sin θ, γ = − sin θ andδ =cos θ, ensuring thatαδ − βγ = 1 (canonical unitary transformation):

BLOCH EQUATIONS 177

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

(E- )/ g

Inte

nsity

(arb

. uni

ts)

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

(E- )/ g

Inte

nsity

(arb

. uni

ts)

(a) Mollow tripletRabi doubletRabi doublet (b)

...

...

? ?

? ?

??

||1, 0〉〉||0, 1〉〉

−~g+~g

||0, 2〉〉

||2, 0〉〉||1, 1〉〉

−2~g

+2~g

−7~g−5~g−3~g

+5~g+7~g

Bose limit

||0, 0〉〉

||7, 0〉〉||6, 1〉〉||5, 2〉〉||4, 3〉〉||3, 4〉〉||2, 5〉〉||1, 6〉〉||0, 7〉〉

...

6

?

?

? ?

?

?

−~g+~g

Fermi limit

−√

2~g

+√

2~g

−√

7~g

+√

7~g

2√

7~ω

2√

2~ω

~ω

Fig. 5.2: Energy diagrams of the two limiting cases of dressed bosons (left) and fermions (right). In the firstcase thenth manifold has constant energy splitting of2~g between all states and couples to the(n − 1)thmanifold by removal of a quantum of excitation with energy~ω±~g which leads to the Rabi doublet, Fig. (a),with splitting2~g. In the second case, each manifold is two-fold with a splitting which increases like a squareroot. All four transitions are allowed, leading to the Mollow triplet, Fig. (b), for high values ofn when thetwo middle transitions are close in energy. The distance from the central peak goes like2~g

√n and the ratio

of peaks is1 : 2. The two lowest manifolds (dashed) are the same in both cases, making vacuum field Rabisplitting insensitive to the statistics.


p = cos θa+ sin θb (5.45a)

q = − sin θa+ cos θb (5.45b)

where

cos θ =∆ +G

√

2∆2 + 8g2 + 2∆G, (5.46)

θ is known as themixing angle.ThenH reads

H = ~ωpp†p+ ~ωqq

†q (5.47)

where

ωp/q = ω ± ∆

2G . (5.48)

The eigenfrequenciesωp/q given by Eqs. (5.48) are plotted in Fig. 5.3.

-10 -5 0 5 10

Fre

quen

cy

∆/g

ωp(∆/g)

ωq(∆/g)

mode b

mode a

Fig. 5.3: Thick lines: Eigenfrequencies of the system Eq. (5.30) as function of the detuning∆ (in units of theinteraction strengthg and withωa set to 0.). At zero detuning, an anticrossing is observed. Atlarge detunings,the bare modes (thin lines) are recovered.

We shall later see how this approach is the archetype of so-called Bogoliubovtrans-formations, first considered in connection with high-density, weakly-interacting Bosecondensates.

For clarity we shall note||i, j〉〉 the dressed states, i.e., the eigenstates of (5.53) withidressed particles of energy~ωp andj of energy~ωq. We call amanifoldthe set of stateswith a fixed number of excitations. In the dressed states basis it reads for the case ofnexitations:

HN = ||n,m〉〉 ; n,m ∈ N with n+m = N . (5.49)

BLOCH EQUATIONS 179

Its energy diagram appears on the left of Fig. 5.2 for manifolds with zero (vacuum),one, two and seven excitations. When an excitation escapes the system while in man-ifold HN , a transition is made to the neighbouring manifoldHN−1 and the energydifference is carried away, either by the leaking out of a cavity photon, or through ex-citon emission into a radiative mode other than that of the cavity, or a non-radiativeprocess. The detailed analysis of such processes requires adynamical study, but as thecavity mode radiation spectra can be computed with the knowledge of only the energylevel diagrams, we shall keep our analysis to this level for the present work. The im-portant feature of this dissipation is that, though such processes involvea or b (ratherthanp or q), they nevertheless still result in removing one excitation out of one of theoscillators. Hence only transitions from||n,m〉〉 to ||n−1,m〉〉 or ||n,m−1〉〉 are allowed,bringing away, respectively,~ωp and~ωq of energy, accounting for the so-called Rabidoublet (provided the initialn andm are nonzero in which case only one transitionis allowed). From the algebraic point of view, this of coursefollows straightforwardlyfrom (5.52) and orthogonality of the basis states. Physically it comes from the fact that,as in the classical case, the coupled system acts as two independent oscillators vibratingwith frequenciesωp/q.

In the case of vacuum field Rabi splitting, a single excitation is shared between thetwo fields, and so the manifoldH1 is connected to the single line of the vacuum mani-fold. In this case there is obviously no possibility beyond adoublet. It is straightforwardto compute the transition amplitudes between the two manifolds by mean of the barestates annihilation operators: The rates, which are proportional to the four componentsof the dressed states, are:

M1 = 〈0, 0|a||1, 0〉〉 = α(∆/g) , (5.50a)

M2 = 〈0, 0|a||0, 1〉〉 = γ = −α(−∆/g) , (5.50b)

M3 = 〈0, 0| b||1, 0〉〉 = β = α(−∆/g) , (5.50c)

M4 = 〈0, 0| b||0, 1〉〉 = δ = α(∆/g) . (5.50d)

The amplitudes of transitionsMi given by Eqs. (5.50) are displayed in Fig. 5.4.

-1

-0.5

0

0.5

1

-10 -5 0 5 10

Tra

nsiti

on a

mpl

itude

∆/g

α = δ = cosθ

γ = -sinθ

β = sinθ

0

0.5

1

-10 -5 0 5 10

Inte

nsity

∆/g

I1 = |α|2 = |δ|2

I2 = |γ|2 = |β|2

0

0.5

1

-10 -5 0 5 10

Inte

nsity

∆/g

I1 = |α|2 = |δ|2

I2 = |γ|2 = |β|2

Fig. 5.4: AmplitudesMi (left) and corresponding intensities|Mi|2 (right) of transitions between the mani-folds of one excitation and the vacuum, as function of the detuning∆ in units of the interaction strengthg.


Their square as this is the physical quantity one is interested in:

I1 = |α(∆/g)|2 , (5.51a)

I2 = |α(−∆/g)|2 . (5.51b)

The physical sense of these results is that when the mode annihilates an excitationin one of the dressed states, it only “sees” it through its weight in the total (or dressed)state, so the strength comes out as proportional to these amplitudes. Intuitively, if the“polariton” becomes more “exciton-like” because of detuning, the cavity emission dis-appears. The intensities degenerate into two lines out of four for the amplitudes becauseof the symmetry of the two modes (here we have two harmonic oscillators, in our casethe two modes are different). The transition rates are antisymmetric with∆, and as onegoes from 0 (from−∞) to 1 (from+∞), the other does the reverse. The emission ofone given mode is therefore two lines, which degenerate at zero detuning (where therates matches) and with the detuning, one vanishes while theother increases.

For instance, the simplest case wherep† andq† creates a coherent superposition ofbare states respectively in and out of phase:

p = (a+ b)/√

2, q = (a− b)/√

2 , (5.52)

which are the eigenstates of zero detuning, with its corresponding diagonalized Hamil-tonian:

H = (~ω − ~g)p†p+ (~ω + ~g)q†q . (5.53)

Back to the general case, we note as before||n,m〉〉 then,m Fock excitations inp, qoscillators and|i, j〉 thei, j Fock excitations ina, b oscillators:

H ||n,m〉〉 = En,m||n,m〉〉 (5.54)

withEn,m = ~(nωp +mωq) . (5.55)

From (5.42) it follows that, in our case (5.45a),

a† = cos θp† − sin θq† (5.56a)

b† = sin θp† + cos θq† , (5.56b)

So if we have||n, 0〉〉 particles, this corresponds in bare states to

||n, 0〉〉 =1√n!p†n||0, 0〉〉 (5.57a)

=1√n!

(cos θa† + sin θb†)n |0, 0〉 (5.57b)

=1√n!

n∑

k=0

(

n

k

)

cosk θ sinn−k θa†kb†n−k |0, 0〉 (5.57c)

=n∑

k=0

√

(

n

k

)

cosk θ sinn−k θ |k, n− k〉 (5.57d)

BLOCH EQUATIONS 181

The general case should be computed along the same lines

||n,m〉〉 =1√n!m!

p†nq†m||0, 0〉〉 (5.58)

(5.59)

to finally obtain

||n,m〉〉 =

n+m∑

ν=0

ν∑

l=0

(−1)l(

n

ν − l

)(

m

l

)

√

(

n+mn

)

(

n+mν

)×

× cosν−l θ sinl θ sinn−ν+l θ cosm−l θ |ν, n+m− ν〉 (5.60)

Exercise 5.8 (∗) Derive Eq. (5.60) above.

This result gives the probability to be measure the bare state |ν, n+m− ν〉 whenthe system is prepared in the dressed state||n,m〉〉:

p(ν) =

∣

∣

∣

∣

∣

ν∑

l=0

(−1)l(

n

ν − l

)(

m

l

)

√

(

n+mn

)

(

n+mν

) cosm+ν−2l θ sinn−ν+2l θ

∣

∣

∣

∣

∣

2

(5.61)

Let us consider transition amplitudes between||n,m〉〉 and||n− 1,m〉〉:

〈〈m,n− 1||p||n,m〉〉 =√n (5.62a)

〈〈m,n− 1||a||n,m〉〉 =√n/(2 cos θ) . (5.62b)

The same appears withq andb.We conclude this Section with the basis of coherent states, that is, we provide the

time evolution of theP representation of the oscillators (cf. 3.82). If we callα thevariable relating to oscillatora andβ that relating tob, the density matrixρ of thecoupled system (5.30) decomposes as

ρ =

∫∫

P (α, α∗, β, β∗, t) |αβ〉〈βα| dαdβ (5.63)

Following the procedure detailed in Sec. 3.3.3, the operator equation (5.30) providesits c–number countepart:

P =iω(α∂α − α∗∂α∗ + β∂β − β∗∂β∗)P

+ig(α∂β − α∗∂β∗ + β∂α − β∗∂α∗)P(5.64)

which can be integrated to yield the exact solution:

P (α, α∗, β, β∗, t) = F( i

2(α∗ + β∗)e−i(ω+g)t,

i

2(−α∗ + β∗)e−i(ω−g)t,

i

2(α− β)e−i(−ω+g)t,

i

2(−α− β)ei(ω+g)t

)

(5.65)

whereF is a differentiable function.


Exercise 5.9 (∗) Check that the partial differential equationaxux + byuy + czuz +dwuw = 0 has solutions of the type

u(x, y, z, w) = F (ya/xb, za/xc, wa/xd) (5.66)

whereF is a differentiable function. Therefore, by diagonalization of (5.64), obtainEq. (5.65).

By matching Eq. (5.65) with the initial conditions, one can obtain the time evolutionof the complete wavefunction. For instance, the important case of two states which aremixtures of coherent and thermal states (in the sense of Section 3.3.5 on page 102), withrespective intensities|α0|2 andn for one oscillator, and|β0|2 andm for the other, leadsto the following evolution of their wavefunction:

P (α, α∗, β, β∗, t) =1

πmexp

(

− |α|2[

cos2(gt)

m+

sin2(gt)

n

]

−α[−α∗

0eiωt cos(gt)

m+

−iβ∗0eiωt sin(gt)

n

]

−α∗[−α0e

−iωt cos(gt)

m+iβ0e

−iωt sin(gt)

n

]

−|α0|2[

1

m

]

)

×

1

πnexp

(

− |β|2[

sin2(gt)

m+

cos2(gt)

n

]

−β[−iα∗

0eiωt sin(gt)

m+

−β∗0eiωt cos(gt)

n

]

−β∗[

iα0e−iωt sin(gt)

m+

−β0e−iωt cos(gt)

n

]

−|β0|2[

1

n

]

)

×

exp(

− α∗β

[

i cos(gt) sin(gt)

m+

−i sin(gt) cos(gt)

n

]

−αβ∗[−i cos(gt) sin(gt)

m+i sin(gt) cos(gt)

n

]

)

(5.67)

This result shows how, due to the coupling, even though it is linear and of the sim-plest kind that can affect two modes, complexcorrelationsbuild up between the twomodes. The main behaviour of the coupled system remains thatof two mixture statesoscillating in their populations, which are represented bythe first two exponentials ofthe above product, with variablesα andβ factoring out, but a third exponential inex-tricably links them. This approach is the basis for justifications of truncated schemes

LINDBLAD DISSIPATION 183

which will be the basis for master and rate equations to be developed in the followingchapters.

5.4 Lindblad dissipation

We have been able to introduce dissipation going from the Rabi picture dealing witha pure state to the Bloch one dealing with a density matrix. Inthis case, though, thedecay was phenomenological, adding a memoryless decay termto the matrix elementreproducing the sought effect. We now describe a popular formalism to introduce dissi-pation in a quantum system, which also gives much insights into the origin and natureof dissipation in a quantum system. We shall illustrate our point on the simple systemof an harmonic oscillator, cf. Eq. (3.40), whose dynamics has already been investigatedfrom many different viewpoints.

The postulates of quantum mechanics do not accommodate wellindirect attempts to“reproduce” dissipation. For instance canonical quantization of the equations of motionof the damped oscillator Eq. (4.1) with methods of Chapter 3 yields:75

[X(t), P (t)] = e−γt[X(0), P (0)] . (5.68)

and the commutation relation and its derived algebra—whichdefine the quantum fieldand such properties as its statistics—are lost in time. The dissipation introduced in thisway washes away the quantum character of the system as well asits dynamics. This isnot a very satisfying picture: an atom which relaxes to its fundamental state should stillremain an atom.

In the previous section where we have investigated the dynamics of two coupledquantum oscillators, we have seen how the excitation of one was transfered to the other.This is a dissipation insofar as the first system is considered over this interval of time.Then the Hamiltonian dynamics brings back the excitation ofthe second oscillator tothe first one and the process continues back and forth cyclically. Here lies the key ideaof a correct model for dissipation in quantum mechanics. It is not a fundamental char-acteristic of a system which needs to be quantised, but it is afeature of its dynamics:the quantum system gives its energy away and takes it back when it couples to anotherstate. Imagine now that the system is coupled not to one, but to numerous other modes,which together form areservoir, so that as the system’s energy is exchanged with thereservoir, each mode has little probability to be the recipient but the system has highprobability to lose its excitation. Once enough time has elapsed so that energy is withhigh probability in the resevoir, it will continue being exchanged in an oscillatory way,as follows from Hamiltonian dynamics which demands that allconfigurations of thesystems be all visited with the same weight. However the system is so insignificant ascompared to the reservoir that the probability of a full return of its initial expenditureis vanishingly small with increasing size of the reservoir.In effect, this accounts for

75The Hamiltonian equations with dissipation beingX = P/m0 andP = −γP −m0ω2X, the corre-spondence principle Eq. (3.9) makesd[X,P ]/dt = XP +PX−XP −XP = −γ[X,P ] which integratesto the result Eq. (5.68) from Eq. (3.17).


dissipation in quantum mechanics in the same way that, in classical physics, the new-tonian dynamics which violates thermodynamics (being reversible) is supressed by lawof large numbers.

We detail the mathematical form in an explicit, simple case now before turning tothe general expression. Consider a single harmonic oscillator a coupled linearly to a setof oscillatorsbi which all merge together to form the bath. All oscillatorsa and thebifollow the algebra of Eq. (3.38). The coupling Hamiltonian reads:

H = ~ωaa†a+

∑

i

~ωib†ibi +

∑

i

~(g∗i ab†j + gia

†bj) (5.69)

We have made the same approximations as in the previous section. The dynamics of thissimple system is clear physically: the systema loses its excitation which is transferred toone of thebi mode through the processab†i weighted by the strength of the couplinggi.The reverse processa†bi where the excitation is destroyed in the modebi and put backin the system exists. It follows, as we have already emphasised, from the Hamiltoniancharacter of this dynamics. Now we proceed to make the approximations which arein order when the set ofbi is treated as a reservoir. Namely, first of all we neglect thedynamics of the reservoir (being in equilibrium) and its direct correlations to the system.We write the total density matrix for the combined system described by Eq. (5.69) as:

(t) = ρ(t) ⊗ R0 (5.70)

whereR0 is the time independent density matrix of bosons at equilibrium:

R0 =⊗

i

(

1 − exp(

− ~ωikBT

)

)

exp(

− ~ωib†ibi

kBT

)

. (5.71)

Note that we have assumed the separability of the density matrix into a product ofthe density matrix of the system and of the reservoir. This iscalled the “Born approxi-mation”. It is physically reasonable for a reservoir, as each mode quickly washes awaythe coherence that it develops with the system through its interaction because it alsointeracts with many other modes of the reservoir. In fact note thatR0 in Eq. (5.71) isalso decorrelated for all modes. The important changes retained are the dephasing andloss of energy of the system alone. The latter now moreover accounts for the entire timedependency. To make it more sensible still, we shift to the interaction picture so thatthe rapid and trivial oscillation due to the optical frequency is removed. From now on,therefore:

a(t) = a−iωat and a†(t) = a†eiωat (5.72)

as detailed in Section 3.1.6 (a(0) anda of the Schrodinger picture being equal.) Theinteracting Hamiltonian become:

H(t) = ~[

a(t)B†(t) + a(t)†B(t)]

(5.73)

where we have introduced the “reservoir operators”:


B(t) =∑

i

gibie−iωit and B†(t) =

∑

i

gib†ieiωit (5.74)

It remains to get rid of the complicated structure of operators Eqs. (5.74) and focuson the dynamics ofρ alone. This can be done by averaging over the degrees of freedomof the reservoir. Let us first obtain the equation of motion for ρ. If we formally integratethe Liouville–von Neumann equation, cf. Eq. (3.33) for, we get:

i~(t) = i~(−∞) +

∫ t

−∞[H(t), ρ(t)] dt (5.75)

where, for a while, we write everywhere the time dependency.We have assumed theinitial condition is att→ −∞.

Exercise 5.10(∗) By inserting the exact result Eq. (5.75) back into Eq. (3.33)(for ),show that the following integro-differential is obtained:

˙(t) = − 1

~2

∫ τ

−∞[H(t), [H(τ), (τ)]] dτ (5.76)

It is enough in this case to limit to order two in the commutator, which yields thedynamics of the populations while neglecting that of the fluctuations. Later in this book,e.g., when we investigate the dynamics of the laser field in weak-coupling microcavities,we shall pursue this kind of expansion of Eq. (5.75) further.They are known asBornexpansions.

Exercise 5.11(∗∗) Carry out the algebra of[H(t), [H(τ), (τ)]] with definitions givenby Eqs. (5.73–5.74) and show that under the approximation ofseparability Eq. (5.70):

ρ = − 1

~2

∫ t

0

[

(a2ρ(τ) − aρ(τ)a)e−iωa(t+τ)〈B†(t)B†(τ)〉 + h.c. (5.77a)

+ (a†2ρ(τ) − a†ρ(τ)a†)eiωa(t+τ)〈B(t)B(τ)〉 + h.c. (5.77b)

+ (aa†ρ(τ) − a†ρ(τ)a)e−iωa(t−τ)〈B†(t)B(τ)〉 + h.c. (5.77c)

+ (a†aρ(τ) − aρ(τ)a†)eiωa(t−τ)〈B(t)B†(τ)〉 + h.c.]

dτ (5.77d)

with, for the reservoir correlators,〈B†(t)B†(τ)〉 =∑

i,j g∗i g

∗j ei(ωit+ωjτ)Tr(R0b

†ib

†j)

and〈B(t)B(τ)〉 =∑

i,j gigje−i(ωit+ωjτ)Tr(R0bibj), and more importantly:

〈B†(t)B(τ)〉 =∑

i

|gi|2eiωi(t−τ)n(ωi) , (5.78a)

〈B(t)B†(τ)〉 =∑

i

|gi|2e−iωi(t−τ)(n(ωi) + 1) (5.78b)

wheren is given by Bose-Einstein distribution, cf. Eq. (3.87) and Eq. (5.71).


Lines(5.77 a&d) cancel exactly because of repeated annihilation on the diagonal den-sity operator Eq. (5.71). Evaluation of Eqs. (5.78) is more involved. The physical ideathat we wish to emphasise over direct attempts towards evaluating the mathematicalexpressions, is that given a time evolution forρ(τ) which is much smaller than thatof e±iωa(t−τ) and〈B†(t)B(τ)〉, 〈B(t)B†(τ)〉 also with rapid dependence of the typee±iωi(t−τ), it is a good approximation to write

ρ(τ) ≈ ρ(t) (5.79)

in Eq. (5.77), which is called theMarkov approximation, since the density matrixρ(t)at timet on the left hand side of Eq. (5.77) loses the dependency on itsvalue at previoustime τ < t, i.e., it has no memory of its past. It is especially clear in the form below:

Exercise 5.12(∗) Combining the arguments given above and using the representation∫

ei(ωi−ωa)(t−τ) dτ = πδ(ωi − ωa), reduce Eq. (5.77) to the following first order dif-ferential, Markovian equation:

ρ = A(aρa† − a†aρ) +B(aρa† + a†ρa− a†aρ− ρaa†) + h.c. (5.80)

where:

A = π∑

i

|gi|2n(ωi)δ(ωa − ωi) , (5.81a)

B = π∑

i

|gi|2(n(ωi) + 1)δ(ωa − ωi) . (5.81b)

Coefficients (5.81) are those that would be obtained by Fermi’s golden rule. Theyexhibit a delta-singularity because of the Markov approximation and the long-time aver-age of the interaction between the system and the reservoir which requires conservationof the energy. By broadening the modes of the reservoir they provide two rates whichwe associate in the following way to the master equation:

ρ =γ

2(aa†ρ+ ρaa† − 2a†ρa) (5.82a)

+ γn(a†aρ+ ρa†a− 2aρa†) . (5.82b)

whereγ = 2πσ(ωa)|g(ωa)|2, σ(ω) being the density of states of the oscillators inthe reservoir at frequencyωa (σ is a smooth function ofγ which is the continuouslimit of ωi). Expression (5.82) is a popular one in quantum optics. Notethat now inthe interaction picture the present form Eq. (5.82) is completely devoid of Hamiltoniandynamics. As we expect and as was the aim of the construction,it precisely describesdissipation, as is investigated further in the exercise below. Before concentrating on thisform, however, let us give the result of the most general possible form for a masterequation under the assumption that the evolution is Markovian (as is the case for manymodels). It is known as theLindblad master equationand reads:

ρ = − i

~[H, ρ] − 1

~

∑

n,m

hn,m(

ρLmLn + LmLnρ− 2LnρLm

)

+ h.c. (5.83)


GoranLindblad with his work “on the generators of quantum mechanicalsemigroups”, Comm. Math. Phys.48(1976), 119, attached his name to termswhich turn the Liouville von-Neuman equation into a dissipative system.

Lindblad is a mathematical physicist and Emeritus Professor of the KTH(“Kungliga Tekniska hogskolan”, the Royal Institute of Technology) inStokholm, from which he retired in 2005.

where has been put back the part from the Schrodinger equation,− i~[H, ρ], and the

term on the right whereLm are operators defined by the system to model the dissipationalong with the constantshn,m. In the case of the harmonic oscillator coupled to the bath,L0 = a andL1 = a†, all other being zero.

Eq. (5.83)—which in our approach derives from the Schrodinger equation after aphysical model of coupling of a small system to a big reservoir—correctly reproducesfeatures of dissipation as can be seen by computing the equation of motions of theobservables of interest in this case. Indeed, consider for instance the average number ofexcitations in the oscillator〈n〉 = 〈a†a〉 = Tr

(

ρa†a)

. Its equation of motion is givenby (we assume we are now in Schrodinger picture to keepa time independent but thecase of the interaction picture follows straightforwardly):

∂〈n〉∂t

= Tr

[(

∂ρ

∂t

)

a†a

]

(5.84a)

= Tr[(γ

2(aa†ρ+ ρaa† − 2a†ρa)

)

a†a]

(5.84b)

+ Tr[(

γn(a†aρ+ ρa†a− 2aρa†))

a†a]

. (5.84c)

The approach of computing equations of motion of operators from the master equa-tion is outlined above. It only remains to simplify the expression (5.84c) using algebraicrelations presented in Chapter 3. Let us carry out explicitly the case of line (5.84b):by cyclic permutation of the trace,76 the density matrix can be factored out to giveTr(

(γ/2)ρ(a†aaa† + aa†a†a − 2a†a†aa))

which give rise to the new operator thatsimplifies further still in terms of quantities already known, for instance by noting that:

a†aaa† + aa†a†a− 2a†a†aa = a†(aaa† − a†aa) + (aa†a† − a†a†a)a (5.86a)

= a†[aa, a†] + [a, a†a†]a (5.86b)

= 2a†a . (5.86c)

76Operators can be permuted cyclically in the trace, i.e., forarbitrary operatorsA,B andC:

Tr(ABC) = Tr(BCA) = Tr(CAB) (5.85)

whereas it is not true, in general, for other permutations, e.g.,Tr(ABC) 6= Tr(BAC).


where the last line follows from Eq. (3.61), so that finally, the total expression simplifiesto γ〈a†a〉 and if the same is carried out for line (5.84c), one gets:

∂〈n〉∂t

= −γ(〈n〉 − n) (5.87)

i.e., the population releases to the Bose distributionn. This integrates immediately to

〈n〉(t) = 〈n〉(0)e−γt + n(1 − e−γt) (5.88)

which is the exact behaviour one would expect for dissipation of an harmonic oscillator:a decay from its initial value (or intensity)〈n〉(0) towards the mean value of the reser-voir n. If the oscillator is initially in vacuum, it gets populatedby thermal contact to thereservoir and comes to equilibrium with it on a timescaleγ−1 whereas on the oppositeif it has more excitations, it loses them to thermalise with the reservoir.

Note how line (5.82a) is linked to the decay in the sense that it empties the mode,whereas line (5.82b) has also the effect of a pump which can populate the mode. TheLindblad terms can be used to that effect in a large number of systems investigated inthe Schrodinger picture.77

Exercise 5.13(∗∗) Show that the master equation (5.82) written for the GlauberPfunction becomes (cf. relations (3.93)):

∂P

∂t=

[

γ

2

(

∂

∂αα+

∂

∂α∗

)

+ γn∂2

∂α∂α∗

]

P (5.89)

Exercise 5.14(∗) Check that the following expression forP satisfies Eq. (5.89) withinitial conditionδ(α− α0) in P space (|α0〉 with Dirac notation).

P (α, α∗, t) =1

πn(1 − exp(−γt)) exp

[

−|α− α0 exp(−(γ/2)t)|2n(1 − exp(−γt))

]

(5.90)

The dynamics of this solution is represented on Fig. (5.5), together with the solutionEq. (5.67) where, modelled after an Hamiltonian, the time evolution is cyclic in time.

5.5 Jaynes–Cummings model

All the previous material has been leading us towards the full quantum treatment of thetwo-level system interacting with a light-mode, where boththe atom (or exciton)andthe photon are quantized, according to Eq. (5.29). The caseswe have been dealing withso far, where the material excitation is modelled as a harmonic quantum oscillator, donot encompass the fermionic limit which is the more important in the case of materialexcitations.

77In the Heisenberg picture, the average of the reservoir operator serves as a fluctuating form whichby the action of the “fluctuation–dissipation” theorem alsoresults in a decay of the averages. There is lessemphasis on the quantum aspect of the decay in this case sincethere is always the possibility or temptationto understand the dynamical averages as semiclassical and the whole formalism becomes one which favoursclassical interpretations or analogies.

JAYNES–CUMMINGS MODEL 189

Isidor IsaacRabi (1898–1988) and BenjaminMollow

Rabi studied in the 1930s the hydrogen atom and the nature of the force binding the proton to the atomic nu-clei, out of which investigations he conceived molecular-beam magnetic-resonance, a new and very accuratedetection method for which he was awarded the exclusive Nobel Prize for Physics in 1944, “for his resonancemethod for recording the magnetic properties of atomic nuclei”. He wanted to be a theorist and had acceptedonly because the invitation was from the prestigious Stern.In his biography he is quoted as saying “Wheneverone of my students came to me with a scientific project, I askedonly one question, ‘Will it bring you nearer toGod?’”

Mollow is presently a Physics Professor at the University ofMassachusets Boston where he studies quantumoptics.

Different physics occurs when such excitations are described by fermionic ratherthan bosonic statistics, only in the nonlinear regime wheremore than one excitationresides in the system, so that higher energy states can be probed, the ground and firstexcited being identical for both Bose and Fermi statistics.With the advent of lasers, suchcases are however not difficult to realize experimentally. In the case of cavity QED thematerial is usually a beam of atoms passing through the cavity, and a single excitationis the independent responses of the atoms to the light field excitation. The simplestsituation is that of a dilute atomic beam where a single atom (driven at resonance sothat it appears as a two-level system) is coupled to a Fock state of light with a largenumber of photons. This case has been described theoretically by Jaynes & Cummings(1963) to yield the model which now bears their name.

The model is similar to the Bloch Equations except that the classical field is now up-graded to a quantum field, and for our single mode, to the Bose annihilation operatora.We rewrite the Hamiltonian (5.29) in the interaction picture and with the approximationof Jaynes and Cummings:

H = ~g(ei∆tσa† + e−i∆tσ†a) (5.91)

where we have noted∆ the detuning in energy between the cavity mode and the atom.Observe that at resonance,∆ = 0, the Hamiltonian is time independent, which allows adirect solution through quite painless algebra. Out of resonance, more mathematics are


Fig. 5.5: Evolution of the center of mass of theP distribu-tion in the complex planeC (horizontal plane) as a functionof timet (vertical axis) of an oscillatora starting in a coher-ent state|α〉 in the cases of the coupling to an other quan-tum oscillator, cf. Eq. (5.67)—yielding beats—and of thecoupling to a reservoir, cf. Eq. (5.82)—converging to thevacuum state. This shows the difference between a Hamil-tonian system and one in presence of dissipation. The os-cillations areRabi oscillationsthis time visualised in thecomplex plane. The closer the center of mass from the cen-tral axis, the greater the decoherence.

involved but in the limit of very high detunings, conservation of the number of photonsand excitation in the atom is recovered and another limitingcase is worthy of interest.

In the resonant case the Hamiltonian (5.91) becomes:

H = ~g(σa† + σ†a) (5.92)

in whicha (the radiation field) is the Bose annihilation operator which removes a photonin the cavity andσ† is the Pauli matrix with transfer the excitation in the atom,so thatin matrix representation:

σ† =

(

0 10 0

)

(5.93)

with |0〉 = (0, 1)T and |1〉 = (1, 0)T . Because the excitation can only be transferedfrom the atom to the cavity or vice-versa, but none is ever lost or created under thedynamics of Eq. (5.92), all the dynamics of a fixed numbern of total excitations iscontained within the manifold

Hn = |0, n〉 , |1, n− 1〉 (5.94)

provided thatn ≥ 1. The associated energy diagrams appear on the right of Fig. 5.2,with two states in each manifold (in our conventions|0, n〉 refers to the bare states withthe atom in the ground state andn photons, while|1, n− 1〉 has the atom in the excitedstate andn − 1 photons) and the total wavefunction|Ψ(t)〉 is a superposition of these


Edwin ThompsonJaynes(1922–1998) and FrederickCummings

Jaynes challenged the mainstream of physics, most notably with his proposition to use Bayesian probabilitiesto formulate a reinterpretation of statistical physics as inferences due to incomplete information. In quantumoptics, he rejected the Copenhagen interpretation which hequalified as “a considerable body of folklore”. Hewrote of Oppenheimer (whom he called Oppy and felt would be anunsuitable Ph. D. advisor to carry outindependent research):

Oppy would never countenance any retreat from the Copenhagen position, of the kind advocated bySchrodinger and Einstein. He derived some great emotionalsatisfaction from just those elements ofmysticism that Schrodinger and Einstein had deplored, andalways wanted to make the world still moremystical, and less rational. This desire was expressed strongly in his 1955 BBC Reith lectures (of which Istill have some cherished tape recordings which recall his style of delivery at its best). Some have seen thisas a fine humanist trait. I saw it increasingly as an anomaly—abasically anti-scientific attitude in a personposing as a scientist—that explains so much of the contradictions in his character.

In quantum optics, he questioned the need of full-quantization of the optical field, e.g., to explain effectssuch as blackbody radiation, spontaneous emission or the Lamb shift, the two latter he claimed could beobtained in the realm of his so-called neoclassical theory,whose fields offer the additional advantage to be“conspicuously free from many of the divergence problems of quantum electrodynamics.” Ironically, he istoday most remembered in quantum optics for the Jaynes & Cummings (1963) model which main merit is tobe an integrable fully-quantized system. This “drosophila” of quantum mechanics was initially derived in aform and with approximations allowing for comparison with aclassical description in favour of which Jayneswas ready to bet, like for the origin of the Lamb shift (the outcome of his bet with Lamb was left undecided).

Frederick Cummings was Jaynes’ student and is now a professor Emeritus at the University of California,Riverside. His interest turned to biophysics in the mid-eighties.

states. If the number of excitations is not fixed, one can still decouple independent dy-namics of|Ψ〉 by projecting the state ontoHn, which, because they are not coupled, canalways be solved independently following the procedure below, and put back togetheragain at the end.

The formal solution of Schrodinger equation reads

|Ψ(t)〉 = exp

(

− i

~Ht

)

|Ψ(0)〉 (5.95a)

= exp(

−ig(σa† + σ†a)t)

|Ψ(0)〉 (5.95b)

We now compute an expression forexp(


. By definition,exp(−ωΩ) =


∑

n≥0(−ω)nΩn/n! with ω ∈ C andΩ ∈ ⊗n≥0Hn. The decomposition into real andimaginary parts of the exponential gives:

exp(


= c− is (5.96)

where

c =

∞∑

n=0

(−1)n(gt)2n

(2n)!(σa† + σ†a)2n (5.97a)

s =

∞∑

n=0

(−1)n(gt)2n+1

(2n+ 1)!(σa† + σ†a)2n(σa† + σ†a) (5.97b)

and we are reduced to algebraic computation of the type(σa† + σ†a)2n:

Exercise 5.15(∗∗) Taking advantage of the algebra ofσ, especially of such propertiesasσ2 = σ†2 = 0, show that

(σa† + σ†a)2n =

(

(aa†)n 00 (a†a)n

)

(5.98)

in the basis of bare states and consequently thatc ands as defined by Eqs. (5.97) havethe following matrix representations:

c =

(

cos(gt√aa†) 0

0 cos(gt√a†a)

)

and s =

0gt√aa†√aa†

a

gt√a†a√a†a

a† 0

, (5.99)

where we noted, e.g.,(√a†a)2n = (a†a)n so as to carry out the series summation

exactly with the new operator defined unambiguously (and as one can check, correctly)on the Fock state basis.

Observe that the time propagator has off-diagonal elements. They correspond tovirtual processes where an odd number of excitation is exchanged between the twofields. The knowledge of Eqs. (5.99) provides the dynamics, including such effects,by direct computation. Let us consider the case of fixedn and the atom in a quantumsuperposition of ground and excited states:

|Ψ(0)〉 = (χg |g〉 + χe |e〉) ⊗ |n〉 (5.100)

which is separable because there is no summation to entanglewith other configurations.The labeln does not intervene directly and one can write:

|Ψ(0)〉 = χg |g, n〉 + χe |e, n〉 (5.101)

which, from Eq. (5.95b), leads to:


|Ψ(t)〉 = (c− is)(χg |g, n〉 + χe |e, n〉) (5.102a)

= χg[cos(√ngt) |e, n〉 − i sin(

√ngt) |g, n− 1〉] (5.102b)

+ χe[cos(√n+ 1gt) |e, n〉 − i sin(

√n+ 1gt) |g, n+ 1〉] . (5.102c)

This dynamics bears much resemblance to the Rabi oscillations of Section 5.2 butthe more pronounced quantum character results in a more complicated dynamics fea-turing virtual transition off-diagonal dynamics of the density matrix.

If the dynamics is constrained from the initial condition tohold in a manifold, e.g.,if the initial state is

|Ψ(0)〉 = |e, n− 1〉 , (5.103)

then the only other state available to the dynamics under Eq.(5.92) is|g, n〉 as one cancheck from the closure relations:

σa† |e, n− 1〉 = |g, n〉 , σ†a |e, n− 1〉 = 0 , (5.104a)

σa† |g, n〉 = 0 , σ†a |g, n〉 = |e, n− 1〉 . (5.104b)

The dynamics is therefore closed inHn where the Hamiltonian can be diagonalizedexactly. For the resonant condition the dressed states for this manifold are split by an en-ergy

√n~g. In the general case, all four transitions between the states in manifoldsHn

andHn−1 are possible, and this would result in a quadruplet in the emitted spectrum. Itis hard to resolve this quadruplet, but it has been done in Fourier transform of time re-solved experiments by Brune et al. (1996). It is simpler to consider photoluminescencedirectly under continuous excitation at high intensity (where the fluctuations of particlesnumber have little effect). In this case, withn ≫ 1, the two intermediate energies arealmost degenerate and a triplet is obtained with its centralpeak being about twice ashigh as the two satellites. This is the Mollow (1969) tripletof resonance fluorescence.We investigate this problem in more details in Section 5.7.

Exercise 5.16(∗) Consider now the out-of-resonance case where the two-leveltransi-tion does not match the cavity photon energy, i.e.,∆ 6= 0 in Eq. (5.91). The Hamiltonianis now time dependent and the formal integration Eq. (5.95a)is not possible anymore.Taking advantage of the closure relations (5.104), consider the ansatz

|Ψ(t)〉 =

∞∑

n=0

[ψg,n+1(t) |g, n+ 1〉 + ψe,n(t) |e, n〉] (5.105)

where the time dependence is in the coefficientsψg,n+1, ψe,n only. Show therefore thatthe Schrodinger equation with Hamiltonian Eq. (5.91) applied on state Eq. (5.105)yields:

ψg,n+1 = −ig√n+ 1ei∆tψe,n

ψe,n = −ig√n+ 1e−i∆tψe,n

(5.106)

Evoking as they do the flopping between two states, those equations are reminiscentof Rabi dynamics.


Exercise 5.17(∗∗) Use a technique of your choice to solve Eqs. (5.105). Show that theresult reads:

ψg,n+1(t) = ei∆t/2

[(

cos(1

2Ωnt

)

− i∆

Ωnsin(1

2Ωnt

)

)

ψg,n+1(0)

− i2g

√n+ 1

Ωnsin(1

2Ωnt

)

ψe,n(0)

]

,

(5.107a)

ψe,n(t) = e−i∆/2[(

cos(1

2Ωnt

)

+ i∆

Ωnsin(1

2Ωnt

)

)

ψe,n(0)

− i2g

√n+ 1

Ωnsin(1

2Ωnt

)

ψg,n+1(0)

]

.

(5.107b)

in terms of the generalised Rabi frequencies:

Ωn =

√

∆2

+4(n+ 1)g2 . (5.108)

Eqs. (5.107–5.108) represent one of the most important result of quantum optics. Theyare the crowning achievement towards which all the results converge, either from sim-plified models such as Bloch optical equations or Rabi flopping between two states,or from more refined theory such as non-rotating wave Hamiltonians which elabo-rate around this dynamics. This general solution connects directly to the resonant casetreated previously from operator algebra. In the detuned case, where2g

√n+ 1 ≪ |∆|,

a serial expansion ofψe,n(t) shows that the atom is inhibited in its transition and re-mains in the same state up to some phase fluctuations induced by the perturbation of theinteraction.78

5.5.1 Excitons in quantum dots

We now show one possible route of extending results of previous sections. We haveinvestigated the limiting cases where the material excitation which couples to the fieldis either a boson, or a fermion. In actual systems, compositeparticles are neither exactlyone or the other. The importance of this distinction can become important in a QuantumDot (QD), where the excitations are located in a tiny region of real space, so that theirwavefunctions overlap appreciably. If the confining potential of the dot is much strongerthan the Coulomb interaction, electrons and holes, which are elementary excitations of

78Explicitly, it is found that

ψg,n+1(t) ≈ exp

„

−i g2(n+ 1)

∆t

«

ψb,n+1(0) and ψe,n(t) ≈ exp

„

ig2(n+ 1)

∆t

«

ψe,n(0) .

(5.109)This can be used a posteriori to define a better ansatz for Eq. (5.105).


the system, will be quantised separately, whereas if Coulomb interactions dominate overthe confinement, one electron-hole pair will bind as an exciton and therefore behaverather like a boson. Here we investigate a model of interactions of light with excitons inQDs of varying size, where their Boson or Fermion character is tuneable.

It is more relevant to carry out the analysis in real space since the QD states arelocalised. We note

ϕene(re) = 〈re|ϕene

〉 and ϕhnh(rh) = 〈rh|ϕhnh

〉 (5.110)

the set of their basis wavefunctions withre andrh the positions of the electron and hole,respectively. Subscriptsne andnh are multi-indices enumerating all quantum numbersof electrons and holes. The specifics of the three-dimensional confinement manifests it-self in the discrete character ofne andnh components. We restrict our considerations todirect bandgap semiconductor with non-degenerate valencebands. Such a situation canbe experimentally achieved in QDs formed in conventional III-V or II-VI semiconduc-tors, where the light-hole levels lie far below, in energy, the heavy-hole ones due to theeffects of strain and size quantization along the growth axis. Therefore, only electron-heavy hole excitons need to be considered. Moreover, we willneglect the spin degreeof freedom of the electron-hole pair and assume all carriersto be spin polarized. Tocarry out the same formalism as presented in the previous sections, we need to build thesecond quantized operator for the QD. We define it as:

X† =∑

ne,nh

Cne,nhe†ne

ς†nh(5.111)

whereene andhnhare fermion creation operators for an electron and a hole in state

∣

∣ϕene

⟩

and∣

∣ϕhnh

⟩

, respectively:

e†ne|0〉 =

∣

∣ϕene

⟩

, h†nh|0〉 =

∣

∣ϕhnh

⟩

, (5.112)

with |0〉 denoting both the electron and hole vacuum fields. The (single) exciton wave-function|ϕ〉 results from the application ofX† on the vacuum. In real space coordinates:

〈re, rh|ϕ〉 = ϕ(re, rh) =∑

ne,nh

Cne,nhϕene

(re)ϕhnh

(rh). (5.113)

At this stage we do not specify the wavefunction (that is, theset of coefficientsCne,nh),

which depends on various factors such as the dot geometry, electron and hole effectivemasses and dielectric constant. Rather, we consider then-excitons state which resultsfrom successive excitation of the system throughX†:

|Ψn〉 = (X†)n |0〉 , (5.114)

The associated normalised wavefunction|n〉 reads

|n〉 =1

Nn|Ψn〉 (5.115)


where, by definition of the normalization constant

Nn =√

〈Ψn|Ψn〉. (5.116)

The creation operatorX† can now be obtained explicitly. We defineαn the non-zeromatrix element which lies below the diagonal in the exciton representation:

αn = 〈n|X†|n− 1〉 , (5.117)

which, by comparing Eqs. (5.114–5.117) turns out to be

αn =Nn

Nn−1. (5.118)

The coefficientsαn can be linked to the coefficientsCne,nh(the latter assuming a

specific value when the system itself is known):

Exercise 5.18(∗∗∗) Show that the normalisation coefficientsN necessary to computethe matrix elementsαn (through Eq. (5.118)), can be computed by the following recur-rent relation:

N 2n =

1

n

n∑

m=1

(−1)m+1βmN 2n−m

m−1∏

j=0

(n− j)2, (5.119)

withN0 = 1 andβm the irreduciblem-excitons overlap integrals,1 ≤ m ≤ n:

βm =

∫

(

m−1∏

i=1

ϕ∗(rei , rhi)ϕ(rei , rhi+1)

)

ϕ∗(rem , rhm)ϕ(rem , rh1)

dre1 . . . dremdrh1 . . . drhm (5.120)

The procedure to calculate the matrix elements of the creation operator is as fol-lows:79 One starts from the envelope functionϕ(re, rh) for a single exciton. Then onecalculates all overlap integralsβm as given by Eq. (5.120), for1 ≤ m ≤ n wheren isthe highest manifold to be accessed. Then the norms can be computed with Eq. (5.119).Finally the matrix elementsαn are obtained as the successive norms ratio, cf. (5.118).

The limiting cases of Bose-Einstein and Fermi-Dirac statistics are recovered in thelimits of large and shallow dots, respectively. This is mademost clear through consid-eration of the explicit case of two excitons (n = 2). Then the wavefunction reads

Ψ2(re1 , re2 , rh1 , rh2) = ϕ(re1 , rh1)ϕ(re2 , rh2) − ϕ(re1 , rh2)ϕ(re2 , rh1) (5.121)

with its normalization constant (5.116) readily obtained as

N 22 =

∫

|Ψ2(re1 , re2 , rh1 , rh2)|2dre1 . . . drh2 = 2 − 2β2 (5.122)

79The numerical computation of theβm andαn values needs to be carried out with great care. Thecancellation of the large numbers of terms involved in Eq. (5.119) requires a very high-precision evaluationof βm.


whereβ2, the two-excitons overlap integral, reads explicitly

β2 =

∫

ϕ(re1 , rh1)ϕ(re2 , rh2)ϕ(re1 , rh2)ϕ(re2 , rh1)dre1 . . . drh2 . (5.123)

This integral is the signature of the composite nature of theexciton. The minussign in (5.122) results from the Pauli principle: two fermions (electrons and holes) can-not occupy the same state. Assumingϕ(re, rh) is normalised,N1 = 1, so accordingto (5.118),

α2 =√

2 − 2β2. (5.124)

Since0 ≤ β2 ≤ 1 this is smaller than or equal to√

2, the corresponding matrix elementof a true boson creation operator. This result has a transparent physical meaning: sincetwo identical fermions from two excitons cannot be in the same quantum state, it is“harder” to create two real excitons, where underlying structure is probed, than twoideal bosons. We note that ifL is the QD lateral dimension,β2 ∼ (aB/L)2 ≪ 1whenL ≫ aB. Thus in large QDs the overlap of excitonic wavefunctions issmall, soα2 ≈

√2 and the bosonic limit is recovered. On the other hand, in a small QD, where

Coulomb interaction is unimportant compared to the dot potential confining the carriers,the electron and hole can be regarded as quantized separately:

ϕ(re, rh) = ϕe(re)ϕh(rh) (5.125)

In this case allβm = 1 and subsequently allαm = 0 at the exception ofα1 = 1. Thisis the fermionic limit whereX† maps to the Pauli matrixσ+.

5.5.2 Gaussian toy model

We now turn to the general case of arbitrary sized QDs, interpolating between the(small) fermionic and (large) bosonic limits. We do not attempt for this conceptual pre-sentation to go through the lengthy and complicated task of the numerical calculation ofthe exciton creation operator matrix elements for a realistic QD. Rather, we consider amodel wavefunction which can be integrated analytically and illustrates some expectedtypical behaviours.

We assume a Gaussian form for the wavefunction which allows evaluation analyti-cally of all the required quantities. This follows from a harmonic confining potential, ashas been considered for instance by Que (1992). As numericalaccuracy is not the chiefgoal of this work we further assume in-plane coordinatesx andy to be uncorrelated toease the computations. The wavefunction reads:

ϕ(re, rh) = C exp(−γer2e − γhr

2h − γehre · rh) (5.126)

properly normalized with

C =

√

4γeγh − γ2eh

π(5.127)

provided thatγeh ∈ [−2√γeγh, 0] with γe, γh ≥ 0. Theγ parameters allow interpo-

lation between the large and small dot limits within the samewavefunction (cf. Sec-tion 4.3.3). To connect these parametersγe, γh andγeh to physical quantities, (5.126)


is regarded as a trial wavefunction which is to minimize the HamiltonianHQD confin-ing the electron and hole in a quadratic potential where theyinteract through Coulombinteraction:

HQD =∑

i=e,h

(

p2i

2mi+

1

2miω

2r2i

)

− e2

ǫ|re − rh|(5.128)

Herepi is the momentum operator for the electron and hole,i = e, h, respectively,me,mh the electron and hole masses,ω the frequency which characterises the strengthof the confining potential,e the charge of the electron andǫ the background dielectricconstant screening the Coulomb interaction. This Hamiltonian defines the two lengthscales of our problem, the 2D Bohr radiusaB and the dot sizeL:

aB =ǫ~2

2µe2(5.129a)

L =

√

~

µω(5.129b)

whereµ = memh/(me+mh) is the reduced mass of the electron-hole pair. To simplifythe following discussion we assume thatme = mh, resulting inγe = γh = γ. The trialwavefunction (5.126) separates asϕ(re, rh) = CΦ(R)φ(r) wherer = re − rh is theradius-vector of relative motion andR = (re + rh)/2 is the center-of-mass position:

Φ(R) =

√

2(2γ + γeh)√π

exp(

−R2[2γ + γeh])

(5.130a)

φ(r) =

√2γ − γeh√

2πexp

(

−r2

[

2γ − γeh4

])

(5.130b)

Eq. (5.130a) is an eigenstate of the center-of-mass energy operator and equating itsparameters with those of the exact solution yields the relationship2γ + γeh = 2/L2.This constraint allows us to minimising (5.130b) with respect to a single parameter,a = −γeh/2 + 1/(2L2), which eventually amounts to minimise4aB/a

2 + aBa2/L4 −

2√π/a. Doing so we obtain the ratio−γeh/γ as a function ofL/aB, displayed in

Fig. 5.6. The transition from bosonic to fermionic regime isseen to occur sharply whenthe dot size becomes commensurable with the Bohr radius. Forlarge dots, i.e., for largevalues ofL/aB, the ratio is well approximated by the expression

−γeh/γ = 2 − (aB/L)2 (5.131)

so that in the limit of big dots whereaB/L → 0, Eq. (5.126) becomesϕ(re, rh) ∝exp(−(

√γere −√

γhrh)2) with vanishing normalization constant. This mimics a free

exciton in an infinite quantum well. It corresponds to the bosonic case. On the otherhand, ifL is small compared to the Bohr radius, withγeh → 0, the limit (5.125) isrecovered withϕ ∝ exp(−γer2

e) exp(−γhr2h). This corresponds to the fermionic case.


0.1 1 100.0

0.5

1.0

1.5

2.0

-eh

/

L/aB

Fig. 5.6: Ratio of parameters−γeh andγ (with γ = γe = γh) as a function ofL/aB. For large dotswhereL ≫ aB, −γeh/γ ≈ 2 which corresponds to the bosonic limit where the electron and hole arestrongly correlated. For shallow dots whereL ≪ aB, −γeh/γ ≈ 0 with electron and hole quantizedseparately. The transition is shown as the result of a variational procedure, with an abrupt transition when thedot size becomes comparable to the Bohr Radius.

The trial wavefunction is of course not exact80 but the exciton operator which ityields is exact, as well as all the intermediates.81

80One can check that (5.126) gives, in the caseγeh → −2√γeγh, an exciton binding energy which is

smaller by only20% than that calculated with a hydrogenic wavefunction, whichshows that the Gaussianapproximation should be tolerable for qualitative and semi-quantitative results.

81 It can be seen, for instance, that the overlap integrals (5.120) take a simple form in terms of multivariateGaussians as function of a matrixA defined below:

βm = C2m

Z

exp(−xTAx) dx

Z

exp(−yTAy) dy (5.132)

where

xT = (xe1 , xe2 , . . . , xem , xh1, xh2

, . . . , xhm ) (5.133a)

yT = (ye1 , ye2 , . . . , yem , yh1, yh2

, . . . , yhm ) (5.133b)

are the2m dimensional vectors which encapsulate all the degrees of freedom of them excitons-complex,andA is a positive definite symmetric matrix which equates (5.120) and (5.132), i.e., which satisfies

xTAx = 2γe

mX

i=1

x2i + 2γh

2mX

i=m+1

x2i + γehxmxm+1

+ γeh

mX

i=1

xixm+i + γeh

m−1X

i=1

xixm+i+1 (5.134)

and likewise fory (to simplify notation we have not written an indexm onx, y andA, but these naturallyscale withβm). The identity for2m-fold Gaussian integrals

Z

exp(−xTAx) dx =πm√detA

(5.135)


Together with (5.127), (5.135) and (5.136), expression (5.132) in footnote 81, pro-vides theβm in the Gaussian approximation. One can see the considerablecomplexityof the expressions despite the simplicity of the model wavefunction. Even a numericaltreatment meets with difficulties owing to manipulations ofseries of large quantitieswhich sum to small values.αn are obtained by exact algebraic computations to freecoefficients from numerical artifacts.

1 3 5 7 9 11 13 150

1

2

3

4

-0.5

n

n

Boson

-1.95

-1.9

-1.8

-1.7

-1.5-1

-0.2Fermion

(a)

1 3 50.0

0.2

0.4

0.6

0.8

1.0

n

n

-1

-0.5-0.2

Fermion

(b)

Fig. 5.7: (a) Matrix elementsαn of the exciton creation operatorX† calculated forn ≤ 15 for variousGaussian trial wavefunctions corresponding to various sizes of the dot. The top curve shows the limit of truebosons whereαn =

√n and the bottom curve the limit of true fermions whereαn = δn,1 . Intermediate

cases are obtained for values ofγeh from −1.95√γeγh down to−0.2

√γeγh, interpolating between the

boson and fermion limit. (b) Magnified region close to the fermion limit. Values displayed are everywheregiven in units of

√γeγh.

Fig. 5.7 shows the behaviour ofαn for different values ofγeh interpolating from thebosonic case (γeh = −2

√γeγh) to the fermionic case (γeh = 0). The crossover from

The problem is now reduced to the determinant ofA, which, being a sparse matrix, also admits an analyticalsolution, though this time a rather cumbersome one. The determinant of the matrixA can be computed as:

detA = γeh2m

mX

k=0

m−kX

l=0

(−1)⌊m/2⌋+kAm(k, l)

„

γeγh

γeh2

«k

. (5.136)

Here we introduced a quantity

Am(k, l) = A′m(k, l) +

mX

i=1

`

A′m−i(k, l− i) −A′

m−i−1(k, l− i)´

(5.137)

and

A′m(k, l) =

p(l)X

η=1

(P

i νlη(i))!

Q

i νlη(i)!

×“ m− l

P

i νlη(i)

”“m − l − P

i νlη(i)

k −P

i νlη(i)

”

(5.138)

with k ∈]0,m], l ∈ [0,m] andp(l) andνη(i) already introduced as the partition function ofl and thenumber of occurence ofi in its ηth partition. For the casek = 0 the finite size of the matrix implies a specialrule which readsAm(0, l) = 4δm,lδm≡2,0.

DICKE MODEL 201

bosonic to fermionic limit can be clearly seen: forγeh close to−2√γeγh, the curve be-

haves like√n, the deviations from this exact bosonic result becoming more pronounced

with increasingn. Forγeh close to−2√γeγh, the curve initially behaves like

√n, the

deviations from this exact bosonic result becoming more pronounced with increasingn.The curve is ultimately decreasing beyond a number of excitations which is smaller thegreater the departure ofγeh from−2

√γeγh. After the initial rise, as the overlap between

electron and hole wavefunctions is small and bosonic behaviour is found, the decreasefollows as the density becomes so large that Pauli exclusionbecomes significant. Thenexcitons cannot be considered as structureless particles,and fermionic characteristicsemerge. Withγeh going to0, this behaviour is replaced by a monatonically decreasingαn, which means that it is “harder and harder” to add excitons inthe same state in theQD; the fermionic nature of excitons becomes more and more important. On Fig. 5.8is displayed a density plot of the emission spectra of the system, as function ofγ (andtherefore as function of size). A transition from the Rabi doublet to the Mollow tripletis observed.

Fig. 5.8: Density plot on logarithmic scale (to discriminate the peaks, their positions and splitting) of spectrumof emission by a QD in strong-coupling with a single cavity mode forγeh = −2

√γeγh, recovering respec-

tively the Rabi doublet (on the left) and Mollow triplet (on the right). In the intermediate region, intricate andrich patterns of peaks appear, split, merge, or disappear.


RobertDicke (1916–1997) was a versatile experimentalist, havingcontributed to the fields of radar physics, atomic physics, quantumoptics, gravity, astrophysics, and cosmology. His contribution to su-perradiance displays his aptitudes in the theoretical fieldas well. Hewrote his autobiography, dated 1975, but it has not been published (itis now possessed by NAS).

He once wrote: “I have long believed that an experimentalist shouldnot be unduly inhibited by theoretical untidiness. If he insists on hav-ing every last theoretical T crossed before he starts his research thechances are that he will never do a significant experiment. And themore significant and fundamental the experiment the more theoreti-cal uncertainty may be tolerated.”

5.6 Dicke model

Closely related to the linear coupling of the previous section lies the Dicke (1954) modelwhich yields qualitatively similar results at low densities. In this model the matter ex-citation gets upgraded to creation operatorJ+ for an excitation of the “matter field”which distributes the excitation throughout an assembly ofN identical two-level sys-tems described by fermion operatorsσi, so thatb† in Eq. (5.30) maps toJ+ with

J+ =

N∑

i=1

σ†i (5.139)

One checks readily thatJ+ andJ− = J†+ thus defined obey an angular momentum

algebra with magnitudeN(N + 1) (and maximumz projection ofJz equal toN ). Inthis case the Rabi doublet arises in the limit where the totalnumber of excitationsµ(shared between the light and the matter field) is much less than the number of atoms,µ ≪ N , in which case the usual commutation relation[J−, J+] = −2Jz becomes[J−/

√N, J+/

√N ] ≈ 1, which is the commutation for a bosonic field. This comes

from the expression of a Dicke state withµ excitations shared byN atoms given asthe angular momentum state|−N/2 + µ〉. Therefore the annihilation/creation opera-torsJ−, J+ for one excitation shared byN atoms appear in this limit like renormalisedbose operators

√Na,

√Na†, resulting in a Rabi doublet of splitting2~g

√N . Such a

situation corresponds, e.g., to an array of small QDs insidea microcavity such that ineach dot electron and hole are quantized separately, while our model describes a singleQD which can accommodate several excitons. The corresponding emission spectra areclose to those obtained here below the saturation limitµ ≪ N , while the nonlinearregimeN ≫ 1, µ ≫ 1 has peculiar behaviour, featuring non-lorentzian emission lineshapes and a non-trivial multiplet structure, like the “Dicke fork” obtained by Laussyet al. (2005).


5.7 Excitons in semiconductors

In the following chapters we pursue the investigation of light-matter coupling in boththe semiclassical and quantum regime, putting more emphasis on specificities of mi-crocavities. To bring forward these results we give now moreelements on the materialexcitations of semiconductors which parallel the exposition of the previous chapter, butfrom a quantum mechanical perspective. At this stage we shall change notations for thefields to follow popular customs, so that for instance,a which was previously a cavitymode (annihilating a photon) will now typically refer to thepolariton. The new notationswill be introduced as we go along.

The second-quantized Hamiltonian of a semiconductor at thefermionic level reads,in real space:

H =

∫

Ψ(r)†(

− ~2

2m∇2 + V (r)

)

Ψ(r) dr (5.140a)

+1

2

∫ ∫

Ψ(r)†Ψ(r′)†e2

|r − r′|Ψ(r′)Ψ(r) dr dr′ (5.140b)

whereΨ(r) is the electron annihilation field operator andV the Coulomb potential. WeexpandΨ in terms ofϕi,k(r) = 〈r|i,k〉 the single-particle wavefunction labelled by thequantum numberk in the ith semiconductor band, in term of theelectron annihilationoperatorei,k:

Ψ(r) =∑

i∈c,v

∑

k

ϕi,k(r)ei,k (5.141)

Because of interactions and correlations, the determination ofϕi,k(r) is a difficulttask, typically solved numerically. The full many-body problem (5.140) can be approxi-mated to an effective single-body problem through the so-called Hartree-Fock approxi-mation, which introduces an effective potentialVeff . The resulting “Schrodinger” equa-tion with HamiltonianHHF = −~2∇2/2m + Veff is nonlinear since the potential de-pends on the wavefunctionϕ, and so the problem remains one of considerable difficulty.Bloch’s theorem however allows a statement of general validity:

ϕi,k(r) ∝ eik·rui,k(r) (5.142)

with u having the same translational symmetry as the crystal.Once all the algebra has been gone through, the semiconductor Hamiltonian (5.140)

becomes in reciprocal space :

H =∑

i∈c,v

∑

k

Ei(k)e†i,kei,k (5.143a)

+1

2

∑

i∈c,v

∑

k,p,q 6=0

V (q)e†i,k+qe†i,p−qei,pei,k (5.143b)

+∑

k,p,q 6=0

V (q)e†c,k+qe†v,p−qev,pec,k (5.143c)


with Ei the dispersion relation for theith band andV (q) the Fourier transform of theCoulomb interaction.

The limit of very low density (in fact in the limit where the ground state is de-void of conduction band electrons) allows some analytical solutions to be obtained afterperforming approximations. One simplification, both conceptual and from the point ofview of the formalism, is the introduction of the hole (fermionic) operatorh as

hk = e†v,−k (5.144)

(spin is also reversed if granted). This allows eliminationof the negative effective massof valence electrons and deal with an excitation as an “addition” of a particle ratherthan annihilation (in terms of valence electrons, the ground state is full of electrons andgets depleted by excitations). Conceptually it replaces a sea of valence electrons by asingle particle, making it easier to conceive the exciton asa bound state. In terms ofelectronsek and holehk, Eq. (5.143) now reads

H =∑

k

[Ee(k)e†kek + Eh(k)h†khk] (5.145a)

+1

2

∑

k,p,q 6=0

V (q)[e†k+qe†p−qepek + h†k+qh

†p−qhphk] (5.145b)

−∑

k,p,q 6=0

V (q)e†k+qh†p−qhpek (5.145c)

with explicit expression for electron and hole dispersion (as a function of their effectivemass):

Ee(k) = Egap +~k2

2m∗e

(5.146a)

Eh(k) =~k2

2m∗h

. (5.146b)

In the low density limit, if one neglects line (5.145b) in theHamiltonian, it can bediagonalised by introducing theexciton operator

Xν(k) ≡∑

p

ϕν(p)hk/2−pek/2+p , (5.147)

with ϕν(p) the Fourier transform of Wannier equation eigenstates (with ν the quantumnumbers, same as for the hydrogen atom; the spectrum of energy we callEν ).

The exciton Hamiltonian becomes:

H =∑

ν,k

EνX(k)X†ν(k)Xν(k) (5.148)

withEνX(k) = Eν + Ee(k) + Eh(k) (5.149)


5.7.1 Excitons as bosons

Excitons behave as true bosons when the commutator of the field operators satisfy therelation:

[Xν(k), X†µ(q)] = δν,µδk,q (5.150)

Direct evaluation of the commutator with explicit expression (5.147) yields

[Xν(k), X†µ(q)] = δν,µδk,q −

∑

p

|ϕ1s(p)|2(c†kcq + h†−kh−q) (5.151)

so that the diagonalisation is legitimate at low densities.Especially,〈[X0, X†0]〉 =

1 − O(Na20), whereN is the density of excitons anda0 is the Bohr radius asso-

ciated withϕ1s. One can therefore treat excitons as bosons with confidence in thelimit Na2

0 ≪ 1.

5.8 Exciton-photon coupling

The polariton discussed in Section 4.4.4.2 can be seen in a simplified but very accuratemodel as the new eigenstates which arise from the coupling oftwo oscillators, to wit,the photon and the exciton. Vividly, the polariton is then seen as the chain process wherethe exciton annihilates, emitting a photon with same energyE and momentumk, whichis later re-absorbed by the medium, creating a new exciton with same(E,k), and so onuntil the excitation finds its way out of the cavity (resulting in the annihilation of thepolariton), or the electron or hole is scattered.

In this part we neglect for brevity spins and other states than 1s for the exciton. Thedipole moment−er of the electron-hole pair couples to the light fieldE and adds thefollowing coupling Hamiltonian to (5.140):

HγX =

∫

Ψ†(r)[−er ·E(r)]Ψ(r) dr (5.152)

The same procedure to obtain (5.145) from (5.140) including(5.152) leads to the fol-lowing exciton-photon coupling Hamiltonian:

H =∑

k

Eγ(k)B†kBk +

∑

k

EX(k)X†kXk +

∑

k

~g(k)(C†kak + Cka

†k) (5.153)

with ~g(k) = µcvϕ1s(0)√

Eγ(k)/2ǫ andµcv being the dipole matrix element dot-ted with electron and hole. Hamiltonian (5.153) can be diagonalised provided thatXoperators obey bosonic algebra Eq. (5.150). In such an approximation, the Hamilto-nian (5.153), bilinear in bosonic operators, is diagonalized with the so-called Hopfieldtransformation:

aLk ≡ XkXk − CkBk, (5.154a)

aUk ≡ CkXk + XkBk, (5.154b)


where the so-calledHopfield coefficientsCk andXk satisfyC2k + X 2

k = 1, so that thetransformation is canonical anda operators follow the bosonic algebra as well. As pre-viously with excitons, Hamiltonian (5.153) reduces to freepropagation terms only

H =∑

k

EU(k)aU†k aU

k +∑

k

EL(k)aL†k aL

k (5.155)

for upper and lower polariton branches, with second quantized annihilation opera-torsaU andaL, respectively. The dispersion relations for these branches are:

EUL(k) =

1

2(EX(k) + Eγ(k)) ± 1

2

√

∆2k + 4~2g(k)2 (5.156)

(theU subscript is associated with the plus sign,L with minus), whereEγ is given byexpression (4.134) andEX by (5.149), and∆k is the energy mismatch, ordetuning,between the cavity and exciton modes:

∆k ≡ Eγ(k) − EX(k) (5.157)

The Hopfield coefficients used to diagonalise this Hamiltonian are most simply ex-pressed as a function of upper polariton dispersion relation (5.156):

Ck =~ΩR

√

(EU(k) − EX(k))2 + ~2Ω2R

, (5.158a)

Xk =EU(k) − EX(k)

√

(EU(k) − EX(k))2 + ~2Ω2R

. (5.158b)

where we introduced theRabi frequencyΩR = 2g(k).

5.8.1 Dispersion of polaritons

Eq. (5.156) is one of the major results of microcavity polaritons physics for the variousconsequences this relation bears on many key issues that we are going to address in thenext chapter. It is plotted in solid lines in Fig. 4.23 on page155 where are also plottedin dashed lines the dispersions for the exciton, Eq. (5.149), and the photon, Eq. (4.134).The first and third of these figures display negative (where bare dispersions cross eachother) and positive (where they do not) detunings, respectively, while the central figuredisplays zero detuning (resonance atk = 0).

As the result of the exciton-photon interaction, there is anavoided crossing (anti-crossing) of energies. The polariton arises as a coherent mixture of the photon and ex-citon states whose fractions are given by Hopfield coefficients (5.158). As already said,the polariton is the true eigenstate of the system, whereas photon and exciton modes aretransient states, exchanging the energy at the Rabi frequency ΩR. In this simple picturethe anticrossing appears however weak the interaction. This can be made more realis-tic, taking into account the broadening of exciton and photon resonances, by adding toEqs. (5.149) and (4.134) imaginary components−iΓX and−iΓγ respectively.ΓX isthe broadening caused by exciton interactions (inter-particle or with phonons), whileΓγ


reflects the finite reflectivity which is inversely proportional to the quality factor of thecavity. At zero detuning eq. (5.156) then becomes

EUL(k) =

1

2(EX(k) + Eγ(k) − iΓX − iΓγ) ±

1

2

√

~2Ω2R − (ΓX − Γγ)2. (5.159)

This expression depends crucially on the sign of the expression below the square root,demonstrating that the physical behaviour of the system depends on the interrelation be-tween the strength of the exciton-photon coupling and dissipation. If~ΩR > (ΓX−Γγ),EU

Lexhibits the energy splitting already encountered, the so-calledRabi splittingwhich

corresponds to thestrong couplingregime, where the correlations between exciton andphoton are important and their interaction cannot be dealt with in a perturbative way. Analtogether new behaviour of the system is expected and should be described in terms ofpolaritons. The termvacuum Rabi splittinghas been introduced by Sanchez-Mondragonet al. (1983) to denote this from Rabi splitting when the oscillation is between two pop-ulated modes (rather than with the vacuum of the other excitation). This terminologyis now generally accepted, though some would refer to “normal mode splitting”. Toput an emphasis on the specificity of microcavities, the denominations “dressed excitonsplitting” or “polariton splitting” have also been used, but are now encountered only forstylistic purposes.

Note that the splitting presented so far relates to the spectrum of energy and is rathera theoretical notion. Experimentally, this splitting relates in more subtle way to split-tings in reflectivity (R), transmission (T), absorption (A)and photoluminescence (PL).Indeed this splitting is in general different in all these cases and is also different fromthe exciton-photon coupling constantg. Yet it can be shown that the general followingrelation holds:

∆ER ≥ ∆ET ≥ ∆EA (5.160)

and alsoΩ ≥ ∆EA. Only the splitting in absorption unambiguously proves strong cou-pling, while the splitting in transmission or reflectivity is a necessary but not sufficientcondition. It might therefore be the most important experimental expression, which wegive here:

∆EA =

√

Ω2 − (Γ2c + Γ2

x)

2(5.161)

along with the PL splitting

∆EPL =

√

2ΩR

√

Ω2R + 4(ΓX + Γγ)2 − Ω2

R − 4(ΓX + Γγ)2 (5.162)

so that it is possible that although in strong coupling regime (where the device wouldexhibit effects expected from polaritons), the splitting cannot be resolved from PL ob-servations. Savona et al. (1998) give an excellent and much more detailed discussion onthese points.

On the other hand, ifΩR < (γX − γγ), the square root becomes imaginary andthus the (real) energy anticrossing disappears. This is theweak couplingregime where


the system can be described in terms of weakly interacting photon and exciton. Nowthe energies are degenerate at some particulark and the broadenings (imaginary partof (5.159)) do differ. More detailed analyses show that withthe reflectivity going tozero, the broadening of the photon mode diverges (the photondoes not remain in thecavity) and the broadening of the exciton mode approaches the bare QW spontaneousemission rate, describing in effect weakly-interacting photons and excitons.

5.8.2 The polariton Hamiltonian

Now that the free-polariton Hamiltonian has been obtained,one can proceed with de-riving next-order or additional processes which will account for the dynamics of po-laritons. Such additions include polariton-polariton interactions (from the underlyingexciton-exciton interaction coming from Coulomb interaction), polariton-phonon in-teraction and possibly such terms as polariton-electron interaction if there is residualdoping, polariton coupling to the external electromagnetic field giving them a chanceto escape the cavity, a term that is ultimately responsible for decay, and its counterpartwhich inject particles, behaving as a pump. Below we providethe expression for theseterms without detailing their derivations. An excellent and thorough account is given bySavona et al. (1998).

In terms of the annihilation and creation operatorsak, a†k for polaritons andbk, b†kfor phonons (wherek is a two dimensional wavevector in the plane of the microcavity),obeying the usual bosonic algebra, our model Hamiltonian inthe interaction represen-tation reads

H = Hpump +Hlifetime +Hpol−phon +Hpol−el +Hpol−pol (5.163)

The usual models considered for these various terms are as follow:

Hpump =∑

k

g(k)(Kpumpa†k +K∗

pumpak) (5.164)

This term describes the pumping of the system by a classical light field of ampli-tudeKpump (a scalar as opposed toak which is an operator).g(k) is the wavevectordependent coupling strength between the two fields. This Hamiltonian is adapted todescribe the resonant pumping or the non-resonant pumping case depending on the ap-proximations performed.

Hlifetime =∑

k

γ(k)(αka†k + α†

kak) (5.165)

Hlifetime describes the linear coupling between the polariton field and an emptyexternal light field responsible for the polariton lifetime. What happens in reality is thatphotons escape the discrete cavity mode into a continuum from where their probabilityof return is zero. This translates as a decay.

αk, α†k are the annihilation-creation operators of the external light field.γ(k) is the

wavevector dependent coupling strength between the two fields.


Hpol−phon =∑

k 6=0,q 6=0

U(k,q)ei~(E(k)+ωq−E(k+q))ta†kbqak+q + h.c. (5.166)

describes the coupling between the polaritona and the phononb fields.U is the Fouriertransform of the interaction potential for polariton-phonon scattering.ωq is the phonondispersion,E(k) is the dispersion of the lower polariton branch.

Hpol−el =1

2

∑

k6=0,p 6=0

q6=0

U el(k,p,q)ei~

(

E(k+q)−E(k)+ ~2

2me(|p−q|2−|p|2)

)

ta†k+qake†p−qep

(5.167)describes the coupling between the polaritona and the electrone fields.ek, e†k are theannihilation-creation operators of the electron field,me is the electron mass,U el is theFourier transform of the interaction potential for polariton-electron scattering.

Hpol−pol =1

2

∑

k6=0,p 6=0

q6=0

V (k,p,q)ei~

(

E(k+q)+E(p−q)−E(k)−E(p))

ta†k+qa†p−qakap

(5.168)describes the polariton-polariton interaction.V is the Fourier transform of the interac-tion potential for this scattering.

This Hamiltonian will be a starting point for Chapters 7 and 8where various ap-proximations are made to single out the resonant dynamics where polariton-polaritoninteractions dominate, or out-of-resonance where relaxations bring the system in quasi-equilibrium with reservoirs as a function of pump and decay.In 9, an extension ofEq. (5.168) will be carried out to include the spin degree of freedom.

5.8.3 Coupling in quantum dots

Although very close in principle, cavity QED (cQED) in atomic cavities and in a solidstate system present many differences.82 A good overview of the cQED with QDs isgiven byImamo glu (2002), which provides in the abstract the main appealing featureof the solid state realisation, namely that “since quantum dot location inside the cavityis fixed by growth, this system is free of the stringent trapping requirements that limitits atomic counterpart” and further comment that “fabricating photonic nanostructureswith ultrasmall cavity-mode volumes enhances the prospects for applications in quan-tum information processing.” Experimental findings by Reithmaier et al. (2004) andYoshie et al. (2004), published as two consecutive letters to Nature magazine, and thatof Peter et al. (2005), have reached the final step of strong coupling (see next Chapter).

82In the words of Weisbuch et al. (1992):

Besides its relying on a much simpler implementation—the solid state system ismonolithic—the effect should lead to useful applications.


Although the exciton-polariton in microcavities as first observed by Weisbuch et al.(1992)—a long time ago and nowadays well understood theoretically as well as finelycontrolled experimentally—had been initially thought to provide the solid state realisa-tion, we have discussed at length how the polaritons represent in fact a more complicatedsystem. One might think that the delay to strip down this complexity to finally achievethe exact counterpart of one atomic-like system coupled to amode of radiation withoutfurther degrees of freedom was merely a technical problem, and that the theory wasalready laid down in twenty years of literature.

Following Imamoglu, this section takes the opposite view that the semiconductorcase brings forward many specificities of its own. The most important being indeed thatin the solid state case one has a much better control of the atomic-like excitation. A QDstuck in the cavity can be kept immobile while in the atomic cavities case, the excita-tions are beams of atoms with much difficulties to single out one atom or to deal withit for prolonged periods of time. In the case of semiconductors, for instance, one canexpect a much better investigation of coherent exchanges between many QDs throughthe radiation field, as is modelled by the Dicke model ofn two-level atoms interact-ing through a single mode of radiation. All variations of these, from phase-mismatchedleading to subradiant configurations where the excitation of two QDs is exchanged backand forth and does not escape, to the cooperative one—in-phase—where the excitationis collectively shared leading to superadiant emission, can be obtained by proper iden-tification of the mapping of QDs in the structure and tuning the excitation accordingly.The model developed by Dicke becomes all-important in this case. These are directionsof research which are just emerging and show how the field of solid-state microcavitieshas a significant future ahead.

6

WEAK-COUPLING MICROCAVITIES

In this Chapter we address the optical properties of microcavities in theweak-coupling regime and review the emission of light frommicrocavities in the linear regime. We present a derivationof the Purcelleffect and stimulated emission of radiation by microcavities, andconsider how this develops towards lasing. Finally we briefly considernonlinear properties of weakly-coupled semiconductor microcavities.Functionality of vertical cavity surface emitting lasers (VCSELs) is alsodescribed.

212 WEAK-COUPLING MICROCAVITIES

6.1 Purcell effect

6.1.1 The physics of weak-coupling

We are interested in this chapter in the so-calledweak-couplingof light with matter inthe sense that the effect of the radiation field can be dealt with as a perturbation on thedynamics of the system. The dynamics we have in mind is typically the spontaneousemission of the system initially in its excited state. As emphasised by Kleppner (1981)in the atomic case, the atom releases its energy because of its interaction with the vac-uum of the optical field, so that if the interaction could be “switched off”, the atomwould remain forever in its excited state. This idea is an extension of one reported muchearlier by Purcell (1946) who was seeking the opposite effect, namely to increase theinteraction so as to speed-up the release of the excitation.83 Intuitively, if the dipole isresonant with the cavity mode, the photon density of states seen by the dipole is in-creased with respect to the vacuum density of states. The spontaneous emission rate istherefore enhanced: the dipole decays radiatively faster than in vacuum and the pho-tons are emitted in the cavity mode. On the other hand, if the dipole is placed out ofresonance, namely in a photonic gap, the photon density of state seen by the dipole issmaller than in vacuum and the spontaneous emission rate is reduced. The Purcell ef-fect therefore perfectly illustrates the role played by an optical cavity which is to locallymodify the photon density of states. The control of spontaneous emission trough thePurcell effect is a way to reduce the threshold of lasers and the effect has been activelylooked for with atoms placed in cavities, for instance by Goyet al. (1983), and morerecently with quantum dots placed in micropillars, micro-disks or photonic crystals, forinstance in the work of Gerard et al. (1998). We review in more details the experimentalrealisations later.

The above description of the emission neglects re-absorption. For dipoles in freespace, one can easily believe intuitively that this effect is weak. In fact it is responsi-ble for the energy shift known as theLamb shiftwhich is indeed orders of magnitudesmaller than the radiative broadening. In quantum electrodynamics, this shift is inter-preted as the influence of virtual photons emitted and re-absorbed by the dipole. Thesituation changes dramatically when the dipole is placed ina cavity. Photons emittedare then reflected by the mirrors and remain inside the cavity. This increases the prob-ability of re-absorption of the photons by the dipole. If theconfinement is so good thatthe probability of re-absorption of a photon by the dipole islarger than its probabilityof escaping the cavity, the perturbative weak coupling regime breaks and instead theso-calledstrong couplingtakes place. This means that the eigenmodes of the coupleddipole-cavity system are no more bare modes but mixed light-dipole modes. Their ener-gies are strongly modified with respect to the bare modes. Thestrong coupling regimeis addressed in details in Chapters 7 and 8, but in discussingweak-coupling, one should

83Purcell was motivated by a practical goal in nuclear magnetic resonance: to bring spins into thermalequilibrium at radio frequencies, with relaxation time forthe nuclear spin in vacuum of order of1021s. Hecalculated that the presence of small metallic particles would, thanks to the Purcell enhancement, be reducedto order of minutes. The much-quoted reference which records this landmark of QED—E. M. Purcell, Phys.Rev.,69, 681, (1946)—is actually a short proceedings abstract.

PURCELL EFFECT 213

bear in mind that strong-coupling is regarded as implying richer physics from the fun-damental point of view84 and that it is typically harder to obtain than the weak-couplingwhich “historically” precedes it in a given system until enough control of the system andits interactions are reached to give preponderance to the quantum hamiltonian dynamicsover the dissipations.

6.1.2 Spontaneous emission

We now give a mathematical derivations of the above ideas, relating spontaneous emis-sion with weak (or perturbative) coupling to the optical field. Our starting point isEq. (3.72) for the electric field operator which we write herein the dipolar approxi-mation and for a given state of polarisation:

E =∑

k

uk

√

~ωk

2ǫ0L3ek(a†k + ak) (6.1)

We recall that photon modes are resulting from the quantization of the electromag-netic field in a box of sizeL. From there we go to the Jaynes-Cummings model (seeSection 5.5) for the multi-mode field operator Eq. (6.1), which turns the coupling termEq. (5.91) into, written directly in the interaction picture:

V =∑

k

√

~ωk

2ǫ0L3(ek · d)e−i(ω0−ωk)t(a†kσ + σ†ak) (6.2)

We consider as initial state the atom in exited state and all photon modes empty,|e, 0k1, 0k2 , · · · , 0kn , · · · 〉. The final states are states with the atom in the ground stateand one photon in one of the final states,|g, 0k1, 0k2 , · · · , 1km , · · · 〉. We also take intoaccount only the terma†kσ of the Hamitonian which destroys the atomic excitation andcreate a photon. The reverse processakσ

† is neglected. This means that we assume thatthe photon is escaping quickly far away from the atom and cannot be reabsorbed. Thematrix element between the initial state and one of the final state therefore reads:

Mkn = 〈e, 0k1, 0k2 , · · · , 0kn , · · · |H |g, 0k1 , 0k2, · · · , 1kn , · · · 〉 =

√

~ωk

2ǫ0L3(ek · d)

(6.3)One can then apply the Fermi golden rule which stands as (we note “at” for atom):

Γat0 =

2π

~

∑

k

|Mk|2δ(E0 − Ek) (6.4)

where the sum is on the set of final states, which becomes a continuum asL→ ∞. Wenow pass to the thermodynamic limit, making the size of the system go to infinity. Thesum is replaced by an integral using the rule:

84It is not so obvious from the experimental point of view that strong coupling is intrinsically better thanweak coupling; all the physics of lasers that is addressed later in this Chapter pertains to the weak coupling.


∑

k

→ 2(

2πL

)3

∫ 2π

0

dφ

∫ π

0

sin θdθ

∫ ∞

0

k2dk (6.5)

The factor 2 in the numerator stands for the two different polarisations,ek · d =d cos θ with θ the angle between the dipole axis and the electric field. Thisgives:

Γat0 =

c

4π2ǫ0d2

∫ 2π

0

dφ

∫ π

0

sin θ cos2 θdθ

∫ ∞

0

δ(E0 − Ek)k3dk (6.6a)

=c

3πǫ0d2

∫ ∞

0

δ(E0 − Ek)k3dk . (6.6b)

We then usek = E/(~c) which yields:

Γat0 =

1

3πc2~3ǫ0d2

∫ ∞

0

δ(E0 − Ek)E3dE (6.7)

which finally gives the usual formula for the emission of an atom in the vacuum:

Γat0 =

ω30

3π~c3ǫ0d2 (6.8)

An approach involving the dynamics—also based on the JaynesCummings Hamil-tonian and known asWigner-Weisskopf theory—also assumes that at timet = 0 theatom is in the excited state and the field modes are in the vacuum state. The state vectortherefore reads:

|ψ(t)〉 = ce(t) |e, 0k〉 +∑

k

cg,k(t) |g, 1k〉 (6.9)

with initial time amplitude of probabilities given byce(0) = 1 andcg,k(0) = 0.We now determine the state of the atom and the state of the light field at some

later time, when the atom starts to emit. We therefore writesthe Shrodinger equa-tion (∂/∂t) |ψ(t)〉 = −(i/~)V |ψ(t)〉 with V given by Eq. (6.2). Projecting this equa-tion on the different basis vectors, one gets the equations of motion for the probabilityamplitudes:

ce(t) = −i∑

k

Mkei(ω0−ωk)tcb,k(t) , (6.10a)

cb,k(t) = −iMkei(ω0−ωk)tce(t) . (6.10b)

We formally integrate Eqs. (6.10b) and substitute the result in Eq. (6.10a) yielding:

ce(t) = −∑

k

|Mk|2∫ t

0

ei(ω0−ωk)(t−t′)ce(t′)dt′ . (6.11)

This expression is still exact. We now perform the so-called“Wigner-Weisskopf”approximation which amounts to the following substitution:

PURCELL EFFECT 215

∫ t

0

ei(ω0−ωk)(t−t′)ce(t′)dt′ ≈ ce(t)

∫ ∞

0

ei(ω0−ωk)t′dt′ (6.12)

This approximation is also known more generally as aMarkov approximation, whichwe shall use repeatedly in next chapters to derive kinetic equations. This approximationconsists physically in neglecting memory effects, that is,to make the time evolutionof a quantity depend on its value at the present time and not onits history (its valuesin the past). Mathematically, it consists in replacingce(t′) by ce(t) in the integral ofEq. (6.12). The second approximation which is performed is to replace the upper boundof the integral by infinity. This approximation is justified if ω0t ≫ 1. We now useCauchy formula, which is:

1

2π

∫ ∞

0

eiωttd =1

2

(

δ(ω) − 1

iπP( 1

ω

)

)

(6.13)

whereP stands for the principal value of the integral.85 Eq. (6.12) becomes

ce(t) =

(

−1

2Γ0 + i∆ω

)

ce(t) (6.14)

whereΓ0 = (2π/~)∑

k |Mk|2δ(E0 − Ek) has the same expression as in Eq. 6.7, and

∆ω = −1

~

∑

k

|Mk|2P(

1

E0 − Ek

)

. (6.15)

This last quantity is the Lamb shift. It is, as already discussed, the renormalisationof the frequency of emission of the atom induced by the re-absorption of the light by theatom after its initial emission. This frequency shift can becalculated using the Wigner-Weisskopf approach whereas it is naturally absent from derivations based on the FermiGolden Rule. This shift is however usually small.

6.1.3 The case of QDs, 2D excitons and 2D electron-hole pairs

This aspect has been treated at length in Chapter 4 and we merely discuss here howthe main results compare with respect to the case of atoms. The QD Hamiltonian andthe procedure which can be used to calculate the decay of a QD excitation is exactlysimilar than the one we have just detailed. The only difference comes from the shapeof the matrix element of coupling.The situation is slightlydifferent for QWs excitonsfor which the decay has been found in Chapter 4 as a solution ofMaxwell equations.The operators describing excitons are bosonic and not fermionic as for atoms. This hashowever no impact on the result since we are dealing with occupation numbers smallerthan one. An other important difference is that a QW exciton with a given in planewavevector is coupled to a continuum of states in a single direction of the reciprocal

85See footnote 8 on page 31.


space, instead of three for atoms or QDs. In the framework of the Fermi Golden rule,the decay rates of a QW exciton having a null wavevector in theplane reads:

ΓQW0 =

2π

~

∑

kz

|MQWkz

|2δ(E0 − Ekz ) (6.16)

with the matrix element of interaction between photons and the QW exciton now being

MQWkz

=µcv

2πa2DB

√

Ekz

ǫLopt. (6.17)

Going to the thermodynamic limit as before, this gives:

ΓQW0 =

2nL

h2c|MQW

E0|2 =

nµ2cvE0

4π2h2cǫ(a2DB )2

. (6.18)

For electron hole pairs in a quantum well the matrix element of interaction simplyreads:

M ehkz

= µcv

√

Ekz

2ǫL3(6.19)

which, with the same procedure as above, yields:

Γeh0 =

nµ2cvE0

h2cǫL2. (6.20)

6.1.4 Fermi’s golden rule

Both inhibition and enhancement of the spontaneous emission are essentially related toFermi’s golden rule.86 For the case of an electric dipoled interacting at pointr andtime t with the light fieldE(r, t), the spontaneous emission rate for an emitter withenergy~ωe, reads

1

τ=

2π

~2|d · E(r, t)|2ρ(ωe) (6.22)

with ρ(ωe) the photon density of states at the energy~ωe of the emitter. In the vacuum,it is given by

ρv(ω) =ω2V n3

π2c3(6.23)

86Fermi’s golden rule states that if|i〉, |f〉 are eigenstates of an HamiltonianH0 subject to a perturba-tionH(t), the probability of transition from an initial state|i〉 to a continuum of final states|f〉 is given bythe formula:

1

τ=

2π

~2| 〈f |H′ |i〉 |2ρf (6.21)

with ρf the density of final states. IfH′ is time independent,ρf becomes an energy-conserving delta function.

PURCELL EFFECT 217

ΡcHΩL ΡvHΩL

dot A dot B

Fig. 6.1: Density of states of the vacuum,ρv (thick dashed line), and of a cavity sin-gle mode (thick solid),ρc, as function ofthe frequencyω. The lines of two emitters,e.g., quantum dots, are sketched in a con-figuration where the emitter is in resonancewith the cavity (A) with an enhanced prob-ability of emission into the cavity mode, orstrongly detuned with no final state to de-cay, resulting in an increase of its lifetime.

In a single-mode cavity, however, with energyωc and quality factorQ, the densityof states becomes a Lorentzian:

ρc(ω) =2

π

∆ωc

4(ω − ωc)2 + ∆ω2c

(6.24)

Both densities are sketched on Fig. 6.1. The localisation ofmodes available for thefinal state (decay) of the emitter around the cavity mode allows to enhance (resp. inhibit)spontaneous emission by tuning (resp. detuning) the emitter with the cavity mode. As aresult of its reduced (resp. increased) lifetime, the line gets correspondingly broadened(resp. sharpened). When the characteristic emission time given by Eq. (6.22) for thevacuum and cavity case are compared, one gets:

Γc

Γ0=

3Q(λc/n)3

4π2Veff

∆ω2c

4(ωe − ωc)2 + ∆ω2c

|E(r)|2|Emax|2

(

d · E(r)

dE

)2

(6.25)

Eq. (6.25) is the central equation behind the Purcell effect. It puts neatly together allthe modifications that the cavity forces on the lifetime of anenclosed emitter:

• The term3Q(λc/n)3/(4π2Veff) depends only on parameters of the cavity, itsquality factorQ, wavelengthλc, refractive indexn and effective volumeVeff .As such it is afigure of meritof the cavity, which quantifies the efficiency ofPurcell enhancement for an ideal emitter coupled in an idealway to the cavity. Itis this quantity which appears in Purcell et al.’s (1946) seminal paper, in honourof whom it is now called thePurcell factor:

FP =3Q(λc/n)3

4π2Veff(6.26)

• The term∆ω2c/[

4(ωe − ωc)2 + ∆ω2

c

]

comes from the density of states of a singlemode in Fermi’s golden rule formula. It shows the effect of the detuning on theefficiency of the coupling, on which it acts through a phase-space filling effect.As this quantity is smaller than one, it contributes always towards inhibition ofemission. In this approach fully focussed on the Purcell effect, it means that thelifetime of the dot can be made arbitrarily long by detuning it farther from the


Fig. 6.2: Time resolved photoluminescence of a quantum dot when placed, (a) in the bulk of the material(GaAs matrix), and (b&c) in a pillar microcavity of the same material, as observed by Gerard et al. (1998).In the case (b) the dot is in resonance with the single mode of the cavity and decays about five times quickerthan when it is coupled to a continuum of modes, case (a). In case (c) the dot still inside the cavity is detunedwith the mode and as a result display only a small enhancementof its lifetime as compared to the bulk case.

cavity. In a real situation, there are other modes to which the emitter couples aswell, especially leaky modes which act as a dissipation. Forinstance emissionthrough the sides of a pillar put a limit to emission inhibition, as observed byBayer et al. (2001) who could partially enhance the situation by coating the sides.Finally, there is always a channel of nonradiative decay, which adds a constantterm to Eq. (6.25) (as is the case in Eq. (1.5) on page 5).

• The term|E(r)|2/|Emax|2 which underlines the importance of fluctuations in thesystems. In the solid state case, a given sample is more stable than its atomiccounterpart as the dot stays fixed at the same position. However the possible con-figurations change from sample to sample and it can require a lot of trials anerrors until a good sample is found where the location, oscillator strength andother properties of the dot are suitable to display nontrivial physics (this is spe-cially true when trying to achieve strong-coupling, as we shall see in the nextchapters). The theoretical fit in Fig. 6.2 for instance were made by averaging overEq. (6.25).

• The last term(d ·E)2/(dE)2 which in some cases or for some authors would fallin the same category as above, as adding an element of randomness weakening theoptimal Purcell enhancement as quantified by the Purcell factor. However recentresults such as those obtained by Unitt et al. (2005) indicate that a determinis-tic pining of the dipole along crystallographic axis allowspolarization-selectivePurcell enhancement.

The Purcell enhancement of spontaneous emission is neatly demonstrated on Fig. 6.2where emission is released according to curve (b) in resonance or (c) out of resonance,displaying much quicker decay in the former case as opposed to the later which is es-sentially the same as without mode coupling (curve (a)).

PURCELL EFFECT 219

An important issue is dimensionality. An atom and its semiconductor counterpart,the quantum dot, is naturally emitting spherical light waves. In a plane wave repre-sentation of light, this means that an atom is intrinsicallycoupled to all possible planewave directions. The decay of such system will be well described by the application ofFermi’s Golden rule. On the other hand, the achievement of Purcell effect or of strongcoupling will require to confine light in the three directions of space, namely to usea three dimensional optical cavity. The case of bulk excitons considered at length inChapter 4 is radically different. In the bulk, an exciton is aplane wave characterised bya well-defined wavevector. This exciton is coupled to a single photon state. The strongcoupling is achieved and eigenstates of the system are exciton-polaritons which only de-cay because the photon ultimately escapes through the edgesof the sample. In QWs thesituation becomes similar again to the atom/dot case. Indeed the transitional invarianceis broken in one direction and kept in the two others. In thesetwo last directions, theone-to-one coupling holds whereas in the third direction, the QW exciton is coupled toa continuum of photonic modes. QW excitons therefore radiatively decay in this direc-tion. The achievement of Purcell effect or of strong coupling in this latter case requiresthe use of a planar microcavity which confines the light in thedirection perpendicularto the QW.

6.1.5 Dynamics of the Purcell effect

We now investigate a toy model illustrating the Purcell effect from a quantum dynam-ical point of view. We consider the case of an atom placed within a tri-dimensionaloptical cavity of volumeVc = L3

c. The fundamental mode of this cavity has resonanceenergy~ωc and is characterised by aQ factorQ = ωc/Γc whereΓc is the width of thecavity mode. We remind that a picturesque definition of the quality factorQ is that itnumbers how many round trips light makes in the cavity beforeleaking out. Formally,the atom is now coupled to a single decaying mode. The Fermi Golden rule cannotbe used anymore. We describe the coherent coupling of the atom with a single cavitymode which is dissipatively coupled to a bath of external photon modes. The system weconsider is sketched on Fig. 6.3.

Fig. 6.3: Sketch of the physics involved in the coupling of one atom in a leaky cavity mode: there is coherentcoupling between states|e, 0〉 and|g, 1〉 through Jaynes-Cummings dynamics, and dissipative coupling to areservoir, bringing the system to the vacuum|g, 0〉. These three states form a basis of states called|1〉, |2〉and|3〉, respectively.

We write the atom-light states involved in coherent/dissipative couplings of the firstmanifold as follow:


|1〉 = |e, 0〉 , |2〉 = |g, 1〉 , and |3〉 = |g, 0〉 . (6.27)

With these notations, the density matrix of the system reads:

ρ =

3∑

i=1

3∑

j=1

ρi,j |i〉〈j| (6.28)

The equation of motion for the density matrix taking into account Lindblad dissipa-tion of Section 5.4 (now for the atom case) gives, in this special where the Hilbert spaceis spanned by the basis|1〉 , |2〉 , |3〉:

∂

∂tρ = − i

~[H, ρ] + Lρ (6.29)

with,

H =

~ω0 M 0M ~ωc 00 0 0

and L =

0 − ωc

2Q 0

− ωc

2Q −ωc

Q 0

0 0 ωc

Q

. (6.30)

It follows for the matrix elements:

ρ11 = iM

~(ρ12 − ρ21) , (6.31a)

ρ22 = −iM~

(ρ12 − ρ21) −ωc

Qρ22 , (6.31b)

ρ12 − ρ21 = iM

~(ρ11 − ρ22) +

[

i(ω0 − ωc) −ωc

2Q

]

(ρ12 − ρ21) , (6.31c)

ρ33 =ωc

Qρ22 . (6.31d)

The Fourier transform of this linear system gives the equations determining theeigenfrequencies, reminiscent of the renormalisations already encountered several timesthroughout this book. They read:

ω± =1

2

(

ω0 + ωc + iωc

Q±√

(

ω0 − ωc − iωc

Q

)2

+ 4(M

~

)2)

(6.32)

One can see that in the general case, the real and imaginary parts of the modes areboth renormalized with respect to the case of the bare states. We now analyse someparticular cases.

First we consider the resonance between the dipole and the cavity mode:ω0 = ωc. Inthat case two different regimes take place, depending on thesign of the quantity whichis below the radical. IfωC/Q > 2M/~, the square root in Eq. (6.32) is imaginary. Thetwo eigenmodes have the same real part but two different imaginary parts, one larger

PURCELL EFFECT 221

and one smaller thanωc/(2Q). This regime is the weak coupling. Ifωc/Q ≫ 2M/~,the expansion of the square root gives

ω+ ≈ ωc + iQ

ωc

(M

~

)2

= ωc + iΓc , (6.33a)

ω− ≈ ωc(1 + i/Q) . (6.33b)

One mode has the energy and decay of the bare cavity mode. The other mode hasthe same energy as the bare mode but the decay constant is enhanced by the presence ofthe cavity, which is the manifestation of the Purcell effect. Using the explicit expressionfor M , the decay of an atom is usually given as:

Γc =Q

ωc

(

M

~

)2

= FPΓ0 (6.34)

whereΓ0 is the free atom decay andFP is the Purcell factor given by Eq. (6.26).One can see thatFP—the Purcell enhancement factor—is proportional to theQ

factor of the cavity and is maximal for small cavity volumes which maximise the overlapbetween the confined photon mode and the atom. Physically, this means that the atomwill emit light only in the cavity mode and will decay much faster than in the vacuum.

On the other hand, ifωc/2 < 2M/~, the radical of Eq. (6.32) is real. In this case, thetwo eigenmodes have the same imaginary part but two different real parts. This regimeis the strong coupling. A photon emitted by the atom does not tunnel out but is virtuallyreabsorbed and re-emitted several times by the atom before it takes a chance to leak out.The eigenmodes of the system are therefore mixed light-matter modes. In the limitingcase whereωc/Q ≪ 2M/~, the splitting between the two eigen frequencies is givenby 2M/~. The strong coupling regime is presented in details in the next Chapters.

In the strong off-resonance case, when|ω0 − ωc| ≫ M/~ and also|ω0 − ωc| ≫ωc/Q, the square root of Eq. (6.32) can be developed keeping only terms contributingto the imaginary part:

ωc

Q±√

(

ω0 − ωc − iωc

Q

)2

+ 4(M

~

)2

≈ (6.35a)

(ω0 − ωc)

(

1 − i

(

ωc

Q(ω0 − ωc)2+

(

4(M/h)2 − (ωc/Q)2)

(ωc/Q)

2(ω0 − ωc)3

))

(6.35b)

In contrast with the other case, the photonic mode thereforeremains unperturbedwhereas the emission from the atom is strongly inhibited.

6.1.6 Case of QDs and QWs

The formalism presented above to calculate the Purcell enhancement factor is valid forany single mode emitter coupled to a single optical mode. Thevalue achieved by theenhancement factor only depends on the dimensionality. Hence the enhancement factorof a QD in a 3D cavity is the same as the one calculated above foran atom. The case of


QWs exciton and electron-hole pairs is however different. We use the formula and wereplace the optical quantization lengthL of the equation by the cavity thicknessLc, andwe find:

F 2DP =

1

4π

λc

nLcQ . (6.36)

This formula has the same form as its zero dimensional counterpart. In usual micro-cavities, the cavity length is of the order of the wavelengthwhich makes an enhancementfactor of the order ofQ/(4π).

Our previous analyses showed that the effect depends on the value achieved by thematrix element of interaction which itself is composed of anintrinsic part (used in theHamiltonian) and of a part induced by the non-ideal overlapping between the cavitymode and the dipole mode. It is easy and instructive to consider as an example the im-pact of the position of a QW in a planar-microcavity. For a cavity surrounded by perfectmirrors, the amplitude of the electric field alongz goes likeE(z) = E0 cos(2πz/L).This is valid if the well is placed at the maximum of the electric field. Otherwise oneshould go back to the formula replacingM by M cos(2πz/L). The Purcell enhance-ment factor therefore becomes:

F 2DP (z) =

1

4π

λc

nLcQ cos2

(2πz

L

)

. (6.37)

One can see that an inappropriate placement of the well inside the cavity can lead toa strong decrease of the Purcell factor and even to a completesuppression of emission.In practice this constraint is not that strong in planar microcavities where a very goodcontrol of the QWs position can be achieved. This constraintis however much strongerfor QDs in 3D cavities. Indeed the position of QDs is only veryweakly controlled bycrystal growth techniques and the quality of overlapping between an optical mode anda QD is often governed by chance, so far.

6.1.7 Experimental realisations

Fig. 6.4: (a) Atom-cavity experiment from Heinzen et al. (1987) for Purcell modification of (b) spontaneousemission with cavity open (i) or blocked (ii).

PURCELL EFFECT 223

Experimental realisations of the Purcell effect have been held back because typicalstrong dipoles emit at high rates (lifetimes of order 1ns) which implies that highQ-factors and small cavities (cavity lifetimes≃ Qc/L) are needed to perturb the photondensity of states (DoS) sufficiently.

For atoms in an optical microcavity, this required the development of cold atomicbeams and suitable continuous-wave tuneable lasers. The first realization of the Purcellexperiment in a larger macrocavity (L = 5cm) by Heinzen et al. (1987) showed afew per cent change in the emission rates, primarily due to the low solid angle in theconfocal cavity mode (Fig.6.4). In the same year, experiments by Jhe et al. (1987) inthe near infrared on plane-plane Au-coated microcavities separated by1.1µm showedthat theµs decay times of Cs atoms could be increased by 60%. Also in the same year,de Martini et al. (1987) demonstrated the same effect for dyemolecules flowing in asolution between two plane-plane dielectric mirrors whosetuneable separation (downto below 100nm) produced photoluminescence lifetime enhancements of up to 300%.

Fig. 6.5: Optically pumped dielectric DBR planar microcavity laser with flowing dye

More recently, similar effects for semiconductor quantum dots embedded in micro-cavities have been demonstrated by Bayer et al. (2001). In 5µm GaAs microdisks theemdedded InAs quantum dot lifetimes could be 300% longer, and corresponding effectshave been seen with quantum dots in 1-20µm diameter micropillars. The difficulty ofgreatly increasing the spontaneous lifetimes in semiconductors is the extra contributionfrom other non-radiative recombination processes, which in general also increase whenpatterning devices into optical microcavities. As a result, such experiments have beencarried out at low lattice temperatures,T ∼ 4K reducing the excited phonon modeoccupations of the solid.

Subsequent improvements in this type of experiment have allowed emission experi-ments on single ions or single semiconductor quantum dots inside microcavities. How-ever the Purcell factors have not been greatly improved, andit remains hard to controlprecisely the spatial position of the emitter within the microcavity—in order to haveslowest emission, the emitter should be in a field minimum. More recent experimentshave shown that photonic crystal defect cavities with semiconductor quantum dots havegreat potential for realising the Purcell effect. This is mainly because the mode volumeis so small that the cavity linewidth is much smaller than thelinewidth of the electronictransition at room temperature, implying that the number ofphoton round trips beforephase scattering inside the solid is what controls the emission characteristics.


atoms

empty

Fig. 6.6: Transmission spectra (around the 3P-3S transition at 589nm) through aL=1.7mm cavity with finesseof 20000 which is (a) empty, and (b) contains sodium atoms at zero detuning∆=0, from (Raizen et al. 1989).

In all the above examples, the cavity-emitter system is in the weak coupling regime.However, it is possible to increase the coupling strength between light and matter suf-ficiently that the normal modes of the system become polaritons (Sections 4.4 & 5.8).The first observations of this so called “vacuum-Rabi splitting” for atomic systems in1989 (Fig.6.6) by Raizen et al. (1989) showed that the atomicdecay rate was halvedon resonance, because the atoms spent half their time in the de-excited state with theenergy in the cavity photon field. Because of the small optical cross section for atoms,much improved cavity finesse was needed before. McKeever et al. (2003) demonstratedemission and lasing from a single atom in the strong couplingregime. Recently in 2004,three realizations were achieved for strong coupling usingsingle semiconductor quan-tum dots: in photonic crystal defect microcavities by Yoshie et al. (2004), in whisperinggallery microdisks by Peter et al. (2005) and planar micropillars byReithmaier et al.(2004) (see Fig. 6.7).

Further progression on such experiments demands the ability to both spectrally tunethe microcavity resonant mode and dipole transition frequency independently, and alsoto control their spatial overlap. A current advantage of solid implementations comparedto atom systems over and above their utility and portabilityis the fixed embedding ofthe emitters—atoms move around or need trapping, and also have to be injected in abeam, hence restricting how much the cavity can be shrunk.

6.2 Lasers

The laser is a direct application of Einstein’s theory of light-matter interaction based ontheA andB coefficients that have been presented in section 4.1.2. Prokhorov (1964)discusses in his Nobel lecture why this “obvious” application—the principle of whichhad been realised by many—took so much time for its realization, and through the skillsof the radio-wave community rather than from optics. The main principle of generat-ing gain in an inverted population by overcoming absorptionwith emission thanks tostimulation was however clear. Moreover, the wide range of possible types of lasers,including semiconductors, was realised early on. In fact inthe Nobel lectures accom-

LASERS 225

Fig. 6.7: (a) Electron micrograph of microcavity pillar containing quantum dots from Reithmaier et al. (2004),with (b) emission spectrum from the single quantum dot tunedacross the cavity resonance by changing thetemperature. (c) Photonic crystal single quantum-dot microcavity from Yoshie et al. (2004) and (d) microdiskmicrocavity from Peter et al. (2005).

panying that of Prokhorov the emphasis is made simultaneously and equally both foratoms by Townes (1964) and for semiconductors by Basov (1964). Those main prin-ciples having to do with the interplay of gain and losses are presented now. Later weturn more specifically to the semiconductor laser, which wasrapidly demonstrated asBasov envisioned it showing that the main ideas were sound, but quickly required in-genious manufacturing to operate efficiently. The description of these elaborations andspecificities are discussed next.

6.2.1 The physics of lasers

Einstein’sA andB coefficients are fundamental parameters of the system. The equilib-rium considerations section 4.1.2 served as a useful particular case to investigate them,but they still apply out-of equilibrium when the system is excited or driven in someway. In the following for simplicity of notations we consider non-degenerate cases sothatB12 = B21 which we will denote simplyB.

If an excited atom can be “induced” to emit a photon by anotherphoton, there is thepossibility to start a chain reaction in a population of inverted atoms with each additionalphoton stimulating another one, in turn stimulating other photons. Quantitatively, in a


CharlesTownes (b. 1915), NikolaiBasov (1922–2001) and AleksandrProkhorov (1916–2002), the 1954Nobel Prize in Physics for “fundamental work in the field of quantum electronics, which has led to the con-struction of oscillators and amplifiers based on the maser-laser principle”.

Townes gained expertise with microwaves techniques from his design of radar bombing systems during theWorld War II. From spectroscopy he gradually went on to develop the concept but also the physical realisationof the maser (he also gave the device its name). In the late 50s, he discussed with Gordon Gould, then a Ph.D.(that was never completed), about theoptical maser, nowadays known under the term that was used for thefirst time by Gould: thelaser. Gould referred to his knowledge of optical pumping to bringthe maser into theoptical window, while Townes and his brother-in-law, Schawlow, were undertaking important founding workon the topic. This sparked one of the most famous patent fight in history which lasted over thirty year andfinally saw Gould victorious over Townes’ design, deemed by court as not eventually working. Gould alsowon his court battles against laser manufacturers and became a multimillionaire from royalties of his patents.Controversy, especially sparked by Townes and Schawlow, still surround Gould as the inventor of the laser.

Prokhorov was Basov’s Ph. D. advisor. They both served in thered army during the war (Prokhorov beingwounded twice) and earned many distinctions from the Sovietunion. Prokhorov was chief editor of the GreatSoviet Encyclopedia since 1969. They made breakthrough in the maser effect simultaneously and indepen-dently from Townes, using a cavity reflecting light at both ends to amplify a microwave beam. Their workdisplays an harmonious mastery of both experimental and theoretical treatments.

population ofn = n1 +n2 atoms (per unit volume),n1 of which are in the ground stateandn2 in the excited state, one hasn1BI photons absorbed by the atoms (withI thephoton energy density) andn2BI emitted by stimulation, henceδnBI photons numberof photons gained per second and per unit volume, withδn = n2 − n1. The numberof photons per unit area exchanged (emitted or absorbed) with those already presentand accounting for a fluxφ in the little distance∆z is δnBI∆z. Going to the limit ofinfinitesimal distance (and flux)dφ = δnBIdz which integrates as a function ofz as

φ(z) = φ(0) exp(δnIBz) . (6.38)

According to the sign ofδn, Eq. (6.38) will display either exponential attenuationor amplification. The conditionδn > 0 is calledpopulation inversionand is realisedwhen there are more atoms in the excited than in the ground state.

The amplification by stimulation results from the interplayof emission and ab-sorption, the former having spontaneous emission in addition to the stimulated chan-nel which in other respects is similar to absorption. To quantify this, we defineIs theradiation energy density which equates spontaneous emission and stimulated emission:

A = IsB (6.39)

LASERS 227

The detailed balanced of the rate equationdn1/dt = −dn2/dt = n2A + (n2 −n1)IB becomes at equilibrium and in terms ofIs:

n2Is + (n2 − n1)I = 0 (6.40)

solving for atomic populations:

n1 =Is + I

Is + 2In and n2 =

I

Is + 2In (6.41)

with n = n1 + n2. These equations show thatn2 < (1/2) < n1 for all values ofI: itis impossible to have more atoms in the excited state than in the ground state by directpumping as the result of the intrinsic nature of light-matter interactions which have achannel ofstimulateddecay in addition of the intrinsic or spontaneous one. The moreintense is the radiating field to excite atoms on the one hand,the more it stimulatesdecay of the atoms already excited on the other.

This shows that atwo-level system, that is, one with only two populations of atomicstates, cannot lase: arbitrary high excitation will only approach the equal populationconfiguration (and displaysoptical transparencyas a beam will cross the material with-out being absorbed or emitted). It is however possible to obtain inversion of populationif there are other transient states. For instance if the finalstate1 of laser radiation haszero lifetime and is always empty, decaying toward the ground state0, however littleis the population of an excited state2 which decays into1, it will realise an inversionof population. Amplification is therefore obtained by turning to at least three-level sys-tems, as we show now.

Consider a population ofn atoms with possible energy states 0, 1 and 2 with en-ergiesE0, E1 andE2 respectively, such thatE0 < E1 < E2. Population of statei isdenotedni.

τ sp

τ nr

τ21

τ 20

τ 1

τ2

1

2

R2

R1

W

Fig. 6.8: Sketch of transition rates of a three-level systemsuitable for amplification by stimulated emission.The lasing transition is 2–1. Level 2 has decay channels intolevel 1 labelled sp for spontaneous emissionand nr for all other nonradiative transitions, and decay channel into the ground state 0 labelled 20. Theyall contribute to a total lifetimeτ2 for this level. Other such transitions are labelled in the same way. Alsomentioned are the pumping ratesR2 populating the excited state andR1 depopulating it. Finally, when theradiation field starts to become important, stimulated emission and absorptionW also enter the picture.


The transition of interest remains the lasing transition, shown in Fig. 6.8 with energyE2 − E1. Level 2 decays into mode 1 with associated lifetimeτ21 and into the groundstate with lifetimeτ20. If there are many channels of decay for a given transition, typi-cally one decay by spontaneous emission with characteristic timeτsp and a nonradiativedecay with timeτnr, then the total lifetime builds up as follows:

τ−121 = τ−1

sp + τ−1nr (6.42)

Without pumping the system quickly decays into its equilibrium state with all atomsin the ground state. Efficient and typical pumping schemes involve pumping the ex-cited state of the laser transition at a rateR1 and depopulating it at a rateR2. The rateequations taking into account these transitions only, are,by definition of the quantitiesinvolved:

dn2

dt= R2 −

n2

τ2(6.43a)

dn1

dt= −R1 −

n1

τ1+n2

τ21. (6.43b)

The steady state solution is readily found in this case and the population differ-enceδn = n2 − n1 reads

δnss = R2τ2

(

1 − τ1τ21

)

+R1τ1 (6.44)

subscript “ss” means “steady state”. Large values ofδnss (and therefore high values ofoptical gain) are obtained when the pumping ratesRi are high and whenτ2 is large so asto build up a high population of excited states relatively tolevel 1, and with small valueof τ1 if R1 < (τ2/τ21)R2. Otherwise the population of level 1 becomes detrimental andit is better if the level is quickly depopulated thanks to theshort lifetime to compensatean inefficientR1.

When the radiation field builds up it triggers transitions between levels 1 and 2by absorption (depopulating 1 for 2) and stimulated emission (inducing the oppositetransition from 2 to 1). The equations now become:

dn2

dt= R2 −

n2

τ2− n2W + n1W (6.45a)

dn1

dt= −R1 −

n1

τ1+n2

τ21+ n2W − n1W. (6.45b)

Observe that the new termδnW cancels in the sum of Eqs. (6.45a). The populationdifference for this case is also obtained readily in the steady state:

∆nss =δnss

1 + τsW(6.46)

whereδnss is the population difference in absence of the radiation field, Eq. (6.44); andτs is the so-calledsaturation time constant:

LASERS 229

τs = τ2 + τ1

(

1 − τ2τ21

)

. (6.47)

Of courseδnss and∆nss coincide whenW → 0.

Exercise 6.1 (∗) Show thatτs is a well behaved time which is always positive. As aresult show that

∆nss ≤ δnss . (6.48)

Eq. (6.47) shows a very important aspect of light-matter interactions as the resultof Einstein processes which is displayed in Fig. 6.9. Stimulated emission is a desirableeffect for amplification (and its coherent properties as theresult of cloning the stim-ulated emitted photon with the stimulating one), but it is detrimental to the populationinversion. When the radiation field becomes important enough, stimulated emission andabsorption dominate, with equal weights as they have equal probabilities. In this casethe dilemma of the two-level system springs up again and strong radiation tends towardsequalisation of populations (this time from above, though).

0

0.5

1

1e-08 1e-06 1e-04 0.01 1 100τsW

∆nss/δnss

Fig. 6.9: Normalised population difference in the steady state∆nss/δss as function of the radiation fieldintensityW . WhenW = τ−1

s , ∆nss has been halved from its optimum valueδnss , the population inversionin absence of radiation.

6.2.2 Semiconductors in laser physics

Semiconductors are important materials in light-matter physics thanks to the radiativerecombination of electrons and holes, although not at equilibrium where their densitiesare too small to produce detectable quantity of light even with high doping.87 It is how-ever easy to operate a semiconductor out of equilibrium by applying an electric voltageto it and generate huge populations of carriers. Indeed a forward-biasedp-n galliumarsenide junction generates strong light in the infrared asreported in the early 60s by

87The first connection of light to semiconductors was made in the beginning of XXth century with cat’swhisker detectors by a collaborator of Marconi, H. J., Round, who reported his finding of a green glow fromSiC in “A note on carborundum” in Elect. World19, 309 (1907).


Hall et al. (1962), Nathan et al. (1962) and Quist et al. (1962).88 Holonyak & Bevacqua(1962) could obtain emission in the visible window89 by using the GaAsP compound.By increasing the pumping of the structure to the point whereelectrons and holes un-dergo an inversion of population, the diode reaches the stage where gain by stimulatedemission overcomes losses, and an input signal on the activeregion is amplified. It re-mains to engineer the device so that this input is levied fromits output to trigger thelaser oscillations. The cavity in this case is provided by the semiconductor crystal it-self whose facets have been cleaved, i.e., terminated alongthe crystal axis to create aperfectly flat endface, perpendicular to the axis of the junction90. When the light gener-ated by recombination of electron and holes gets to this surface, it is partially reflectedback by internal reflection. The reflectivity is consequently quite low for such lasers,about 30% (the facets can be coated for better reflection).

These preliminary diode sources were not efficient lasers asthe active region whereelectron and hole recombine is spread-out across the junction with great losses andrequiring significant threshold currents to compensate. Only the short pulse regime ispossible before melting the device. A solution was envisioned on how to constrain car-riers effectively, theoretically by Kroemer (1963) and later realised in the Ioffe instituteby Alferov: the double heterostructure (DH). It consists ofa thin region of semiconduc-tor with a small energy gap sandwiched between two oppositely doped semiconductorswith a wider bandgap. When forward biased, carriers flow intothe active region and re-combine more efficiently because of the potential barriers of the heterostructure confinethe carriers to the active region. Practical and soon efficient operation was achieved andthe device became one key element in the computer and information era, with maybeits most significant impact in the data storage with optical reading of CD and DVDtypes of optical disks (still widely used today for these applications.)91 These structuresthat are now calledclassical heterostructuresrely on the profile of the energy bandsfor providing potential traps for the carriers. Their size vary in the range between a fewhundred ofµm and a fewmm. The idea was pushed forward by reducing further stillthe area of localisation to the point where size-quantisation plays a role, opening theway toquantum heterostructures, quantum wells, quantum wires and quantum dots.

These various schemes of lasing with semiconductors are sketched on Fig. 6.10.Lasing with a simple junction is a brute force approach whichrequire high threshold

88Biard and Pittman could prevail on the observation of radiation by a junction while working on GaAsdiodes, dating it to 1961. Not expecting light emission, they noticed it using an infrared microscope. Theircontribution has been acknowledged for the record by patentissues (for which they received $1 each) al-though their oldest publication on this topic under the title “GaAs Infrared Source” By Biard, Bonin, Carr andPittman, is also dated 1962 (in the PGED Electron Device Conference.)

89Bringing radiation emission from semiconductors “in the visible spectrum where the human eye sees”,to quote Holonyak, made his paper the most quoted of Applied Physics Letter.

90The idea to use the crystal itself as the cavity comes from Hall who polished the surfaces. He shared itwith Holonyak who was planning to use an actual external cavity. Holonyak had the further idea to cleave thecrystal. It proved too difficult however and Hall came first with the semiconductor laser.

91Other applications of semiconductor lasers for public use include fiber-optics communication, laserprinters, laser surgery, barcode readers, laser pointers.. . They are also useful for research and the military.

LASERS 231

ZhoresAlferov (b. 1930) and HerbertKroemer (b. 1928)

The 2003 Nobel prize-winners (with Kilby), enabled semiconductors the means to revolutionise laser physicsand shape the era of telecommunications.

Kroemer, now a Professor at University of California, SantaBarbara, proposed the concept of the doubleheterostructure in 1963. A major publication on this topic was rejected for Applied Physics Letters and pub-lished in Proc. IEEE instead. His favourite saying—as claimed in his Nobel lecture—is “If in discussing asemiconductor problem, you cannot draw an energy band diagram, then you don’t know what you are talkingabout.”

Alferov’s name is an icon for the Ioffe institute in Saint Petersburg, where he worked from 1953, and asits director in the period between 1987 and 2004. A gifted administrator, he managed to save the instituteinfrastructures from disaster in the 1990s. In 1971 he received the USA Franklin’s institute gold medal forhis pioneering works on semiconductor heterolasers. In itshistory, the Franklin medals has been awarded tofour Russian physicists: Kapitsa, Bogoliubov, Sakharov and Alferov, all but Bogoliubov having received theNobel prize later on.

currents.92 Consequently the device can only be operated in pulsed mode with muchloss in the conversion. By confining the carrier in the activeregion with the heterostruc-ture potential trap (simulating the action of aquasi-electric field—a term coined byKromer—a feast that no genuine external field can achieve), the lasing could be operatedfor threshold currentsJth reduced by two orders of magnitude, a trend that has contin-ued by following the road towards evermore quantisation, asis illustrated in Fig. (6.11)where a further two orders of magnitude gain on the current threshold is attained withQDs as the most recent realization.

The main advantage of edge-emitting lasers is that the size of the active regionallows one to store significant amount of energy as compared to the microscopic orsmaller volumes involved in diodes where the active region lies in a quantum het-erostructure. For this reason, the light-matter interaction must be compensated by con-straining the photons inside the active region. This is achieved by shifting the opticalaxis from the plane of the structure (where photons travels afew times over millime-ters) to the growth axis (where photons travel back and forththousands of times overa micron). The mirror that, in the case of the edge emitting laser is provided by in-dex contrast, must now have reflectivity of the order of99.9%. This is possible only

92Holonyak & Bevacqua (1962) reports11 × 103A/cm2 as a threshold to superlinear emission andlinewidth narrowing.


Fig. 6.10: Sketch ofsemiconductor lasers: a) the p-n junction where electron-hole recombination at theinterface serve as the active population; this scheme is more viable for LED operation,b) the edge-emittinglaser where the active region is confined by an heterostructure andc) the VCSEL where localisation is pushedto the quantum limit and emission made from the surface.

Fig. 6.11: Evolution of the threshold currentJth of semiconductor lasers, from Alferov (2001).

with dielectric mirrors, namely Bragg mirrors, that form a microcavity which confinethe active region. Such structures are known as VCSELs (for Vertical-Cavity Surface-Emitting Laser)

6.2.3 Vertical-Cavity Surface-Emitting Lasers

Until the late 1970s, semiconductor lasers exclusively used the stripe geometry withcavity lengths longer than 100 microns. However with the production of high-qualityintegrated DBR mirrors, it became possible to rotate the orientation of the emissionso that it emerged normal to the growth layer planes. The principal advantage of this

LASERS 233

surface emission geometry of Vertical Cavity Surface Emitting Lasers (VCSELs) is theability to make and test large numbers of lasers on a single wafer without having tocleave the wafer into individual lasers with facets, with a large reduction in the costof quality control, manufacture and ease of packaging devices. Another advantage isthe high spatial quality and non-astigmatic laser emission, which makes it easy to alsomatch to optical fibres. Subsequently it was realised that very fast modulation could beobtained in VCSELs with low power consumption.

However in order to get these devices to work, the hundred-fold reduction in cavitylength (and hence round trip gain) has to be recovered in decreased round trip loss (andhence the need for very high mirror reflectivities). This became possible with the intro-duction of lattice-matched DBR designs producing 99% to 99.9% intensity reflectioncoefficients. The cavity lifetime typically can reachτc = 1ps (i.e.Q values up to 1000)which is similar to that in conventional stripe lasers in which the photons reflect off thecavity mirrors hundreds of times less often.

In order to efficiently pump the device, and retain single mode operation, the cur-rent transport and optical emission within the planar microcavities has to be laterallyconfined. Typically this is achieved in two ways: either by etching a circular mesa withan annular top contact injecting holes into the upperp-type material, or by using an Al-rich layer within the active region which is oxidised from anoutside trench to producean insulating annulus which forces current through the centre of the device. Howeverother fabrication procedures can give the same effect including growing in buried holesand proton bombardment. The oxide technology has some useful advantages in that aswell as confining the injection current, it also confines photons (since the central corehas a higher refractive index that then aluminium oxide annulus), which results in betteroverlap of the light and electron-hole pairs.

Typically to get maximum gain, quantum wells or quantum dotsare used as theactive material, and placed within the structure at the peakof the antinodes of the in-tracavity electric field. To overcome the round trip loss, the gain needs to be as high aspossible and thus the carrier density in the active region isuniversally above the ioniza-tion threshold for excitons. Hence these devices operate without excitonic contributionsto their gain spectrum, distinguishing them from the polariton lasers discussed later.

One of the advantages of VCSELs is that their active volume can be made verysmall. Typical small mode areas can be 1-10µm2, and active cavity material volumescan be as low asV =0.05µm3 which is about 100 times smaller than conventional stripelasers. Hence the threshold in a VCSEL is correspondingly small since the thresholdcurrent is approximatelyIth = eV Nt/τc when the cavity loss rate is small, controlledby the need for carrier densities above the transparency valueNt ≃ 1018cm−3. Thresh-olds below 10µA have recently been reported for quantum dot VCSELs by Zou etal.(2000).

The growth of such multilayers is now at a highly sophisticated level, with specialdesigns being able to simultaneously control the electrical transport through the DBRlayers (using specific doping profiles that provide useful band-bending), to control thephoton confinement (through the oxide apertures), to control the thermal management(by matching the thermal impedence mismatches through the DBR stacks) and to con-


Fig. 6.12: Light output curve for a quantum dot VCSEL, as reported by Zou et al. (2000).

trol the electrical carrier spatial distributions to maximise the gain. The resulting over-laps with the optical fields can be of order 5% in the vertical direction (overlap withQWs at centre of cavity) and up to 80% in the lateral direction, leading to confinementfactors of 4% or so.

The power conversion (or wallplug) efficiency of VCSELs is also very high whenpumped above threshold, because little voltage drop existsacross thep − n junctionexcept at the active region, leading toη > 50%. However as the current increases, thetransverse mode of the VCSEL has a tendency to switch, due to spatial hole burning andcurrent spreading. To prevent this, structures can be designed which are anti-guiding andoperate well at higher power. In addition, another switching can appear at higher powerdue to the birefringence of the devices, originating eitherfrom shape anisotropies ofintrinsic strain in the layers. The polarisation output of these devices is always linear,but can switch axes between two near degenerate cavity modesas the temperature ofthe device rises. Full polarisation dynamics of VCSELs havebeen discussed by Gahlet al. (1999).

We now provide a more thorough discussion of the Boltzmann equations (i.e. semi-classical rate equations) following an analysis by Bjork &Yamamoto (1991) and Bjorket al. (1993), which is modified to provide a clear comparisonwith the inversionlesspolariton lasers discussed in Chapter 8. We define generallya population densityNxpumped at a rateP , producing laser emission from stateN0 through the steady-staterate equations:

LASERS 235

VdNxdt

=P

~ω− ΓnrV Nx −N0R(Nx) − S(Nx) = 0 (6.49)

dN0

dt= −Γ0N0 +N0R(Nx) + S(Nx) = 0 (6.50)

whereN0 is the number of photons in a conventional laser, and the number of polaritonsin a matter-wave laser. For a conventional semiconductor laser of active volumeV , thespontaneous,S, and stimulated,R, scattering rates are

S = βΓsV Nx (6.51)

R = βΓsV (Nx −Nt) (6.52)

with non-radiative decay rateΓnr = 1/τnr + (1 − β)Γs, whereΓ0 is the photon cavityescape rate,Γs is the spontaneous emission rate, andτnr is the non-radiative lifetime.The fraction of photons that are spontaneously emitted intothe lasing mode isβ, whichis a key parameter for enhancing the emission from microcavities.

These equations can be simply solved to yield the output power as the pump rate isincreased, giving a lasing threshold,

Pth =~ωΓ0

β(1 + Γnr/Γs)

(

1 +βΓsV Nt

Γ0

)

(6.53)

Bjork & Yamamoto (1991) have shown how this threshold behaves in a verticalmicrocavity laser in the weak coupling regime, and emphasized that the values of bothβ andΓnr are critical to low threshold action. Typically in a semiconductor Fabry-Perotlaser, thresholds are around 10mW in a 300µm long device. This contrasts with VCSELswith optimisedβ∼10−2, allowing thresholds below 1mW (Fig.6.13).

0.2

0.1

0.0

-0.1

-0.2

Rat

e (p

s-1)

1x1018

0.50

Carrier density (cm-3

)

R

S(a)

Nt10

-6

10-4

10-2

100

102

Out

put p

ower

(m

W)

10-4

10-2

100

102

104

Pump Power (mW)

(b)

Fig. 6.13: VCSEL characteristics: (a) Spontanteous (S) and stimulated (R) scattering rates from excitons tophotons as the carrier density increases.τnr=50ps,Γ−1

0 =30ps,Γ−1s =3ps,β=3.6×10−3, V =0.15µm3.

(b) Output vs. input power for the conditions in (a) (solid),for τnr = 50ns (dashed), and for a Fabry-Perotlaser withτnr>10ns,Γ−1

0 =3ps,Γ−1s =3ns,β=3×10−5, V =60µm3 (dotted).

The form of these equations is the same for most lasers, and isunderpinned by thedeep relationship between spontaneous and stimulated emission through the Einstein


coefficients of a transition. The result is that population inversion is a necessary con-dition for lasing, and in a semiconductor laser this requires a sufficient carrier densityNt to bleach the absorption. At these densities above the Mott density, screening by theCoulomb interaction and phase-space filling is sufficientlystrong that the electron-holeplasma screens out the exciton binding, and thus exciton lasing is impossible. The situ-ation is completely altered when the polariton pair scattering discussed in Chapter 7 isused to feed energy into a lasing transition which is in the strong-coupling regime.

One further important characteristic of the VCSEL is the possibility of turning iton and off at high frequencies. Typically this rate is controlled by the relaxation oscil-lation frequency,fr, the natural oscillation rate between cavity photons and electronicexcitations produced by the coupled equations (6.50):

fr =Γs2π

√

1

Γsτc

(

I

Ith− 1

)

(6.54)

For typical parameters this frequency exceeds 10 GHz in VCSELs compared to factorsof 10 smaller for conventional lasers.q

We note also here, there has been some discussion about the extent to which coher-ence in the electronic system can play a role in semiconductor lasers, and the coherencein the system remains difficult to calculate due to the many body nature of the problem.

6.2.4 Resonant cavity LEDs

Recently, a popular alternative to VCSELs for strong light emission which is incoher-ent has been the resonant-cavity light-emitting diode (RCLED). Because the gain of awavelength-thick semiconductor layer is typically small,high quality mirrors are re-quired to produce effective lasers, and these require sophisticated fabrication of manyprecise semiconductor layers. On the other hand, for many applications, the coherenceof the light emission from the device is not so important, andit is the efficiency and di-rectionality that is key. Typical applications are in display technologies, in xerography,in bio-photonics, and in general lighting applications. Inan RCLED, the semiconductoremitting layer is clad with low-finesse mirrors (unbalancedso they have smaller reflec-tivity on the top side), which modify the angular emission pattern, and suppress emis-sion into in-plane waveguide modes which do not escape efficiently from the sample.With these modifications, the LED efficiencies can exceed 50%and make such emit-ters competitive even with incandescent lighting. The riseof such devices has trackedthat of the GaN-based technologies so that UV/blue emitterscan now be tuned acrossthe visible spectrum using either phosphor-based light conversion or In-doping of theemitting GaN to produce ranges of colours.

Typically the design of RCLEDs is similar to VCSELs. To enhance the emission,the activep − n junction is placed at the antinodes of a microcavity whose length isclose to a few optical wavelengths. This ensures that the linewidth of the cavity modeset by theQ-factor is as large as possible, while the angular emission is beamed asmuch as possible in the surface normal direction. In thick cavities of refractive indexn, the fraction of resonances that exist within the light cone(from which light canescape) is given by1/2n2, and this emission is shared between the number of possible

LASERS 237

microcavity resonances (see Fig.6.14). This also shows intuitively that the enhancementwill be better for active LEDs with a narrower spectral width.

Fig. 6.14: (a) Cavity dispersion (curves) and LED active emission spectrum (shaded box) for (a) wavelength-scale and (b) thick microcavity, showing different emission patterns inside the light cone (dashed).

The simplest design for a RCLED uses a buried conducting DBR and and a metalmirror as the top contact, with the light extracted through the substrate (which thereforehas to be transparent). In such a structure the extraction efficiency can exceed 20%. Theremainder is emitted into guided modes (which are absent as they are below cut-off inhalf wavelength thick cavities) or remains trapped in the substrate.

6.2.5 Quantum theory of the laser

In the above sections we have developed theories of the laserdealing with averagevalues both of populations of atomic (or carrier) populations and of the radiation fieldintensity (number of photons).

A full quantum treatment of the laser requires some approximations which one canconsider in some depth in the classical or semi-classical theories, such as multimodeoperation, spatial inhomogeneities, temporal drifts, inhomogeneous broadenings and soon. The mathematical complications brought by dealing withoperators in the quantumcounterpart makes it awkward to draw a dear parallel. All that pertain to approximationswhich concern the average populations can be stripped from aquantum perspective asthey appear as classical averages anyway. The most simple Hamiltonian of interactionis sought anda posteriori investigations show that for most purposes it is enough toconsider:

H = ~g

aσ†u(r)ei(ω0−ω)(t−t0) + aσu(r)e−i(ω0+ω)(t−t0) + h.c.

(6.55)

in the interaction picture, wherea is the photon annihilation operator,σ the atom annihi-lation operator,u(r) the normalised cavity mode function andg the interaction strengthderived in Chapter 4. We have already considered a system close to this one to derivethe Bloch equations, but an important approximation has been retained here which hints


at the difference in the laser case, namely the rotating waveapproximation of the secondterm.

The density matrix of the system is the combined atom (carrier)–photon field sys-tem. At initial time in absence of correlations between them, it reads:

ρ(t0) = ρA(t0) ⊗ ρF(t0) , (6.56)

with ρA (resp.ρF) the density matrix for the atom (resp. photon field). The state ofρF isobtained by tracing other the atomic variables. If we write the Liouville–Von Neumannequation of motion as its Born expansion to infinite order before doing so, we get:

ρF(t) = ρF(t0) + TrA

(

∞∑

n=1

1

(i~)r

∫ t

t0

∫ t1

t0

· · ·∫ tr−1

t0

[H(t1), [H(t2), [. . . [H(tr), ρ(t0)] . . .]]]dt1dt2 · · · dtr)

(6.57)

To successfully describe laser action—even in the simplestsetting—with Eqs. (6.55)and (6.57), one must carry out the algebra to high order in thecommutators. One gets

[H, ρ] = ~g[σa†ρFu∗(r) − σ†ρFau(r)] (6.58)

and successively, up to fourth order commutator which yields

[H, [H, [H, [H, ρ]]]] = (~g)4|u(r)|4[σ†σ(aa†aa†ρF + 3aa†ρFaa† + h.c.)

− 4σσ†(a†aaa†ρFa+ h.c.)] (6.59)

To proceed one considers the evolution from initial atomic position both in groundand excited state. For instance in the excited state, Eq. (6.59) becomes, tracing overatomic variables:

TrA([H, [H, [H, [H, ρ]]]]) = (~g)4|u(r)|4(

aa†aa†ρF

+ 3aa†ρFaa† − 4a†aaa†ρFa+ h.c.

)

(6.60)

Up to now we have dealt with a single atom interacting with a single mode of thecavity mode. The latter approximation is reasonable but thesingle atom does not de-scribe a conventional laser,93 which properly involves an assembly of atoms as its ac-tive media. Rigorous but heavy methods have been developed,for instance by LambJr. (1964) or Scully & Lamb Jr. (1967). Even textbooks specialising on this topic findit difficult to attain to such feats of meticulousness. Mandel & Wolf (1995) proposeas a shortcut—which a posteriori proves to be essentially equivalent—to consider theeffects of an assembly of atoms by considering a coarse-grained lifetime average with

93The necessity to consider an assembly of atoms in the model ofa laser is made more stringent from thefact that there exist single-atom lasers which would dispense from these requirements, demanding in exchangea more thorough considerations of its quantum mechanical features.

LASERS 239

probability distributionP (∆t) = e−∆t/T2/T2 whereT2 is the lifetime of the excitedstate (level 2), and multiplied byR2 the pumping rate of this level. Also the equationis averaged spatially over the active medium. An equation ofmotion is obtained for thegain mechanism (since the atom was in its excited state). Thesame procedure can bestarted again for the atom initially in its ground state, yielding another master equation.The sum of which (since there is no coherence between the atoms) provides the finalmaster equation for the photon field which reads:

∂ρF

dt= −1

2A[aa†ρF − a†ρFa+ h.c.] − 1

2C[a†aρF − aρFa

† + h.c.]

+1

8B[aa†aa†ρF + 3aa†ρFaa

† − 4a†aa†ρFa+ h.c.] (6.61)

where the above mentioned derivation (introducing quantities such asη(r) the densityof active atoms andη1 that of loss atoms) provides coefficients:

A = 2(R2/N)(gT2)2

∫

η(r)|u(r)|2 dr , (6.62a)

B = 8(R2/N)(gT2)4

∫

η(r)|u(r)|4 dr , (6.62b)

C = 2(R1/N)(gT2)2

∫

η1(r)|u(r)|2 dr . (6.62c)

Coefficients (6.62) characterise gain, nonlinearity and losses of the laser, respec-tively. Observe thatB is of the order of the square ofA andC.

Eq. (6.61) is a typical single-mode laser master equation. Coefficients would varyfor other systems derived under other approximations (or more rigorously derived) butthe main principles remain with nonlinear terms displayingsuch asymmetric repartitionabout the density operator. This results in coupling the diagonal elements to off-diagonalelements and plays a role in the coherence of the field. As for the diagonal elements,their equation of motion is readily obtained by dotting the master equation to get theequation of motion ofp(n, t) = 〈n| ρF |n〉 as:

∂p(n, t)

∂t= −A(n+ 1)

(

1 − B

A(n+ 1)

)

p(n, t) +An(

1 − B

An)

p(n− 1, t)

+C(n+ 1)p(n+ 1, t) − Cnp(n, t)

(6.63a)

≈ − A(n+ 1)

1 + (B/A)(n + 1)p(n, t) +

An

1 + (B/A)np(n− 1, t)

+C(n+ 1)p(n+ 1, t) − Cnp(n, t) .

(6.63b)

Eq. (6.63) is a rate equation similar to those already encountered in the classical theoryof lasers but this time describing the flow in probability space rather than for averages(the later can of course be obtained by summing the weighted probabilities). In the sameway, detailed balance can be applied to obtain the steady state solution of Eq. (6.63)from the compensation of configurations differing by one photon. This is seen clearly


in Eqs. (6.63) where a term involvingA cancels with the term involvingC of the op-posing sign. The two other terms also cancel in this way; theyare in fact equivalentsubstitutingn for n+ 1:

A(n+ 1)

1 + (B/A)(n+ 1)p(n, t) = −C(n+ 1)p(n+ 1, t) , (6.64a)

An

1 + (B/A)np(n− 1, t) = −Cnp(n, t) . (6.64b)

The corresponding transitions are sketched on Fig. 6.15

An1+(B/A)n

p(n − 1)

|n〉

|n − 1〉

|n + 1〉

A(n+1)1+(B/A)(n+1)

p(n)

Cnp(n)

C(n + 1)p(n + 1)

Fig. 6.15: Flow of probabilities between configurations with n − 1, n andn + 1 photons. At equilibrium,steady state is established by detailed balancing of the neighbouring terms which equate each other throughsubstitutionsn→ n+ 1.

Eq. (6.64) can be solved by recurrence, yieldingp(n) knowingp(n− 1) through:

p(n) =A/C

1 + (B/A)np(n− 1) (6.65)

which repeated application yields

p(n) = p(0)

n∏

i=1

A/C

1 + i(B/A)(6.66)

with p(0), the starting point, being determined by normalisation condition∞∑

n=0

p(n) = 1 (6.67)

Exercise 6.2 (∗) Show that the polynomial expansion for the equation of motion of〈n〉 =∑

n np(n) derived from Eq. (6.63b) is of the type:

d〈n〉dt

= α〈n〉 − β〈n2〉 + γ (6.68)

Analyse this equation providing physical meaning of the parametersα, β and γ andlink them to microscopic parameters.

LASERS 241

0

0.02

0.04

0.06

0 20 40 60 80

Pro

babi

lity

n

Coherent

Laser

<n>=40

Fig. 6.16: Statisticsp(n) of a laser given by Eq. (6.66) as compared to the Poisson distribution of a coherentstate. CoefficientsA/C = 1.2 andB/C = .05 result in an average number of photons〈n〉 = 40. Evenabove threshold a laser still has large deviation from the ideal coherent case.

In the semiclassical theory, the linewidth is obtained fromthe Fourier transformof 〈E(t)〉 whereE is the photon field operator so that, in Schrodinger picture:

〈E〉(t) =√

~ω/2e0V sin(kz)Tr(

ρ(t)(a− a†))

(6.69a)

∝ sin(kz)

∞∑

n=0

√n+ 1ρn,n+1(t)e

iνt (6.69b)

whereρn,n+1 = 〈n| ρ |n+ 1〉 is the upper diagonal element of the density matrix.Conversely to diagonal elements, the off-diagonal elements do not form a closed setof equations and couple to all other elements of the density matrix (including diagonalelements), showing that the dynamics of coherence which is of a quantum character ismore complicated than the dynamics of population which is ofa classical character.

Dotting Eq. (6.66) with|n〉 and|n+ 1〉, one gets for the equation of motion:

ρn,n+1 = −[

(

A−B(n+3

2))

(n+3

2) +

1

8B +

ν

Q(n+

1

2)

]

ρn,n+1

+(

A−B(n+1

2))

√

n(n+ 1)ρn−1,n (6.70)

+ν

Q

√

(n+ 1)(n+ 2)ρn+1,n+2 .

High above threshold wheren assumes high values over whichρ varies smoothly,ρn,n+1 can be approximated by its neighbour valueρn,n, for which recursive closedrelations are known, cf. Eq. (6.66). This turns Eq. (6.2.5) into:


ρn−1,n ≈ ν

Q

√

(A−Bn)(A−B(n+ 1))ρn,n+1 (6.71a)

ρn+1,n+2 =Q

ν

√

(A−B(n+ 1))(A−B(n+ 2))ρn,n+1 (6.71b)

Injecting back this expression in Eq. (6.2.5) gives

ρn,n+1 = −1

2Dρn,n+1 (6.72)

with

D ≈ 1

2

A

〈n〉 (6.73)

From the Fourier transform of Eq. (6.69), the lineshape of the laser turns out to be:

S(ω) =|〈E(0)〉|2

(ω − ν)2 + (D/2)2(6.74)

that is, it is a Lorentzian centered on the laser transition with width D. The notablefeature is thatD varies inversely with the photon field intensity: the laser has a verynarrow line as a result of the photon compression in phase space, an effect first realizedby Schawlow and Townes. This is however more of a theoreticallimit as other factorsbroaden the line much beyond the value given by Eq. (6.73).

6.3 Nonlinear optical properties of weak-coupling microcavities

By placing a material inside a microcavity, its nonlinear optical properties are enhanced.The first enhancement arises simply from the enhancement in internal optical intensitydue to the finesse, which thus reduces the external thresholdlight intensity to get a cer-tain nonlinear optical response. The advantage of using a microcavity is that the builduptime for the optical field is short, as well as the transit time, and hence the device opera-tion remains nearly as fast as that of the intrinsic nonlinear material. Another advantageis that the refractive part of the nonlinear response is converted into a transmission non-linearity due to the optically-induced spectral shifting of the cavity modes.

The nonlinear optical process may arise directly from occupation of the upper stateof two-level systems (in atoms, or in semiconductors) whichdepends on the fermionicstatistics of electrons. Or it may arise from Coulomb interactions between optically ex-cited states (such as the exchange interaction). It also is typically divided into ‘real’ and‘virtual’ processes, which correspond to whether switching optical energy ultimatelyends up absorbed inside the medium (the former), or the energy remains within theoptical field (the latter). In effect, the ‘virtual’ processis a transient state inside themedium which reemits the optical energy (for instance if an atom is strongly excitedoff-resonantly), but transiently produces nonlinear responses as above. Such a processwill be intrinisically faster than a ‘real’ process, in which the absorbed energy mustbe lost (e.g through recombination or spontaneous emission) before the next switchingevent can take place.

NONLINEAR OPTICAL PROPERTIES OF WEAK-COUPLING MICROCAVITIES 243

Fig. 6.17: (a) Microcavity with enhanced optical intensitywithin nonlinear medium, (b) Optically-inducedspectral shift of cavity modes from refractive index changeproduces a change in transmission at a near-resonant wavelength.

The use of microcavities in these applications has been studied since the 1980s,typically in semiconductor Fabry-Perot interferometers,for possible ultrafast opticalswitching elements. A typical example is a gold-coated semiconductor slab containinga quantum well that is optically-excited to the long wavelength side of the exciton reso-nance. The creation of a virtual population of excitons blueshifts the exciton resonance,thus changing the refractive index at the cavity resonance and producing an enhanced ul-trafast response. In general the problem with such devices is that the nonlinear responsefrom a small volume microcavity is limited.

6.3.1 Bistability

New effects occur when the cavity mode can be spectrally shifted by more than thecavity linewidth. In this case, optical bistability can occur in which there are conditionsfor which two stable states of the cavity transmission exist, ‘high’ and ‘low’. The ideais to set up the cavity response and the pump laser tuning in such a way that an increasein incident optical power spectrally shifts the cavity closer into resonance with the ex-citation laser. This further increases the power fed into the cavity, and thus providesa positive feedback which clamps the transmission to maximum. The reverse situationoccurs as the incident power is reduced, in that the internaloptical field is sufficientlystrong so that the cavity resonance remains closer to incident laser wavelength than ex-pected from the incident optical power alone, until a critical minimum power at whichthe whole effect switches off. Two regimes are possible for bistability in microcavities,with the nonlinear response primarily either absorptive ordispersive, as shown by Gibbs(1985). Bistability is also observed in the atom-filled microcavity system in the strongcoupling regime, in which case measurements of the transmission at higher power be-come distorted and shifted, as seen in Fig. 6.18(c) by Gripp et al. (1997).

A bistable optical response can also be seen in polarisationswitching in VCSELs.In such lasing microcavities, only one mode lases at any time, however perturbations(for instance in current injection, or incident light) can switch the lasing between twoorthogonally-polarizednearly-degenerate lasing wavelengths. The balance between thesestates is controlled by the spin relaxation of the excitons inside the active quantum wellregions, as well as strain within the fabricated pillar microcavities.


Fig. 6.18: (a) Optically-induced spectral shift of cavity modes locks cavity to input laser wavelength at highpower, (b) transmitted intensity response vs input intensity showing the region of bistability (shaded). (c)Transmission through strongly-coupled atom-cavity showing hysterisis as the incident light is tuned, fromGripp et al. (1997).

For any microcavity system which is bistable, there remain the transverse degrees offreedom which allow optical pattern formation within the microcavity. At one extreme,this can lead to the formation of spatial solitons, in which the light within a region of themicrocavity which is switched ‘high’ suffers nonlinear diffraction in such a way that thelateral shape of the resonant optical mode within the regionis preserved. In other cases,2D grating patterns can emerge either statically, or in a constantly changing dynamicpattern evolution. The exact response depends critically on the illumination conditions,the cavity length and mode spectrum, and the boundary conditions, as discussed byHachair et al. (2004).

Fig. 6.19: Pattern formation in electrically-contacted microcavities: (a) spatial soliton in liquid-crystal micro-cavity as observed by Hoogland et al. (2002) and (b) pattern formation and (c) seven stable spatial solitons insemiconductor VCSELs reported by Hachair et al. (2004).

A number of realisations of pattern formation within microcavities have been demon-strated including (a) atoms on resonance (though this is notin a microcavity but ex-tended over cm lengths), (b) liquid crystals within planar microcavities, and (c) semi-conductor quantum wells in large area VCSELs (Fig.6.19). Ingeneral, pattern formationis a sensitive phenomena and thus perturbed strongly by imperfections in the micro-cavity properties. Use of this phenomena, for instance for switching of pixels, is thusproblematic.

CONCLUSION 245

6.3.2 Phase matching

One further use for nonlinear microcavities has been to act as optimised optical fre-quency doubling devices. By carefully controlling the Bragg reflector mirror stack, it ispossible to produce a microcavity which is resonant at bothω and2ω, with a selectablephase difference between the two per round trip due to different DBR penetrations. Thiscan thus act as a phase matching device, when non-critical phase matching is difficult.

An equivalent use of microcavities has been as a pulsed photodiode to measure ultra-short optical pulses, using two photon absorption to generate adcelectrical current evenat small input intensities. By surrounding the active region of a two-photon photodiodewith a microcavity tuned to the input wavelength, the electrical current measured, whichdepends on the peak field of the pulses, is amplified by 104 while the short cavity lengthensures minimal broadening of the temporal response.

6.4 Conclusion

In this chapter, a basic overview of emission from microcavities in the weak couplingregime shows a number of their benefits including lower thresholds, fast response, andcontrollable emission characteristics. In the next chapter we show how these are modi-fied in the strong coupling regime.

7

STRONG COUPLING: RESONANT EFFECTS

This chapter presents experimental studies performed on planarsemiconductor microcavities in the strong coupling regime. The firstsection reviews linear experiments performed in the 90s which haveevidenced linear optical properties of cavity exciton-polariton. Thechapter is then focused on experimental and theoretical studies ofmicrocavity emission resonantly excited. We mainly describeexperimental configuration at which the stimulated scattering wasobserved due to formation of a dynamical condensate of polaritons.Pump-probe and cw experiments are described as well. Dressing of thepolariton dispersion and bistability of the polariton system because ofinter-condensate interaction are discussed. The semi-classical and thequantum theories of these effects are presented and their resultsanalyzed. The potential for realisation of devices is also discussed.

246

OPTICAL PROPERTIES BACKGROUND 247

7.1 Optical properties background7.1.1 Quantum well microcavities

In 1992, the strong-coupling regime was first identified by Weisbuch et al. (1992) insemiconductor microcavities. In fact their goal had been tooptimise the superradiantemission of quantum wells inside microcavities, and the splitting in reflection that theyobserved was not really expected, because it was previouslythought that the light-mattercoupling was too small for strong-coupling. The correct identification of this as strongcoupling led to a number of investigations of the emission characteristics of these de-vices.

Fig. 7.1: (a) Strong coupling reflection spectra in a planar semiconductor microcavity, and (b) normal incidentpolariton energies vs cavity detuning (scanning across sample), as observed by Weisbuch et al. (1992).

Besides the complete formulation of such a multilayer structure presented in Section2.7, there are several simple models for the strong couplingthat are appropriate for theintuition they provide. A sharp exciton (or atomic) transition produces a characteristicresonant absorption and dispersion lineshape (Fig. 7.2a).If this is inserted into a mi-crocavity, then the total round trip phase as a function of wavelength acquires an extracontribution (Fig.7.2b), which means that there are now three resonant conditions.

The upper and lower resonant conditions occur where the absorption is small and sohave narrow linewidths, while in this picture the central constructive condition remain-ing at the resonance energy occurs with strong absorption and is not observed. Thissimple picture of net refractive index corresponds to the non-local dielectric suscepti-bility model presented in Section 4.4.2 so that excitons feel the optical field from thecavity together with the polarisation from all other excitons around them.

The key characteristic of the semiconductor microcavity inthe strong coupling

248 STRONG COUPLING: RESONANT EFFECTS

Fig. 7.2: (a) Dispersion and absorption of resonance, producing (b) net microcavity round trip phase,φ,without (dashed) and with (solid) resonant medium. The lower (•) and upper () polaritons are at energieswhereφ = 2π.

regime is the dispersion relation. This maps how the resonant polariton modes shift within-plane wavevector (or angle of incidence). The derivation for the multilayer (Eqs. 1.9,2.147 and 4.134) can be simply realised from the resonant condition on the wave vectorperpendicular to the planar cavity mirrors (see Sections 1.3, 2.7 and 4.4.4.2). While theangular dispersion neark = 0 can be expanded quadratically giving a very light polari-ton mass, the full dispersion is often crucial for the effects reported and is the solutionof Eqs. (4.130), producing the characteristic shape in Fig.7.3. We have termed the cen-tre of this dispersion ak-space polariton trap as the energy of lower polaritons hereisbelow that of all other electronic excitations, and scattering out of the trap is difficultfor polaritons ifkBT < Ω/2.

Fig. 7.3: Angular dispersion of a typical semiconductor microcavity at zero detuning, (a) on a log scaleshowing the constrast between light polaritons and heavierexcitons and (b) showing the critical regionsaround the polariton trap.


7.1.2 Variations on a theme

Quantum well microcavities which incorporate InGaAs microcavities exhibit the clear-est polaritonic features because the strain within the InGaAs energetically splits theheavy- and light-hole excitons so that only the simplej=3/2 heavy-hole polaritonsare resolved. Even in this case the spin-degeneracy and residual lattice strain along[110] produces complicated polariton interactions. Microcavities in which GaAs quan-tum wells are incorporated have narrower linewidths (typically below 0.1meV) becauseof the eliminated alloy disorder, however the light-hole excitons with a third of the os-cillator strength are only a few meV to higher energy and alsostrongly couple to thecavity mode, producing a more complicated polariton dispersion. Microcavities whichincorporate GaN are even more complicated, since there areA,B andC excitons whichall couple to the cavity mode with different polarisation dependences.

Besides microcavities which use quantum wells for the excitonic coupling to thecavity mode, it is also possible to use wavelength-thickness layers of bulk semicon-ductors. Because binary semiconductors do not have alloy disorder, their excitoniclinewidths can be narrow, although larger thicknesses needto be used to overcome theweaker oscillator strength, see for instance the discussion by Tredicucci et al. (1995).Both reflectivity and luminescence show similar strong coupling to QW microcavities,with clear Rabi splittings of typically 4meV, as well as extra polaritonic modes from thequantisation of the exciton centre of mass within the finite thickness layer (Fig.7.4).

Fig. 7.4: Angular dispersion of aλ/2 GaAs bulk semiconductor microcavity showing the strong couplingand additional center of mass polariton modes, from Tredicucci et al. (1995).


It is not even strictly necessary to use a microcavity to produce such polaritonicdispersions, since unwrapped the microcavity looks like a periodic array of quantumwells.

First theoretical study of the “Bragg arranged quantum wells” has been undertakenby Ivchenko et al. (1994), who showed that the Bragg arrangement leads to an ampli-fication of the exciton-light coupling strength proportional to the number of quantumwells. Nowadays, this effect is also exploited in 2D and 3D resonant photonic crystals.

Experimentally, “Bragg-arranged quantum wells” have beenexplored by severalgroups, see for instance the publications by Hubner et al. (1996) and Prineas et al.(2002). They show many analogous features to semiconductormicrocavities. Howeverit is in practise harder to produce many quantum wells (up to 100 are needed in GaAs)of all exactly the same thickness, spacing and composition,and this is even harder inother material systems. However even in a small number of closely spaced quantumwells, polaritonic effects can be observed, as reported by Baumberg et al. (1998).

Typically it is useful to study different detuning conditions of the microcavity, wherethe detuning is the energy difference between the normal incidence uncoupled cavitymode and the exciton energy∆ = ωC − ωX at θ = 0. One way this is achievedis by increasing the growth variation between different areas of the wafer (typicallyby eliminating the wafer rotation in the growth reactor), which produces an increasingthickness of the cavity length across the wafer. Hence it is possible to find areas in whichzero detuning is present, and either side of this detunings greater or less than zero. Theweaker dependence of the QW energy on the well width means that this method is quiteeffective. Another possibility is to use temperature to control the detuning, since the ex-pansion of the lattice shifts both exciton energy and cavitymode to lower energy, thoughthe exciton shifts about three times faster. This is see on Fig.7.5 from the work of Fisheret al. (1995). At temperatures above 100K the thermal ionization of excitons becomessufficient to broaden the excitons in III-V semiconductors and wash out strong coupling,limiting the effective range of tuning. However this technique has been frequently used,often to tune localised excitons in quantum dot microcavities.

However there are situations in which one would like to remain in a specific posi-tion on the sample and tune the cavity mode or exciton energy.Tuning of the excitonenergy is possible using either electric or magnetic applied fields. By growing the mi-crocavity in ap–i–n device, for instance with the DBR mirror stacks doped as in aVCSEL, a vertical electric field can be applied which produces a quantum-confinedStark shift of the exciton energy. Up to 35 kV/m, the oscillator strength of the excitontransition decreases by only 30%, while the excitons red-shift by 20meV, allowing themto be scanned across the cavity mode to demonstrate the strong coupling regime, as onFig.7.6 from Fisher et al. (1995). On the other hand, magnetic fields split the heavy-holeexciton into spin-up and spin-down components, and the magneto-exciton can have alarger oscillator strength due to its more compact binding.This allows a weak to strongcoupling transition to be observed with applied magnetic field, as reported by Tignonet al. (1995). In addition although the magneto-splitting is only 1meV forB=10T andhence less than the Rabi splitting, the individual spin-down and spin-up polaritons canbe resolved using circularly-polarized light, as reportedby Fisher et al. (1996).


Fig. 7.5: Temperature tuning of an InGaAs microcavity by Fisher et al. (1995).

7.1.3 Motional narrowing

Another effect of strong coupling is to change the effect of disorder in the exciton andphoton modes. This arises because the length scale over which polaritons average overdisorder can be different to the lengthscales of disorder inthe components. Typically ex-citons, even in high quality quantum wells, are localised onthe 10-100nm lengthscales,on the order of 10 Bohr radii. The unavoidable variation in the width of the quantumwells (so called monolayer fluctuations, which are different on the two sides of the quan-tum well) means that there is a population of excitons with different energies in differentspatial locations. When these excitons are all coupled to the same cavity mode, then theresulting polariton averages over all their energies producing an inhomogeously broad-ened polariton much narrower than the exciton distribution. This effect has been termed“motional narrowing”, as see in Section 4.4.5. Another way to see this effect is that thelower energy polariton has a linewidth given by the imaginary part of the dielectric con-stant which is much reduced further away from the centre of the exciton distribution.Measurements of the polariton linewidth in the strong coupling regime indeed show thiseffect, with a reduced inhomogeneous distribution for polaritons compared to excitons,see Fig.4.26 on page 158.

7.1.4 Polariton emission

One of the first observations concerning the polariton radiative emission from InGaAsquantum wells in GaAs semiconductor microcavities was thatalthough the photolu-minescence mapped onto the predicted dispersion relation,the intensity of this lumi-nescence was rather different from the typical thermalisedemission seen from barequantum wells. Similar results were seen in CdTe-based microcavities, for instance by


Fig. 7.6: Electric field tuning of an InGaAs microcavity by Fisher et al. (1995).

Muller et al. (1999) (see Fig.7.7). The reason for this is that the excitons generatedimmediately after non-resonant excitation with a pump laser relax quickly to the high-k part of the lower polariton dispersion (often termed the ‘exciton reservoir’). Theircooling to lower energies and lowerk, and particularly into the polariton trap, is thenrestricted by the need to lose large amounts of energy with very little simulataneousreduction ink. Very few quasiparticles exist within the semiconductor that can removethis combination of energy and momentum, and hence the exciton-polaritons collect atthe “bottleneck” region in the vicinity of the trap (Fig. 7.3b). Hence instead of the great-est luminescence intensity emerging at the lowest energies, much more luminescenceemerges from this bottleneck spectral region. Note also that unlike quantum wells, theluminescence spectrum is also angularly dependent due to the polariton dispersion.

From angular measurements of the luminescence as a functionof detuning (whichchanges the depth of the trap) and temperature, one can estimate that more than fiveacoustic phonon scattering events are needed to cool a carrier into these 3meV polaritontraps (for GaAs-based microcavities), which is significantly slower than the radiative

NEAR-RESONANT-PUMPED OPTICAL NONLINEARITIES 253

Fig. 7.7: Emission intensity and emission rate as a functionof the emission angle from a non-resonantlypumped CdTe-based microcavity, from Muller et al. (1999).

lifetime in the bottleneck region. Hence in the linear regime, emission from strong-coupled semiconductor microcavities is reduced rather than enhanced, besides beingstrongly angle-dependent.

In 1998, experiments such as those of Le Si Dang et al. (1998) and Senellart &Bloch (1999) began to show that the bottlenecked luminescence from the trap stateswas highly nonlinear with the injected laser power. Full understanding of this behaviourrequired an overview of the scattering processes availableto exciton-polaritons, whichwe treat in the next Chapter.

7.2 Near-resonant-pumped optical nonlinearities

7.2.1 Pulsed stimulated scattering

In quantum wells, many experiments have shown how the injection of excitons or freeelectron-hole pairs leads to changes in the exciton absorption spectrum. These resultfrom scattering processes between excitons (generally repulsive) and between excitonsand free carriers (which can ionize the exciton). However the experimental difficultyin studying excitons within a quantum well is that the optically accessible states arenot distinguishable by changing the angle of incidence (dueto their almost flat disper-sion), so that the inhomogeneous broadening dominates. Equally problematic is thatdespite best efforts to grow smooth-walled atomically flat interfaces, the disorder fromroughness and alloying of quantum wells produces excitonicstates which are at mostdelocalised over a few hundred nanometres, many times the exciton Bohr radius (∼15nm in GaAs) but much less than the optical wavelength. Hence these exciton statesemit and absorb in all directions. Thus it is not possible to directly observe exciton col-lision processes using conventional spectoscopy in quantum well (or bulk) samples, andone has to resort to indirect methods.


On the other hand, the clear dispersion of exciton-polaritons in semiconductor mi-crocavities allows polaritons at different angles and energies to be distinguished. Henceit became possible to perform resonant nonlinear experiments on microcavities, bypumping excitons at one angle and measuring how fast they scatter into other states.

The first pump-probe experiments on semiconductor microcavities were performedwith two beams under normal or quasi normal incidence with asan underlying objec-tive to modify the system using an intense pump pulse and to record the resulting po-larisation by measuring the reflection, transmission, absorption or scattering of a weakprobe pulse. The main goal of such preliminary investigations, such as by Jahnke et al.(1996) (an extensive list of references is given by Khitrovaet al. (1999)) was to elucidatethe mechanisms responsible for the loss of the strong-coupling regime. After this earlystage, the understanding of nonlinear optical properties of microcavities has progressedconsiderably. This progress has been mainly due to use of advanced spectroscopy tech-niques, allowing one to tune the angle, energy and time delaybetween pulses indepen-dently. The breakthrough came from an experiment performedby Savvidis et al. (2000)and discussed at length below. This experiment has evidenced the bosonic behaviour ofcavity polaritons. It has also shed much light on the main mechanisms governing opti-cal nonlinearity in microcavities. An avalanche of experimental and theoretical worksfollowed that of Savvidis revealing rich and deep physical phenomena. Most of theseresults are now being discussed.

One first indication of the peculiarities of the polariton interactions was how themeasured optical nonlinearities depended not just on the energies of the polaritons, buton their full dispersion, see the discussion by Baumberg et al. (1998). This confirmedthat angular tuning and position tuning (in which the cavitymode energy varies acrossthe sample due to a low-angle wedged thickness variation) were not equivalent. In 2000,the group at Southampton first reported experiments by Savvidis et al. (2000) whichdefinitively showed that the scattering of polaritons was influenced by pre-existing pop-ulations of polaritons—in other words that scattering could be a stimulated process.While it is well known that photons can stimulate photon emission, the process of stim-ulated scattering is much less studied.

By injecting a pump pulse at a particular angle of incidence (k) and energy (ω),the time-resolved evolution of the scattering of polaritons can be tracked using a weakbroadband probe pulse to measure the reflection spectrum at different times (Fig.7.8),as done by Savvidis et al. (2000). For particular conditions, reflectivities much largerthan 100%—corresponding to extremely large amplifications—were observed, reach-ing 10 000%. These gains persisted only while the pump-injected polaritons remainedinside the microcavity. Moreover the gain of the seeded probe pulse is extremely sen-sitive to the incident pump angle—termed themagic angle(see Fig.7.9)—and pumppower. These features are the signature of the polariton pair scattering process shownin Fig.7.9(c) in which two polaritons injected by the pump have exactly the right (k,ω)to mutually scatter sending one down to the bottom of the trap(atk=0, often called the“signal”) and the other to2k (the “idler”).

Three clear new features are shown in this experiment:


Fig. 7.8: (a) Microcavity dispersion, showing pump pulse injecting polaritons at (k,ω) and scattering to otherstates on the dispersion, (b,c) Reflection spectrum before and as pump pulse arrives showing strong gain onthe lower polariton atk=0, and (d) Time response of the gain, as published by Savvidis et al. (2000).

Fig. 7.9: (a) Resonant gain (atk=0) as the pump angle is varied, (b) exponential observed pump powerdependence of the gain and (c) schematic polariton pair scattering, from Savvidis et al. (2000).

• polaritons can scatter strongly from each other providing that both energy con-servation and momentum conservation can be simultaneouslysatisfied in the twoquasiparticles collision,

• polariton scattering can be enhanced by occupation of the final state. In otherwords, that polariton scattering can be stimulated, as expected for bosons,

• polaritons are stable at the bottom of the polariton trap, incomparison to theirlifetime (governed by their escape from the cavity).

While all bosons should possess the particle statistics which produce this effect,it has not been observed for excitons in semiconductors. This is because the excitondispersion is normally so flat that many other processes can scatter excitons (e.g. disor-der, phonons), and hencek is not a good quantum number for excitons; a macroscopic


population of a single quantum state is unlikely for excitons. On the other hand, for cav-ity polaritons, because of the tendency for bosons to occupythe same state, the gainsmeasured can exceed 106cm−1, larger than in any other material system.

In the language of nonlinear optics, such processes are saidto beparametric scat-tering processes, and are commonly observed for parametric downconversion (wherea photon at 2ω transforms into two photons atω + ǫ andω − ǫ). Polariton scatter-ing in this experiment is equivalent to a four wave mixing process (or near-degenerateparametric conversion) where two pump polaritons create a signal and idler polariton,which emerge from the sample at different angles. Because this description via fourwave mixing only deals with the incident and emitted photons, it describes nothing ofthe solid state coherence within the semiconductor microcavity and is thus a limited toolfor understanding polariton scattering and coherence.

In a similar way, using the exciton and cavity photon basis for understanding po-lariton scattering is also limiting. For zero cavity detuning, both the lower and upperpolaritons are composed of half a photon and half an exciton,however the scatteringproperties of these polaritons are completely different, due to their different energiesand the density of states into which they can scatter. In an exciton/photon basis the onlydifference is the sign with which their wavefunctions are combined.

Further evidence for the coherent nature of the lower polariton signal state atk=0is provided by coherent control experiments which show how the signal polaritons am-plified by a first seed pulse may be destroyed by a subseqently oppositely-phased resetpulse. Such experiments have been done by Kundermann et al. (2003).

Fig. 7.10: Stimulated polariton scattering peak gain as a function of temperature for different microcavitysamples, from Saba et al. (2003).

Stimulated scattering at the magic angle has been observed in many different semi-conductor microcavities, with single or multiple quantum wells, of different materials,and in patterned mesa microcavities (which have different dispersions with the extratransverse mode). The main effect of using different materials is to change the tem-perature at which the stimulated scattering process switches off (see Fig.7.10 and itsdiscussion by Saba et al. (2003)). The current model of temperature-dependence is thatbecause of the parametric process, both signal and idler polaritons together (in a joint


coherent state) generate the stimulation (see Section 7.2.2). Scattering of the idler po-laritons, which occurs at elevated temperatures thus destroys the polariton stimulation.If the idler is too close energetically to the electron-holecontinuum, then the fast scat-tering (very similar to that of excitons) resumes, which destabilises the idler polaritons.This motivates the current experimental push to building strong coupling microcavitieswhich are based on ZnSe, GaN and ZnO semiconductors since these are predicted toprovide strong stimulated scattering at room temperature.This would open the way tobuilding more complex optoelectronic devices (such as coherent interferometers andswitches) from semiconductor microcavities.

Fig. 7.11: Stimulated scattering processes in (a,b) photonic wires, from Saba et al. (2003), and (c) from twopump beams launched at equal angles either side ofk=0.

Stimulated scattering also occurs in a variety of geometries. For instance, when aplanar microcavity is patterned into photonic wires, quantization perpendicular to thewire produces a nested series of lower polariton dispersioncurves fork along the wire(see Fig.7.11 and discussion by Dasbach et al. (2003)). These produce a new range ofpossibilities for stimulated scattering, involving more than one branch of the dispersion,with final idler states existing at lower energies thus reducing the scattering which con-strains the temperature of operation. Stimulated scattering of polaritons is not limitedto the magic angle condition in which the two initial polaritons are at the same (k,ω).It can also be observed for polaritons which are in initiallydifferent states, for instanceon either side near the bottom of the polariton trap (Fig.7.11b), see the discussion byRomanelli et al. (2005). All these schemes have suggested novel ways in which to effi-ciently produce entangled photons as part of signal and idler beams.

One clear signature of the stimulated scattering process (to be discussed in detailin the theory of Section 7.3) is the rigid blue-shift of the whole lower polariton disper-sion as it becomes macroscopically occupied. This results from a self-scattering term tothe energy, but it has the effect of modifying dynamically the tuning of incident lasersand dispersion as scattering occurs. This rigid energy shift of the polariton dispersionis only the first order effect, and is proportional to the total population of polaritons.


Fig. 7.12: (a) Blue shifted lower polariton disperion (dashed to solid thin lines) subsequently produce stimu-lated scattering. The macroscopic signal and pump polariton occupation (•) generates new off-branch polari-tons (), observed at the indicated output angle (dash-dot vertical) as (b) new peaks in emission (arrows).

A second order term means that occupation of the dispersion changes the shape of thedispersion—a highly nonlinear process. One effect of this is that when pump and sig-nal polaritons become macroscopically occupied, new scattering processes appear forwhich one of the final states is off-branch (see Fig.7.12 and the discussion by Savvidiset al. (2001)). The polariton dispersion is distorted, and produces a flat region aroundk=0, whose onset also signals the destabilisation into spatial solitons (mentioned in Sec-tion 6.3.1). Thus there remain many confusing and novel features about the stimulatedscattering process in both space and time that need to be further explored.

7.2.2 Quasimode theory of parametric amplification

In this section, we address the theories of CW parametric scattering in the dynamicregime. This identifies, at each moment in time, the transient eigenstates of the pairpolaritons which independently experience the gain or loss. We assume a slowly varyingpolariton amplitude (a reasonable approximation for narrow spectral linewidth cavities),and also work in the limit of negligible pump depletion (i.e.at low probe powers). Inthis case the equations governing the slowly-varying envelope of signal (S) and idler(I) can be written

∂S

∂t= −γSS − ΛI∗ (7.1)

∂I∗

∂t= −γII∗ − Λ∗S (7.2)

whereΛ(t) = iV P 2(t)eiνt accounts for the coupling. HereV is the exchange in-teraction between polaritons,P is the dynamic pump polariton occupation, andν =2ωP − ωS − ωI is the frequency mismatch from the magic angle condition. Welookfor solutions corresponding to gain:S, I∗ ∝ eqt. Solving the determinant of Eq.(7.2)produces the two solutions for the damping:


Fig. 7.13: Quasimode calculations as a function of real time(ps) for (a) eigenvaules ofM,N , (b) mixingparameterψ, (c) fractional amount of signal and idler components inM , (d) dynamics of eigenmodesM ,Nand (e) of signal and idler, when the probe pulse is att = −1ps, pump att = 0ps.

γ± = −γS + γI2

±√

α2 + |Λ|2 (7.3)

with α = (γS − γI)/2. These solutions are time dependent, withq± < 0 away fromthe pump pulse corresponding to the individual damping of signal and idler. They re-pel strongly when the pump arrives, to produce transient gain (q+ > 0, Fig.7.13a).The eigenvectors of these solutions correspond to the two mixed modes (M,N ) whichexperience these gains.

M = C

−eiφS + eψI∗

(7.4)

N = C

eψS + eiφI∗

(7.5)

where we have definedsinhψ = α/ |Λ|, φ = arg(Λ), and the normalisationC =1/

√1 + e2ψ. This mixed complex transformation of the signal and idler is controlled

by the phase mismatch,φ = νt, and a mixing parameter,ψ(t). The modeM is ampli-fied when the pump pulse arrives, while the modeN is de-amplified. The gain of thesemodes is given byq± = λ±Λ coshψ, with the average damping,λ = (γS +γI)/2.Theincident probe couples into both modes, giving new instantaneously decoupled dynam-ical equations:


∂M

∂t= q+M − CeiφSprobe(t) (7.6)

∂N

∂t= q−N − CeψSprobe(t) (7.7)

In the vicinity of the pump pulse, the modesM,N contain roughly equal admixturesof the signal and idler (Fig.7.13c): in other words, when thepump is present, the truemodes of the system are notS, I butM,N . The dynamics of the quasi-uncoupled modesand the signal and idler are shown in Fig.7.13d,e for a probe pulse which is 1ps beforethe pump, and with damping of signal and idler,γs,i=0.2,0.4meV corresponding to theexperiments.

From these equations it can be seen that the amplification of the population of po-laritons in theM -mode is roughly given by:

∣

∣

∣

∣

Mout

Min

∣

∣

∣

∣

2

= exp 2q+T = exp 2 |Λ|T = exp 2V IpumpT (7.8)

whereT is the pulselength andIpump is the pump power. This recovers the experimentalresult. It is also not what might be intuitively expected from a pair scattering processwhich in an uncoupled system would have a gain proportional to thesquareof the pumpintensity. The completely mixed nature of signal and idler polaritons is what makes theparametric amplification so sensitive to dephasing of the idler component.

7.2.3 Microcavity parametric oscillators

While the multiple effects of stimulated scattering are clearest for pulsed excitation, theyare also observed in continuous wave (CW) excitation. A pumpbeam incident at themagic angle first generates spontaneous parametric pairs tosignal and idler states, whichthen act as the seed for further stimulated scattering, see Fig.7.14 and its discussionby Baumberg et al. (2000). After this threshold (where the signal polariton populationexceeds unity), scattering then proceeds exponentially with pump power until saturationoccurs. A set of spectra obtained in this configuration by Stevenson et al. (2000) atvarious angle of detection and at different powers is displayed on Fig. (7.15), where thefeatures atk = 0 completely dominate above threshold.

The system behaves as amicro-parametric oscillator(µOPO), an integrated equiv-alent to the cm- to m-scale bulk parametric oscillators (normally based on parametricdownconversion which is a three-photon and not a four-photon process). Typical thresh-olds for this micro-OPO device are in the 10mW range, although the physical gainlength is some10 000 times smaller than conventional OPOs.

The output of theµOPO is coherent, narrow spectral linewidth (≈ 1GHz), andemitted into a narrow angular beam (width∼ 5). While the signal output phase is in-dependent of the pump laser phase, the sum of signal and idlerphases is locked to thatof the pump (as in a normal parametric oscillator). However there are some peculiarnovel features of theµOPO system. One of these is that because of the energy shiftspossible in the lower polariton dispersion as it becomes occupied, the device can organ-ise itself to optimise the stimulated scattering. Thus, while pulsed stimulated scattering


Fig. 7.14: (a) Geometry for micro-parametric oscillator with (b) spectrum atk=0 and (c) power dependence,from Baumberg et al. (2000).

only occurs close to a magic angle, in theCWcase the device adjusts to produceµOPObehaviour over a wide range of pump angles (Fig.7.16).

One result of the twin photon production of signal and idler is that they are quantummechanically correlated, or entangled. This can be most simply understood from theirsimultaneous origin with correlated phases from collidingpump polaritons, even thoughthey emerge with different energies in different directions. Such correlations can beextracted from experiments in which the two beams are mixed with a local oscillator ontwo balanced photodiodes, as has been done by Messin et al. (2001). Theoretically suchexperiments can only slightly (by a few %) circumvent the quantum noise limit, due tothe degradation of the perfect polariton correlation when they convert into photons onexiting the sample.

More recently, there have been proposals by Ciuti (2004) andSavasta et al. (2005)for generating more useful entangled photon pairs from semiconductor microcavity po-lariton pair scattering, using geometries in which both signal and idler are lower inenergy than the exciton reservoir. However it remains a challenge to generate brighthigh-efficiency correlated photon beams from these devices.

The effect of disorder is getting increasing attention in recent research both in theresonant experiments studied in this Chapter but also in theoff-resonant case exposedin Chapter 8. In the case of theµOPO, a pseudo-periodic potential due to strain—which is present in every sample from CdTe to GaAs—results inlocal differences inthe refractive index of the microcavity. Due to this disorder—or so-called “photonicpotential”—the formation of the signal of theµOPO is strongly influenced by the min-


Fig. 7.15: Spectra observed by Stevensonet al. (2000) in the resonant CW pumpingof a microcavity, as a function of power.The system is below threshold in (a), ap-proching threshold in (b) and increasinglyabove threshold in (c), (d) and (e). Eachfigure displays spectra collected at differentangles. The strong feature around1.456eVis induced by the laser. Below threshold, ex-citations relax to states close to the pump.Above threshold strong emission is ob-served at the signal and idler states cor-responding to0 and 32, respectively.Thek = 0 emission quickly becomes by farthe dominant one (the importance of idler isseen in the insets).

ima of the potential wells. Sanvitto et al. (2006) recently reported that when the poweris increased, the signal occupies different regions in realspace.94 This might prove to bea key point in the formation of OPO and related physics.95 For instance, Sanvitto et al.(2005) reported that locally, theQ factor could increase up to30 000 from a nominalvalue of10 000 just by restricting to a region of 5µm2.

In a final review of microcavity parametric scattering, we note that recently aµ-OPO regime has been observed in the weak-coupling regime by Diederichs & Tignon(2005). In these devices, three microcavities are stacked such that their cavity photonscan mix between them, while quantum wells still provide the nonlinear scattering pro-cess. Instead of relying on the distorted dispersion of the lower branch polaritons, thethree photon cavity mode branches provide a phase matched photon-stimulated emis-sion for all pump, signal and idler atk=0. Hence in a similar way to the strong-coupled

94Disorder is an open door to richer physics in relatively wellunderstood systems. In the case of the OPO,for instance, “bistability” (discussed in further detailsin Section 7.4.3) manifests strikingly by “switching”the OPO on and off and results in depopulating some regions asothers get populated under the influence ofpotential traps in the disorder.)

95Effect and potential importance of disorder in the case of spontaneous condensation will be discussedin Chapter 8 on page 289. The intimate links between these twolimits as regard to disorder are not yet fullyunderstood.

RESONANT EXCITATION CASE AND PARAMETRIC AMPLIFICATION 263

Fig. 7.16: (a) Power dependence of signal and idler near the magic angle, and (b) threshold for signal atdifferent pump angles, from Butte et al. (2003).

microcavities, these cavities lower the emission energiesfor the participating modes be-low that of the dissipative excitons, however at the expensehere of returning to photonand not polariton quasiparticle scattering processes.

7.3 Resonant excitation case and parametric amplification

This section presents the theoretical description of microcavity emission for resonantexcitation. We focus on the experimental configuration of this Chapter where stimu-lated scattering is observed due to formation of a dynamicalcondensate of polaritons.Pump-probe and cw experiments are both described. Dressingof the polariton disper-sion because of inter-condensate interaction is discussedas well as its main consequencewhich is the bistable behaviour of this system. The semi-classical and the quantum the-ories of these effects are presented and their results analysed.

7.3.1 Semi-classical description

We describe parametric amplification experiments using rate equations.96 The advan-tages of such a description with respect to the parametric amplifier model (classical orquantum), which will be presented next, is that it allows us to account for stimulatedscattering and to include easily all types of interactions affecting exciton-polariton re-laxation. Its disadvantage is that dispersion dressing of polaritonic energies—an impor-tant feature of parametric amplification—cannot be easily accounted for in this model.In the resonant configuration, one can single out states where energy-momentum trans-fer are very efficient and which dominate the dynamics. We assume the simplest caseof a three-level model:97 the ground orsignal state, thepump stateand theidler state(cf. Section 7.3.4). The names arise from similar physics innonlinear optics. The main

96Rate equations of populations are closely linked to so-called “Boltzmann equations”, which will bestudied in Chapter 8 where relaxation of polaritons will be the central theme of study.

97In the complete picture including all states to which we shall return in chapter 8, the generic Boltzmannequation (see footnote 96) for a state of wavevectork is given by Eq. (8.35) on page 301. The equation to besolved describing the polariton dynamics is formed by the ensemble of Boltzmann equations written for allallowed values of the in-plane wavevector. It can be solved numerically, choosing suitable initial conditions.These conditions are, for a pump-probe experiment,n0(0) = nprobe, nkp (0) = npump andPk = 0. For


loss processes for these states are radiative losses and elastic scattering processes drivenby disorder, which can both be included in the same loss constant even if their nature isvery different. The radiative loss means disappearance of the particles, while the disor-der scattering implies transfer of a particle towards otherstates that are neglected in thismodel. The interaction with phonons in this framework is similar to the disorder inter-action, and it can also be included as a loss with its appropriate constant. The phononcontribution is often negligible at low temperatures in typically-used cavities. It maycause a significant broadening of the polariton states in experiments performed at hightemperatures. The broadening, or loss parameter, can be written as:

1

Γk=

xk∆ + Γphonons

+(1 − x)

Γc(7.9)

where∆ is the exciton inhomogeneous broadening,Γphonons is the phonon-inducedbroadening andΓc is the cavity photon broadening. At low temperature,Γphonons ≪∆. Moreover, in most of the cavity samples studied experimentally, ∆ ≈ Γc, whichyieldsΓ0 ≈ Γp ≈ Γi = Γ. In this framework, the system can be described by a set ofthree coupled equations:

n0 = P0 − Γn0 −1

2αn0ni(np + 1)2 +

1

2α(n0 + 1)(ni + 1)n2

p (7.10a)

np = Pp − Γnp + 2αn0ni(np + 1)2 − 2α(n0 + 1)(ni + 1)n2p (7.10b)

ni = Pi − Γni −1

2αn0ni(np + 1)2 +

1

2α(n0 + 1)(ni + 1)n2

p (7.10c)

where

α =2π

~

|M |2πΓ/2

whereM is the polariton-polariton matrix element of interaction which is here approx-imately equal to one fourth of the exciton-exciton matrix element of interaction. Thissystem of equations can be easily solved numerically. Moreover, if one considers the cwexcitation case,P0 = P2p = 0, this givesn0 = n2p. The system (7.10) thus becomes:

n0 = −Γn0 −1

2αn0(np + 1)2 +

1

2α(n0 + 1)n2

p +1

2αn2

p (7.11a)

np = P − Γnp + 2αn20(np + 1)2 − 2α(n0 + 1)2n2

p (7.11b)

7.3.2 Stationary solution and threshold

In the stationary regime,n0 = np = 0. Before proceeding further with the formalism,we have to discuss how to define correctly the threshold condition for amplification in

cw experiments, these initials conditions arenk(0) = 0 andPkp(t) = P0. In Chapter 8, cylindrical symme-try of the distribution function will be assumed. In the parametric amplification experiments of interest in thisChapter, the resonant excitation conditions break this symmetry and a two-dimensional polariton distributionfunction should be assumed. However, just below and above the amplification threshold a good description ofthe ground-state population can be performed assuming onlythe three states of the text.


the stationary case. Very often, it is believed that a good empirical criterion is that thepopulation of a given state reaches one. Indeed, the evolution equation for the ground-state population formally reads:

n0 = Win(n0 + 1) −Woutn0 . (7.12)

Win is supposed to include all channels used by incoming polaritons andWout allchannels for their departure from the ground state. The+1 in brackets corresponds tothe spontaneous scattering process,n0 in brackets describes the stimulated scatteringand−Woutn0 the loss term. Therefore, the conditionn0 = 1 means that the stimulationterm is as large as the spontaneous scattering term and that the amplification threshold isreached. This point of view is, however, quite misleading. The equation for the ground-state population can indeed be rewritten as

n0 = n0(Win −Wout) +Win (7.13)

Thus, the threshold is given by the conditionWin −Wout = 0. Eq. (7.13) yields in thiscase:

−Γ − αn0(np + 1)2 + α(n0 + 1)n2p = 0 (7.14)

which implies:

n0 =αn2

p − Γ

α(2np + 1). (7.15)

n0 is a population so it should be positive or zero. In the lattercase

np =

√

Γ

α. (7.16)

Below threshold,np ≈ P/Γ, which gives:

Pthres = Γ

√

Γ

α= Γ

~Γ

2|M | (7.17)

Using the conventional threshold condition leads to a similar formula for the thresh-old power (see Exercice 7.1). It is noteworthy that using twoapparently independentthreshold conditions, one recovers exactly the same value of the amplification thresh-old. Assuming an exciting laser spot size of 50 microns,~Γ = 1meV and for the typicalGaAs parameters,Pthres ≈ 106Γ ≈ 50µW. This is in good agreement with experimen-tal data.

Exercise 7.1Assumingnp ≫ n0, find the solution of the system (7.11). Find the thresh-old assuming as a threshold conditionn0 = 1.

7.3.3 Theoretical approach: quantum model

Our starting point is the Hamiltonian (5.163). We neglect interactions with phonons orfree carriers. The framework used historically to describethis configuration, e.g., by


Louisell et al. (1961) for the general problem and by Ciuti etal. (2000) or Ciuti et al.(2001), for microcavities, is the one of the Heisenberg formalism rather than the densitymatrix approach. We will therefore follow this trend.

To obtain the equation of motion for polariton operatorsak anda†k, we write theHeisenberg equation:

i~dakdt

= [ak, H ] = ELP(k)ak +∑

k,k′′

Eintk,k′,k′′a

†k′+k′′−kak′ak′′ + P (k) (7.18a)

i~da†kdt

= [a†k, H∗]= E∗

LP(k)a†k −∑

k,k′′

Eintk,k′,k′′ak′′ak′a†k′+k′′−k + P (k) (7.18b)

whereELP is the lower-polariton branch dispersion relation

Eintk,k′,k′′ =

1

2

(

Vk′,k′′,k−k′ + Vk′,k′′,k′′−k

)

(7.19)

andP (k) the polarisation amplitude induced by an external pumping field.

7.3.4 Three-level model

A three-level model has been proposed by Ciuti et al. (2000) for the description of theSavvidis-Baumberg experiment described in Section 7.2.1.Its starting point is Eqs. (7.18),considering only the three most important states, namely the pumped statekp, theground or signal statek0 and the idler state2kp. The authors assumed these three statesto be coherently and macroscopically populated. In other words, they assumed thesestates to behave as classical coherent states and they replaced the operatorsa0, akp ,a2kp and their adjoint byc-numbers. This Ansatz was proposed in the 1950s by Bo-goliubov (1947) to describe superfluids (see also Bogoliubov’s (1970) textbook). Hediagonalised a Hamiltonian equivalent to Eq. (5.163), considering the existence of amacroscopically occupied ground state (the superfluid). Heassumed that only interac-tions involving the ground state were important and also proposed to neglect fluctuationsof the ground state because of its macroscopic occupation. His argument is that for theground state[a, a†] ≫ N whereN is the ground state population. Therefore, the non-zero value of the commutator can be neglected and the ground-state operators can bereplaced by complex numbers. Ciuti et al. proposed a similarapproximation but threecondensates instead of one were assumed.

In this Section, we consider that only the pumped state operators reduce to com-plex numbers, keeping the operator nature of signal and idler. In this framework, thesystem (7.18) can be reduced to just three equations:

−i~a0 = ELP(0)a0 + Einta†2kpP 2kp + Pprobe(t) , (7.20a)

−i~Pkp = ELP(kp)Pkp + EintP∗kpa0a2kp + Ppump(t) , (7.20b)

−i~a†2kp = ELP(2kp)a†2kp + E∗inta0P

2kp (7.20c)

where


ELP(0) = ELP(0) + 2V0,kp,0|Pkp |2 , (7.21a)

ELP(kp) = ELP(kp) + 2Vkp,kp,kp |Pkp |2 , (7.21b)

ELP(2kp) = ELP(2kp) + 2V2kp,kp,0|Pkp |2 . (7.21c)

and

Eint =1

2

(

Vkp,kp,kp + Vkp,kp,−kp) . (7.22)

The advantage of this formalism with respect to rate equations of populations inSection 7.3.1 is that it allows one to account for the energy renormalisation processesdriven by inter-particle interactions. Here a blue shift ofthe three states consideredis induced by the pump intensity. Replacing all operators bycomplex numbers, thisequation system can be solved numerically for any pump and probe configuration.

We now consider the steady-state excitation case where a stationary pump of fre-quencyωp excites the system, without a probe. This pump drives the pump polarisationgiven by:

Pkp(t) = Pkpeiωpt (7.23)

with Pkp ∈ C. The system of Eqs. (7.21) reduces to two coupled equations:

−i~a0 = ELP(0)a0 + Einta†2kpP 2kpe

2iωpt (7.24a)

−i~a†2kp = −E∗LP(2kp)a†2kp − Einta0P

∗2kp e

−2iωpt (7.24b)

We define:

ω0 =1

~ℜ(ELP(0)) , ωi =

1

~ℜ(ELP(2kp)) ,

Γ0 =2

~ℑ(ELP(0)) , Γi =

2

~ℑ(ELP(2kp)) .

and introduce the two rescaled quantities:

a0 = a0e−iω0t , a†2kp = a†2kpe

iωit

andβ = |β|e2iϕp = EintP

2kp .

The two previous equations become:

−i ˙a0 = −Γ0

2a0 + βa†2kpe

i(2ωp−ω0)t (7.26a)

−i ˙a†2kp = −Γi

2a†2kp + β∗a0e

i(2ωi−ωp)t (7.26b)

This equation is nothing but a quantum mechanical equation for parametric processesfirst written and solved by Louisell et al. (1961). Replacingall quantum operators inthis equation system by complex numbers is equivalent to treating the classical para-metric oscillator studied in the last century by Faraday andLord Rayleigh, as has been


pointed out by Whittaker (2001). This equation system has been widely studied in re-cent decades. It can be solved in the Heisenberg representation in the time domain, asdetailed in the expositions of Mandel & Wolf (1961) and Louisell et al. (1961) or in thefrequency domain, as discussed by Loudon (2000) and in the case of microcavities byCiuti et al. (2000).

For simplicity, we assume that the resonance conditions aresatisfied and that theloss coefficients are the same for signal and idler:

ω0 + ωi − 2ωp = 0 , Γ0 = Γp = Γi = Γ .

Eqs. (7.26) become

˙a0 = −Γ

2a0 + iβa†2kp , (7.27a)

˙a†2kp = −Γ

2a†2kp + iβ∗a0 . (7.27b)

Eq. (7.27a) implies

˙a†2kp =1

iβ( ˙a0 +

Γ

2) , (7.28a)

¨a0 + Γ˙a0 +(Γ2

4+ |β|2

)

a0 = 0 (7.28b)

The solutions of the characteristic equation associated with Eq. (7.28b) are

r± = −Γ

4± |β| (7.29)

The solutions of the system (7.28) are

a0(t) = e−Γ2 t(

a0 cosh(|β|t) − ia†2kp sinh(|β|t)ei2ϕp)

, (7.30a)

a2kp(t) = e−Γ2 t(

a†2kp cosh(|β|t) + ia0 sinh(|β|t)e−i2ϕp)

, (7.30b)

Note that right-hand side of Eq. (7.30a) is back in terms ofa rather thana (since att = 0,operators coincide).

For t > 1/|β|, cosh andsinh can be approximated by exponentials with positiveargument. Therefore:

a0(t≫ |β|−1) =1

2e(|β|−Γ/2)t

(

a0 − ia†2kpei2ϕp

)

, (7.31a)

a2kp(t≫ |β|−1) =1

2e(|β|−Γ/2)t

(

a†2kp + ia0e−i2ϕp

)

. (7.31b)

The “particle number” operator for the signal is:

a†0(t)a0(t) =1

4e(2|β|−Γ)t

(

a†0a0 +a†2kpa2kp − i(a†0a†2kpei2ϕp −a2kpa0e

−2iφp))

(7.32)

TWO-BEAM EXPERIMENT 269

If the signal and idler states are initially in the vacuum state, the average number ofparticles is therefore:

〈a†0(t)a0(t)〉 = 〈0, 0| a†0(t)a0(t) |0, 0〉 =1

4e(2|β|−Γ)t . (7.33)

However

〈a0(t)〉 = 〈0, 0| a0(t) |0, 0〉 = 0 . (7.34)

Eqs. (7.33)–(7.34) show that a ground state, initially symmetric in the phase space,will have its population growing exponentially while its amplitude remains zero. Thisshows that the symmetry of the ground state is not broken by the pumping laser. Toillustrate our purpose we consider that the system is initially in a state other than thevacuum.

7.3.5 Threshold

The threshold condition to stimulated scattering is given by Γ = 2|β|, that is,

|Pkp |2 =~Γ

2Eint(7.35)

With such a pump polarisation, the energy shift of the signalat threshold is equal to thepolariton linewidth. This theoretical result is in a good agreement with available experi-mental data. It is instructive to compare the criterion (7.35) with the threshold conditionobtained in Section 7.3.2 from a population rate equations (Boltzmann equations).

The relation between the pumping power and the coherent polarisation isΓ|Pkp |2 ≈P and the threshold condition for the pump power is thus:

P =~Γ2

2Eint(7.36)

If one assumes, as in Section 7.3.2, that the broadeningΓ is independent of the wavevec-tor, and thatEint ≈ |M |, the polariton-polaritonmatrix element of interaction, Eq. (7.36)becomes:

Pthres = Γ~Γ

2|M | (7.37)

This value is exactly the same as the one obtained in Section 7.3.2, illustrating theequivalence of the semi-classical and quantum models in this aspect.

7.4 Two-beam experiment

7.4.1 One beam experiment and spontaneous symmetry breaking

We assume that a cw pump laser excites the sample, together with an ultrashort probepulse which seeds the probe state. Therefore, att = 0, the probe state is a coherent


state|α0〉 with α0 = |α0|eiϕ0 . The idler state is initially unpopulated (vacuum state).The initial state of the signal⊗idler system is denoted|α0, 0〉. With such initial states:

〈a†0(t)a0(t)〉 =1

4(1 + |α0|2)e(2|β|−Γ)t , (7.38)

〈a0(t)〉 =1

2α0e

(|β|−Γ/2)t . (7.39)

The phaseϕ0 of the order parameter does not depend on the value of the pumpphaseϕp and is determined by the probe phase. We define the coherence of the systemas

η =|〈a0(t)〉|2

〈a†0(t)a0(t)〉(7.40)

We find:

η =|α0|2

1 + |α0|2(7.41)

This coherence is constant for any phase relationship between pump and probe. It isclose to one if the probe introduces a coherent seed population much larger than one. Inthis case, the symmetry of the system is broken by the probe.

We have seen in the previous paragraph that the wavefunctionof the initially sym-metrical system will remain symmetrical during its temporal evolution. Now we aregoing to artificially break this symmetry, assuming that theinitial state is a coherentstate characterised by a small but finite amplitude. Since the signal and idler are nowcompletely identical, we consider that they are both initially in a coherent state withthe same amplitudeα0 = |α|eiϕ0 , α2kp = |α|eiϕ2kp but different phases. The averagesignal polarisation and population are:

〈a0(t)〉 =1

2e(|β|−Γ/2)t

(

α0 − iα∗2kp

)

=1

2|α|e(|β|−Γ/2)teiϕ0

(

1 − ei(2ϕp−ϕ0−ϕ2kp ))

. (7.42)

The signal polarisation strongly depends on the phase relation between pump, probeand idler. Namely, it vanishes if

2ϕp − ϕ0 − ϕ2kp = 0 mod 2π (7.43)

and achieves its maximum if

2ϕp − ϕ0 − ϕ2kp = π mod 2π (7.44)

This last equation is the phase-matching condition for parametric oscillation to takeplace. A similar equation can be written for the signal population:

〈a†0(t)a0(t)〉 =1

4e(2|β|−Γ)t

(

1 + |α0|2 + |α2kp |2 − i(α∗0α

∗2kpe

i2ϕp − α2kpα0e−i2ϕp)

)

=1

4e(2|β|−Γ)t

(

1 + |α|2(

1 − cos(2ϕp − ϕ0 − ϕ2kp ))

)

. (7.45)

One can see that the population has a minimum (but does not vanish) if condi-tion (7.43) is fulfilled, while it has a maximum if condition (7.44) is fulfilled.


The coherence then reads:

η =2|α|2(1 − cos(2ϕp − ϕ0 − ϕ2kp))

1 + 2|α|2(1 − cos(2ϕp − ϕ0 − ϕ2kp))(7.46)

An initial coherent state is expected to appear because of the system fluctuations. Itis hard to describe such fluctuations theoretically and to quantify |α|. It is, however, clearthat the system will choose to grow on the most “favourable” fluctuation, respecting theconstructive phase-matching condition (7.44). It is essential to note that the phase ofthe signal and idler is not fixed by the phase of the pumping laser together with thephase matching condition, as was proposed by Snoke (2002). One can see that onlythe quantity is actually fixed. Therefore, there is a well defined phase relation betweensignal and idler but all the values of the signal phase are equivalent for the system.This signal phase is not a priori determined by the pump phase, but it is randomlychosen by the system from experiment to experiment. Choosing its phase, the system“breaks its symmetry”. This symmetry breaking effect is common to the laser phasetransition, superconducting phase transition, and Bose Einstein Condensation (BEC). Tosummarise, it is a common feature of phase transitions induced by the bosonic characterof the particles involved. This is not a BEC, however, because, as is the case with lasers,it is an out-of-equilibrium phase transition, so that, for example, a chemical potentialcannot be defined.

7.4.2 Dressing of the dispersion induced by polariton condensates

As already mentioned, stimulated scattering experiments have shown new emissionpeaks surprisingly far from the polariton dispersion. This“off branch emission” is in-duced by strong interactions taking place between macroscopically populated states,which are the pump, signal and idler states. Interaction between these states is not onlya perturbation in the sense that it leads to a dressing of the polariton dispersion. Ciutiet al. (2000) have provided the theoretical interpretation. In this Section we briefly sum-marise this theory. We shall consider all scattering processes which involve two macro-scopically populated states as initial states, and, as finalstates, one state on the polaritonbranch and one off-branch state. We shall require conservation of energy and wavevec-tor. As an example of such a transition, one can consider scattering events having twopump polaritons as initial states. The wavevector and energy conservation laws give inthis case:

kp, kp → k, 2kp − k (7.47a)

2ELP(kp) = ELP(k) + Eoffpp (2kp − k) (7.47b)

Eq. (7.47b) defines a new dispersion branchEoffpp . Appearance of a polariton on

this branch is possible because the corresponding scattering event is fast enough (seeSection 7.2.1). Four other branches corresponding to signal-pump, pump-idler, signal-signal and idler-idler scattering can be defined. The observation of off-branch emis-sion, which can be associated with the existence of macroscopically populated polari-ton sates, is a characteristic feature of phase transitionsof weakly interacting bosons.


Its experimental observation confirms once again that microcavity polaritons are quasi-particles suitable for the observation of collective bosonic effects.

7.4.3 Bistable behavior

As discussed in the previous section, an important feature of the resonant excitationscheme is the renormalization of the polariton energies. This renormalization is ob-served in the polariton emission, but it plays also a key rolein the absorption of thepump light. Two situations can be distinguished. If the laser is below the bare polari-ton energy, the absorption is simply reduced by the pump induced blue shift. If thelaser is above the bare polariton energy, the pump gets closer to the absorption energybecause of the blue shift which in turn increases the shift which enhances the absorp-tion and so on. There are two different regimes. At low pumping the pump energy re-mains above the renormalized polariton energy. At higher pumping the polariton energyjumps above the pump energy which results in a dramatic increase of the population ofthe pumped state. The threshold between the two regimes is called a bistable thresholdsince it comes from the existence of two possible polariton populations and energies forthe same pump energy and intensity. Bistability in stronglycoupled microcavities waspredicted in 1996 by Tredicucci et al. (1996) and observed eight years later by Baaset al. (2004). This threshold yields a very abrupt jump of thepopulation of the pumpedstate and can initiate the parametric scattering process. The coexistence of two differentnonlinear physical effects (bistability and stimulated parametric scattering) make thisconfiguration extremely rich to analyze as shown by Gippius et al. (2004) and Whit-taker (2005). In the following we present the formalism describing the pumping of asingle state which leads to bistability only.

We can use Eqs. (7.20b) and (7.23) describing the dynamics ofthe pump state, butwithout the coupling term with idler and signal, in other words the nonlinear term of theHamiltonian reduced toa†kpa

†kpakpakp :

˙Pkp = i(ωkp − ωp + iΓkp)Pkp + i2

~Vkp,kp,kp |Pkp |2Pkp +Ap (7.48)

This last quantity should be zero in the stationary regime. Therefore multiplying (7.48)by its complex conjugated and replacingPkp by the population of the pump state,and|Pkp |2 by the pump intensity, one gets:

[

(

(ωkp − ωp) +2

~Vkp,kp,kpNp

)2

+ Γ2kp

]

Np = Ip (7.49)

The plotNp versusIp is shown in Fig. 7.17 from Baas et al. (2004). The dashedpart of the curve is unstable. The plot clearly exhibits the hysteresis cycle taking placewhen the pump intensity is successively increased and decreased. The position of thetwo turning points can be found from the conditiondIp/dNp = 0 which yields:

3(2

~Vkp,kp,kp)

2N2p + 4(ωkp − ωp)

2 + Γ2kp = 0 (7.50)

Bistability takes place if there are two positive differentsolutions for the quadratic equa-tion (7.50) which gives the following condition:


Fig. 7.17: Bistability of the polariton amplifier

ωp > ωkp +√

3Γkp (7.51)

Eq. (7.51) says that the only condition to get bistability isto pump about one linewidthabove the bare polariton energy. Of course in a real situation, the pumping cannot be toohigh in energy since it would require an enormous pump intensity to reach the bistablethreshold. If Eq. (7.51) is fulfilled the solutions for the turning points read:

Np =2(ωp − ωkp) ±

√

(ωp − ωkp)2 − 3Γ2

kp

6~Vkp,kp,kp

(7.52)

The solution with the minus sign corresponds to turning point with the higher pump-ing, namely the one which can be found increasing the pumpingpower. The solutionwith the plus sign corresponds, on the other hand to the turning point which can befound decreasing the pumping power. Therefore the− solution of Eq. (7.52) can beinjected into Eq. (7.49) in order to find the pumping threshold intensity.

8

STRONG COUPLING: POLARITON BOSE CONDENSATION

In this chapter we address the rich physics revolving about the notion ofa Bose-Einstein condensation of exciton polaritons, for instancepolariton lasing. From these discussions it is clear that Bosecondensation at room temperature is a practical goal, and could be usedfor of a new generation of opto-electronic devices. The way toward thisbreathtaking perspective and the most serious obstacles onthis way,which are not yet overcome, are addressed in this Chapter.

INTRODUCTION 275

8.1 Introduction

As discussed in chapter 5, cavity polaritons, although theyare a mixture of excitonsand photons, behave as bosons in the low-density limit. One would therefore expectthem to exhibit bosonic phase transitions such as Bose-Einstein condensation (BEC).This fascinating possibility would represent the first clear example of Bose condensa-tion in a solid state system.98 The effective realisation of this effect also opens the wayto the realisation of apolariton laserwhich we shall introduce as a new type of coher-ent light emitter.Imamo glu & Ram (1996) were the first to point out how the bosoniccharacter of cavity polaritons could be used to create an exciton-polariton condensatethat would emit coherent laser light. The buildup of a ground-state coherent populationfrom an incoherent exciton reservoir can be seen as a phase transition towards a Bosecondensed state, or as a polariton-lasing effect resultingfrom bosonic stimulated scat-tering. This conceptual proposal was followed in 2000 by theobservation of polaritonstimulated scattering in resonantly-pumped microcavities as shown in chapter 7. A po-lariton laser is, however, different from a polariton parametric amplifier. In the formercase, the system is excited non-resonantly—optically or electronically—resulting in acloud of electrons and holes that form excitons, which subsequently thermalize at theirown temperature mainly through exciton-exciton interactions. They reduce their kineticenergy by interacting with phonons and relax along the lowerpolariton branch. Theyfinally scatter to their lower-energy state, where they accumulate because of stimulatedscattering. The coherence of the condensate therefore builds up from an incoherent equi-librium reservoir and the associated phase transition can be interpreted as a BEC. Oncecondensed, polaritons emit coherent monochromatic light.As the light emission by apolariton quasi-condensate is spontaneous, there is no population inversion conditionin polariton lasers, absorption of light does not play any role and ideally there is nothreshold for lasing. Concerning this latter point, the argument is that it is sufficient tohave two polaritons in the ground state to create a condensate, which will subsequentlydisappear with the emission of two coherent photons. Moreover, because of the smallpolariton mass, critical temperatures larger than 300 K canbe achieved. All these char-acteristics as a whole make polariton lasers ideal candidates for the next generation oflaser-light emitting devices.

8.2 Basic ideas about Bose-Einstein condensation

8.2.1 Einstein proposal

A fascinating property of bosons is their tendency to accumulate in unlimited quantity ina degenerate state. Einstein (1925) made an insightful proposition based on this propertyin the case of an ideal Bose gas which led him to the predictionof a new kind of phasetransition. Let us considerN non-interacting bosons at a temperatureT in a volumeRd,whereR is the system size andd its dimensionality. The bosons are distributed in energyfollowing the Bose-Einstein distribution function:

98Superconductivity works in the BCS limit and as such does notqualify as Bose condensation.

276 STRONG COUPLING: POLARITON BOSE CONDENSATION

Einstein (1879–1955) andBose(1894–1974) shortly after the publication of “Plancks Gesetz und Lichtquan-tenhypothese” in Zeitschrift fur Physik,26, 178 (1924), written by Bose (1924) and translated by Einstein,who would latter complement it with the concept of what is nowknown as “Bose-Einstein” condensation,although Bose did not take part in this aspect of the theory.

fB(k, T, µ) =1

exp

[

E(k) − µ

kBT

]

− 1

(8.1)

wherek is the particled–dimensional wavevector,E(k) is the dispersion function of thebosons,kB is Boltzmann constant andµ is the chemical potential, which is a negativenumber if the lowest value ofE is zero.−µ is the energy needed to add a particle to thesystem. Its value is given by the normalisation condition for the fixed total number ofparticlesN ,

N(T, µ) =∑

k

fB(k, T, µ) . (8.2)

Before going to the thermodynamic limit,99 it is convenient to separate the groundstate from the others:

N(T, µ) =1

exp

[

− µ

kBT

]

− 1

+∑

k,k 6=0

fB(k, T, µ) . (8.3)

In the thermodynamic limit, the total polariton density is given by bringing the sumto converge into an integral over the reciprocal space:

n(T, µ) = limR→∞

N(T, µ)

Rd= n0 +

1

(2π)d

∫ ∞

0

fB(k, T, µ)dk (8.4)

where

99The thermodynamic limit is the limiting process by which thesystem size and the number of particlesincrease indefinitely but conjointly so that the density remains constant.

BASIC IDEAS ABOUT BOSE-EINSTEIN CONDENSATION 277

n0(T, µ) = limR→∞

1

Rd1

exp

[

− µ

kBT

]

− 1

(8.5)

If µ is nonzero, the ground-state density vanishes. On the otherhand, the integral onthe right-hand side of Eq. (8.4) is an increasing function ofµ. So, if one increases theparticle densityn in the system, the chemical potential also increases. The maximumparticle density which can be accommodated following the Bose distribution functionis therefore:

nc(T ) = limµ→0

1

(2π)d

∫ ∞

0

fB(k, T ) dk (8.6)

This function can be calculated analytically in the case of aparabolic dispersion re-lation. It converges ford > 2 but diverges ford ≤ 2, i.e., in two or less dimension(s), aninfinite number of non-interacting bosons can always be accomodated in the system fol-lowing the Bose distribution, the chemical potential is never zero and there is no phasetransition. In higher dimensions, however,nc is a critical density above which it wouldseem no more particles can be added. Einstein proposed that at such higher densities theextra particles in fact collapse into the ground state, whose density is therefore given by:

n0(T ) = n(T ) − nc(T ) . (8.7)

This is a phase transition characterised by the accumulation of a macroscopic num-ber of particles—or equivalently by a finite density—in a single quantum state. Theorder parameter is the chemical potential, which becomes zero at the transition.

8.2.2 Experimental realization

This proposal was not immediately accepted and understood by the scientific commu-nity, principally because of Uhlenbeck’s thesis, wherein it was argued that BEC was notrealistic because it would not occur in a finite system. The interest in BEC saw a revivalin 1938 with the first unambiguous report of Helium-4 superfluidity. A few months afterthis observation, London first proposed the interpretationof this phenomenon as a man-ifestation of BEC. This link between BEC and He-4 superfluidity marked the beginningof an impressive amount of scientific activity throughout the 20th century, which is stillbeing pursued. Another early field where Einstein’s intuition found potential and practi-cal applications is superconductivity. However, the link was only properly understood inthe 1950s with the advent of the Bardeen, Cooper, and Schrieffer (BCS) theory. In bothcases (He and superconductivity) the total particle density is fixed. It is thus possible todefine a critical temperatureTc given by the solution of

nc(Tc) = n (8.8)

A remarkable indication of the validity of Einstein’s hypothesis (as was immediatelypointed out by London) is that its direct application yieldsa BEC critical temperature of3.14K for He, very close to the experimental value of 2.17K. However a major difficultyis that these two systems are strongly-interacting, in factalready in their liquid phase,


Pyotr LeonidovichKapitza (1894–1984) and FritzLondon (1900–1954)The pioneers of quantum hydrodynamics.

Kapitsa worked in the Cavendish Laboratory in Cambridge with Ernest Rutherford, whom he called“the crocodile”. He had one carved on the outer wall of the Mond Laboratory, now the emblem of the presti-gious research center. The following year he visited SovietUnion where he was held forcefully to continue hisresearch. He died the only member of the presidium of the Soviet Academy of Sciences who was not a mem-ber of the communist party. He was awarded the Nobel prize in 1978 for his work in low-temperature physics.

London’s (1938) proposition to root superfluidity in Bose condensation was highly controversial athis time, if only because he based his argument on an ideal gaswhereas a fluid is a strongly interactingsystem. He, more than any other, took the Bose condensation seriously, even at the time when Einsteinhimself seemed to accept its rebuttal by the scientific community.

and therefore particles interactions are expected to play afundamental role thus makingthem poor realisations of Einstein’s ideal gas. Consequently, the objective of most the-oretical efforts of the 1940s-1960s was to describe condensation of strongly-interactingbosons. Stimulated emission of light and laser action is also induced by the bosonicnature of the particles involved—the photons—which do not interact. This, however,means that they cannot self-thermalise and a photon assembly represents fundamentallya non-equilibrium system. Consequently, laser action is a non-equilibrium phase transi-tion which cannot be directly interpreted as a BEC. In fact, the first clear manifestationof condensation in a weakly-interacting Bose gas was performed recently by Andersonet al. (1995) with trapped alkali atoms. This discovery, crowned by the 2001 Nobel prizefor physics, has given a strong revival to this field.

8.2.3 Modern definition of Bose-Einstein condensation

Research on BEC was extremely intense in the period 1938-1965, especially on the the-oretical side, where it allowed for many deep advances in understanding. In particular,these efforts led to a new definition of the BEC criterion. BECis now associated withthe appearance of a macroscopic condensate wavefunctionψ(r), which has a nonzeromean value〈ψ(r)〉:

〈ψ(r)〉 =√

ncond(r)eiθ(r) , (8.9)

SPECIFICITIES OF EXCITONS AND POLARITONS 279

〈ψ(r)〉 is the order parameter for this phase transition. It is a complex number withan amplitude—the square root of the condensate density—anda phase. The systemHamiltonian is invariant under an arbitrary phase change ofψ(r) (a property referred toasglobal gauge invariance). However, at the phase transition this symmetric solutionbecomes unstable and the system breaks this symmetry by choosing a specific phasethat is assumed throughout the whole condensate, which is therefore completely phase-coherent. Penrose & Onsager (1954) proposed the following criteria for BEC:

〈ψ†(r)ψ(r′)〉 −−−−−−−→|r−r′|→∞

〈ψ(r′)〉∗〈ψ(r)〉 (8.10)

which is now generally accepted. Its significance has emerged gradually through theefforts of many theorists. Goldstone (1961) and Goldstone et al. (1962) advanced theidea of spontaneous symmetry breaking, Yang (1962) termed the phenomenon “off-diagonal long-range order” (ODLRO) and Anderson (1966) emphasised the notion ofphase coherence. The superfluid velocity can be defined from Eq. (8.9) as

mvs(r, T ) = ~∇θ(r, T ) . (8.11)

A system is therefore “superfluid” if two arbitrary spatial points are connected by aphase-coherent path, allowing for frictionless transport, i.e., no scattering.

8.3 Specificities of excitons and polaritons

Depending on their density and on temperature, excitons behave as either a weakly-interacting Bose gas, a metallic liquid, or an electron-hole plasma. It has been under-stood by Moskalenko (1962) and Blatt et al. (1962) that excitons remain in the gas phaseat low densities and low temperatures, and are therefore good candidates for observa-tion of BEC in the way envisioned by Einstein. At that time there were no experimentalexamples of BEC of a weakly interacting gas and a great deal ofresearch effort wasdedicated to the problem of exciton BEC. A number of theoretical works on excitoniccondensation and superfluidity have appeared, with major publications such as those byKeldysh & Kozlov (1968), Lozovik & Yudson (1975, 1976a, 1976b), Haug & Hanamura(1975) and Comte & Nozieres (1982). In most of these, the fermionic nature of excitoncomposite quasiparticles is also addressed. The starting point of these models is a sys-tem of degenerate electrons and holes of arbitrary densities that is treated in the spiritof the BCS theory. A key point of all the formalisms that have been developed is thatthey assume an infinite lifetime for the semiconductor excitations. In other words, thesetheories are looking for steady-state solutions of the Schrodinger equation of interactingexcitons. It is indeed clear that to have enough time to Bose-condense, excitons musthave a radiative lifetime much longer than their relaxationtime. Thus, the use of “dark”(uncoupled to light) excitons seems preferable. This is thecase for bulk Cu2O paraex-citons, whose ground state spin is 2, or of excitons in coupled quantum wells, wherethe electron and hole are spatially separated. These two systems have been subject tointense experimental studies which have sometimes claimedachievement of excitonBEC or superfluidity, see for instances the publications by Snoke et al. (2002), Butov


et al. (2001) and Butov et al. (2002). However, careful analysis by Tikhodeev (2000)and Lozovik & Ovchinnikov (2002) showed that clear evidenceof excitonic BEC hasnot yet been achieved in such systems. The difficulties of Bose-condensing excitonsare twofold. The first reason is linked to the intrinsic imperfections of semiconductors.Because of an unavoidable structural disorder, dark excitons non-resonantly excited areoften trapped in local minima of the disorder potential and can hardly be considered asfree bosons able to condense. The second source of difficulties is connected with theproblem of detection of the condensed phase. The clearest signature of exciton Bosecondensation should be the emission of coherent light by spontaneous recombination ofcondensed excitons. Such emission is a priori forbidden fora system of dark excitons.

On the other hand, “bright” excitons directly coupled to light might also be goodcandidates for condensation, despite their short lifetimes. In bulk semiconductors thiscoherent coupling gives rise to a polarisation wave that canbe considered from a quan-tum mechanical point of view as a coherent superposition of pure exciton and photonstates (polaritons). Bulk polaritons are stationary states that transform into photons onlyat surfaces. Polaritons also being bosons, they can, in principle, form condensates thatwould emit spontaneously coherent light. Typical dispersion curves of bulk polaritonsare shown in Fig. (4.14). In the vicinity of the exciton-photon intersection point, thedensity of states of polaritons is strongly reduced and the excitonic contribution to thepolariton is decreased. One should note that strictly speaking, ak = 0 photon doesnot exist, and that consequently thek = 0 polaritonic state of the LPB does not existeither. The polariton dispersion has no minimum so that a true condensation processis strictly forbidden. Polaritons accumulate in a large number of states in the so-calledbottleneck region. The situation is drastically differentin microcavities. The cavity pre-vents the escape of photons and allows the formation of long-lifetime cavity polaritons.Conversely to the bulk case, the in-plane cavity polariton dispersion exhibits a well-defined minimum located atk = 0, but since they are two-dimensional quasiparticlesthey cannot exhibit a strict BEC phase transition, but rather a local condensation orso-calledKosterlitz-Thouless phase transitiontowards superfluidity. They have, more-over, an extremely small effective mass aroundk = 0, allowing for polariton lasingat temperatures that could be higher than 300K. Experimental discovery of stimulatedscattering of polaritons in microcavities (see chapter 7) has proved that a microcavityis probably a very suitable system to observe effects linkedto the bosonic nature ofpolaritons, and probably BEC. The recent paper by Kasprzak et al. (2006) shows thatBEC of polaritons is indeed possible in CdTe based microcavities at temperatures upto about 40K. Much experimental and theoretical effort followed Imamoglu’s proposalof a polariton laser. We shall describe in detail these efforts throughout the rest of thischapter. A fundamental peculiarity and difficulty of a microcavity is the finite polaritonlifetime that may be responsible for a strongly non-equilibrium polariton distributionfunction. The relaxation kinetics of polaritons plays a major role in this case.

8.3.1 Thermodynamic properties of cavity polaritons

In this section we discuss thermodynamic properties of microcavity polaritons consid-ered as equilibrium particles, i.e. particles having an infinite lifetime. Even though this


approximation is very far from reality, mainly governed by relaxation kinetics, it isinstructive to examine the BEC conditions in this limiting case. We shall, moreover,assume that polaritons behave as either ideal or weakly-interacting boson gas, so thatthe following analysis is valid only in the low-density limit.

As mentioned in Section 8.2.1, the critical condensation density is finite for nonzerotemperature ifd > 2. However, this density diverges in the two-dimensional case. Thus,a non-interacting Bose gas cannot condense in an infinite two-dimensional system, andthe same statement turns out to be true when interactions aretaken into account. A rig-orous proof of the absence of BEC in two dimensions has been given by Hohenberg(1967). An equivalent statement known as the Mermin-Wagnertheorem after the workof Mermin & Wagner (1966), asserts that long-range order cannot exist in a system ofdimensionality lower than two. Finally, it has been shown that spontaneous symmetrybreaking does not occur in two dimensions, see for instance the discussion by Cole-man (1973). However, a phase transition between a normal state and a superfluid statecan still takes place in two dimensions as predicted by Kosterlitz & Thouless (1973)in the framework of theXY spin model. Such a second-order phase transition is for-bidden for ideal bosons, but according to Fisher & Hohenberg(1988) can take placein systems of weakly-interacting bosons such as low-density excitons or polaritons, asshown by Fisher & Hohenberg (1988). The case of excitons has been especially inves-tigated by Lozovik et al. (1998) and Koinov (2000). In the next Section, we introducethe Kosterlitz-Thouless phase transition and its application to the cavity polariton sys-tem. We describe the effect of interactions on bosons through a presentation of theBogoliubov formalism as guide. The problem of local condensation is addressed inSection 8.3.3.

8.3.2 Interacting bosons and Bogoliubov model

In order to explain the properties of superfluid He, a phenomenological model was de-veloped by Landau (1941) (extended in 1947) who introduced an original energy spec-trum, displayed in Fig. 8.1. This spectrum is composed of twokinds of quasiparticles:phononsandrotons. These quasiparticles are collective modes in the “gas of quasipar-ticles” and are associated with the first and second sound. Such a spectrum introduced“by hand” allowed Landau to describe most of the peculiar properties of superfluidHe. Bogoliubov’s work of 1947 was a real breakthrough. He presented a microscopicdescription of the condensed weakly-interacting Bose gas.As we shall see below, heshowed how BEC is not much altered in a weakly-interacting Bose gas, somethingwhich was not obvious at the time. He also showed how interactions completely alterthe long wavelength response of a Bose gas. He recovered qualitatively the spectrumassumed by Landau for the quasiparticle dispersion relation. Most of the further theo-retical developments in the field are based on the Bogoliubovapproach.

Bogoliubov considered the Hamiltonian describing an interacting Bose gas:

H =∑

k

E(k)a†kak +1

2

∑

k,k′,q

Vqa†k+qa

†k′−qakak′ (8.12)


Lev DavidovichLandau (1908–1968) and Nikolai NikolaevichBogoliubov (1909–1992) provided the mostlasting phenomenological and microscopical interpretations of superfluidity.

Landau made numerous contributions to theoretical physics, especially in Russia where the “Landau School”is still referred to. Alexey Abrikosov, Lev Pitaevskii, Isaak Khalatnikov are among his most famous students.He put special emphasis on broad qualifications for a physicist as opposed to narrow expertise. A childprodigy, he said he couldn’t remember a time when he was not familiar with calculus. He kept a list ofphysicists whom he graded, with his contemporaries Bohr, Heisenberg and Schrodinger falling into the firstcategory along with Newton, but he made an exception for Einstein who he ranked in a superior category ofhis own. He put himself in the second category. He was the victim of a severe car accident in January 1962that would ultimately claim his life. He abandoned his research activities during this last period. He receivedthe 1962 Nobel Prize in Physics for his work on superfluidity.

Bogoliubov published his first paper at 15. He was one of the first to have studied nonlinearities inphysical systems at a time where absence of computers made them forbidding. In the late 40’s and 50’she studied superfluidity, successfully taking into accountthe nonlinear terms that others had deemed toocomplicated. He introduced the Bogoliubov transformationin Quantum Field Theory. In the 60’s he turnedhis interest to quarks in nuclear physics.

with a†k, ak the creation and annihilation bosonic operators andVq the Fourier trans-form of the interaction potential for the boson–boson scattering. His objective was todiagonalise this Hamiltonian making reasonable approximations.

At zero Kelvin an ideal Bose gas should be completely condensed. Bogoliubov as-sumed that interactions are only responsible for weak condensate depletion. In otherwords, most system particles are assumed to be still inside the condensate. This im-plies〈a0〉, 〈a†0〉 ≈

√N0 and therefore

[a0, a†0] ≪ 〈a†0a0〉 (8.13)

whereN0 is the condensate population. Bogoliubov proposed to neglect the conden-sate fluctuations and to replace the operatorsa0, a†0 by complex numbersA0, A∗

0. Thecondensate is thus treated classically as a particle reservoir. The second Bogoliubovapproximation was to keep only the largest contributions inthe interacting part of theHamiltonian. The largest contributions are those which involve the condensate. There-fore, one keeps only the terms which involvea0 two times or more. Eq. (8.12) becomes:


Fig. 8.1: The energy spectrum of liquid Helium II, showing phonons ask → 0 and rotons as the highk dip.

H = V0N20 +

∑

k,k 6=0

(

E(k) +N0(Vk + V0))

a†kak+

+N0

∑

k

Vk

(

a†ka†−kaka−k

)

(8.14)

The Bogoliubov approximations conserve off-diagonal coupling terms that inducethe appearance of new eigenmodes. Then, a change of basis is made through the trans-formations (5.37) which, according to this procedure,diagonalises the Hamiltonian (8.12)into

H = V0N20 +

∑

k 6=0

EBog(k)α†kαk (8.15)

as a function of new operatorsαk, α†k, chosen to remain bosonic operators, and with, as

an all-important consequence,

EBog(k) =√

E(k)[E(k) + 2N0Vk] . (8.16)

This is Eq. (8.16) which justifies the Landau spectrum. WhenVk is given by the Fouriertransform of Coulomb potential, the spectrum assumes the shape displayed in Fig. 8.1.It is also called theBogoliubov spectrum. The unperturbed dispersion is recovered ifN0

vanishes. On the other hand, if one considers the existence of a condensate, there followsfrom a quadratic unperturbed dispersion

E(k) =~2k2

2m(8.17)

the renormalised spectrum neark = 0 equal to


EBog(k) ≈ ~k

√

N0v0m

, (8.18)

This is the linear part of the Landau spectrum, which displays features of phononsquasiparticles with sound velocity

vs =

√

V0N0

m(8.19)

Considering a large polariton population ofN0 = 106 and a system size of 100µ,the typical bogolon velocity in a GaAs microcavity would be5×105m.s−1. In the sameway one can estimate the reciprocal space region where the bogolon spectrum is linearas the set ofk values such that

k <2

~

√

mN0V0 . (8.20)

This corresponds to an angular width of about 3 degrees in theabove-mentionedconditions.

8.3.3 Polariton superfluidity

Superfluidity is a property deeply associated with BEC and atfirst glance it seems thatone of these properties cannot exist without the other. Thisis not exactly true. BEC islinked with the appearance of a Dirac function atk = 0 in the distribution function ofbosons. The Fourier transform of this Dirac function gives the extension in the directspace of the condensate wavefunction, which is infinite and constant. BEC thereforemeans the appearance of a homogeneous phase in direct space.This homogeneity im-plies superfluidity. Particles can move throughout space along a phase-coherent, dissi-pationless path. Superfluidity means that statistically, two points in space are connectedby a phase-coherent path, even if the whole space is not covered by a phase-coherentwavefunction. As a conclusion, a superfluid state can occur without the existence ofstrict BEC. This is the kind of state which arises in two dimensions where a strict BECis forbidden. We now describe qualitatively how this phase transition takes place, andwe estimate the Kosterlitz–Thouless (KT) transition temperatureTKT for the polaritoncase.

At temperatures higher than the critical temperatureTKT, the superfluid numberdensityns is zero, but local condensation can take place. Condensate droplets can havequite large sizes as we shall see later, but they are characterised by an exponentiallydecreasing correlation function and are not connected together. Free vortices preventlong-range ordering, i.e., percolation of the quasi-condensate droplets. However, oncethe critical temperatureTKT is reached, single vortices are no longer stable. They bind,forming pairs or clusters with the totalwinding number(or verticity) equal to zero, al-lowing for a sudden percolation of the quasi-condensate droplets which therefore forma superfluid. For temperatures slightly belowTKT, the superfluid number density is pro-portional toTKT with a universal coefficient (see the discussion by Nelson & Kosterlitz(1977)):


ns =2mkBTKT

π~2, (8.21)

wherem is the bare polariton mass atq = 0. The pairs of vortices remain well be-low TKT and the correlation function is not constant, but decreasesas a power of thedistance. The superfluid wavefunction has thus a finite extension in reciprocal space, andconsequently is not a BEC wavefunction. Complete homogeneity and true long-rangeorder can be achieved only atT = 0, where vortices disappear. BelowTKT normal andsuperfluid phases coexist. The normal fluid can be characterised by a densitynn and avelocityvn, while the superfluid has a densityns and velocityvs. The total fluid densityis:

n = nn + ns , (8.22)

wherenn can be calculated following, for instance, Koinov (2000). Despite the absenceof BEC in two dimensions, the energies of quasiparticles (bogolons) in the superfluidphase are still given by the Bogoliubov expression

EBog(k) =√

E(k)[E(k) + 2µ] . (8.23)

We need to know the chemical potential of interacting polaritons to calculate thequasiparticle dispersion. With the meaning previously given for the chemical potential,when one considers added particles going into the ground state, the associated interac-tion energy yields:

µ = NV0 (8.24)

Once the quasiparticle dispersion is known, one can use the famous Landau for-mula to calculate the normal mass density. This formula reads for a two dimensionalsystem:100101

nn =1

(2π)2

∫

E(k, T, µ)(

− ∂fB(Ebog(k, T, 0))

∂Ebog(k)

)

dk . (8.25)

Note that the Bose distribution function Eq. (8.1) enteringthis expression is taken atzero bogolon chemical potential, while the nonzero polariton chemical potential is stillpresent in the bogolon dispersion relation (8.23). Equations (8.23–8.25) yieldns(T, n).Its substitution into Eq. (8.22) allows one to obtainTKT(n) and therefore to plot apolariton phase diagram. Such a phase diagram is shown in Fig. 8.2.

Solid lines (a–d) show the critical concentration for a KT phase transition accord-ing to the above-mentioned procedure, and calculated for typical microcavity structures

100The derivation of Landau formula requires current conservation law and therefore is exact for particleswith a parabolic spectrum only. In the case of cavity polaritons with a non-parabolic dispersion, Eq. (8.25)remains a good approximation at low temperatures, where theexciton-like part of the low polariton dispersionbranch is weakly populated. Keeling (2006) has shown that athigher temperatures, the critical density for theKT transition is lower than is predicted by Eq. (8.25). See Appendix C for more details.

101The integral in Eq. (8.25) can be computed approximately in the case of a parabolic dispersionif µ/(kBT ) ≫ 1, as shown by Fisher & Hohenberg (1988). This approximation is for example well satisfiedfor excitons having critical temperature in the Kelvin range. It is no more the case for polaritons because oftheir small masses. Therefore numerical integration has tobe carried out.


Fig. 8.2: Phase diagrams for GaAs (a), CdTe (b), GaN (c), and ZnO (d) based microcavities at zero detuning.Vertical and horizontal dashed lines show the limits of the strong-coupling regime imposed by the excitonthermal broadening and screening, respectively. Solid lines show the critical concentrationNc versus tem-perature of the polariton KT phase transition. Dotted and dashed lines show the critical concentrationNc forquasi condensation in a 100µm and in a one meter lateral size systems, respectively. The thin dashed line(upper right) symbolizes the limit between vertical cavitysurface emitting laser (VCSEL) and light-emittingdiode regimes.

based on GaAs (a), CdTe (b), GaN (c), and ZnO (d). In all cases we assume zero de-tuning of the exciton resonance and the cavity photon mode. For GaAs- and CdTe-based microcavities we have used the parameters of the samples studied by Senellart &Bloch (1999), Senellart et al. (2000) and Le Si Dang et al. (1998). Parameters of modelGaN and ZnO microcavity structures have been given by Malpuech, Di Carlo, Kavokin,Baumberg, Zamfirescu & Lugli (2002) and Zamfirescu et al. (2002). The latter structurehas only hypothetical interest, since strong coupling has not yet been achieved experi-mentally in ZnO-based cavities. Vertical and horizontal dashed lines show the approxi-mate limit of the strong-coupling regime in a microcavity that come from either excitonscreening by a photo-induced electron-hole plasma or from temperature-induced broad-ening of the exciton resonance. Below the critical density,if still in the strong-couplingregime, a microcavity operates as a polariton diode emitting incoherent light, while inthe weak-coupling regime the device behaves like a conventional light-emitting diode.Above the critical density, in the weak-coupling regime, the microcavity acts as a con-ventional laser. Thin dotted lines in Fig. 8.2 indicate the limit between the latter two


phases that cannot be found in the framework of our formalismlimited to the strong-coupling regime. One can note that critical temperatures achieved are much higher thanthose which can be achieved in exciton systems. In existing GaAs- and CdTe-based cav-ities these temperatures are high enough for experimental observation of the KT-phasetransition under laboratory conditions, but do not allow one to produce devices workingat room temperature. Record critical temperatures ofTKT=450K and 560K for GaN-and ZnO-based model cavities are given by extremely high exciton dissociation ener-gies in these semiconductors. It is interesting to note thateven aboveTKT(n), n − nn

does not vanish, which reflects the existence of isolated quasi-condensate droplets. Aswe show below, these droplets can reach substantial size, even above the Kosterlitz-Thouless temperature or density. Their properties may dominate the behaviour of realsystems in some cases.

At high excitonic densities the behaviour of the polariton liquid can be affectedby fermionic effects linked to the composite nature of the exciton. This may result inthe crossover from the quasi-condensed or superfluid phase to the quantum correlatedplasma similar to the gas of electronic pairs in a superconductor. This phase is com-monly referred to as BCS from the first letters in the names of Bardeen, Cooper andSchrieffer, who proposed a theoretical model to explain thesuperconductivity in termsof collective phenomena in a stongly correlated gas of composite bosons, calledCooperpairs. The group of Littlewood in Cambridge has published a seriesof papers basedon the Dicke model (see Section 5.6), that was generalised byEastham & Littlewood(2001) and applied to the study of disorder and structural imperfections, for instanceby Marchetti et al. (2004), since the model is especially suited to that purpose. Keelinget al. (2005) have calculated the phase diagram displayed onFig. 8.3 where the regimesof BEC, BCS and weak-coupling regimes (termed “BEC of photons” by the authors)can be distinguished. The only parameter which govens the phase boundaries in thismodel is the mean field, which characterises the strength of interparticle interactions inthe electron-hole-exciton-polariton plasma. Experimentally, the BCS phase of exciton-polaritons has not been observed so far.

8.3.4 Quasi-condensation and local effects

In this Section we define a rigorous criterion for boson quasi-condensation in finite-sizesystems. For the sake of simplicity we neglect here all kindsof interactions betweenparticles. Let us consider a system of sizeR. The particle density is given by

n(T,R, µ) =N0

R2+

1

R2

∑

k

k>2π/R

fB(k, T, µ) (8.26)

whereN0 is the ground-state population. We define the critical density as the maximumnumber of bosons which can be accommodated in all the states but the ground state:

nc(R, T ) =1

R2

∑

k

k>2π/R

fB(k, T, 0) (8.27)


Fig. 8.3: BCS-BEC crossover caused by the internal structure of polaritons, as studied by Keeling et al.(2005). The thermodynamics of composite polaritons is investigated through the critical temperature of con-densation. At low density there is no deviation from the internal structure. At higher density, the system shiftto BCS-like behaviour with associated critical temperature now comparable to the Rabi splitting. The phasediagram shown and obtained by Keeling et al. (2004) displaysthe phase boundary in a mean-field theory(solid line) and phase boundaries taking into account fluctuation corrections for different values of the photonmass. Temperature is plotted in units ofg

√n with g the coupling strength andn the density of excitons.

The quasi-condensate density is thus given byn0 = n− nc. In this case, formally,the chemical potentialµ is always strictly negative, but it approaches zero, allowingone to put as many bosons as desired in the ground state, whilekeeping the concen-tration of bosons in all other states finite and limited bync. The concentration (8.27)can be considered as the critical concentration for local quasi-Bose condensation intwo-dimensional systems. Further, we shall refer toTc defined in this way as the criti-cal temperature of Bose condensation in a finite two-dimensional system. On the otherhand, it appears possible, knowing the temperature and density, to deduce the typicalcoherent droplet size, which is given by the correlation length of the quasi-condensate.

From a practical point of view, experiments are performed onsamples having a lat-eral size of about 1cm. Electron-hole pairs are generated bylaser light with a spot area ofabout 100µm. These electron-hole pairs rapidly (typically on a timescale less than 10ps)form excitons, which relax down to the optically active region, where they strongly in-teract with the light field to form polaritons. Excitons thatform polaritons have a finitespatial extension in the plane of the structure, but they areall coupled to each other vialight, as illustrated for instance in the applications developed by Malpuech & Kavokin(2001) or Kavokin et al. (2001). The polariton system thus covers the whole surfacewhere excitons are generated. If the KT critical conditionsare not fulfilled, but if typi-cal droplets sizes are larger than the light spot size, the whole polariton system can betransiently phase-coherent and thus exhibits local BEC. Aswe shall show below, thissituation is the most likely to happen in current optical experiments performed at lowtemperature.

Let us underline at this point an important advantage of polaritons with respect toexcitons weakly coupled to light for the purposes of BEC or superfluidity. Individual

HIGH-POWER MICROCAVITY EMISSION 289

excitons in real structures are subject to strong localisation in inevitable potential fluc-tuations that prevent them from interacting and forming condensed droplets. Polaritonsare basically delocalized, even though the excitons forming them could be localised, apoint emphasised in the work of Keeling et al. (2004) and Marchetti et al. (2006). Thisis why their interactions are expected to be more efficient and bosonic behaviour morepronounced. The dotted and dashed lines in Fig. 8.2(a–d) show the critical concentra-tion for local quasi-condensation in microcavity systems of 100µm and 1m lateral size,respectively. In the high-temperature (high-concentration) limit, the critical concentra-tions are very similar for both lateral sizes and they slightly exceed critical concentra-tions of the KT phase transition. This means that in this limit the KT transition takesplace before the droplet size reaches 100µm. Conversely, in the low-temperature (low-concentration) limit the KT curve is between the transitioncurves of the 100µm and1m size systems. This shows that droplets at the KT transition are larger than 100µmbut smaller than 1m. Since the typical laser spot size is of about 100µm, this meansthat local Bose condensation takes place before the KT transition at low pumping. Adetailed analysis could allow one to obtain the percolatingdroplet size versus tempera-ture, which is beyond the scope of our present discussion.

Note finally that inhomogeneous broadening of the exciton resonance leads to broad-ening of the polariton ground state in the reciprocal space.Any broadening in the re-ciprocal space is formally equivalent to localization in the real space. Such a localiza-tion, present even in an infinite microcavity, allows for quasi-condensation, in principle.Agranovich et al. (2003) have recently studied disordered organic semiconductors proneto exhibit such effects.

Experimental results such as those of Richard, Kasprzak, Romestain, Andre & LeSi Dang (2005) evidenced localisation of the condensate in real space due to the disor-der, whereas no superfluid behavior—like linearisation of the excitation spectrum—wasobserved. The impact of localisation is therefore extremely strong and the followingqualitative picture can be drawn. In presence of disorder, the lowest energy states arelocalised states. In a non interacting boson picture, only the lower energy state shouldbe filled giving rise to a fully localised condensate. This picture changes drastically inpresence of interactions. Indeed, once a localised state starts to be filled, it blue-shiftsbecause of polariton-polariton interaction. The chemicalpotential increases and reachesthe energy of another localised state which in turn starts tobe populated and blue-shifts.The system therefore assumes an assembly of strongly populated localised states, allwith the same chemical potential. This situation which doesnot exhibit any superflu-idity is known as aBose Glass. See for instance the discussion by Fisher et al. (1989).Once the chemical potential reaches the value of the localisation energy, a delocalisationof the condensate occurs and a standard KT phase transition can take place as shown byBerman et al. (2004). In realistic systems one should therefore expect two successivebosonic phase transitions when increasing the density.

8.4 High-power microcavity emission

In Chapter 7 we have explored the optical nonlinearities produced by strong polaritoninteractions when they are resonantly injected on the lowerpolariton dispersion. How-


ever, around the same period of research it became clear thatpeculiar effects are alsoobserved when the strong-coupling microcavities are pumped non-resonantly at higherpower. In this case, the pump photons inject electron-hole pairs at high energy into thecontinuum absorption of the quantum wells, and these carriers rapidly bind and relaxonto the exciton dispersion at highk. We have discussed above how these excitons canapproach easily to the edge of the lower polariton trap neark = 0, but then there is a bot-tleneck for their further relaxation. As the density of excitons is increased by pumpingharder, suddenly an entirely new regime of microcavity emission appears.

Such possible properties were suggested byImamo glu & Ram (1996) andImamo gluet al. (1996) for stimulated scattering of excitons into polaritons atk = 0, based onthe bosonic nature of the final state. However most experiments in III-V microcavitiesshowed limited evidence for this effect due to the competingproblem of exciton ion-ization at high carrier densities. More recent measurements on III-V microcavities un-der near-resonant pumping of the lower polariton branch at high angles by Deng et al.(2003) showed how different the polariton laser emission isto a conventional photonlaser.

Fig. 8.4: CdTe microcavity studied by Le Si Dang et al. (1998)and its (a) reflectivity and (b) PL for a rangeof excitation densities at 4K.

The experiment by Le Si Dang et al. (1998) which shows this dramatic new regimemost clearly uses CdTe microcavities with large Rabi splittings (23 meV) and exci-tons stable to higher densities due to their larger binding energy (25 meV). As thepump power is increased the lower polariton luminesence emerging at normal inci-dence (k = 0) shifts to higher energy and increases in magnitude faster than the pumppower (Fig.8.4). From our previous discussion about polariton-induced blue-shifts tothe dispersion, this provides compelling evidence for the build-up of large polaritonpopulations, while the narrowing of the emission line and its beaming into a narrowangular range indicates the coherent nature of this population. Unfortunately these mi-

HIGH-POWER MICROCAVITY EMISSION 291

crocavities are complex to fabricate and no further devicesof this performance levelhave yet been grown.

Fig. 8.5: Far-field angular emission from Richard, Kasprzak, Romestain, Andre & Le Si Dang (2005) for(a,b) below and (c,d) above the stimulation threshold. (a,d) display far-field emission in theθx,y Fourierplane, (b,e) display the emission measured in theE, θx plane.

What remains unclear is the extent to which disorder on the tens of microns scaleaffects the process of polariton condensation into a macroscopic coherent state. For in-stance, observations of the angular emission of these CdTe microcavities by Richard,Kasprzak, Romestain, Andre & Le Si Dang (2005) show that forsome positions on asample, the coherent emission is in the form of an annular ring above the pump thresh-old (Fig.8.5). Similar observations have been observed in III-V microcavities for thecondition of near-resonant pumping of the exciton reservoir by Savvidis et al. (2002).However also clearly visible in the II-VI experiments is theincrease in angular intentsityfluctuation in this condition, which has also been observed in III-V microcavities in thespatial domain. Hence one of the key theoretical questions is how a polariton conden-sate can emerge, and what its properties will be in a situation of weak spatial inhomo-geneities.

Another interesting question has been if the scattering of excitons from their reser-voir into the polariton trap can be enhanced to allow efficient polariton condensationin III-V microcavities. One possible route has been to weakly dope the quantum wellwith electrons to provide a lighter quasi-particle which absorbs more energy from re-laxing excitons, as proposed by Malpuech, Kavokin, Di Carlo& Baumberg (2002)and Malpuech, Di Carlo, Kavokin, Baumberg, Zamfirescu & Lugli (2002). Experimen-tally, although some promising results have been achieved,for instance in the work ofLagoudakis et al. (2003) and Bajoni et al. (2006), it appearsthat the speed-up from theexciton-electron scattering is insufficient to surmount the polariton bottleneck before


exciton ionization is also produced.Experiments in wide-bandgap semiconductors look most promising at this time, but

are developing as the technology for their fabrication matures. We should also mentionthe possibility of hybrid microcavities which contain organic semiconductors or dyelayers in the strong coupling regime, such as studied by Lidzey et al. (1998). While theyshow typical strong coupling features in reflectivity, and even in photoluminescenceand electroluminescence, as reported by Tischler et al. (2005), their nonlinear proper-ties have proved to be dominated by other states in the systemand as yet no polaritonscattering has been identified, see the discussion by Lidzeyet al. (2002) and Savvidiset al. (2006). Currently this is believed to be due to the localised nature of the excitonsin these systems which reduces the exciton-exciton scattering interaction. A number ofother realisations of strong coupling with organic semiconductors also look potentiallyinteresting, including plasmon-exciton modes on noble metal surfaces, as discussed byBellessa et al. (2004). Thus a wide variety of systems in which excitons are coupled toelectromagnetic modes can produce conditions suitable forpolaritonic optical nonlin-earities.

8.5 Threshold-less polariton lasing

We have already introduced the polariton laser as an optoelectronic device based on thespontaneous emission of coherent light by a polariton Bose condensate in a microcavity.In order to motivate the full description of such a device, wepresent a simple analysisthat allows a direct comparison with the standard Boltzmannequations (6.50) describingconventional lasing (Chapter 6). As we have seen in discussions of parametric scattering(Chapter 7), a key ingredient of scattering into the polariton trap is polariton-polaritonscattering. This is true not only for polaritons in the vicinity of the polariton trap, butalso for polaritons which are at highk, and thus predominantly excitonic (Fig.8.6).

Fig. 8.6: Dispersion relations of polariton laser, showingpair scattering of excitons atk1,2 feeding energyinto the polariton trap atk‖=0.

Our central premise to build this simple model is that the exciton andk = 0 polaritonpopulations are mutually coupled only by Coulomb pair scattering. Acoustic phononshave too small a velocity to couple polaritons to excitons (energy-momentum conser-vation is impossible) while at the temperatures discussed here, both the optic phonons

THRESHOLD-LESS POLARITON LASING 293

0.50

(c) T=300K

0.50

Nx (×1018

cm-3

)

(b) T=100K

R S

1.0

0.5

0.0

Par

amet

ric S

catte

ring

Rat

es (

ps-1

)

0.50

(a) T=30K

Fig. 8.7: Exciton density dependence of the integrated spontaneous (S) and stimulated (R) scattering ratesfor Ω=7meV and InGaAs QW parameters.

and the exciton statesωLO∼36meV (in GaAs) above thek = 0 polaritons, are unpopu-lated. However the exciton-exciton interaction is sufficiently strong to potentially allowthe excitons to mutually thermalise (as will be discussed inthe full model we developlater in this chapter).

Our model (using the same equations 6.50) couplesN0 polaritons with an excitonreservoir of densityNx, assumed uniform over the polariton active mode volumeV . Thestrong coupling splits the polariton states atk = 0 by the Rabi frequencyΩ, producinga classical trap for polaritons in momentum space of depthΩ/2 and FWHM width~c∆k <

√6εΩEex (for effective sample dielectric constantε). Pair scattering of two

excitons atk1, k2 can deposit a polariton intok0 = 0 and a higher momentum excitoninto k3 (Fig. 8.6). For pair scattering to occur both energy and momentum must beconserved but, because of the quadratic shape of the excitondispersion, this requiresexcitons with a sufficiently highk necessitating a large enough density or temperature.

From k1 + k2 = k3 andE1 + E2 = ELP + E3 whereEi = Eex + ~2|ki|22M , and

ELP = Eex−Ω/2, these produce the constraint

~2k1.k2/M = Ω. (8.28)

whereM is the exciton mass. The total scattering rate depends on theoccupation ofthe states involved. Here the stimulated scattering discussed in Chapter 7 shows that theN0 polaritons behave as bosons. The pair scattering rateΓPS = V [(1 + N0)f1f2(1−f3) − N0f3(1−f1)(1−f2)] where the first term is the final-state enhanced scatteringinto theN0 polariton condensate and the second term is the re-ionisation out of this trap.The Coulomb coupling constantV ∝ e2/aBε whereaB is the exciton Bohr radius. TheFermi functionsfi = 1/(1 + e(Ei−Ef )/kBT ) provide an approximate distribution forthe reservoir of excitons at temperaturekBT , and Fermi levelEf . Extracting theN0-dependent andN0-independent parts, this can be written in the form


N0R(Nx) + S(Nx) =

∫

dk1

∫

dk2 ΓPS(k1, k2) (8.29)

This integral is computationally tractable and its behaviour can be easily understood.For low temperatures compared to the trap potential depth (kBT <Ω/2), pair scatteringonly occurs ifEf > Ω/2, implying that a sufficient number of excitons are needed topopulate states at high enoughk for which energy-momentum conservation becomespossible. For small exciton densities, no pair scattering is possible until the tempera-ture is raised enough to populate these higherk states. In the low temperature regime,f3 =0 and soR=S implying that only pair scatteringinto the polariton trap can occur,this being stimulated by occupation of thek = 0 state. As the temperature increases,the greater range of populatedk1 andk2 increases the pair scattering although ioniza-tion of polaritons out of the trap is also now possible from very high-k excitons, thusreducingR compared toS (Fig.8.7c).

Full evaluation of the scattering integrals (Eq.8.29) allows the rate equations to besolved for the polariton emission [Fig.8.8(a,b)]. At low densities the pair scatteringbehaves quadratically as expected from a two-body incoherent process, so that

S = cN2x (8.30)

R = c(N2x − d

Γ0

ΓnrNx) (8.31)

whered allows for polariton ionization. Below the threshold,

Pth = ~ωV Γnr√

Γ0/c (8.32)

the output powerPout = ~ωV cP 2

Γ2nr

corresponds to spontaneous pair scattering, whileabove threshold,Pout = P − Pth (Fig.8.8). A more complicated behaviour is seen ifionization of polaritons becomes significant, giving rise to an intermediate regime witha second threshold. It is thus clear that theL−I curve characteristic differs from normallasers, and gives scope for judicious tailoring for practical applications.

The polariton laser threshold is calculated for two different polaritons trap depths[Fig.8.8(c,d)] corresponding to the conditions for InGaAs(Ω=7meV) and CdTe (Ω=20meV) QW-based microcavities. The different curves correspond to different pair scat-tering rates, with the solid line corresponding to rates measured in time-resolved exper-iments. The minimum laser threshold is found for a temperature kBT ∼ Ω/2 whichensures enough excitons are available in the high-k states for strong pair scattering.Such simple estimates show that it should be possible to obtain thresholds at input pow-ers significantly below 1mW. The lasing threshold is higher in II-VI microcavities withlarge Rabi splittings because the deeper trap potential requires a comparable increase inexciton Fermi energy. In contrast to previous discussions about the influence of acousticphonons in semiconductor microcavities, temperature plays a rather different role here:it modifies the occupation of the exciton dispersion and thuscontrols the pair scattering.

Observation of the expected quadratic dependence at low powers is experimentallycomplicated by changes in the emission angular width. A key question is the actual

THRESHOLD-LESS POLARITON LASING 295

10-6

10-4

10-2

100

102

Out

put p

ower

(m

W)

102

101

100

10-1

10-2

Pump Power (mW)

(a) Ω=7meV

10K 30K 100K 300K

P

P2

102

101

100

10-1

10-2

Pump Power (mW)

30K 100K 200K 300K

(b) Ω=20meV

102 4 6 8

1002 4 6

Temperature (K)

(d) Ω=20meV

0.1

1

10

Thr

esho

ld p

ower

(m

W)

102 4 6 8

1002 4

Temperature (K)

(c) Ω=7meV

Fig. 8.8: Output power vs. power absorbed in the microcavityat different exciton temperatures for microcav-ities of (a) InGaAs QWs and (b) CdTe QWs. At higher temperatures a two threshold behaviour is seen. Belowthreshold, a quadratic behaviour is seen. (c,d) Corresponding temperature dependence of the threshold, forrelative pair scattering rates of (from top to bottom) 0.01,0.1,1,10. The solid line is the equivalent temperatureof the trap potentialΩ/2, while the dashed line is the exciton binding energy.

temperature of the exciton distribution compared to the lattice temperature. For everyphoton emitted, the excitons are heated byΩ/2, however this is generally much less thanthe energy provided to the lattice as photo- or electrically-injected carriers relax to formexcitons at the bandedge. Further experiments are thus required to verify the expectedexciton density and temperature, and have recently been reported (see previous section).

In a polariton laser, absorption at the emission wavelengthis extremely weak, evenat low exciton densities. Ionization of polaritons from their trap is prevented by the


Fig. 8.9: Comparison of the key features of VCSELs and polariton lasers (“plasers”), showing the differentcontributions of the spontaneous and final-state stimulated scattering ratesS, R as the reservoir density isincreased. In VCSELs, the emitted photons can be reabsorbedby the electronic states while in plasers, polari-tons excited in the trap simply decay again without being ionized into the exciton reservoir (for low enoughtemperature).

fast relaxation of high energy excitons forming a ‘thermal lock’ (Fig.8.6). The strongcoupling regime allows the large reservoir of exciton states providing gain to exist atenergies above the radiating polariton. Pair scattering separates the processes of absorp-tion and emission through control of the excitonk-distribution, in contrast to conven-tional lasers for which absorption and emission are intimately linked. Because of thisscheme, gain does not require large carrier densities to invert the carrier population, andthe strict condition for inversion is avoided. VCSELs operate by stimulation of photonsacross the electron-hole transition, while the polariton laser operates by stimulation ofpolaritons across the in-plane dispersion (rapidly leading to photon emission). Becauseof the large number of possible stimulated pair transitions, the ‘inversion’ in a polaritonlaser must be defined using the integrated exciton densityNx rather than the occupa-tion at eachk. In spite of this, the exciton density at threshold is several times less thanthe transparency densityNt. Laser action can occur as soon as more than one polaritonbuilds up in the trap, and the pair scattering rate becomes stimulated by this final stateoccupation. This in turn depends on the pair scattering strength, the exciton density, thecavity finesse and the polariton dispersion, all of which canbe tuned by device design.

The polariton laser works well until the buildup of carriersproduces excessivebroadening of the exciton transition resulting in absorption at the polariton energy whichcollapses the strong coupling and destroys the polariton trap.

This simple model makes a number of assumptions for the sake of simplicity, whichcan only be checked in the full quantum approach which is detailed in the rest of thischapter. However they provide a first intuition as to the polariton laser compares toconventional VCSELs, and highlight the nature of the inversionless operation.

FORMATION OF POLARITON CONDENSATES: SEMI-CLASSICAL PICTURE. 297

8.6 Kinetics of formation of polariton condensates: semi-classical picture

As pointed out in the previous section, the condensate of cavity polaritons is an equilib-rium state of the polaritonic system for a wide range of external parameters. However,polaritons have a finite lifetime in microcavities and are therefore non-equilibrium par-ticles. Relaxation of polaritons in the steepest zone of thepolariton dispersion, andtherefore towards the ground state, is a slow process compared to the polariton life-time. The slow energy relaxation of polaritons combined with their fast radiative decayresults in the bottleneck effect. The photoluminescence mainly comes from the “bottle-neck region” and the population of the ground state remains much lower than what onecould expect from the equilibrium distribution function (see Section 7.1.4). A suitableformalism to describe polariton population dynamics is thesemi-classical Boltzmannequation. This formalism has been widely used, particularly by Tassone & Yamamoto(1999), Malpuech, Kavokin, Di Carlo & Baumberg (2002), Malpuech, Di Carlo, Ka-vokin, Baumberg, Zamfirescu & Lugli (2002) and Porras et al. (2002). It has provensuccessful when its results have been compared with experimental data, for instance inthe work of Butte et al. (2002). We now discuss qualitatively the main features of polari-ton photoluminescence, and show how the semi-classical Boltzmann equations can beused to describe polariton relaxation. A major weakness of this approach is, however,that it only allows calculation of the populations of polaritonic quantum states. All otherquantities of interest, such as the order parameter and various correlation functions, arebeyond its scope and a derivation involving quantum features of the system must beundertaken and will be presented later.

8.6.1 Qualitative features

Typical dispersion curves of bulk polaritons are shown in Fig. 4.14. Excitons created byan initial laser excitation relax along the lower polaritondispersion, which is essentiallythe bare exciton dispersion, except near the exciton-photon resonance. In this region,the polariton density of states is strongly reduced and the excitonic contribution to thepolariton is decreased. One should note that strictly speaking, a photon withk = 0 doesnot exist, and that consequently, thek = 0 polaritonic state of the LPB does not existneither. The polariton dispersion has no minimum and polaritons accumulate in a largenumber of states in the bottleneck region, already mentionned, from where the light ismainly emitted. In this respect, PL experiments performed on the bulk can be viewedas being influenced by the polaritonic effect. More simply, the bulk bottleneck effect isinduced by the sharpness of the energy/wavevector region where an exciton can emita photon considering energy and wavevector conservation conditions. In microcavities,the polariton dispersion is completely different and the LPB has a minimum atk = 0.A bottleneck effect still arises because of the sharpness ofthis minimum, but light isclearly emitted from the whole polariton dispersion including the ground state. The po-laritonic effect is from this point of view much clearer thanin the bulk, as the PL signalcomes from polariton modes, which are easily distinguishable from bare exciton andphoton modes. States which emit light in a strongly-coupledmicrocavity are polaritonstates, despite the localisation effect. The consequencesare twofold. First, PL gives di-rect access to the polariton dispersion as pointed out by Houdre et al. (1994). Second,


a theoretical description of PL experiments should accountfor the polariton effect andthe particle relaxation should be described within the polariton basis.

The initial process is a non-resonant optical excitation oran electrical excitation ofthe semiconductor. At our level, the differences between the two kinds of excitation areonly qualitative. This excitation generates non-equilibrium electron-hole pairs whichself thermalise on a picosecond time scale. A typical temperature of this electron-holegas is of the order of hundreds or even thousands of Kelvin. This electron-hole gasstrongly interacts with optical phonons and is cooled down to a temperature smallerthanωLO/kb on a picosecond time scale. During these few picoseconds excitons mayform and populate the exciton dispersion. The exact ratio between excitons and electron-hole pairs and their relative distribution in reciprocal space is still the object of intenseresearch activity, for example from the work of Selbmann et al. (1996) or from Gu-rioli et al. (1998). For simplicity we choose to completely neglect these early-stageprocesses. Rather, we choose to consider as an initial condition the direct injection ofexcitons in a particular region of reciprocal space. We assume that the typical time scaleneeded to achieve such a situation is much shorter than the typical relaxation time of po-laritons within their dispersion relation. Therefore, ourobjective is to describe the relax-ation of particles (polaritons) moving in a dispersion relation composed of two branches(the upper polariton mode and the lower polariton mode), as shown in Fig. 8.10(a). Wemoreover assume for simplicity that the upper branch plays only a minor role sinceit is degenerate with the highk LPB, and that polaritons only relax within the LPB.The peculiar shape of the dispersion relation plays a fundamental role in the polaritonrelaxation kinetics.

The LPB is composed of two distinct areas. In the central zone, excitons are coupledto the light. In the rest of reciprocal space, excitons have awavevector larger than thelight wavevector in the vacuum and are therefore dark as shown on Fig. 8.10(a). In theactive zone, the polariton lifetime is mainly associated with radiative decay and is of theorder of a few picoseconds. In the dark zone, polaritons onlydecay non-radiatively witha decay time of the order of hundreds of picoseconds. The darkzone has a parabolicdispersion associated with the heavy-hole exciton mass, which is of the order of thefree electron mass. The optically active zone is strongly distorted by strong exciton-light coupling. The central part of this active zone can be associated with a very smalleffective mass (about10−4m0). This mass rapidly increases with wavevector to reachthe exciton mass at the boundary between the optically active and dark zones.

The physical processes involved in polariton relaxation towards lower energy statesare:

Polariton-acoustic phonon interaction:Interactions between excitons and acousticphonons are much less efficient than optical phonon-excitoninteractions. Each relax-ation step needs about 10ps and no more than 1meV can be exchanged. About 100-200ps are therefore needed for a polariton to dissipate 10-20meV of excess kinetic en-ergy and to reach the frontier zone between dark and active areas. This relaxation timeis shorter than the particle lifetime within the dark zone and some thermalisation cantake place in this region of reciprocal space. Once polaritons have reached the edge of


Fig. 8.10: The polariton dispersion re-lation and dynamics of relaxation. (a)Bare exciton and bare cavity photon dis-persions (dashed line), and polariton dis-persion (solid line) as function of thewavector (×107m−1). In the “dark zone”,the exciton wavevector is larger than thelight wavevector in the media. (b) Sketchof the polariton relaxation within thelower polariton branch showing the “bot-tleneck” where polaritons accumulate. Asketch of polaritons relaxation due to directpolariton-polariton interaction has alreadybeen shown on Fig. 8.6 in a special casewith population of the ground state as a re-sult. (c) Sketch of polariton-electron scatter-ing process (the curve on the right hand sidedisplays the free electron dispersion.).

the active zone they still need to dissipate about 5-10meV toreach the bottom of thepolariton trap. This process assisted by the acoustic phonon needs about 50ps, which isat least ten times longer than the polariton lifetime in thisregion. Therefore, polaritonscannot strongly populate the states of the trap. The distribution function takes largervalues in the dark zone and at the edge of the active zone than in the trap. It cannotachieve thermal equilibrium values because of the slow relaxation kinetics. This effecthas been called thebottleneck effectby Tassone et al. (1997), since it is induced by theexistence of a relaxation “neck” in the dispersion relation. Such a phenomenon does nottake place in a single QW with a parabolic dispersion. The energy difference betweenthe dark-active zone frontier is only 0.05meV and a dark exciton can reach the groundstate by a single scattering event.


Polariton-polariton interaction: This elastic scattering mechanism is a dipole-dipoleinteraction with a typical time scale of a few ps. It is very likely to happen because ofthe non-parabolic shape of the dispersion relation. Each scattering event can providean energy exchange of a few meV. It is the main available process allowing populationof the polariton trap and overcoming the “bottleneck effect”. All experimental resultsin this field confirm that polariton-polariton interactionsstrongly affect polariton relax-ation. However, the polariton-polariton interaction doesnot dissipate energy and doesnot reduce the temperature of the polariton gas. If one considers the process sketchedin Fig. 8.10(b), one polariton drops into the active zone where it will rapidly decay,whereas the other polariton gains energy and stays in a long-living zone. Altogether,this process heats the polariton gas substantially and may generate a non-equilibriumdistribution function as we shall see later.

Polariton-free carrier interaction This scattering is sketched in Fig. 8.10(c). As statedabove, optical pumping generates hot free carriers, which may interact with polaritons.Actually, the formation time of excitons or of strongly-correlated electron-hole pairs ismuch shorter than the polariton lifetime. It is reasonable to assume that they do not playa fundamental role in polariton relaxation. However, a freecarrier excess may exist inmodulation-doped structures, or may even be photo-inducedif adapted structures areused, as has been done by Harel et al. (1996) and Rapaport et al. (2000). A large free-carrier excess destroys excitonic correlations. However,at moderate density it can keeppolaritons alive and provide a substantial relaxation mechanism. The polariton-electroninteraction is a dipole-charge interaction and the associated scattering process has a sub-picosecond timescale. Electrons are moreover quite light particles in semiconductors(typically 4–5 times lighter than heavy-hole excitons). Anelectron is therefore able toexchange more energy by exchanging a given wavevector than an exciton. This aspect isextremely helpful in providing polariton relaxation in thesteepest zone of the polaritondispersion. An electron-polariton scattering event is moreover a dissipative process forthe polariton gas. It may be argued that the electron system can be heated by such aninteraction. This is only partially true, and we assume thatthe two-dimensional electrongas covering the entire sample represents a thermal reservoir. In this framework, anelectron gas plays a role similar to acoustic phonons in polariton relaxation, but with aconsiderably enhanced efficiency. This efficiency allows insome cases the achievementof a thermal polariton distribution function, as proposed theoretically by Malpuech,Kavokin, Di Carlo & Baumberg (2002) and later investigated experimentally by Qarryet al. (2003), Tartakovskii et al. (2003) and more recently by Perrin et al. (2005) andBajoni et al. (2006).

8.6.2 The semi-classical Boltzmann equation

The classical Boltzmann equation describes the relaxationkinetics of classical particles.In reciprocal space this equation reads:

dnk

dt= Pk − Γknk − nk

∑

k′

Wk→k′ +∑

k′

Wk′→knk′ (8.33)


Ludwig Boltzmann (1844–1906) developed the statistical theory ofmechanics and thermodynamics. As such he fathered Boltzmann’sconstant kB, Maxwell-Boltzmann distribution and the Boltzmannequation to describe the dynamics of an ideal gas, among others.

Suffering from depression he committed suicide, some say be-cause of the unsatisfying response about his work. His denseandintricate production was later lightened and disseminatedby Ehrenfest(who also committed suicide).

wherePk is the generation term, due to optical pumping or to any otherphysical pro-cess,Γk is the particle decay rate, andW is the total scattering rate between the statesand due to any kind of physical process. Uhlenbeck & Gropper (1932) first proposed toinclude the quantum character of the particles by taking into account their fermionic orbosonic nature. Eq. (8.33) written for fermions reads:

dnk


∑

k′

Wk→k′(1 − nk′) + (1 − nk′)∑

k′

Wk′→kn′k (8.34)

whereas for bosons it is:

dnk


∑

k′

Wk→k′(1 + nk′) + (1 + nk′)∑

k′

Wk′→kn′k (8.35)

Eqs (8.34) and (8.35) are called thesemi-classical Boltzmann equations. The main taskto describe the relaxation kinetics of particles in this framework is to calculate scatter-ing rates. One should first identify the main physical processes which provoke scatteringof particles. Then, scattering rates can be calculated using the Fermi Golden Rule. Thisprocedure is usable only if the scattering processes involved are weak and can be treatedin a perturbative way. Interactions should provoke scattering of particles within their dis-persion relation and not provoke energy renormalisation. For example, the coupling ofparticles with the light should be a weak coupling, only responsible for a radiative de-cay. In a strongly-coupled microcavity one cannot describerelaxation of excitons usinga Boltzmann equation. One should first treat non-perturbatively the exciton-photon cou-pling giving rise to the polariton basis. Then, polaritons weakly interact with their envi-ronment. This weak interaction provokes scattering of polaritons within their dispersionrelation and Eq. (8.35) can be used. The scattering rates canindeed be calculated in aperturbative way (Fermi Golden Rule) because they are induced by weak interactions.

In a semiconductor microcavity the main scattering mechanisms identified are:

• Polariton decay (mainly radiative),• Polariton-phonon interactions,• Polariton-free carrier interactions,


• Polariton-polariton interactions,• Polariton-structural disorder interactions.

The calculation of the rates of these scattering mechanismsare presented in ap-pendix B.

8.6.3 Numerical solution of Boltzmann equations, practical aspects

Despite its very simple mathematical form, Eq. (8.35) is very complicated to inte-grate102 and one must use numerical methods. We detail below one possible such ap-proach:

The phase space, in our case the reciprocal polariton space,must first be discretized.As mentioned previously, it is reasonable to assume cylindrical symmetry for the distri-bution function. The elementary cells of the chosen grid should reflect this cylindricalsymmetry, and therefore these cells should be annular. The cell numberi, C(i) shouldcontain all states with wavevectorsk satisfyingk ∈ [ki, kk+1[. Various choices of scalefor the ki (linear, quadratic or other) have been used in the literature already quoted.The most important requirement being that the distributionfunction does not vary tooabruptly from cell to cell, one should use small cells in the steep zone of the polaritondispersion, whereas very large cells can be used in the flat excitonic area. The natureof the polariton dispersion makes therefore the choice of a nonlinear grid a much bet-ter candidate. In all cases, one state requires a particularattention over all the others:the ground state, especially if one wishes to describe “condensation-like phenomena”,namely a discontinuity of the polariton distribution function. If such a discontinuitytakes place the actual size of the cells plays a role. We cannot choose infinitely smallcells numerically. This means that one cannot solve numerically the Boltzmann equa-tions in the thermodynamic limit in the case of Bose condensation. What can be done isto account for a finite system sizeR. The spacing between states becomes finite (of theorder of2π/R). The grid size plays a role, but it is no longer arbitrary butrelated to areal physical quantity. In such a case the cell size should follow the real state spacing inthe region where the polariton distribution function varies abruptly.

8.6.4 Effective scattering rates

The total scattering rate from a discrete state to another discrete state is the sum of allthe scattering rates:

Wk→k′ =wk→k′

S= W phon

k→k′ +W polk→k′ +W el

k→k′ (8.36)

We now need to calculate two kinds of transition rate. The first is the transition ratebetween a discrete initial statek and all the states belonging to a cell of the grid, indexedby the integeri.

W outk→k′

k′∈C(i)

=∑

k′∈C

wk→k′

S(8.37)

We pass to the thermodynamic limit, changing the sum to an integral:

102Hilbert worked on the problem of integrating Boltzmann equation but without success.


W outk→k′

k′∈C(i)

=S

(2π)2

∫

k′∈C(i)

wk→k′

Sdk′ =

1

(2π)2

∫

k′∈C(i)

wk→k′ dk′ . (8.38)

Here the integration takes place over final states. The cellshave cylindrical symme-try, which means that the scattering rate does not depend on the direction ofk. The totalscattering rate towards celli is therefore the same for any state belonging to cellj. So:

If k ∈ C(j), W outi→j = W out

k→k′

k′∈C(i)

. (8.39)

One also needs to calculate the number of particles reachinga state from the cellC(i):

W ink→k′

k′∈C(i)

= W ini→j =

1

(2π)2

∫

k′∈C(i)

wk→k′ dk . (8.40)

Here, as opposed to Eq. (8.38), the integration takes place over initial statesk.If one wishes to describe condensation in a finite-size system, the ground-state cell

is constituted by a single state and no integration takes place when the final state is theground state:

W outk→0 = W out

i→0 =wout

k→0

S. (8.41)

This scattering rate is inversely proportional to the system size.In this framework, Eq. (8.35) becomes:

dnidt

= Pi − Γini − ni∑

j

Wi→j(1 + nj) + (1 + nj)∑

j

Wj→inj (8.42)

It is really an ensemble of coupled first-order differentialequations which can beeasily solved numerically. One should point out that despite the cylindrical symme-try hypothesis, and despite the one-dimensional nature of the final equation, all two-dimensional scattering processes are correctly accountedfor.

8.6.5 Numerical simulations

The bottleneck region of the LPB corresponds to the transition from the exciton-liketo the photon-like part of the dispersion. Exciton-excitonscattering has been proposedas an efficient relaxation process for polaritons. It remains, however, an elastic scatter-ing process which does not dissipate the total polariton energy. It may allow stimulatedscattering to take place, but as we shall see below, it hardlyallows the achievement of athermal distribution. Currently, we see two possible ways to suppress the bottleneck ef-fect and to achieve polariton lasing. First, in future GaN, ZnSe or ZnO-based cavities atroom temperature, if the strong-coupling regime still holds, acoustic phonon relaxationshould be much more efficient than in presently-available cavities at helium temper-ature. The case of CdTe cavities is already much more favourable than that of GaAssamples. Theoretical calculations conducted by Porras et al. (2002) show that a thermaldistribution of cavity polaritons with large ground state occupation can be achieved inthese systems as shown in Fig. 8.11(a).


Fig. 8.11: a) Distribution function of polaritons calculated by Porraset al. (2002) solving thesemi-classical Boltzman equation in a CdTe microcavity (Rabi splitting 10meV). Polariton-phononand polariton-polariton interaction are taken into account. The lattice temperature is 10K. Thepolariton reservoir is assumed to be thermalized with an excess temperature computed consis-tently. The pumping power ispx = 1, 2, 5, 8, 15 × 1010cm−2/100ps, from bottom to top.b) distribution function of polaritons at 10K calculated by Malpuech, Kavokin, Di Carlo & Baumberg (2002)when non-resonantly pumped with a power of 4.2W/cm2. Results are shown for (a) polariton-acoustic phononscattering (dotted), (b) as (a) plus polariton-polariton scattering (dashed) , and (c) as (b) plus polariton-electronscattering (solid). The thin dotted line shows the equilibrium Bose distribution function with zero chemicalpotential.

This is confirmed by recent experimental achievements (see previous chapter). Analternative (complementary) way could be to use n-doping ofmicrocavities which isexpected to allow efficient electron-polariton scatteringwithin the photon-like part ofthe dispersion. The advantage of this scattering mechanismwith respect to previouslydiscussed ones is that the matrix element of electron-exciton scattering is quite large.Also, an electron has a much lighter mass than a heavy-hole exciton. Thus, the energyrelaxation of a polariton from the bottleneck region to the ground state requires fewerscattering events than for polariton-polariton or polariton-phonon scattering. These ad-vantages have been found theoretically to be strong enough to restore fast polaritonrelaxation and allow the polaritons to condense into their trapped state.

In order to illustrate this, we consider a GaAs based microcavity containing a sin-gle quantum well. We also take into account a finite system sizeR which is assumedto be given by a 100µm excitation spot size. This is practically done by consideringa spacing of2π/R between the ground state and the first excited states, whereas theremaining reciprocal space states are assumed to vary continuously. Fig. 8.11(b) showspolariton distributions calculated by Malpuech, Kavokin,Di Carlo & Baumberg (2002)by solving the complete set of Boltzmann equations for allk states. Taking into accountonly the acoustic phonon scattering (curvea in Fig. 8.11), a thermal distribution func-tion is seen only beyondk = 2 × 104cm−1 (the bottleneck region) where polaritonsaccumulate. Equilibrium is reached 10ns after the start of the non-resonant pumping,leaving an equilibrium polariton density of2.5 × 1010cm−2. Including both polariton-polariton and polariton-acoustic phonon scattering processes (curveb) shows partialrelaxation of the bottleneck and a flat polariton distribution. However, the equilibriumpolariton density in the cavity remains the same, close to the saturation density for ex-


citons (about5 × 1010cm−2). The distribution function near the polariton ground stateapproaches one. This result is in excellent agreement with experimental results obtainedby Senellart & Bloch (1999), Butte et al. (2002), Tartakovskii et al. (1999), Tartakovskiiet al. (2000) and Emam-Ismail et al. (2000). This shows that the amplification thresholdfor the distribution function at the trap state is reached when the strong-coupling regimeis likely to be suppressed. To obtain an increase in the population of the lowestk stateone has to increase the population of a large number of states, requiring a large densityof excitons. This is due to the flat shape of the distribution function which comes fromthe nature of the polariton-polariton scattering process (each scattering event increasesthe population of the high-k states; relaxation of polaritons from these states is thenassisted only by phonons and is slow).

The radiative efficiency, which we estimate as the ratio of the concentration of pho-tons leaving the cavity within a cone of less than 1 to the pumping intensity, is thusfound to be only 1.7%. When a small free electron density of10 × 10cm−2 is takeninto account (curvec in Fig. 8.11b), a huge occupation number of the lowest energystate of more than104 is achieved. This system thus acts as a polariton laser, in whichscattering of polaritons injected at highk by optical or electrical pumping is stimulatedby population of low-k states. In such a situation the light power emitted in a cone of1 is 3.3W/cm2 and the efficiency of the energy transfer from pump to emittedlight isabout 80%. The light emitted by the cavity is much more directional and comes froma smaller number of states than in caseb. The equilibrium polariton density in the cav-ity is now 1.25 × 109cm−2, i.e., 20 times lower than in cases (a, b). Pump powers atleast forty times stronger can be used before the strong to weak coupling threshold isreached. The thin dotted line in Fig. 8.11 shows the equilibrium polariton density froma Bose distribution function plotted for zero chemical potential. It follows quite closelycurvec, which clearly demonstrates that a thermodynamic equilibrium is practicallyachieved for this value of the chemical potential, which is asignature of Bose conden-sation of polaritons. Another interesting feature is foundin Fig. 8.12, which shows theradiative efficiency of the cavity as a function of input power with, (a), and without, (b),polariton-electron scattering. In the first case, the emission rises quadratically up to thethreshold, while in the second case it is much larger and increases linearly. The dottedline on curve (b) marks the excitation conditions for which the strong-coupling regimecollapses because of the bleaching of the excitons. A ground-state population largerthan one is achievable within strong coupling, especially using microcavities made oflarger bandgap semiconductors, such as CdTe, as investigated by Le Si Dang et al.(1998), Richard, Kasprzak, Andre, Romestain, Dang, Malpuech & Kavokin (2005) andKasprzak et al. (2006). Fig. 8.12 shows the results of a simulation performed by Porraset al. (2002) for a CdTe structure. A thermal exciton bath of variable temperature wasassumed. At low pumping a bottleneck effect is clearly visible which vanishes at higherpumping allowing a large ground state population building up.


Fig. 8.12: Radiative efficiency versus power absorbed in themicrocavity considered at 10K, for (a) a dopedcavity,ne = 1010cm−2 and (b) an undoped cavity. The dotted part of the curve (b) corresponds to a calcu-lated exciton density> 5 × 1010cm−2.

8.7 Kinetics of formation of polariton condensates: quantum picture in theBorn-Markov approximation

In this section we use standard procedures developed in quantum optics which we adaptto describe the relaxation kinetics and condensation kinetics of cavity polaritons. Ourgoal is to provide a self-consistent microscopic derivation starting with the polaritonHamiltonian. Polaritons are modelled as weakly interacting bosons moving within thelower polariton branch. They can in principle interact withphonons and free carriers(we will limit ourselves to the interaction with electrons). Furthermore we include inthe Hamiltonian coupling to an external light field which allows for a self consistentdescription of pumping and radiative lifetimes. Then, the procedure used can be sum-marised as follows. We write the equation of motion for the density matrix of the system(von Neumann equation) and perform some approximations. Inall cases we will allowfor the Markov approximation, canceling memory effects. Ina second step we performthe so-called Born approximation which allows us to decouple the density matrices ofdifferent systems and eventually different polariton states. This Born approximation willbe applied in any case to decouple the polariton density matrix from the phonon den-sity matrix, the electron density matrix and the external free photon density matrix. Allsystems except the polariton one are considered as reservoirs and we trace (average)over them. At the end of this procedure we will find a master equation for the densitymatrix of the whole polariton system (Eq. 8.45) which will depend only on polaritonoperators and on semi-classical scattering rates. This equation will be our starting pointfor further approximations on two different levels. First we describe the case where wefully apply the Born approximation to the polariton system.Namely, we decouple den-sity matrices of all polariton states. Then, we trace over all polariton states. This obtainsthe semi-classical Boltzmann equations, which are therefore found to be rigorously jus-tified. A partial trace applied only on excited states allowsus to get a master equationfor the ground state density matrix, depending only on semi-classical quantities which

FORMATION OF POLARITON CONDENSATES: QUANTUM PICTURE 307

in turn can be calculated solving the Boltzmann equation. Aswe will see this last equa-tion can be formally solved and we will discuss in details thenumerical results whichemerge from this model. However, we can already say that in this framework, spon-taneous coherence build-up cannot take place which shows the need to relax some ofthe approximation performed. This alternative approach ispresented in the next Sectionwhere we partially relax the Born approximation keeping thecorrelations between theground state and the excited states. As a result we get a quantum Boltzmann masterequation which describes the dynamics of the ground state statistics. The numerical so-lution of the coupled Boltzmann-Master equation is presented. It demonstrates that thespontaneous coherence build-up takes place in the polariton system.

8.7.1 Density matrix dynamics of the ground state

The procedure we are going to outline is familiar in quantum optics in open systemswhere it has been used by numerous authors. The account by Carmichael (2002) isespecially readable and insightful. Shen (1967) and Zel’dovich et al. (1968) provide arather more historical approach.

However, our case presents additional features with no counterpart in pure quantumoptics. This is essentially because polaritons:

• are massive particles with a non-parabolic dispersion• are self-interacting• have a finite lifetime.

In what follows, we calculate the evolution of the density matrix of the systemρ(t)under the influence of the polariton Hamiltonian (5.163). The tools required to followthe derivation have been introduced in chapter 3. First, we iterate the Liouville-vonNeumann Eq. (3.33) to obtain

iρ = [H, ρ(−∞)] +

∫ t

−∞[H(t), [H(τ), ρ(τ)]] dτ (8.43)

Then we perform a Born approximation between the polariton field and the phonon,electron and photon fields. The density matrix of the system therefore reads:

ρ(t) = ρpol(t)ρphonρelργ (8.44)

The polariton density matrix evolves in time, while the phonon and electron sub-systems are considered as thermal baths at the lattice temperature.ργ describes thephoton vacuum of the electromagnetic field outside of the cavity (to which is coupledthe electromagnetic field from within the cavity).ρphon, ρel andργ are kept equal totheir equilibrium value. We then trace over phonon, electron and photon states. Aftersome lengthy but straightforward algebra, this partial trace gives results in a Lindbladform for the dissipative processes, i.e., with terms of the kindLL†ρ+ ρLL† − 2L†ρL,Eq. (5.83), (we now write simplyρ instead ofρpol), multiplied by time-dependent co-efficients and with operatorsL depending only on polariton operators. It is at this stagethat the Markov approximation is invoked: the populations are assumed to vary slowly


with time and are taken out of the integral in Eq. (8.43) att = τ . The remaining prod-uct of exponentials is then integrated to give the energy-conserving delta functions. Theequation of motion for the polariton density matrix then becomes:

ρ = Lpumpρ+ Llifetimeρ+ Lphonρ+ Lelρ+ Lpolρ (8.45)

whereL are superoperators, that is, an expression constraining normal operators to acton the density matrix. For practical purposes they can be understood as notational conve-nience. Because of the equation they derive from, they are sometimes calledLiouvillian.This is a convenient name as following from the linearity of Liouville equation one canassociate to each part of the Hamiltonian a corresponding Liouvillian which affects theevolution of the density matrix. We define them now. The pump Liouvillian reads:

Lpumpρ =1

2

∑

k

Pk

(

2akρa†k − a†kakρ− ρa†kak

)

(8.46)

where

Pk =2π

~|g(k)|2|Kpump| . (8.47)

The lifetime Liouvillian reads

Llifetimeρ =1

2

∑

k

Γk

(

2a†kρak − aka†kρ− ρaka

†k

)

(8.48)

where

Γk =2π

~|γ(k)|2 (8.49)

is the radiative coupling constant of the state to the external photon field. This result isobtained assuming that the photon modes are empty and therefore unable to replenishthe corresponding polariton mode. Only their quantum fluctuations are playing a role,namely bringing a perturbation at the origin of the transition.

There is a clear symmetry between Eqs. (8.46) and (8.48), reversing the orderingof a anda†. Note that each expression is equal to its hermitean conjugate.

The Liouvillian of interaction with phonons reads

Lphon = − 1

2

∑

k′

∑

k′ 6=k

W phonk′→k(2a†kak′ρa′†k ak + a′†k aka

†kak′ρ+ ρa′†k aka

†kak′)

− 1

2

∑

k′

∑

k 6=k′

W phonk→k′(2a

†k′akρa

†kak′ + a†kak′a′†k akρ+ ρa†kak′a′†k ak)

(8.50)

where

W phonk′→k

=2π

~

∑

q

|U(q)|2(1 + nq)δ(E(k′) − E(k) ∓ ωq) , (8.51)

W phonk→k′ =

2π

~

∑

q

|U(q)|2(ξ± + nq)δ(E(k′) − E(k) ∓ ωq) (8.52)


wherenq = 〈b†qbq〉 is the phonon distribution function given by the Bose distribution,andξ+ = 1 (with corresponding+ sign in the delta function) corresponds to emissionof a phonon while the caseξ− = 0 corresponds to absorption of a phonon (ifE(k) <E(k′)).

The electronic LiouvillianLel is the same asLphon but for the transition rates whichget replaced with

W elk→k′ =

2π

~

∑

q

|U el(q,k,k′)|2neq(1 − ne

q+k′−k)×

× δ(

E(k′) − E(k) +~2

2me(q2 − |q + k − k′|2)

)

, (8.53)

with neq the Fermi distribution function.

Exercise 8.1 ∗ Derive the LiouvillianLpol for the polariton-polariton scattering in theapproximation where it can be dealt with perturbatively. Observe the similarity withexpressions Eqs. (8.46), (8.48), (8.50). Relate it to the Lindblad form, i.e., what is theoperatorL of Eq. (5.83) in this case?

At this level, making the trace on polariton states in Eq. (8.45) makes appear fourthorder correlators for the phonon and electron terms and eveneighth order correlatorsfor the polariton-polariton terms. These correlators havenow to be decoupled in orderto get a closed set of kinetic equations.

For simplicity we consider only polariton scattering with acoustic phonons. Othercases can be related easily to the following derivation. We now perform the Born ap-proximation on the polariton system itself. This means thatwe factorise the densitymatrix of the polariton system asρ(t) = ρ0(t)ρk1(t) · · · ρk(t) · · · . This implies thatcorrelators can be decoupled in the following way:

〈a†kaka†k′ak′〉 = 〈a†kak〉〈a

†k′ak′〉 (8.54)

if k 6= k′. This quantity is of course the product of polariton populations. We shall notethese with uppercase letters, to separate them from phonon populations that we notewith lower case:

〈a†kak〉 = Nk, 〈b†kbk〉 = nk . (8.55)

In this framework, we make the trace over all polariton states but the ground state.Using this procedure, we get as foretold the semi-classicalBoltzmann equations whichdescribe the dynamics of the polariton distribution function:

dNk

dt= Pk − ΓkNk −Nk

∑

k′

Wk→k′(1 +Nk′) + (1 +Nk′)∑

k′

Wk′→kN′k (8.56)

The dynamics of the ground state population is also governedby this equation butwe also end up with a master equation for the ground state density matrix which reads:


ρ0 =1

2Win(t)(2a†0ρ0a0 − a0a

†0ρ0 − ρ0a0a

†0)

+1

2(Γ0 +Wout(t))(2a0ρ0a

†0 − a†0a0ρ0 − ρ0a

†0a0) (8.57)

where

W in(t) =∑

k

Wk→0Nk and W out(t) =∑

k

W0→k(1 +Nk) , (8.58)

the former is the total scattering rate toward the ground state and the latter the totalscattering rate from the ground state toward excited states. This equation is completelysimilar to the one usually accepted to describe a single modelinear amplifier—an ex-haustive description of which is given by Mandel & Wolf (1995)—except for the timedependence of the transition rates Eqs. (8.58). Eq. (8.57) can nevertheless be solved us-ing the Glauber–Sudarshan representation of the density matrix. The derivation and thediscussion of this solution have been given by Rubo et al. (2003) and Laussy, Malpuech,Kavokin & Bigenwald (2004b). The result appears below:

Exercise 8.2 (∗∗) Show that the solution of Eq. (8.57) with initial conditionP (α, α∗, 0) =δ(α− α0) is

P (α, α∗, t) =1

πm(t)exp

(

− |α−G(t)α0|2m(t)

)

(8.59)

in terms of time dependent parameters:

G(t) = exp

[

1

2

∫ t

0

(W in(τ) −W out(τ)) dτ

]

, m(t) = G(t)2∫ t

0

W in(τ)G(τ)2 dτ

(8.60)

The exact solution Eqs. (8.59)–(8.60) shows that in the Born-Markov approxima-tion, the state of the condensate is that of a thermalised coherent state, parametersGandm relating to the relative importance of the coherent and thermal fractions, re-spectively. The quantities of interest that we can derive directly from the solution arethe occupation numberN0 (ground state population), the order parameter〈a0〉, theground state statisticsp0(n) = 〈n| ρ |n〉 and the second order coherenceg(2). Thesecan be obtained from the complete solution by integrating their equations of motion(derived from Eq. (8.57)). First, equations for the scalar quantities (we use for conve-nienceη = 2 − g(2) rather thang(2) directly):

N0 = (Win −Wout − Γ0)N0 +Win (8.61a)

˙〈a0〉 =1

2(Win − (Wout + Γ0))〈a0〉 (8.61b)

∂t(ηN20 ) = 2(Win − (Wout + Γ0))ηN

20 (8.61c)

Second, the set of coupled differential equations for the statistics:


p0(n) = (n+ 1)(Wout + Γ0)p0(n+ 1)

− [n(Wout + Γ0) + (n+ 1)Win]p0(n0) + nWinp0(n− 1) (8.62)

The quantities〈a0〉4 andηN20 are both extensive and proportional toN2

0 . They aresurprisingly described by the same equation. Whereas〈a0〉 depends on off-diagonal el-ements of the density matrix,ηN2

0 depends only on diagonal elements. The equation ofmotion for the normalised (intensive) quantitiesη which describes the diagonal coher-ence and〈a0〉2/N0 which describes the off-diagonal coherence are given by:

η = −2Winη

N0(8.63)

∂

∂t

( 〈a0〉2N0

)

= −Win

N0

〈a0〉2N0

(8.64)

8.7.2 Discussion

The set of equations (8.61)–(8.63) is particularly simple and the meaning of the variousterms is transparent. Eq. (8.61) is inhomogeneous, it is composed of a spontaneous scat-tering term and of a stimulated scattering term. The equations for the order parameterand for the quantityη are both homogeneous and governed by the stimulated terms.This means that the ground state initially empty or in a thermal state will stay thermalforever even if a large number of particles comes in. Howevera coherent seed in theground state can be amplified if stimulated scattering takesplace. As we will see in thenext Section, this set of equations allows us to describe thetransfer of a large number ofincoherent reservoir particles within a ground state having a high degree of coherence.This means that the total coherence of the system can increase. However the coherencedegree of the ground state itself can only decay in time as onecan see on Eqs. (8.63),except if the ground state population becomes infinite whichcan be the case only ininfinite systems.

In the steady state regime the equilibrium value forN0 is:

N0(∞) =Win

Wout(∞) + Γ0 −Win(∞)(8.65)

and the system is in thermal state with zero diagonal and off-diagonal coherence. Thecharacteristic decay constant of the order parameter is called the phase diffusion coeffi-cient which is equal to the emission linewidth of the ground state:

D =1

2

[

Win(∞) −Wout(∞) − Γ0

]

a =Win

2N0(∞)(8.66)

The energy broadening of the condensate is no longer given bythe radiative lifetimebut by the balance between incoming and outgoing scatteringrates. In the low temper-ature limitWout is small andWin ≈ Γ0 which allows one to recover the well-knowndiffusion coefficient from laser theory

D ≈ Γ0

2N0(8.67)


However, one should notice that the polariton-polariton interaction which is not in-cluded in the former equations plays an important role in thephase diffusion coefficient,as discussed by Laussy et al. (2006) and in more details later.

8.7.3 Coherence dynamics

Before presenting numerical results, we give a qualitativeanalysis of the polariton con-densate formation. The kinetics are characterised by a transient regime, during whichthe polaritons come to the condensate, after being excited in somek 6= 0 state att = 0.Their relaxation rate depends nonlinearly on the pumping intensity. For strong enoughpumping the stimulated scattering of polaritons into the condensate flares up at a timet >0, so that the in-scattering rate increases drastically and becomes much greater thanthe out-scattering rate. In the time domain, whereWin(t) > Wout(t) + Γ0, the solu-tion becomes unstable. This instability allows the condensation to happen in a coher-ent quantum state. The formation of the condensate witht 6= 0 implies breaking thesymmetry of the system, which cannot happen spontaneously in the framework of theformalism used. Therefore, to study the possibility of coherence build-up we introducean initial seed (a coherent state with small average number of polaritons). This ini-tial coherence can survive and be amplified for the high relaxation rates, as long as atime window exists in whichWin(t) > Wout(t) + Γ0. After the steady-state regime isreached the point〈a0〉 becomes stable again, since the rates reach the time-independentvaluesWin(∞) andWout(∞), with Wout(∞) + Γ0 > Win(∞). However, the differ-ence between the stationary rates is very small, inversely proportional to the systemarea, which corresponds to a large stationary number of condensed polaritons. If thecoherence is formed its decrease due to spontaneous scattering is slow in large cavities.

As before, we consider the cavity containing1010 electrons/cm−2. The coefficientsWin(t)andWout(t) are extracted from the Boltzmann equation. We model the following ex-periment: att = 0 an ultrashort laser pulse generates a coherent ground statecontaininga variable polariton number (the seed). At the same time, an incoherent nonresonant cwpumping is turned on. Three pumping densities (0.8, 8 and 160W/cm2) are considered.In all cases the strong-coupling regime is maintained. Fig.8.13 shows the evolution ofthe ground-state population for different pumping densities and a seed withN0 = 102

particles. Table 8.2 gives the parameters obtained in the steady-state regime, namelyground-state populations, the ratio of populations of the ground state and the first ex-cited state, the ratio of the ground state population and thetotal populations, and thechemical potentialµ = kBT ln(1 − 1/N0). The steady-state distribution functions arefound to be very close to the Bose distribution function.

Table 8.1 Parameters used for numerical computations.

Cavity photon Non radiative Heavy hole Electronlifetime lifetime exciton mass mass

8ps 1ns 0.5m0 0.07m0

Fig. 8.14 shows the evolution of two second-order coherenceparameters of the sys-tem.η has already been defined as the ratio of coherent polaritons in the ground state


Fig. 8.13: Time dependence of the Bose-condensate occupation numberN0. The pumping densities are 0.8,8, and 160W/cm2 for curves (a), (b) and (c), respectively. The evolution of the seed at the initial stage isshown in the inset.

Fig. 8.14: Ground state coherenceη (dashed lines)and total system coherenceχ (solid line) for pumpingpower of.8, 8 and 160W/cm2 respectively.

over the total number of polaritons in the entire system, andχ is the ratio of coherent po-laritons in the ground state over the total number of polaritons in the ground state. Theyare linked byη = χn0/

∑

q nq. Ground state coherenceχ is maximum at initial time,which corresponds to the artificial introduction of a coherent seed. The relevant physicalquantity att = 0 is η, which is vanishingly small. Fig. 8.14a is the regime below thresh-old for coherent amplification, where the seed coherence is rapidly washed out and thatof the entire system remains zero. Fig. 8.14c in contrast displays the regime where the


Table 8.2 Main equilibrium values obtained numerically.

Pumping densityN0 N0/N1 N0/N −µ(W/cm2) (µeV)

0.8 7.7×103 23.5 0.04 568 2.7×105 510 0.59 1.6160 5.8×106 5800 0.95 0.07

coherent seed is coherently amplified: a macroscopic numberof particles populate thecoherent fraction of the initially small seed. Fig. 8.14b displays an intermediate regime.

Fig. 8.14c which shows quasi-condensation of polaritons, is characterised by thebuildup of the order parameter and by the amplification of theinitial seed coherentstate. The whole system coherence also strongly increases from 0 to more than 90%. Atintermediate pumping densities the values of the steady-state coherence degree dependnoticeably on the seed’s characteristics.

8.8 Kinetics of formation of polariton condensates: quantum picture beyond theBorn-Markov approximation

8.8.1 Two-oscillator toy theory

To gain insight into the mechanisms at work, we first revert toa toy model which reducesall the relevant physics to its bare minimum. Later we give the full picture suitable todescribe a realistic microcavity.

Since dimensionality is not an issue because it is not the accommodation of a pop-ulation in phase-space but dynamical effects that are responsible for populating theground state, we describe the system by a zero-dimensional two-oscillator model, oneoscillator representing the ground state, the other an excited state (or assembly of ex-cited states combined as a whole). We also neglect inter-particle interactions, which willclearly show that efficient relaxation is required (conserving particle number), but thatintrinsic inter-particle interactions are not necessary.The number of polaritons in theentire system fluctuates, but we shall see that the correlations implied by conservationof polaritons in their relaxation is at the heart of our mechanism. One can reconcilethe conservation of particles with a fluctuation in their total number through an inter-pretation in terms of a pulsed experiment, where a laser injects periodically in time afluctuating number of particles in the system. Each relaxation taken separately involvesan exact and constant number of particles, while observed results are averaged overpulses and thus echo an overall fluctuating population.

We label the states as 1 and 2. There is only one parameter to distinguish them whichis the ratioξ of the rate of transitionsw1→2 andw2→1 between these states:

ξ ≡ w2→1

w1→2(8.68)

Thesew are constants, especially they have no time dependence coming from pop-ulations included in these scattering terms. We assumeξ > 1 which identifies state 1 asthe ground state (i.e., state of lower energy), since from elementary statistics:


w2→1

w1→2= e(E2−E1)/kT (8.69)

with Ei the energy of statei (by definition of the ground state,E2 > E1) andT is thetemperature of the system once it has reached equilibrium.

The Hamiltonian for the two-oscillator system coupled through an intermediary os-cillator (depicting a phonon) in the rotating wave approximation reads in the interactionpicture

H = V ei(ω1−ω2+ω)ta0a†1b+ h.c. (8.70)

wherea1, a2 and b are (time independent) annihilation operators with bosonic alge-bra for first oscillator (ground state), second oscillator (excited state) and the “phonon”respectively, with free propagation energy~ω1, ~ω2, ~ω and coupling strengthV . Car-rying the same procedure as previously for, i.e., evaluating the double commutatorgives

∂tρ = − 12 [w1→2(a

†1a1a2a

†2ρ+ ρa†1a1a2a

†2 − 2a1a

†2ρa

†1a2)

+w2→1(a1a†1a

†2a2ρ+ ρa1a

†1a

†2a2 − 2a†1a2ρa1a

†2)] , (8.71)

after the Markov approximation ((τ) ≈ (t)) and factorisation of the entire systemdensity matrix into ρρph, with ρ, ρph the density matrices describing the two oscil-lators and the phonons respectively. Of course at this stagethe whole construct is veryclose to our previous considerations, but note that no Born approximation is made onρso that correlations between the two oscillators are fully taken into account. The transi-tion rates are given by:

w1→2 = 2π|V |2〈b†b〉/(~ω2 − ~ω1) (8.72a)

w2→1 = 2π|V |2(1 + 〈b†b〉)/(~ω2 − ~ω1) (8.72b)

We now obtain from (8.71) the equation for diagonal elements

p(n,m) ≡ 〈n,m|ρ|n,m〉 , (8.73)

wherep(n,m) is the joint probability distribution to haven particles in state 1 andmin 2. This equation reads

∂tp(n,m) = (n+ 1)m[w1→2p(n+ 1,m− 1) − w2→1p(n,m)]

+ n(m+ 1)[w2→1p(n− 1,m+ 1) − w1→2p(n,m)] . (8.74)

This equation for a probability distribution parallellingthe Boltzmann equation is theQuantum Boltzmann Master Equation (QBME) for the two-oscillators model.

The QBME can be derived rigorously from a microscopic Hamiltonian, as wasdone by Gardiner who pioneered this approach in a series of papers on quantum ki-netics started by Gardiner & Zoller (1997). However in the two-oscillator toy model,the physical picture is so straightforward that one hardly needs this approach (that is in-vestigated below). In a similar spirit, Scully (1999) applied fixed number of particles to


the case of Bose condensation of atoms. Coming back to our twomodes, and focusing,for instance, on the first term on the right hand side, one seesthat it expresses how theprobability can be increased to have(n,m) particles in states(1, 2) through the processwhere starting from(n + 1,m − 1) configuration, one reaches(n,m) by transfer ofone particle from state 1 to the other state. This is proportional ton + 1, the numberof particles in state 1 and is stimulated bym − 1 the number of particles in state 2 towhich we add one for spontaneous emission, whence the factor(n + 1)m. We repeatthatw1→2 andw2→1 are constants and should not be confused with thebosonic transi-tion ratedefined asw1→2(1 +m) andw2→1(1 +n) to account in a transparent way forstimulation. Our present discussion will be clarified by being explicit.

For all quantitiesΩ which pertain to a single state only, say the ground state (sowecan writeΩ(n)), it suffices to know the reduced probability distribution for this state,i.e., for the ground state:

p1(n) ≡∞∑

m=0

p(n,m) (8.75)

and vice-versa, i.e., for excited statep2(m) ≡∑∞n=0 p(n,m). So that indeed

〈Ω(n)〉 =

∞∑

n=0

∞∑

m=0

Ω(n)p(n,m) =

∞∑

n=0

Ω(n)p1(n) (8.76)

We will soon undertake to solve exactly equation (8.74) but to delineate its qualityin explaining how coherence arises in the system we first showthat if we make theapproximation to neglect correlations between the two states, i.e., if we assume thefactorisation

p(n,m) = p1(n)p2(m) (8.77)

then the system at equilibrium will never display any coherence, i.e., in accord with ourprevious discussion, both states will be in a thermal state no matter the initial conditions,the transition rates or any other parameters describing thesystem. Indeed putting (8.77)into (8.74) and summing overm, we obtain:

∂tp1(n) = p1(n+ 1)w1→2(n+ 1)(〈m〉 + 1)

− p1(n)(

w2→1(n+ 1)〈m〉 + w1→2n(〈m〉 + 1))

+ p1(n− 1)w2→1n〈m〉(8.78)

with 〈m〉 ≡ ∑mmp2(m) the average number of bosons in state 2. At equilibrium thedetailed balance of these two states gives the solution

p1(n+ 1) =〈m〉

〈m〉 + 1

w2→1

w1→2p1(n) (8.79)

The same procedure for state 2 yields likewise

p2(m+ 1) =〈n〉

〈n〉 + 1

w1→2

w2→1p2(m) (8.80)


0

0.5

1

0 5 10 15 20

<n>

θ=<n>/(1+<n>)

0

1

2

3

0 1 2 3 4kBT/−hω

<n>=θ/(1-θ)=(e−hω/kBT-1)-1

θ=e-−hω/kBT

Fig. 8.15: “Thermal parameter”θ as a function of〈n〉 (left figure) and temperature (right figure, dashed line).θ varies in the interval[0, 1[ and tends towards unity with increasing particle number. The average number ofparticle is also displayed as function of the temperature onthe right figure (solid line).

with the notational shortcuts

θ ≡ 〈n〉〈n〉 + 1

, ν ≡ 〈m〉〈m〉 + 1

(8.81)

equations (8.79) and (8.80) read after normalisation

p1(n) = (1 − νξ)(νξ)n (8.82a)

p2(m) = (1 − θ/ξ)(θ/ξ)m (8.82b)

so that〈n〉 ≡∑np1(n) = νξ/(1 − νξ) which inserted back into (8.81) yields

ξ =θ

ν(8.83)

or, written back in terms of occupancy numbers and transition rates:

w2→1

w1→2=

〈n〉〈n〉 + 1

〈m〉 + 1

〈m〉 (8.84)

which give in eq. (8.79), (8.80):

p1(n+ 1) =〈n〉

〈n〉 + 1p1(n) and p2(m+ 1) =

〈m〉〈m〉 + 1

p2(m) (8.85)

achieving the proof that both states are (exact) thermal states under the hypothesis (8.77)that we will now relax. This will give rise to a likewise regime where both states arethermal states, but also to another regime where the excitedstate (state 2) is still ina thermal state, but the ground state (state 1) is non-thermal (and in some limit, hasthe statistics of a coherent state). This is possible if one takes into account correlationsbetween states. In our case these correlations come from theconservation of particlenumber, so that the knowledge of particle number in one statedetermines the number in


other state. In fact observe how the QBME connects elements of p(n,m) which lie onantidiagonals of the plane(n,m). One such antidiagonal obeys equation

n+m = N (8.86)

whereN is a constant, namely, the distance of the antidiagonal to the origin from thegeometrical point of view, and the number of particles from the physical point of view.One such antidiagonal is sketched on Fig. 8.16. The equationcan be readily solved ifonly one antidiagonal is concerned, i.e., if there are exactly N particles in the system. Inthe case where the particle number in the system is only knownwith some probability,one can still decouple the equation onto its antidiagonal projections, solve for themindividually and add up afterwards weighting each antidiagonal with the probability tohave the corresponding particle number. We hence focus on such one antidiagonalNwhoseconditionalprobability distribution is given byd(n|N) ≡ p(n,N − n), withequation of motion given by (8.74) as:

∂td(n|N) = (n+ 1)(N − n)[w1→2d(n+ 1|N) − w2→1d(n|N)]

+n(N − n+ 1)[w2→1d(n− 1|N) − w1→2d(n|N)](8.87)

The equation is well behaved within its domain of definition0 ≤ n ≤ N since itsecures thatd(n|N) = 0 for n > N . This also ensures unicity of solution despite therecurrence solution being of order2, for d(1|N) is determined uniquely byd(0|N),itself determined by normalisation.

The stationary solution is obtained in this way (or from detailed balance):

d(n+ 1|N) =w2→1

w1→2d(n|N), (8.88)

with solution

d(n|N) = d(0|N)

(

w2→1

w1→2

)n

(8.89)

whered(0|N) is defined for normalisation as

d(0|N) =ξ − 1

ξN+1 − 1, ξ ≡ w2→1

w1→2(8.90)

Technically solving ford resembles the procedure already encountered to solve theequation under assumption (8.77). However we are now payingfull attention to correla-tions between the two states, which turns detailed balancing (8.79) and (8.80) into oneof an altogether different type (8.89). This gives by weighting (8.89) the solution to theQBME:

p(n,m) =ξ − 1

ξn+m+1 − 1ξnP (n+m) (8.91)

whereP (N) ≡

∑

n+m=N

p(n,m) (8.92)

is the distribution of total particle number, i.e., the probability to haveN particles in theentire system.P (N) is time independent since the microscopic mechanism involved


n

m

p(n,m)

antidiagonaln+m=7

Fig. 8.16: p(n,m) steady state solution from Laussy, Malpuech & Kavokin (2004) in the case where thedistribution function for the number of particles in the entire systemP (N) is a gaussian of mean (and vari-ance)15 andξ = 1.2. One “antidiagonal”,n+m = 7, is shown for illustration. The projection on then-axisdisplays a coherent state whereas the projection on them-axis displays a thermal state.

conserves particle number for any transition (one can also check that∂tP (N) = 0).This allows us to derive the statistics of separate states:

p1(n) = ξn∞∑

N=n

ξ − 1

ξN+1 − 1P (N)

p2(n) = ξ−n∞∑

N=n

ξ − 1

ξN+1 − 1ξNP (N)

(8.93)

Observe how then dependence of the sum index prevents trivial relationship betweenp1

and p2 of the kind p1(n) = p2(N − n). Also the asymmetry between ground andexcited state is obvious from (8.91). It is this feature which allows two states with drasticdifferent characteristics, typically a thermal and a coherent state. Indeed,p1 (resp.p2) isthe product of a sum with an exponentially diverging (resp. converging to zero) functionof ξ. In both cases, the sum of positive terms is a decreasing function of n, so thatclearly no coherence can ever survive in the excited state whose fate is always thermalequilibrium, or at least, in accord with our definitions,

p2(n) > p2(n+ n0) for all n, n0 in N (8.94)

For p1 however,ξn diverges withn which leaves open the possibility of a peak notcentred about zero in this distribution, while it can still be a decreasing function if the


sum converges faster still. It is toP (N) to settle this issue, which as a constant of motionis completely determined by the initial condition. The solution for the case whereP (N)is a gaussian of mean (and average) 15 is displayed in Fig. 8.16.p(n,m) is in this casemanifestly not of the typep1(n)p2(m) and there is always coherence in the system. Inthe next section we investigate the more interesting situation where coherence does notexist a-priori in the system.

8.8.1.1 Growth of the condensate at equilibriumBy growth at equilibrium we meanthat, still in the approximation of infinite lifetime, coherence can arise when one lowerstemperature, i.e., increasesξ, in a system where initially all states are thermal states. Inthis case the initial condition for the system is the thermalequilibrium

p(n,m) = (1 − θ)(1 − ν)θnνm (8.95)

whereθ, ν are the thermal parameters for ground and excited states respectively. Theylink to 〈n〉, the mean number of particles in ground state, through

〈n〉 =θ

1 − θ, (8.96)

or, the other way around,

θ =〈n〉

1 + 〈n〉 . (8.97)

Similar relations hold forν andm. This is one possible steady solution of (8.74) and wediscuss how it arises from (8.91) below. The thermal state iswildly fluctuating. Once ina while, thermal kicks transfer in the state one or many particles, which however do notstay for long before the state is emptied again or replaced byother, unrelated particles.This accounts for the chaotic, or incoherent, properties ofsuch a state. This essentiallyempty but greatly fluctuating statistics brings no conceptual problem for little popula-tions, but one might enquire whether it is conceivable to have a thermal distributionwith high mean number. This is possible for a single state butnot for the system as awhole. A macroscopic population can distribute itself in a vast collection of states sothat each has thermal statistics, constantly exchanging particles with other states anddisplaying great fluctuations, but as expected from physical grounds, the whole systemdoes not fluctuate greatly in its number of particles. Therefore we expectP (N), thedistribution of particles in theentire system, to be peaked about a nonzero value, typ-ically to be a gaussian of mean and variance equal toN . In the pure Boltzmann case,this results from the central limit theorem since the total number of particles is the sumof a large number of uncorrelated random variables, and thusis itself a Gaussian ran-dom variable. Remembering our previous definitions, this however does not qualify thesystem as a coherent emitter, since the statistics must refer to asingle state, not to a vastassembly of differing emitters. Thus, not surprisingly, coherence arises when asinglequantum mode models or copies features of a macroscopic system, typically its pop-ulation distribution. The two-oscillator system which is arather coarse approximationto a macroscopic system will however display very clearly this mechanism. In the limit


Fig. 8.17: Ground state distributionp1(n) for ξ = θ

ν≈ 1.26 with 3

particles in excited state and 17 inground state, both in thermal equi-librium, g(2)(0) = 2.

Fig. 8.18: Same withξ raisedto 1.5. Distribution is non-thermal,especiallyp1(1) > p1(0), thoughthe distribution is then always de-creasing.g(2)(0) ≈ 1.89.

Fig. 8.19: Same withξ → 0.The distribution is that for the en-tire system, p1(n) = P (N),cf. (8.102). Yet it is far from co-herent in this model.g(2)(0) →1.745.

whereξ ≫ 1 it is already visible from (8.93) thatp1(n) ≈ P (n), so that the statistics ofthe entire system indeed serves as a blueprint for the groundstate (and it alone, excitedstates being always decreasing as already shown). At equilibrium, with two thermalstates, the distribution for the whole system reads:

P (N) =∑

n+m=N

p(n,m) = (1 − θ)(1 − ν)θN+1 − νN+1

θ − ν(8.98)

This exhibits a peak at a nonzero value provided that

ν + θ > 1 (8.99)

This criterion refers to a first necessary condition: there must be enough particles in thesystem. The fewest particles available so that (8.99) is fulfilled, is two. This minimumrequired to grow coherence fits nicely with the Bose-Einstein condensation picture (oneneeds at least two bosons to condense). It is not a necessary condition, though; also thedynamical aspect is important as shown by the key role ofξ. Indeed if the system issteady in configuration (8.95),ξ is not a free parameter but is related toθ andν by:

ξ =θ

ν(8.100)

and in this case the distribution of ground state

p1(n) = (1 − θ)(1 − ν)ξn∞∑

N=n

θN+1 − νN+1

ξN+1 − 1

ξ − 1

θ − ν(8.101)

reduces by straightforward algebra top1(n) = (1 − θ)θn., i.e., as should be for con-sistency, the ground state is in a thermal state, independently of the value ofθ (i.e.,no matter what is the number of particles in the ground state). This can come as asurprise, but it must be born in mind that this two-oscillator model is an extreme sim-plification which cannot dispense with some pathological features, namely, the abil-ity to sustain a thermal macroscopic population, an abilitythat we understand easily


since the ground state accounts for half of the system! It is expected that with in-creasing number of states, dimensionality will forbid suchan artifact. Also the shapeof P (N) hardly resembles a gaussian (see Fig. (8.19)) but already inthis limiting caseit is able to display a peak at a nonzero value provided there are enough particles.With increasing number of states, the central limit theoremwill turn this distributioninto an actual gaussian. Once again,P (N) is time independent because the relaxationmechanism conserves particle number, which results in correlations between the twostates. By increasingξ to ξ′, one might search new values ofθ, ν, sayθ′, ν′, so thatθ/(1 − θ) + ν/(1 − ν) = θ′/(1 − θ′) + ν′/(1 − ν′) (conservation of particle number)andξ′ = θ′/ν′. This is possible if one allowsP (N) to change, in which case the twonew states are also thermal states. IfP (N) is constrained by correlations induced bystrict conservation of particle number—so that the uncertainty is not shifted as the sys-tem evolves—then (8.100) breaks down and this allows (8.91)to grow a coherent statein the ground state. This process is illustrated on Figs. 8.17–8.19, starting frow ther-mal equilibrium and lowering temperatures (increasingξ). In the two-oscillator model,coherence grown out of thermal states cannot come much closer to a gaussian than il-lustrated in Fig. 8.19 where is displayed the limiting caseξ → 0 for which the groundstate distribution reduces toP (N), cf. Eq. (8.98),

p1(n) = (1 − θ)(1 − ν)θn+1 − νn+1

θ − ν, (8.102)

which is obvious on physical grounding (one particular realisation is the one for whichw1→2 = 0 and thus with all particles eventually reaching the ground state) and rein-forces our understanding of Bose condensation as the groundstate distribution functioncoming close to the macroscopic distribution, with complete condensation correspond-ing to identification ofp1(n) with P (N).

8.8.1.2 Growth of the condensate out of equilibriumThe previous case holds inan equilibrium picture and for that matter refers to coherence buildup in systems likecold atom BEC. To address the polariton laser case, it is necessary to extend the two-oscillator model with the additional complication of finitelifetime τ of particles instate 1, with a balance in the total population provided by a pump which injects par-ticles in state 2 at a rateΓ. (We will not crucially need a finite lifetime in the excitedstate and thus neglect it, which is a good approximation in a typical microcavity wherethe radiative lifetime drops by a factor of ten to a hundred inthe photon-like part of thedispersion.) Although the QBME can be readily extended phenomenologically to takethese into account,

p(n,m) = (n+ 1)m[w1→2p(n+ 1,m− 1) − w2→1p(n,m)]

+ n(m+ 1)[w2→1p(n− 1,m+ 1) − w1→2p(n,m)]

+1

τ(n+ 1)p(n+ 1,m) − 1

τnp(n,m)

+ Γp(n,m− 1) − Γp(n,m)

(8.103)


the couplings between different particle numbers forbid solving this new equation alongthe same analytical lines as previously, though the numerical solution can be obtainedstraightforwardly. Introducing〈m〉n the mean number of polaritons in the excited stategiven that there aren in the ground state, i.e.,

〈m〉n p0(n) =∞∑

m=0

mp(n,m) (8.104)

with p0(n) ≡∑m p(n,m) is the reduced ground state statistics, we obtain, by averag-ing eq. (8.74) over excited states, an equation for the ground state statistics only:

∂tp1(n) = (n+ 1)(

w1→2(〈m〉n+1 + 1) + 1/τ)p1(n+ 1)

−

n(

(〈m〉n + 1)w1→2 + 1/τ)

+ (n+ 1)〈m〉nw2→1

p1(n)

+ n〈m〉n−1w2→1p1(n− 1) , (8.105)

However in this out-of-equilibrium regime, the excited state is not as important asin the equilibrium case where it must be thermal and whose configuration is of utmostconsequence on the ground state. Thus we can dispense from the actual distributionof the excited state and use simply the mean〈m〉n obtained from

∑

mmp(n,m) =〈m〉np2(n). In this case (8.103) can be decoupled to give an equation forp1(n) alone,and in the “dynamical” steady state, the detailed balance reads:

p1(n+ 1) =w2→1〈m〉n

w1→2(〈m〉n+1 + 1) + 1/τp1(n) (8.106)

Up to this point it is still exact, and also in the out-of-equilibrium regime we grantthe conservation of particle number as the origin of correlations between the two states,but because of lifetime and pumping, it can now be secured only in the mean, leadingus to the following approximation for〈m〉n:

〈m〉n = N − n (8.107)

The pump, which has quantitatively disappeared from the formula, is implicitly takeninto account through this assumption, since even though particles have a finite lifetime,their number is constant on average. In the coherent case,p1(n) is a poisson distributionwith maximum atN − Nc, so that this dependency ofN on the pump is in this caseN = τP +Nc.

When the population has stabilised in the ground state by equilibrium of radiativelifetime and pumping, it is found in a coherent state ifN > Nc with Nc the criticalpopulation defined by:

Nc =1

τ(w2→1 − w1→2)(8.108)


obtained from (8.106) and (8.107) with the requirement thatp1(1) > p1(0). If thispopulation is exceeded, coherence builds up in the system along with the population,which stabilises at an average given by the maximum of the gaussian-like distribution:

〈n〉 = nmax = N −Nc (8.109)

obtained fromp1(n) = p1(n + 1); so that effectively ifN < Nc there is no suchgaussian and coherence remains low with a thermal-like state whose maximum is forzero occupancy. IfN > Nc the state is a gaussian whose mean increases with increasingdeparture of population from the critical population. Thus, the more the particles, theless the particle number fluctuations of the state, and the better its coherence.

On Fig. 8.20 is displayed the numerical solution of eq. (8.103) for p1(n, t), whereparameters (see legend) have been chosen so thatN exceeds (8.108) and therefore growsome coherence from an initially empty ground state (cf. Fig. 8.20-a). The coherenceis maintained for infinite times and the statistics for the ground state occupancy tendstowards a gaussian-like function neatly peaked about a highvalue (cf. Fig. 8.20-b). Wedefine acoherence degreeequal to2 − g(2)(0), so chosen to be 0 for a genuine thermalstate and 1 for a genuine coherent state. In the case where (8.108) is exceeded, thecoherence degree of ground state quickly reaches unity (Fig. 8.20-c).

On Fig. 8.21 is displayed the counterpart situation where parameters (see legend) re-sult in a sub-critical population so that the steady state isthermal, as shown on Fig. 8.21-b. The dynamics ofp(n), starting from vacuum, is merely to grow this thermal state(cf. Fig. 8.21-a) and the coherence degree remains low (cf. Fig. 8.21-c).

We have varied the temperature (throughw2→1) for simplicity but kept all other pa-rameters constant. This is not very convenient experimentally as this requires adjustingthe pumping. The total number of particlesN can be expressed as a function of otherparameters as follows: the rate equation of particle numberin ground and excited stateare

∂tn = −γn− w1→2n(N − n+ 1) + w2→1(n+ 1)(N − n) , (8.110a)

∂t(N − n) = P − (n+ 1)(N − n)w2→1 + n(N − n+ 1)w1→2 . (8.110b)

Going to the steady state yields the relation

N =P (γ2 + w2→1(γ − P ) + w1→2(γ + P ))

γ(w1→2(γ + P ) − w2→1P )(8.111)

which provides an equivalent set of parameters to cross the threshold as a function ofPor γ, which is more relevant experimentally.

8.8.2 Coherence of polariton laser emission

The extension from the two-oscillator model to a realistic microcavity is straightforwardas far as the formalism is concerned, but the resulting equations cannot be tackled di-rectly as previously. As before, the pumping and finite lifetime should not be neglected,but for the relaxation, one can still consider phonon-mediated scattering only (neglect-ing polariton-polariton scatterings). Since all these processes are dissipative, we seek a


Fig. 8.20: System configuration suitable forcoherence buildup:N = 350, w2→1 =10−5 (arb. units),w1→2 = 0.75 × 10−5

(arb. units) and1/τ = 20×10−5 (arb. units).All units have the same dimension of an in-verse time. (a) is a density plot for the timeevolution of p(n), starting from vacuum itquickly evolves towards a coherent state. (b) isthe projection ofp(n) in the steady state. (c)is the time evolution of the normalised coher-ence degreeη = 2 − g(2)(0): full coherenceis quickly attained.

Fig. 8.21: System configuration unable to de-velop coherence. Parameters are the same asfor Fig. 8.20 exceptw2→1 = .95 × 10−5

(arb. units) corresponding to a higher tempera-ture. (a) Starting from vacuum the ground statesteadies in a thermal state for which a projec-tion (b) is shown. (c) The normalised coher-ence degree remains low.


master equation of the Lindblad type for the density matrixρ of polariton states in thereciprocal space:

∂tρ = (Lpol−ph + Lτ + Lpump)ρ (8.112)

HereL are Liouville super-operators which describe respectively scattering (throughphonons), lifetime and pumping. The polariton-polariton scattering would add a unitary(non-dissipative) contributionLpol−pol = − i

~[Hpol−pol, ρ]. In the following we under-

take the derivation ofLpol−ph from the microscopic HamiltonianHpol−ph for polariton-phonon scattering. Exactly similar procedures can be carried forLτ andLpump to yield:

Lτρ = −∑

k

1

2τk(a†kakρ+ ρaka

†k − 2akρa

†k) (8.113a)

Lpumpρ = −∑

k

Pk

2(aka

†kρ+ ρa†kak − 2a†kρak) (8.113b)

with τk the lifetime,Pk the pump intensity in the state with momentumk andak theBose annihilation operator for a polariton in this state. The expression for the lifetimecomes from the quasi-mode coupling of polaritons with the photon field outside thecavity in the vacuum state (thereby linking spontaneous emission with the perturbationfrom vacuum fluctuations). We later neglect finite lifetime elsewhere than in the groundstate, where it is typically several orders of magnitude shorter because of the dominantphoton fraction. In our simulations, pumping injects excitons 10meV above the bottomof the bare exciton band which we model by nonzero values ofPk for a collection ofk-states normally distributed about a high momentum mean value. Expression (8.113b)describes an incoherent pumping provided by a reservoir which pours particles in thesystem but does not allow their coming back. Its effect is thus merely to populate thesystem with incoherent polaritons, which will relax towards the ground state where theymight join in a coherent phase before escaping the cavity by spontaneous emission (thelight thus emitted retaining this coherence).

We pay special attention toLpol−ph which contains the key-ingredients of our re-sults. We repeat here the interaction picture polariton-phonon scattering term from thepolariton Hamiltonian:

Hpol−ph =∑

k,q 6=0

Vqei~(Epol(k+q)−Epol(k)−~ωq)tak+qa

†kb

†q + h.c. (8.114)

with Vq the interaction strength,Epol the lower polariton-branch dispersion,~ωq thephonons dispersion andaq, resp.bq, the Bose annihilation operator for a polariton,resp. a phonon, in stateq.

As previously, we computeLpol−ph starting with Liouville-Von Neumann equationfor polariton-phonon scattering,

∂t = − i

~[Hpol−ph, ] (8.115)

where is the density matrix for polaritons and phonons. Its usefulpart, namely thepolariton density matrix, is obtained by tracing over phonons,ρ ≡ Trph. The density


matrix for phonons is combined as a reservoir in equilibriumwith no phase coherencenor correlations withρ. To dispense from this reservoir we write the equation for toorder two in the commutator and trace over phonons,

∂tρ(t) = − 1

~2

∫ t

−∞Trph[Hpol−ph(t), [Hpol−ph(τ), ]] dτ (8.116)

We define

Ek,q(t) ≡ Vqei~(Epol(k+q)−Epol(k)−~ωq)t (8.117)

and for convenience we write

H(t) ≡∑

k,q

Ek,q(t)ak+qa†kb

†q (8.118)

so thatHpol−ph = H+H†. Operators are time independent. Because the phonon densitymatrix is diagonal,[Hpol−ph(t), [Hpol−ph(τ), ]] reduces to[H(t), [H†(τ), (τ)]]+h.c.,which halves the algebra. Also the conjugate hermitian follows straightforwardly, so weare left only with explicit computation of two terms, of which the first reads:

[H(t),H†(τ)] =∑

k,q 6=0

∑

l,r6=0

[Ek,q(t)ak+qa†kb

†q, E∗

l,r(τ)ala†l+rbr(τ)] (8.119)

which, taking the trace over phonons and callingθqρ ≡ Trph(b†qbq), becomes:

Trph[H(t),H†(τ)(τ)] =∑

k,l,q 6=0

Ek,q(t)E∗l,q(τ)θq(ak+qa

†kala

†l+qρ(τ) − ala

†l+qρ(τ)ak+qa

†k) (8.120)

Solving numerically this equation is a considerable task, which however has alreadybeen carried out for a similar equation by Jaksch et al. (1997), using Quantum Monte-Carlo simulations. We prefer to make further approximations to reduce its simulationto a level of complexity of the same order as for Boltzmann equations: we take intoaccount correlations between ground state and excited states only, neglecting all cor-relations between excited states. This is legitimated by the fast particle redistributionbetween excited states and their rapid loss of phase correlations. Physically this meansthat if a particle reaches the ground state, its absence is felt to some extent in the col-lection of excited states in a way which ensures particle number conservation. On thecontrary, redistribution of particles between excited states will be seen to obey the usualBoltzmann equations which pertain to averages only. Formally, we thus neglect termslike 〈ak1a

†k2ak3a

†k4〉 if ki involve nondiagonal elements in the excited state. For non-

vanishing terms, we further allow〈ak1a†k1ak2a

†k2〉 = 〈ak1a

†k1〉〈ak2a

†k2〉 if neitherk1


nork2 equal0, while otherwise we retain the unfactored expression. Terms from (8.120)featuring the ground state are:

∑

k 6=0

Ek,−k(t)Ek,−k(τ)∗θk(a0a†0a

†kakρ(τ) − aka

†0ρ(τ)a0a

†k) (8.121a)

+∑

k 6=0

E0,k(t)E0,k(τ)∗θk(a†0a0aka†kρ(τ) − a0a

†kρ(τ)a

†0ak) (8.121b)

Recall this expression (8.121) is one part of the term insidethe time integral whichgives∂tρ(t) evolution. Since

Ek,q(t)Ek,q(τ)∗ = |Vq|2 exp(

− i

~(Epol(k + q) − Epol(k) − ~ωq)(t− τ)

)

the time integration would yield a delta function of energy (times−i/~) if ρ in (8.121)wasτ -independent. This delta would itself provide selection rules for allowed scatteringprocesses through the sum overk. Thatρ(τ) time evolution is slow enough as comparedto this exponential to mandate this (Markov) approximationcan be checked throughevaluation of the phonon reservoir correlation time, which, when the reservoir has abroad-band spectrum as in our case, is short enough to allow the approximation ofρ(τ)by ρ(t). In this case, (8.121b) vanishes as a non-conserving energyterm. Gatheringother terms similar to (8.120) eventually gives (from now onwe do not writeρ timedependence anymore, which ist everywhere):

∂tρ = − 1

2

∑

k 6=0

W0→k(a†0a0aka†kρ+ ρa†0a0aka

†k − 2a0a

†kρa

†0ak) (8.122a)

− 1

2

∑

k 6=0

Wk→0(a0a†0a

†kakρ+ ρa0a

†0a

†kak − 2a†0akρa0a

†k) (8.122b)

where

W0→k ≡ 2π

~|Vk|2θkδ(Epol(k) − Epol(0) − ~ωk) (8.123a)

Wk→0 ≡ 2π

~|Vk|2(1 + θk)δ(Epol(k) − Epol(0) − ~ωk) (8.123b)

We call p(nk) the diagonal of the polariton density matrix, i.e., the dotting of ρwith |nk〉 = |n0, nk1 , · · · , nki , · · · 〉 the Fock state withnki polaritons in stateki:

p(nk) ≡ 〈· · · , nki, · · · , nk1 , n0|ρ|n0, nk1 , · · · , nki , · · · 〉 (8.124)

This is the probability that the system be found in configuration nk whose equationof motion is the master equation obtained from (8.122) as


p(nk) = −∑

k

(W0→kn0(nk + 1) +Wk→0(n0 + 1)nk)p(nk)

+∑

k

W0→k(n0 + 1)nkp(n0 + 1, . . . , nk − 1, . . .) (8.125)

+∑

k

Wk→0n0(nk + 1)p(n0 − 1, . . . , nk + 1, . . .)

This is the counterpart to (8.74), which also parallels closely the Boltzmann equationwith which it shares the same transition rates (8.123) givenby Fermi’s golden rule, andso it represents the QBME for polariton lasers. We include back (8.113) in (8.122) (notethat we could have done this at any moment) and, following thesame spirit, we do notsolve it for the entire joint probabilityp(nk) but average over all excited states toretain the statistical character for the ground state only.Excited states will be describedwith a Boltzmann equation, thus with thermal statistics. Calling

p0(n0) ≡∞∑

i=1

∞∑

nki=0

p(n0, nk1 , nk2 , · · · , nkj , · · · ) (8.126)

the ground state reduced probability (the sum is over all states but the ground state,cf. (8.75)), and

〈nk〉n0p0(n0) ≡

∑

nk1,nk2,···

nkp(nk) (8.127)

cf. (8.104), we get the ground state QBME equation:

p0(n0) = (n0 + 1)(Wn0+1out + 1/τ0)p0(n0 + 1)

−(

n0(Wn0

out + 1/τ0) + (n0 + 1)Wn0

in

)

p0(n0) (8.128)

+ n0Wn0−1in p0(n0 − 1)

with rate transitions now function of the ground state population numbern0:

Wn0

in (t) ≡∑

k

Wk→0〈nk(t)〉n0(8.129a)

Wn0

out(t) ≡∑

k

W0→k(1 + 〈nk(t)〉n0) (8.129b)

while for excited states, in Born-Markov approximation, weindeed recover the Boltz-mann equations:

˙〈nk〉 = 〈nk〉∑

q 6=0

Wk→q(〈nq〉+1)− (〈nk〉+1)∑

q 6=0

Wq→k〈nq〉, k 6= 0 (8.130)

Inclusion ofLτ andLpump for the above adds−〈nk〉/τk + Pk to this expression. Ob-serve that in this case, the transition rates are constants.


Cast in this form, eq. (8.128) has the same transparent physical meaning in termsof a rate equation for the probability of a given configuration, much like the usual rateequations for occupation numbers in the framework of the Boltzmann equations. Thedifference is that transitions from one configuration to a neighbouring one occur at rateswhich depend on the configuration itself, through the population of the state.〈nk〉n0

is afunction ofn0 that we estimate through a first order expansion about〈n0〉. This impliesthat fluctuations of excited states are proportional (with opposite sign) to fluctuations ofground state:

〈nk〉n0≈ 〈nk〉〈n0〉 +

∂〈nk〉n0

∂n0

∣

∣

〈n0〉(n0 − 〈n0〉) (8.131)

〈nk〉〈n0〉 is given by a Boltzmann equation. Since the derivative does not depend onn0

(it is evaluated at〈n0〉), we compute it by evaluation of both sides at a known value,for instancen0 = N with N the total particle number in the entire system, ground andexcited states together. This gives

∂〈nk〉n0

∂n0

∣

∣

∣

∣

∣

〈n0〉=

〈nk〉〈n0〉 −N

(8.132)

since〈nk〉N = 0 (no particles are left in excited states when they are all in the groundstate). With the knowledge of (8.131) and (8.132) which are known from semi-classicalBoltzmann equations, this is now only a matter of numerical simulations.

8.8.3 Numerical simulations

Parameters used are for a CdTe microcavity of 10µm lateral size with one QW and aRabi splitting of 7 meV at zero detuning. The size corresponds to the light spot radiusreported by Deng et al. (2002). This is an important parameter as correlations increasewith decreasing size of the system. Scattering is mediated by a bath of phonons at atemperature of 6 K and with a thermal gas of electrons of density 1011 cm−2 accountedfor to speed up relaxation. This is below the exciton bleaching density. The cavity isinitially empty and pumped non-resonantly fromt = 0 onwards.

Figure (8.22) displays the ground state population normalised to its steady statevalueneq and the coherent fractionχ. Starting from zero for the vacuum state, coher-ence steadily rises in the system as more polaritons enter the ground state. Interestingly,even though the dynamics can give rise to a temporary decrease in the number of groundstate polaritons, the coherence does not drop in echo but continues its ascent. This is bet-ter understood with Fig. (8.23) where a density plot forp0(n0) is shown and where onecan observe how the coherent statistics is reached through atightening of the numberof states with high probability of occupancy about the average. The functionp0(n0),which at first varies wildly, quickly flattens with a large number of particles in a ther-mal state, then a nonzero maximum appears and the statisticsevolves as gaussian-liketowards a poisson distribution of mean〈n0〉 in the steady state. The polariton densityof 1010cm−2 achieved in the simulation is more than one order of magnitude smallerthan the strong/weak coupling transition density in CdTe. In Fig. (8.25) is displayed thesteady state coherence ratio and population as a function ofpumping. It exhibits a clear


Fig. 8.22: Time evolution of normalised groundstate population〈n0(t)〉/neq (dotted line) andof normalised coherence ratioη = 2 − g(2)

(solid line), starting from the vacuum (no coher-ence): both population and coherence buildup areobtained in the timescale of hundreds of ps.

Fig. 8.23: Density plot ofp0(n0) (ground statepolaritons distribution) as a function of time fora realistic microcavity (with parameters given inthe text). Darker colors correspond to smaller val-ues. Initial state is the vacuum. A coherent stateis obtained in the timescale of hundreds of ps.

Fig. 8.24: Projections ofp0(n0) of previousfigure at three various times: still in a thermalstate (exponential decay) close to initial times,≈ 75ps; in a fully grown coherent state (Gaus-sian) in the steady state and in the intermediateregion where coherence is building-up,≈ 100ps.

Fig. 8.25: Ground state population normalized to pumppower〈n0〉/P (solid line) and normalised coherence de-greeη in the steady state (dashed) as a function of thereduced pumpingP/Pthreshold; we findPthreshold =26W/cm−2 .

threshold behavior. These results published by Laussy, Malpuech, Kavokin & Bigen-wald (2004a) can be complemented by another run of simulations published as Laussy,Malpuech, Kavokin & Bigenwald (2004b).

8.8.4 Order parameter and phase diffusion coefficient

8.8.4.1 Formalism and discussionThe phase diffusion coefficientD is the decayconstant of the order parameter〈a0(t)〉 = 〈a0(0)〉eiω0te−Dt whereω0 is the free prop-agation energy of the considered field. This quantity has been introduced in Chapter 6and discussed in Section 8.7.2. It is directly related to thelaser linewidth and inverselyproportional to its coherence length. For usual lasers, this phase diffusion coefficient


is due to the finite ratio which exists between spontaneous and stimulated emission oflight.D is therefore inversely proportional to the number of photons in the lasing mode.This number is finite is any finite size system and in any realistic system.

For polariton lasers, as we have seen in Sections 8.3.1 and 8.3.2, this decay constanthas, for non interacting particles, the same physical origin as for normal lasers. It is dueto the spontaneous scattering toward the ground state whichprovokes a dephasing (seeequations 8.74 and 8.75). This dephasing is responsible fora broadening of the emissionlinewidth. The emission spectrum can indeed be calculated through:

I(ω, t) =1

πℜ∫ +∞

0

〈a†0(t+ τ)a0(t)〉eiωτ dτ (8.133)

This quantity becomes independent oft in the steady state to give

I(ω, t) = ℜ∫ +∞

0

〈a†0(τ)a0(0)〉eiωτ dτ =DN0/π

(ω − ω0)2 +D2. (8.134)

This is the classical Lorentzian spectrum. This aspect has been widely discussed inlaser theory. However polaritons are interacting particles and this may lead to a substan-tial shift of the emission energy and to an increase of the linewidth. Formally this shiftand broadening originates from the terms

∑

k Vka†0a0a

†kak of the Hamiltonian. These

do not invoke any real scattering and they do not modify either the populations or thestatistics. They are calleddephasing terms. Their impact has been analysed by Porras &Tejedor (2003) in the steady state and by Laussy et al. (2006)on the polarisation. Porras& Tejedor considered the broadening of a thermal state. Fig.8.26 shows the full widthat half maximum of the ground state line versus pumping power. The width first de-cays above the threshold because of the decrease of dephasing induced by spontaneousscattering. However, the dephasing induced by interactions increases proportionally tothe ground state population and becomes dominant for large pumping powers. In whatfollows we give an intuitive derivation which gives the sameresults as the quantumformalism developped by Laussy et al. (2006).

Fig. 8.26: Ground state occupancy of a polariton laser versus pumping power (dashed line) and Full width athalf maximum of the ground state emission (solid line), as predicted by Porras & Tejedor (2003).


We consider again a polariton system withN particles in all andN0 in the groundstate. We also defineN1 = N − N0 the number of particles in the excited states.The interparticle interaction constant is supposed to be constant and is given byV .Therefore, the average interaction energy of the ground state is simply given byV NN0.The energy per particle is equal to the energy shift of the line:

Eshift =V NN0

N0= V N (8.135)

This shift is a rigid shift of the complete dispersion, if oneneglects the dependenceof V on the wavevector.N however is a fluctuating quantity governed by a statisticaldistribution. The uncertainty of the number of particles leads to an uncertainity in theenergy. We are going to separate the uncertainty fromN1 and the uncertainty fromN0.If one considers a large ensemble of independent states, allhaving thermal statistics anda small average number of particles, the statistics of the total number of particles in thissystem is Poissonian. This is a consequence of the central limit theorem. Therefore, thenumber of particles in excited states follows a Poissonian distributionP1(n1). The rootmean square of this distribution isσ1 =

√N1. The emission line resulting from the

interaction between the ground states and the excited states is therefore Poissonian witha width at half maximumV 2

√2 ln 2

√N1. On the other hand, the uncertainty of the

number of particles in the condensate is sensitive to the statistics of exciton-polaritonsin this state. For a state having the thermal statistics,σ0 =

√

N20 +N0 ≈ N0. The

associated line has the shape of a thermal distribution, namely exponential on the highenergy tail, and with an abrupt cutoff corresponding to the casen0 = 0. Such statisticsyields an asymmetric broadening as shown on Fig. 8.26. The total line is in this case asuperposition of two very different functions. If however the ground state is coherentthe uncertainty in the number of particles for the coherent statistics is onlyσ0 =

√N0.

In a general case, the condensate is in a mixed state having both coherent and thermalfractions. Its statistics is described by the second order coherence functiong2(0), whichvaries from 2, for the thermal state, to 1 for the coherent state. The root mean square ofsuch statistics is given byσ0 =

√

N0 +N20 (g2(0) − 1). If g2(0) is close enough of 1,

the associated statistics is well described by a Gaussian. Making the assumption that thetwo random variablesN0 andN1 are not correlated, the statistics of their sum is givenby a Gaussian of meanN0 +N1 and of root mean squareσ =

√

σ20 + σ2

1 . Now, if weneglect the spontaneous broadening which is very small ifN0 is large, the linewidth ofthe emission line is given by:

∆E ≈ V 2√

2 ln 2√

N1 +N0 +N20 (g2(0) − 1) (8.136)

Now, we suppose thatN0 can be measured experimentally. We use Eq. (8.135)to getN1 as a function ofN0 andEshift. The knowledge of these two quantities—Vcombined with a measurement of∆E—finally allows one to deduce the second ordercoherence as:

g2(0) = 1 +1

8 ln 2

(

∆E

VN0

)2

− Eshift

V N20

(8.137)


This formula has of course a weak precision ifσ1 is larger thanσ0. In the oppositelimit it is probably very accurate and represents a clear anddirect way to measureg2(0).This technique can only be applied to the case of interactingparticles. It is special tothe polariton system with respect to a photon system.

We consider a typical CdTe system, having a lateral size of 10microns, contain-ing 105 particles, mainly in the ground state having itself coherent statistics. In thisframework, the emission linewidth is∆E = 7µeV which corresponds to a phase dif-fusion time of about 200ps. Bose condensation is a phase transition linked with thespontaneous symmetry breaking of gauge invariance, that is, with the appearance of awell defined phase in the system. Such a phase transition manifests itself in the sponta-neous appearance of a non-zero, long lived order parameter.The observation of such anorder parameter is difficult if the measurements are performed on a purely circularly-polarized polariton state. On the other hand, if the cavity is pumped by unpolarizedlight and is isotropic, aσ+ and aσ− condensate should spontaneously appear. Thesetwo condensates are a priori uncorrelated. Each of them is characterised by a randombut well-defined phase. The superposition of these two circularly polarized coherentstates results in a linearly polarized emission. Again, if the microcavity is isotropic,the in-plane direction of this linear polarisation changesrandomly from experiment toexperiment. The detection of this spontaneous buildup of anin-plane linear polarisa-tion with a polarisation axis changing from experiment to experiment would representa clear evidence of a spontaneous symmetry breaking effect in the system. Once thispolarisation has appeared, the dephasing smears out the phase of the two independentcondensates and will result in a decay of the linear polarisation degree. This statementwhich is here qualitatively described will be proven more rigorously in Chapter 9.

8.9 Semiconductor luminescence equations

In this section we discuss an alternative view of polariton lasing based on the considera-tion of fermion-like electrons and holes instead of the boson-like exciton polaritons. Theobservations of Pau et al. (1996) claiming experimental evidence for the boser action,i.e., polariton lasing, have been reproduced by the group ofKhitrova and Gibbs and pub-lished with a theoretical model by Kira et al. (1997) in conceptual opposition with theboson approach started with the so-called “boser” ofImamo glu et al. (1996). These au-thors criticised the bosonic approach for approximating “the full electron-hole Coulombinteraction in the many-body Hamiltonian [. . . ] by including only some aspects of theinterband attractive part” and proposed instead a full quantum picture starting from atwo-band Hamiltonian which includes consistently Coulombinteractions and correla-tions between electrons and holes. They obtained a complex set of coupled equations forthe polarisationPk ≡ c†v,kcc,k, the population operatorsc†c,kcc,k and the photon anni-hilation operatorsBq for photons with zero in-plane wavevector(q‖ = 0, q). These area special case of a general theory to describe light-matter interaction in semiconductorsdeveloped by the authors, which they call the “semiconductor luminescence equations”.With the dynamical decoupling scheme (explained in details, e.g., by Binder & Koch(1995))—retaining correlators〈B†

qPk〉 and〈B†qbq′〉 and electronfek ≡ 〈cc,k†cc,k〉, hole

SEMICONDUCTOR LUMINESCENCE EQUATIONS 335

fhk ≡ 1−〈cv,k†cv,k〉 populations—a closed set of equations is obtained, which howeverremains very complicated and requires essentially numerical treatment:

i~∂t〈B†qPk〉 =(−Ec(k) − Ev(k) − ~ωq + EG)〈B†

qPk〉 (8.138a)

+ (fek + fhk − 1)Ω(k, q) (8.138b)

+ fekfhkΩSE(k, q) (8.138c)

i~∂t〈B†qBq′〉 =~(ωq′ − ωq)〈B†

qBq′〉 (8.139a)

+ iEquq〈Bq′PH〉 + iEq′ u∗q′〈B†qPH〉 (8.139b)

i~∂tfe(h)k = 2iℑ[µ∗

cv(k)〈PkD〉] (8.140)

The detailed expressions for the various quantities involved need not concern us here,but let us instead list the physical meaning of each quantity, which yields the gist ofthis model:Ec,v(k) are the dispersions for conduction/valence electrons (renormalisedby interactions and structure details like the QW confinement factor, in a way we donot reproduce),EG the gap energy,Eq the radiation field vacuum amplitude,u(r) theeffective cavity mode wavefunction andµcv the dipole matrix element [Kira et al. (1997)provide a full discussion]. More importantly,

Ω(k, q) ≡ µcv(k)〈B†qE〉 +

∑

k′

V (k′ − k)〈B†qPk′〉 , (8.141)

ΩSE(k, q) ≡ iµcv(k)Equq , (8.142)

which involve the Coulomb matrix elementV and the quantised radiation fieldE ∝∑

q Bquq, and provide the source for field-particle correlations (cf. (8.138c)). Sincefekf

hkΩSE is nonzero in presence of excitations (electrons and holes), it triggers〈B†

qPk〉even if they are initially zero. We note thatPk is the amplitude for an interband transitionof the electron, so that〈B†

qPk〉 is the photon-mediated polarisation. ThusΩSE, whichdrives the emission through electron-hole recombination,is interpreted as aspontaneousemissionterm. In the same way,Ω on line (8.138b) provides stimulation (the secondterm in (8.141) being the renormalisation of stimulation tothe first “bare” stimulationterm).

From these equations, one can compute the normal incidence luminescence spec-trum as the time variation of the light field intensity:

I(k) ∝ ∂t〈B†kBk〉 . (8.143)

This, however, demands numerical simulations, which the authors have performedonly in 1D and found to be in agreement with experimentalistsobservations. Two ap-pealing features of such numerical approaches are the possibility to artificially neutralisesome contributions, e.g., the stimulation term (8.138b), and to gain access to some ex-perimentally awkward or poorly defined quantities, as for instance〈[X,X†]〉. For a true


boson, this quantity is exactly one, cf. Eq. (3.39). The computations have shown how-ever that this quantity varies from 0.7 down to 0.3 as a function of increasing densities(from ≈ .5 to ≈ 3 × 1011cm−2), leading to the conclusion that the Boson picture isnot valid, and that Pau et al.’s (1996) explanation in terms of “close-to-boson” polari-tons was mistaken. The more convincing result from this workhowever comes fromthe other numerical latitude, namely the artificial switch-off of stimulation, which ineffect amounts to discarding strong coupling. Doing so, they observed the persistenceof the splitting, which was previously claimed as evidence of strong coupling. The cor-rection of the semiconductor luminescence equation and thebosonic picture remains anunresolved issue.

Fig. 8.27: PL spectra reported by Kira et al. (1997), experimental (left column) and theoretical as computedwith (8.143) solving numerically eqs. (8.138—8.140) (right column). The dashed curves were obtained bydiscarding line (8.138b) which is reponsible for strong coupling, in absence of which the emitted spectraare clearly seen to emit at the bare cavity and “exciton” modes. The cavity detuning of 3meV is ultimatelyresponsible for the observed splitting in the high pumping regime (top of figure) where the upper polaritonbranch is emitting superlinearly. The shaded curves are theexcitonic absorption spectra, the exciton emissionis not sensitive to the dressing of the exciton, even when theexciton resonance is bleached.

CLAIMS OF EXCITONS AND POLARITONS BOSE-EINSTEIN CONDENSATION 337

8.10 Claims of excitons and polaritons Bose-Einstein condensation

Finally we crown this chapter with the state of the art of the experimental findings,reporting positive outcome to the quest of solid state BEC. Indeed, very recently a num-ber of groups have found that thermalisation of polaritons on the lower polariton branchis possible in conditions of positive cavity detuning. In II-VI microcavities, Kasprzaket al. (2006) could obtain a direct fitting of the experimentally-observed polariton oc-cupations using the Bose-Einstein distribution above and below threshold, as seen inFig. 8.28, and show that this is different to Maxwell-Boltzmann statistics, i.e., strongevidence of the bosonic nature of the polaritons. In addition, it is clear that despite thestrong spatial fluctuations and disorder-induced localisation of the emission, the differ-ent emitting “hot” spots are coherently locked in phase, i.e., they are part of the samestate. The transition to this polariton BEC state is found around 20K, where the polari-ton temperature is close to the lattice temperature. Similarly, in III-V microcavities newevidence is emerging that polariton BECs are indeed thermalised at low temperatures,whether for positive cavity detuning when excited at resonant energies of the excitons,as done by Deng et al. (2006), or using localised stress to form a real-space trap forpolaritons, as done by Balili et al. (2006). All these new experiments clearly show theappearance of bosonic-induced coherence in the semiconductor microcavity system inthe strong coupling regime, and pave the way to coherent matter wave monolithic de-vices.

8.11 Further reading

Griffin et al. (1995) edited a proceeding volume for the first Levico conference on BECin 1993, held before its experimental realisation two yearslater and announced “hotoff the press” at the following conference. Its lucky timingand contributions from emi-nences of the time made it an important publication in the field. The review papers giveinsightful and personal accounts by pioneers of the field while brief reports provide aninteresting historical snapshot of the time. It also includes some unusual systems for thecondensed matter physicist, such as BEC of mesons or Cooper pairing in nuclei. Griffinremembers the conference, now an historical one, and the story of this book in J. Phys.B: At. Mol. Opt. Phys.37 (2004).

More pedagogical and unified texts have flourished since. Especially notable is theexcellent text by Pitaevskii & Stringari (2003). Pethick & Smith (2001) provide a goodintroductory description of the atomic case which became the “hero” of BEC as its firstexperimental realization, and a source of inspiration for the condensed matter commu-nity. Another useful proceedings is the Enrico Fermi’s Varenna summer school volumeedited by Inguscio et al. (1999).

Moskalenko & Snoke (2000) provide a good overview of theoretical work (withmany discussions of experimental results) related to BEC insemiconductors. The firstchapter opens with “Many people seem to have trouble with the concept of an exci-ton. . .” and gives a gentle introduction to an otherwise involved exposition. The contentindeed borrows a lot from research papers of the Russian literature. It is therefore a use-ful window to the non-Russian speaking reader into the extensive amount of theoreticalwork made by the Soviet school on this topic.


Fig. 8.28: Far-field angular emission below and above threshold showing the formation of a polariton Bose-Einstein condensate, as claimed by Kasprzak et al. (2006). The cover illustration is based on thek spacecounterpart of this figure.

An enduring classic dealing with superfluidity is the textbook by Khalatnikov (1965)which cover all the main characteristics of Bose gas and liquids, especially the excitationspectrum, hydrodynamics and kinetics of the problem. The discussion of the Landauspectrum and its two-liquid model is one of the best expositions. The Bogoliubov modelis reviewed in full details in the review by Zagrebnov & Bru (2001).

9

SPIN AND POLARISATION

In this Chapter we consider a complex set of optical phenomena linkedto the spin dynamics of exciton-polaritons in semiconductormicrocavities. We review a few important experiments whichreveal themain mechanisms of the exciton-polariton spin dynamics andpresentthe theoretical model of polariton spin relaxation based onthe densitymatrix formalism. We also discuss the polarisation properties of thecondensate and the superfluid phase transitions for polarizedexciton-polaritons. Finally, theoretically predicted optical spin-Hall andspin-Meissner effects are described.103

103We acknowledge enlightening discussions with I. Shelykh, Yu. Rubo and K. Kavokin who contributedsubstantially to the content of this Chapter.

340 SPIN AND POLARISATION

When optically created, polaritons inherit their spin and dipole moment from the ex-citing light. Their polarisation properties can be fully characterized by a Stokes vectoror—using the language of quantum physics—a pseudospin accounting for both spin anddipole moment orientation of a polariton. From the very beginning of their creation in amicrocavity, polaritons start changing their pseudospin state under the effect of effectivemagnetic fields of various nature and due to scattering with acoustic phonons, defects,and other polaritons. This makes pseudospin dynamics of exciton-polaritons rich andcomplex. It manifests itself in non-trivial changes in the polarisation of light emitted bythe cavity versus time, pumping energy, pumping intensity and polarisation. Experimen-tal studies of the polarisation properties of microcavities have given evidence of a set ofunusual effects (giant Faraday rotation, polarisation beats in photoluminescence, polar-isation inversion in the parametric oscillation regime etc.). Linear optical effects (likeFaraday rotation) have been interpreted in terms of the theory of non-local dielectricresponse (developed in Chapter 3). The analysis of the data of nonlinear optical experi-ments require more substantial theoretical effort. The experiments unambiguously indi-cate that energy and momentum-relaxation of exciton-polaritons are spin-dependent, ingeneral. This is typically the case in the regime of stimulated scattering when the spinpolarisations of initial and final polariton states have a huge effect on the scattering ratebetween these states. It appears that critical conditions for polariton Bose-condensationare also polarisation-dependent. In particular, the stimulation threshold (i.e. the pumpingpower needed to have a population exceeding 1 at the ground state of the lower polari-ton branch) has been experimentally shown to be lower under linear than under circularpumping. These experimental observations have stimulatedtheoretical research towardunderstanding the mutually dependent polarisation- and energy-relaxation mechanismsin microcavities.

9.1 Spin relaxation of electrons, holes and excitons in semiconductors

The concept of spin was introduced by Dutch-born US physicists Samuel AbrahamGoudsmit and George Eugene Uhlenbeck in 1925. In the same year the Austrian theoristWolfgang Pauli proposed his exclusion principle which states that two electrons cannotoccupy the same quantum state. Later on, it was understood that this principle applies toall particles and quasi-particles with a semi-integer spin(fermions) including electronsand holes in semiconductors. On the other hand, it is not valid for particles (quasi-particles) having an integer spin, in particular for excitons.

Spins of electrons and holes govern the polarisation of light generated due to theirrecombination. The conservation of spin in photoabsorbtion allows for spin-orientationof excitons by polarized light beams (optical orientation), an effect which manifests it-self also in the polarisation of photoluminescence.σ+ andσ− circularly-polarised lightexcitesJ = +1 and−1 excitons, respectively. Linearly-polarized light excites a linearcombination of+1 and−1 exciton-states, so that the total exciton spin projection onthe structure axis is zero in this case. Optical orientationof carrier spins in bulk semi-conductors was discovered by a French physicist George Lampel in 1968. In quantumwells, this has been extensively studied since the 1980s by many groups. For good re-

SPIN RELAXATION OF ELECTRONS, HOLES AND EXCITONS IN SEMICONDUCTORS341

GeorgesUhlenbeck (1900-1988),H. A. Kramers andS. A. Goudsmit (1902-1978) on the left. WolfgangPauli (1900-1958) on the right.

Uhlenbeck is mainly known for his introduction (with Goudsmit) of the concept of spin of the elec-tron, although he favoured and was most active in statistical physics, where lies his work on quantum kinetic(see on page 301). Known to privilege rigour to originality,one of his student comments about him that “Hefelt that something really original one did only once—like the electron-spin—the rest of one’s time one spenton clarifying the basics.”

Pauli’s first talk in Sommerfeld’s “Wednesday Colloquium” impressed the venerable professor so much thathe entrusted the young student to prepare an article on relativity for the “Encyklopadie der mathematischenWissenschaften.” He produced a 237 pages article which impressed Einstein himself who reviewed the workas “grandly conceived.” He remained widely praised for the mastery and clarity of his expositions, withhis two review articles on quantum mechanics in the Handbuchder Physik known as the “Old and NewTestament.” Oppenheimer called the later “the only adult introduction to quantum mechanics.” Howeverhe published little as compared to his actual scientific production, especially as he aimed for a thoroughunderstanding. He onced commented “The fact that the author thinks slowly is not serious, but thefactthat he publishes faster than he thinks is inexcusable.” Many of his own results were confined to privatecorrespondence, like the equivalence of Heisenberg and Schrodinger pictures (in a letter to Jordan), the timeenergy uncertainty relation (to Heisenberg), or the neutrino to rescue energy conservation in radioactivity.He almost never cared about recognition. He developed the “Ausschliessungsprinzip” or exclusion principlein 1924, already known as the Pauli principle when it earned him the Nobel prize in 1945. Despite a strongopposition to the initial idea of the spin of the electron (which Kronig never published after his idea wasridiculed by Pauli), he formalised the concept of spin following the young theory of Heisenberg, culminatingwith Pauli matrices. He later laid the foundations of quantum field theory with the spin-statistics theorem,linking bosons and fermions to integer and half-integer spins, respectively. After his first marriage to a dancerbroke down in less than a year and when his mother committed suicide, he suffered a deep personal crisis andstarted to drink. He consulted the psychologist Carl Jung (who first delegated a female assistant thinking thatPauli’s problem were linked to women) for whom he detailed over a thousand dreams and established a deeprelationship as a strong believer in psychology. He wrote that “in the science of the future reality will neitherbe ‘psychic’ nor ‘physical’ but somehow both and somehow neither.” Numerous anecdotes are in circulationabout him, like the “Pauli effect” dooming an experiment if he was in its vicinity. He was also known forhis severity and scathing comments, but also for his wit, illustrated by his famous remark “This isn’t right.This isn’t even wrong.” One of the most brilliant theorists of all times, Albert Einstein described him as hisintellectual successor during his Nobel celebration in Princeton.


views we address the reader to the famous volume “Optical orientation” edited by Meier& Zakharchenia (1984).

Fig. 9.1: Polarisation of optical transitions in zinc-blend semiconductor quantum wells. Red, blue and greenlines showσ+, σ− and linearly polarized transitions, respectively.

An exciton is formed by an electron and hole, i.e., by two fermions having projec-tions of the angular momenta on a given axis equal toJez = Sez = ± 1

2 for an electronin the conduction band with S-symmetry andJhz = Shz + Mh

z = ± 12 ,± 3

2 for a holein the valence band with P-symmetry (in zinc-blend semiconductor crystals). The stateswith Jhz = ± 1

2 are formed if the spin projection of the holeShz is antiparallel to the pro-jection of its mechanical momentumMh

z . These states are calledlight holes. If the spinand mechanical momentum are parallel, the heavy holes withJhz = ± 3

2 are formed (seeSection 4.2). In bulk samples, atk = 0 the light and heavy hole states are degenerate.However, in quantum wells the quantum confinement in the direction of the structuregrowth axis lifts this degeneracy so that energy levels of the heavy holes lie typicallycloser to the bottom of the well than the light-hole levels (See Fig. 9.1).

The ground state exciton is thus formed by an electron and a heavy-hole. The to-tal exciton angular momentum104 J has projections±1 and±2 on the structure axis.Bearing in mind that the photon spin is0 or ±1 and that the spin is conserved in theprocess of photoabsorbtion, excitons with spin projections±2 cannot be optically ex-cited. These are spin-forbidden states that, since they arenot coupled to light, are alsocalleddark states. This vivid terminology extends to states which couple to light (withspin projection±1) calling thembright states. Since, in quantum microcavities,J = 2states are not coupled to the photonic modes, we shall neglect them in the followingconsideration but it should be born in mind, however, that insome cases these dark

104 In this text as is common practise in the field the total exciton angular momentum is referred to forconvenience as the excitonspin.

SPIN RELAXATION OF ELECTRONS, HOLES AND EXCITONS IN SEMICONDUCTORS343

states still come into play. They can for instance be mixed with the bright states by anin-plane magnetic field or be populated due to the polariton-polariton scattering.

GeorgesLampel Professor at the Ecole Polytechnique, Paris and Boris P.Zakharchenia (1928-2005).

Lampel is the father of “optical orientation”, an importantbranch of crystal optics studying the opticallyinduced spin effects in solids. One of Lampel’s PhD studentswas Claude Weisbuch, who was the first toobserve the strong light-matter coupling in semiconductormicrocavities. At present, Lampel and Weisbuchwork in the same laboratory of the famousEcole Polytechnique in Paris.

Zakharchenia was a PhD Student of Evgenii Gross and participated in the early works on the Wannier-Mottexcitons in CuO2. Later on, he headed a laboratory of the Ioffe institute in Saint-Petersbourg which publishedthe pioneering works on what is now called “spintronics”. Zakharchenia is also known for his theory on thePushkin’s duel place.

The polarisation of exciting light cannot be retained infinitely long by excitons.Sooner or later they lose polarisation due to inevitable spin and dipole moment relax-ations. Excitons, being composed of electrons and holes, are subject to mechanisms ofspin-relaxation for free carriers in semiconductors. The main important ones being:

1. Elliot (1954)–Yaffet mechanism involving the mixing of the different spin wave-functions withk 6= 0 as a result of thekp interaction with other bands. In quantumwells this effect plays a major role in the spin relaxation ofholes and can inducetransitions between the optically active and dark exciton states|+1〉 → |−2〉 and|−1〉 → |+2〉.

2. D’yakonov & Perel (1971b) mechanism caused by the spin-orbit interaction in-duced spin splitting of the conduction band in non-centrosymmetric crystals (likezinc-blend crystals) and asymmetric quantum wells (Dresselhaus and Rashbaterms respectively) atk 6= 0. This mechanism is predominant for the electronsand also leads to transitions between the optically active and dark exciton states,|+1〉 → |+2〉 and|−1〉 → |−2〉.

3. The Bir-Aronov-Pikus (BAP) mechanism first published by Pikus & Bir (1971),involving the spin-flip exchange interaction of electrons and holes. For excitonsthe efficiency of this mechanism is enhanced, as the electronand the hole forma bound state. The exchange interaction consists of so-called “short-range” and


“long-range” parts of the Coulomb interaction105. The short range part leads tothe coupling between heavy hole (hhe) and light hole excitons (lhe) and thus issuppressed in the quantum wells where the degeneracy of lhe and hhe is lifted.The long range part leads to transitions within the optically active exciton dou-blet106.

4. Spin-flip scattering between carriers and magnetic ions in diluted magnetic semi-conductors. As an example, paramagnetic semiconductors containing Mn2+ ions(spin5/2) allow for efficient spin relaxation of electrons, heavy- and light-holesflipping their spins with the magnetic ion spins.107

Vladimir IdelevichPerel (b. 1922) and Mikhail IgorevichDyakonov (b. 1939) in 1976.

Dyakonov and Perel personalise the excellence of the Soviettheoretical physics. Their names are associ-ated to numerous physical effects and theories, including the non-radiative recombination and spin-relaxationmodels, spin-Hall effect among others. Their elegant and seemingly simple analytical models helped under-standing of extremely complex phenomena of modern solid state physics. The famous “Tea seminar” createdby Dyakonov and Perel still run at the Ioffe institue of SaintPetersburg every thursday at 5pm.

105For details see E.L. Ivchenko, Optical spectroscopy of semiconductor nanostructures, Alpha, Harrow(2005), pages 252-253.

106The ability of the long-range part of the exchange interaction to couple the exciton doublet can lead tothe inversion of the circular polarisation in time-resolved polarisation measurements, as shown by Kavokinet al. (2004a).

107The spin-flip scattering is discussed at length by P. A. Wolff, in Semiconductors and Semimetals, editedby J. K. Furdyna and J. Kossut, Diluted Magnetic Semiconductors, Vol. 25. (Academic Press, Boston, 1988).

MICROCAVITIES IN THE PRESENCE OF MAGNETIC FIELD 345

In a key paper, Maialle et al. (1993) have shown that the third(BAP) mechanism ispredominant for the quantum confined excitons in non-magnetic semiconductors. Thelong-range electron-hole interaction leads to the longitudinal-transverse splitting of ex-citon states, i.e., energy splitting between excitons having a dipole moment parallel andperpendicular to the wavevector. This splitting is responsible for rapid spin-relaxation ofexcitons in quantum wells. For description of exciton-polaritons in microcavities it hasa very important consequence: the dark states can be neglected, which allows us to con-sider an exciton as a two level system and use the well-developed pseudospin formalismfor its description. From the formal point of view the exciton can be thus described bya2×2 spin density matrix which is completely analogical to the spin density matrix forelectrons.

Exercise 9.1 (∗) In 1924, British physicists Wood & Ellett (1924) reported anamazingpolarisation effect: they measured the circular polarisation degree of fluorescence ofthe mercury vapour resonantly excited by a circularly polarized light. In their initialexperiments a high degree of circular polarisation was observed while it significantlydiminished in later experiments. They wrote: “It was then observed that the apparatuswas oriented in a different direction from that which obtained in the earlier work, andon turning the table on which everything was mounted throughninety degrees, bringingthe observation direction East and West, we at once obtaineda much higher value ofthe polarisation.” Explain this effect.

9.2 Microcavities in the presence of magnetic field

A magnetic field strongly affects excitons in quantum wells,and thus it also affectsexciton-polaritons in microcavities. One can distinguishbetween three kinds of linearmagneto-exciton effects in cavities:

• The energies of electron and hole quantum-confined levels change as a functionof the magnetic field, following the so-calledLandau fan diagram. As a result,the energies of electron-hole transitions increase by

δE = (l + 1/2)~ωc , (9.1)

where the cyclotron frequencyωc = eB/(µc) depends on the reduced effective

mass of electron-hole motionµ = mem‖h/(me+m

‖h) and magnetic fieldB.m‖

h isthe in-plane hole mass,me is the electron effective mass andl = 0, 1, 2, · · · is theLandau quantum index. Landau quantization of exciton energies takes place dueto the magnetic field effect on the orbital motion of electrons and holes. This effectis polarisation independent. In microcavities, Landau quantisation results in theappearance of a fine-structure of polariton eigenstates andgives the possibility oftuning of different Landau levels into resonance with the cavity mode, as observedby Tignon et al. (1995).

• Zeeman splitting of the exciton resonance. Excitons with spins parallel or antipar-allel to the magnetic field have different energies. The splitting is given by:

∆E = µBgB, (9.2)


whereµB ≈ 0.062meV/T is the Bohr magneton andg is the excitong-factorwhich depends on the materials composing the quantum well and the magneticfield orientation with respect to the QW. In the following we shall consider theso-calledFaraday geometry, i.e. the magnetic field parallel to the wavevector ofincident light and, consequently, normal to the QW plane (normal incidence case).Note thatg can be positive or negative; in most semiconductor materials it variesbetween−2 and2, and in GaAs/AlGaAs QWs it changes sign as the QW widthchanges. Exciton Zeeman splitting leads to an effect known as resonant Faradayrotation, i.e., rotation of the polarisation plane of linearly polarised light passingthrough a QW in the vicinity of the exciton resonance. In microcavities, this effectis strongly amplified due to the fact that light makes a seriesof round trips, eachtime crossing the QW before escaping from the cavity. This isto be discussed indetail below.

• An increase of the exciton binding energy and oscillator strength due to theshrinkage of the exciton wavefunction in a magnetic field. This effect is importantfor strong enough magnetic fields, for which the magnetic lengthL =

√

c~/(eB)is comparable to the exciton Bohr diameter (typically about100A). An increaseof the exciton oscillator strength in a magnetic field enhances the vacuum-fieldRabi splitting.

In the rest of this Section we consider the three above mentioned effects in detail.Figure 9.2 shows the relative exciton radiative broadening, Rabi splitting and period

of Rabi oscillations (i.e. oscillations in time-resolved reflection due to beats betweentwo exciton-polariton modes in the cavity) measured experimentally and calculated witha variational approach of Berger et al. (1996). One can see that the oscillator strengthincreases by about 80% with the magnetic field changing from 0to 12T, which resultsin the Rabi splitting increasing by more than 30% and a corresponding decrease in theperiod of Rabi oscillations. Note that due to the exciton diamagnetic shiftδE − EB

(with δE given by Eq. (9.1) andEB being the exciton binding energy, the energy of po-lariton ground state shifts up. This shift is parabolicallydependent on the magnetic fieldat low fields. On the other hand, due to the increase of the exciton oscillator strengthand resulting increase of the Rabi-splitting, the polariton ground state energy is pusheddown. In realistic cavities, the latter effect dominates atlow fields and the ground stateenergy shifts down, while in the limit of strong fields, when the magnetic lengths be-comes comparable to the exciton Bohr radius, the ground state is expected to start mov-ing up. Moreover, due to the increase of the Rabi-splitting,the exciton polariton dis-persion gets slightly steeper near the ground state, so thatthe polariton effective massdecreases with the increase of the magnetic field.

9.3 Resonant Faraday rotation

Faraday rotation is rotation of the electric field vector of alinearly polarised light wavepropagating in a medium in the presence of a magnetic field oriented along the lightpropagation direction (see Chapter 2). Consider the propagation of linearly (X) po-larised light, whose initial polarisation is described by aJones vector

RESONANT FARADAY ROTATION 347

Fig. 9.2: Relative exciton radiative broadening, Rabi-splitting and period of Rabi-oscillations, i.e., oscilla-tions in time-resolved reflection due to beats between two exciton-polariton modes in the cavity, measuredexperimentally from a GaAs-based microcavity with InGaAs/GaAs QWs (points), and calculated (lines), fromBerger et al. (1996).

(

10

)

=1

2

(

1−i

)

+1

2

(

1−i

)

(9.3)

here the upper and lower components of the vectors correspond to the electric fieldprojections in thex andy directions, respectively, and the two terms on the right-handside describeσ+ (right-circular) andσ− (left-circular) polarised waves, respectively. Ifthe structure is placed in a magnetic field, the transmissioncoefficients forσ+ andσ−

polarised waves become different, so that the transmitted light can be represented as

t =1

2A exp(iφ+)

(

1i

)

+1

2C exp(iφ−)

(

1−i

)

(9.4)


where the amplitudesA andC and phasesφ+ andφ− coincide in the absence of thefield, but may be different in the presence of the field.t is the polarisation vector ofelliptically polarised light, which can be conveniently represented as a sum of waveslinearly polarised along the main axes of the ellipse:

t =i

2(B − C) exp(iψ)

(

sinφ− cosφ

)

+1

2(A+ C) exp(iψ)

(

cosφsinφ

)

(9.5)

whereψ = (ϕ+ + ϕ−)/2 andϕ = (ϕ− − ϕ+)/2 is theFaraday rotation angle.The transmission coefficient of the structure for light detected inX-polarisation is

Tx =1

4|A exp(iϕ+) + C exp(iϕ−)|2 (9.6)

and inY -polarisation it is

Ty =1

4|A exp(iϕ+) − C exp(iϕ−)|2 (9.7)

In σ+ andσ− polarisation the amplitude of light transmitted across thequantumwell at the exciton resonance frequency is given by

tσ+,σ− = 1 +iΓ0

ωσ+,σ−

0 − ω − i(γ + Γ0)(9.8)

whereωσ+,σ−

0 is the exciton resonance frequency in the two circular polarisations,whose splitting in a magnetic field is referred to as exciton Zeeman splitting,Γ0 isthe exciton radiative decay rate andγ is the exciton non-radiative decay rate.

Hereafter we shall neglect the exciton inhomogeneous broadening. The polarisationplane of linearly polarised light passing through the QW rotates by the angle

φ =1

2

[

arctan(ω−

0 − ω)Γ0

(ω−0 − ω)2 + (γ + Γ0)2

− arctan(ω+

0 − ω)Γ0

(ω+0 − ω)2 + (γ + Γ0)2

]

≈ (ωσ−

0 ωσ+

0 )Γ0

(γ + Γ0)2. (9.9)

In the case of a microcavity, the Faraday rotation can be greatly amplified. Let us firstanalyze the expected effects in the framework of ray optics.Consider a cavity-polaritonmode as a ray of light travelling backwards and forwards inside the cavity within itslifetime. At the anticrossing point of the exciton and photon modes the lifetime of cavitypolaritonsτ is

τ =1

2(κ+ γ)(9.10)

whereκ is the cavity decay rate, dependent on the reflectivity of theBragg mirrors. Theaverage number of round trips of light inside the cavity is

N =τc

2ncLc=Q

2, (9.11)

wherenc is the cavity refractive index andLc is its length,Q is the quality factor of thecavity (see Chapter 2). In standard quality GaAs-based microcavities this factor reaches

SPIN RELAXATION OF EXCITON-POLARITONS IN MICROCAVITIES: EXPERIMENT349

a few hundred. While circulating between the mirrors the light accumulates a rotation,before escaping the cavity. The amplitude of the emitted light can be found as

E = t1 + t1r1eiφ + t1r

21e

2iφ + · · · =t1

1 − r1eiφ(9.12)

wherer1 andt1 are the amplitude reflection and transmission coefficients of the Braggmirror, respectively. The angle of resulting rotation of the linear polarisation is

θ = arg(E) = arctanr1 sinφ

1 − r1 cosφ. (9.13)

Note that this consideration neglects reflection of light bya QW exciton, since theamplitude of the QW reflection coefficient is more than an order of magnitude smallerthan the transmission amplitude.

To observe a giant Faraday rotation a peculiar experimentalconfiguration is needed.In the reflection geometry only a very small rotation of the polarisation plane can beobserved (Kerr effect). Actually, the reflection signal is dominated by a surface reflec-tion from the upper Bragg mirror which does not experience any polarisation rotation.To observe the giant effect predicted by Eq. (9.13), either the measurement should becarried out in the transmission geometry, which would implyetching any absorbing sub-strate, or a pump-probe technique should be used to introduce the light into the cavity atan oblique angle and then to probe emission at the normal angle. The Faraday rotationdescribed above is a linear optical effect induced by an external magnetic field appliedto the cavity. There exists also an optically induced Faraday rotation in microcavitieswhich is a nonlinear effect having a strong influence on the polarisation of emission ofthe cavities. It will be considered in detail below.

Exercise 9.2 (∗∗∗) Describe the resonant Faraday rotation of TE-polarized light inci-dent on a QW at oblique angle. A magnetic field is oriented normally to the QW plane.

9.4 Spin relaxation of exciton-polaritons in microcavities: experiment

The spin dynamics of cavity polaritons has been experimentally studied by measure-ment of time-resolved polarisation from quantum microcavities in the strong couplingregime. At the beginning of the XXIst century, a series of experimental works appearedin this field, which reported unexpected results. Let us briefly summarize the most im-portant of them.

In the experimental work by Martin et al. (2002), the dynamics of the circular po-larisation degreec of the photoemission from the ground state of a CdTe-based micro-cavity was measured.c was determined as

c =I+ − I−

I+ + I−=N+k=0(t) −N−

k=0(t)

Nk=0(t)(9.14)

whereI± denotes the measured circularly-polarised intensities,N±k=0 represent the

population of the ground state (k = 0) with polaritons having spin projections on


the structure axis equal to±1, respectively,Nk=0(t) = N+k=0(t) + N−

k=0(t). Thepump pulse was circularly polarized and centered on the energy above the stop-bandof the Bragg mirrors composing the microcavity (non-resonant excitation). In the linearregime, when the stimulated scattering of the exciton-polaritons did not play any role,an exponential decay of the circular polarisation degree ofphotoemission was observed.However, above the stimulation threshold,ρc exhibited a non-monotonic temporal de-pendence. At positive detuning, the initial polarisation of ≈ 30% first increased upto ≈ 90% and then showed damped oscillations (see Fig. 9.3). For negative detuning,the polarisation degree started from a positive value≈ 50%, rapidly decreased down tostrongly negative values and then increased showing attenuated oscillations with a pe-riod of about 50ps. As we show in the next Section this effect is caused by the TE-TMsplitting of polariton states withk 6= 0.

Fig. 9.3: Experimentally measured temporal evolution of the photoluminescence of a CdTe-based microcavityexcited by circularly-polarized light at the positive detuning, upper polariton branch (a,δ = 10meV) andnegative detuning, lower polariton branch (b,δ = −10meV). The filled circles/solid line (open circles/dashedline) denote theσ+ (σ−) emission. The deduced time evolution of the circular polarisation degree for positiveand negative detunings is shown in (c) and (d), respectively. The inset shows the maximum value of thepolarisation degree at 20ps in the negative detuning case. From Martin et al. (2002).


In the experiments carried out by the group of Y. Yamamoto andpublished by She-lykh et al. (2004), the microcavity was pumped resonantly atan oblique angle (≈50).The dependence of the intensity and of the polarisation of light emitted by the groundstate on pumping polarisation and power was studied, with all experiments were carriedout in the continuous pumping regime. The pump intensity corresponding to the stim-ulation threshold appeared to be almost twice as high in the case of circular pumpingthan in the case of linear pumping as shown in Fig. 9.4 (a,b). Fig. 9.4 (c) shows thedependence of the circular polarisation degree of the emitted light on the pumping in-tensity for different pumping polarisations. For both polarisations, c ≈ 0 is observedfar below the threshold andc ≈ 0.9 near threshold. This indicates that spin relaxationis complete at low excitation density, and that stimulated scattering into a definite spincomponent (say, spin-up) of the ground state takes place at high densities. Above thresh-old, c remains large in the case of a circular pump. In case of a linear pump, however,c decreases sharply at high pumping intensities. This effectwas interpreted in terms ofinterplay between ultra-fast scattering and spin-relaxation of exciton-polaritons due toTE-TM splitting (see also below).

The polarisation dynamics of polariton parametric amplifiers was the focus of ex-perimental research carried out by several groups. The parametric amplifier is realizedif polaritons are created by resonant optical pumping closeto the inflection point ofthe lower dispersion branch at the “magic angle” (see Chapter 7). In this configurationthe resonant scattering of two polaritons excited by the pump pulse toward the signal(k = 0) and the idler state is the dominant relaxation process. Thescattering can bestimulated by an additional probe pulse used to create the seed of polaritons in the sig-nal state, or it can be strong enough to be self-stimulated.

In the experiments carried out by Lagoudakis et al. (2002), the polarisation of theprobe pulse was kept right-circular, whereas the pump polarisation was changed fromright- to left-circular passing through elliptical and linear polarisation. Consequently,the spin-up and spin-down populations of the pump-injectedpolaritons were variedwhile the pump intensity was kept constant. The two circularcomponents of polarisa-tion of light emitted by the ground state and four in-plane linear components (vertical,horizontal and the two diagonal ones) were detected as functions of the circular polarisa-tion degree of the excitation. To briefly summarise the results of these measurements—shown in Fig. 9.5 (a,b,c): in the case of a linearlypolarizedpump pulse and circularly-polarized probe pulse the observed signal was linearly polarized, but with a plane ofpolarisation rotated by 45 degrees with respect to the pump polarisation. In the case ofellipticallypolarized pump pulses the signal also became elliptical, while the directionof the main axis of the ellipse rotated as a function of the circular polarisation degree ofthe pump. In the case of a purely circular pump, the polarisation of the signal was alsocircular, but its intensity was half that found for a linear pump. The polarisation of theidler emission emerging at roughly twice the magic angle showed a similar behaviour,although in the case of a linearlypolarized pump the idler polarisation was rotated by90 degrees with respect to the pump polarisation. As we show below, the rotation of thepolarisation plane in this experiment is a manifestation ofthe optically induced Faradayeffect: the imbalance of populations of spin-up and spin-down polariton states produces


Fig. 9.4: a) Emission fromk‖ ≈ 0 polaritons vs. pump power under the circular pump. The two circular-polarisation components of the emission and their total intensity are plotted. b) Same as a) but for the linearpump. c) Circular polarisation degreec vs. pump power with circular (triangles) and linear (circles) pumps.From Shelykh et al. (2004).

a spin-splitting of the polariton eigenstates. This imbalance has been introduced by theelliptically polarized pump. The whole effect, referred toas “self induced Larmor pre-cession” will be discussed in the next section.

The polarisation dynamics in a parametric oscillator without probe has been con-sidered in the experimental work by the Toulouse group. Quite surprisingly, it was ob-served that for a linear pump, the polarisation of the signalis also linear, but rotated by90 (see Fig. 9.6). This effect is connected with the anisotropyof the spin-dependentpolariton-polariton scattering.

An interesting polarisation effect in a microcavity was observed in 2002 by Wolf-gang Langbein (Fig. 9.7) who illuminated a microcavity witha spot of linearly polarizedlight (say, along thex-axis) and detected spatially and temporally resolved emission


Fig. 9.5: Experimental (a-c) and theoretical intensities of circularly (a, d) and linearly (b, c, e, f ) polarizedcomponents of the light emitted by a microcavity ground state as a function of the circular polarisation degreeof the pumping light. From Kavokin et al. (2003).


Fig. 9.6: Experimentally measured degree of linear polarisation of emission from the ground exciton-polariton state in a microcavity measured under linearly polarized pumping at the magic angle. Differentcurves correspond to different pumping intensities. One can see that the linear polarisation degree quicklybecomes negative which corresponds to 90 of rotation of the polarisation plane with respect to the pumpinglight polarisation. From Renucci et al. (2005a).

from the sample in co- and cross-linear polarisations. In the cross-polarisation he ob-served a characteristic cross (Langbein cross), showing that the conversion of co- tocross-linear polarisation is efficient within the quartersof the reciprocal space separatedby x- andy-axes, but not in the vicinity of the axes. These beautiful images appeardue to precession of the pseudospin of propagating exciton-polaritons in the effectivemagnetic field induced by TE-TM splitting (see next Section).

9.5 Spin relaxation of exciton-polaritons in microcavities: theory

As we mentioned in Section 9.1, the main mechanisms for spin relaxation of exciton-polaritons in the linear regime are the transitions within the optically active doublet dueto the longitudinal transverse TE–TM splitting of exciton-polaritons. Consequently, thedark states can be neglected in most cases and an exciton-polariton state with a givenin-plane wavevectork can be treated as a two level system. It can be described bythe2× 2 density matrixρk which is completely analogous to the spin density matrix ofan electron.

It is convenient to decompose the polariton pseudospin density matrix as:

ρk =Nk2

1 + Sk · σσσ (9.15)

where1 is the identity matrix,σσσ is the Pauli-matrix vector108, Sk is the meanpseu-dospinof the polariton state characterized by the wavevectork. It describes both theexciton spin state and its dipole moment orientation (see Fig. 9.8).

108ThePauli matrix vectoris a vectorσσσ = (σx, σy , σz) whose componentsσi are matrices, namely thePauli matrices (3.12) on page 79.

SPIN RELAXATION OF EXCITON-POLARITONS IN MICROCAVITIES: THEORY 355

Fig. 9.7: Spatial imaging of polariton propagation (lineargrey-scale, all images from W. Langbein in Pro-ceedings of 26th International Conference on Physics of Semiconductors, (Institute of Physics, Bristol, 2003),p.112.). Top: Co-linearly polarized polaritons. Bottom: cross-linearly polarized polaritons.

The pseudospin is the quantum analog of the Stokes vector (see Section 2.2): themean value of the pseudospin operator coincides with the Stokes vector of partially po-larized light in our notation. Note that here and further we use the basis of circularlypolarized states, i.e., associate the states having definiteSz with the polariton radiativestates with their spin projection on the structure axis equal to ±1. Their linear com-binations correspond to eigenstates ofSx andSy yielding linearly polarized emission.The pseudospin parallel to thex-axis corresponds toX-polarized light, the pseudospinantiparallel to thex-axis corresponds to Y-polarized light and the pseudospin orientedalong they-axis describes diagonal linear polarisations.

For non-interacting polaritons the temporal evolution of each density matrix (9.15)is governed by its Liouville-von Neumann equation (3.30)

i~∂tρk = [Hk, ρk] (9.16)

where the HamiltonianHk reads in terms of the pseudospin:

Hk = En(k) − gµBBeff,k · Sk (9.17)

HereEn(k) is the energy of then-th polariton branch,g is the effective polaritong-factor,µB is the Bohr magneton andBeff,k is an effective magnetic field. Unlike real


Fig. 9.8: Poincare sphere with pseudospin (identical to the Stokes vector in this case, Cf. Fig. (2.2) onpage 30). The equator of the sphere corresponds to differentlinear polarisations, while the poles correspondto two circular polarisations.

magnetic fields the effective field only applies to the radiatively active doublet and doesnot mix optically active and dark states. We do not consider here effects of real magneticfield on the polariton spin dynamics.

Thex- andy-components ofBeff,k are nonzero if the exciton states whose dipolemoments are oriented in, say,x- andy-directions have different energies. This alwayshappens for excitons having non-zero in-plane wavevectors. The splitting of excitonstates with dipole moments parallel and perpendicular to the exciton in-plane wavevec-tor is calledlongitudinal-transverse splitting(LT-splitting, see Chapter 4 for the bulk po-lariton LT-splitting). The longitudinal-transverse splitting of excitons in quantum wellsis a result of the long-range exchange interaction. It can bedescribed by the reducedspin Hamiltonian of Pikus & Bir (1971):

Hex =3

16

|φex(0)||φbulk

ex (0)|2 ~ωLTf(k)

k

[

k2 (kx − iky)2

(kx + iky) k2

]

, (9.18)

where~ωLT is the longitudinal-transverse splitting of a bulk exciton(see Chapter 4)andφex(0), φbulk

ex (0) are bulk and QW exciton envelope functions, respectively, takenwith equal electron and hole coordinates. The form-factorf(k) is given by

f(k) =

∫∫

Ue(z)Uhh(z)e−k|z−z′|Ue(z

′)Uhh(z′) dzdz′ (9.19)

with Ue(z), Uhh(z) being electron and heavy hole envelope functions in normal to QWplane direction. Off-diagonal terms of the Hamiltonian (9.17) lead to polariton spinflips and thus create an effective in-plane magnetic fieldBeff,k referred below as theMaialle field. If no other fields are presentBeff,k = BLT,k. The Maialle field is in

SPIN RELAXATION OF EXCITON-POLARITONS IN MICROCAVITIES: THEORY 357

general not parallel tok but makes with thex-axis twice the same angle ask. Theeffective magnetic field is zero fork = 0 and increases as a function ofk followinga square root law at largek. For more details on the longitudinal-transverse splittingthe reader can refer to the work of Tassone et al. (1992). In microcavities, splitting oflongitudinal and transverse polariton states is amplified due to the exciton coupling withthe cavity mode. Note that the cavity mode frequency is also split in TE- and TM- lightpolarisations (see Chapter 2). The resulting polariton splitting strongly depends on thedetuning between the cavity mode and the exciton resonance and, in general, dependsnon-monotically onk. Fig. 9.9 shows the TE-TM polariton splitting calculated for aCdTe-based microcavity sample for different detunings between the bare cavity modeand the exciton resonance. For these calculations, polariton eigenfrequencies in twolinear polarisations have been found numerically by the transfer matrix method. Onecan see that the splitting is very sensitive to the detuning,and may achieve 1meV, whichexceeds by an order of magnitude the bare exciton LT splitting.

Fig. 9.9: Longitudinal-transverse polariton splitting calculated for detunings: +10meV, 0meV, -10meV, -19meV. Solid lines: lower polariton branch, dashed lines: upper polariton branch. From Kavokin et al.(2004b).

Using the pseudospin representation, one can rewrite the Liouville-von Neumannequation for the density matrix as

∂tSk =gµB

~Sk × Beff,k (9.20)

The Maialle field thus induces precession of the pseudospin of the ensemble ofcircularly polarized polaritons and can cause oscillations of the circular polarisation de-gree of the emitted light in time as was observed experimentally by Martin et al. (2002)(Fig. 9.3) and conversion from linear to circular polarisation observed by the Yamamotogroup (Fig. 9.4). In the latter case, the pseudospin of exciton-polaritons excited at theoblique angle rotates about the Maialle field, while in the ground state (k = 0) the ro-tation vanishes. Thus, linear to circular polarisation conversion depends on the ratio ofthe rotation period at the oblique angle and energy relaxation time of exciton-polaritons,


i.e., the time needed for a polariton to relax to the ground state. At low pumping inten-sity, the relaxation time is longer than the rotation period, at some intermediate pumpingtwo times coincide and at high pumping the relaxation time becomes shorter than theperiod of pseudospin rotation. This explains the non-monotonic dependence of the cir-cular polarisation degree of emission on the pumping intensity in the linear pumpingcase (Fig. 9.4).

The precession of the polariton pseudospin about the Maialle field resembles theprecession of electron spins about the Rashba effective magnetic field109. The physi-cal origin of these two fields is, however, different. The Rashba field is created by thespin-orbit interaction in asymmetric quantum wells, whereas the Maialle field appearsbecause of the long-range electron-hole interaction and TE-TM splitting of the cavitymodes. The orientation of these two fields is also different.The Rashba field is perpen-dicular to the electron wavevector, whereas the Maialle field is neither perpendicularnor parallel to the exciton (polariton) wavevector, in general.

9.6 Optical spin Hall effect

In high quality microcavities, exciton-polaritons can propagateballistically, i.e., with-out scattering, over a few picoseconds. This presents an opportunity of observing thepseudospin precession of individual exciton polaritons under effect of the Maialle field.The pseudospin precession in the ballistic regime leads to the optical spin Hall effectdetailed by Kavokin, Malpuech & Glazov (2005).

The spin Hall effect is the appearance of a spin flux due to the direct current flow ina semiconductor. This effect predicted by D’yakonov & Perel(1971a) found its experi-mental proof only recently in the work of Kato et al. (2004). It has a remarkable analogyin semiconductor optics, namely, in Rayleigh scattering oflight in microcavities. Thespin polarisation in a scattered state can be positive or negative dependent on the ori-entation of the linear polarisation of the initial state andon the angle of rotation of thepolariton wavevector during the act of scattering. Very surprisingly, spin polarisationsof the polaritons scattered clock-wise and anti-clock-wise have different signs.

Consider a semiconductor microcavity in the strong-coupling regime. We supposethat one ofk 6= 0 states of the lower polariton dispersion branch is resonantly excitedby linearly polarized light (herek = (kx, ky) is the in-plane polariton wavevector). TheRayleigh scattered signal comes from the quantum states whose wavevector is rotatedwith respect to the initialk by some nonzero angleθ. We study polarisation of thescattered light as function ofθ andk taking into account the pseudospin rotation undereffect of the Maialle field (9.20).

Let us assume that light is incident in the (x ,z)-plane, and is polarized along thex-axis (TM-polarisation). It excites the polariton state with a pseudospin parallel tothex-axis. As the pseudospin is parallel to the effective field, it does not experienceany precession at the initial point (see Fig. 9.10(b)). Consider now the scattering actwhich brings our polariton into the state(k′x, k

′y) with k′x = k cos θ andk′y = k sin θ.

109The Rashba effect predicted in 1984 (Y.A. Bychkov, E.I. Rashba, JETP Lett., 39, 78 (1984)) has stimu-lated development of spintronics, a new area of semiconductor physics studying propagation of spin-polarizedelectrons.

OPTICAL SPIN HALL EFFECT 359

Fig. 9.10: (a) experimental configuration allowing for observation of the optical spin Hall effect: linearlypolarized light is incident at an oblique angle, circular polarisation of the scattered light is analyzed, (b) ori-entations of the effective magnetic field induced by the LT splitting of the polaritons for different orientationsof the in-plane wavevector, (c) pseudospin of polaritons created by the TM polarized pump pulse at the pointshown by the large circle, arrows show the pseudospins of polaritons just after the Rayleigh scattering act andthe effective field orientation.

Following the classical theory of Rayleigh scattering, we assume that the polarisationdoes not change during the scattering act, so that at the beginning the pseudospin of thescattered state keeps oriented inx-direction (Fig. 9.10(b)). As the effective field is nomore parallel to the pseudospin, it starts precessing. Due to precession, the pseudospinacquires az-component and circular polarisation emerges. The polarisation of emissiondepends on the ratio between the period of precession (givenby the TE-TM splitting ofthe polariton state) and the polariton lifetime. If both quantities are of the same order, apeculiar angle-dependence of the circular polarisation ofscattered light appears.

Fig. 9.11 shows schematically the dependence of the circular polarisation degree oflight scattered by the cavity in different directions. Darkareas correspond to the right(left) circularly polarized light. One can notice inequivalence of clockwise and anti-


Fig. 9.11: Schematic diagram of the angular dependence of circular polarisation of emission of the micro-cavity. Dark areas correspond to the right-(left-) circularly polarized light.

clockwise scattering: if the spin-up majority of polaritons (right-circular polarisation)dominate scattering at the angleθ, the signal at angle−θ is mostly emitted by spin-downpolaritons (left-circular polarisation), and vice-versa. In order to obtain the polarisationdistribution in scattering of TE-polarized light (incident electric field in theyz-plane)one should simply interchange areas Fig. 9.11.

Rotation of the polariton pseudospin around the Maialle field is responsible for ap-pearance of the Langbein cross (Fig. 9.7). In his experiment, in x, −x, y, and−y direc-tions, X-polarized light corresponds to one of the eigenstates of the system (TE- or TM-polarized). The effective field in these points is parallel (antiparallel) to the pseudospinof the exciting light, thus no precession takes place and thepolarisation is conserved.On the other hand, within the quarters, the effective field isinclined at some angle tothe psedospin, so that the precession takes place, and the emission signal in cross-linearpolarisation appears.

9.7 Optically induced Faraday rotation

Thez-component ofBeff,k splitsJ = +1 andJ = −1 exciton states. It would be zeroin the absence of polariton-polariton interactions. However, in the nonlinear regime itcan arise due to the difference of concentrations of spin-upand spin-down polaritons,which leads to the optically induced Faraday rotation in microcavities. To show this, letus first consider the connection of the pseudospin formalismwith the second quantiza-tion representation.

If the polariton concentration is lower than the saturationdensity, the polaritonsbehave as good bosons, and thus a pair of bosonic annihilation operatorsak,↑, ak,↓can be introduced to describe the polariton quantum states having a wavevectork. Theoccupation numbers of spin up, spin down polariton states and z-component of thecorresponding pseudospin can be found from

OPTICALLY INDUCED FARADAY ROTATION 361

9.4.17Nk↑ = Tr[

ρka†k,↑ak,↑

]

= 〈a†k,↑ak,↑〉 (9.21)

Nk↓ = Tr[

ρka†k,↓ak,↓

]

= 〈a†k,↓ak,↓〉 (9.22)

Sk,z =1

2

[

〈a†k,↑ak,↑〉 − 〈a†k,↓ak,↓〉] (9.23)

To find the dynamics of in-plane components of the pseudospinone should intro-duce the bosonic operators for linear polarized polaritonsas follows:

ak,x = 1√2

(

ak,↑ + ak,↓)

, ak,−x =1√2

(

ak,↑ − ak,↓)

, (9.24)

ak,y = 1√2

(

ak,↑ + iak,↓)

, ak,−y =1√2

(

ak,↑ − iak,↓)

. (9.25)

Knowing the dynamics of these operators,Sk,x andSk,y are expressed as:

Sk,x =1

2

[

〈a†k,xak,x〉 − 〈a†k,−xak,−x〉]

= ℜ〈a†k,↓ak,↑〉 (9.26)

Sk,y =1

2

[

〈a†k,yak,y〉 − 〈a†k,−yak,−y〉]

= ℑ〈a†k,↓ak,↑〉 (9.27)

We shall consider the polariton pseudospin dynamics in the nonlinear regime. Let usstart from the simplest case where all the polaritons are in the same quantum state, i.e.,form a “condensate”, so that the scattering to the other states is completely suppressed.Of course, this is true only at zero temperature. The generalform of the interactionHamiltonian reads

H = ε(a†↑a↑ + a†↓a↓) + V1(a†↑a

†↑a↑a↑ + a†↓a

†↓a↓a↓) + 2V2(a

†↑a↑a

†↓a↓) (9.28)

where the index corresponding to the polariton statek in the reciprocal space is omit-ted.110 In Eq. (9.28),ε is the free polariton energy while the matrix elementsV1 andV2

correspond to the forward scattering of the polaritons in the triplet configuration (paral-lel spins) and in the singlet configuration (antiparallel spins). If the polariton-polaritoninteractions were spin-isotropic, i.e., the Hamiltonian (9.28) were covariant with re-spect to the linear transformation of the operators (9.24),the matrix elements would beinterdependent, so that

V1 = V2 . (9.29)

However, this situation is not realized in semiconductor microcavities, where the majorcontribution to polariton-polariton interaction is givenby the exchange term, so that thepolariton-polariton interaction is in fact anisotropic (dependent on mutual orientationof spins of interacting polaritons) and|V1| ≫ |V2|. This anisotropy manifests itself

110The Hamiltonian without notation shortcut reads

H = ε(a†k,↑ak,↑ +a†

k,↓ak,↓)+V1(a†k,↑a

†k,↑ak,↑ak,↑ +a†

k,↓a†k,↓ak,↓ak,↓)+2V2(a†

k,↑ak,↑a†k,↓ak,↓)


experimentally in the optically induced splitting of the spin-up and spin-down polaritonstates, which has been experimentally measured as a function of the circular polarisationdegree of the excitation (directly linked to the imbalance of populations of spin-up andspin-down states).

The Hamiltonian (9.28) conservesN↑ andN↓ but not the in-plane components ofthe pseudospin. It is straightforward to directly evaluatethe commutator of this Hamil-tonian with the operators governing linear polarisation. This commutator is zero only ifthe condition (9.29) is satisfied, which is not the case experimentally. The equation ofmotion for〈a†↓a↑〉 can be obtained from Eq. (9.16) and (9.28) to read

∂t〈a†↓a↑〉 =i

~2(V1 − V2)

[

〈a†↓a↓a†↓a↑〉 − 〈a†↑a↑a↑a↓〉

]

(9.30)

In the mean field approximation, the fourth-order correlators in the right side of (9.30)can be decoupled and Eq. (9.30) can be transformed into an equation of precession of thepseudospin in an effective magnetic fieldBint oriented along the structure growth axis.The absolute value of the field is determined by the difference between the populationsof spin-up and spin down polaritons:

gµB|Bint| = 2(V1 − V2)(N↓ −N↑) . (9.31)

Experimentally, the effect manifests itself as a rotation of the polarisation plane of emis-sion if the σ+ andσ− populations are imbalanced, i.e., in case of optically inducedFaraday rotation.

To summarize, in general the effective magnetic field actingon the polariton pseu-dospin has two components,Beff,k = BLT,k + Bint,k. The in-plane component is gov-erned by the TE-TM splitting of the cavity modes and the long range electron-hole in-teraction in the exciton. It is concentration independent and leads to the beats betweenthe circularly polarized components of the photoemission.The component parallel tothe structure growth axis arises because of the anisotropy of the polariton-polariton in-teraction and depends on the imbalance between spin-up and spin-down polaritons. Itleads to the Faraday rotation of the polarisation plane of light propagating through thecavity.

9.8 Interplay between spin and energy relaxation of exciton-polaritons

The linear model developed above neglects the inelastic scattering of polaritons leadingto their relaxation in reciprocal space and their energy relaxation. Interpretation of non-linear polarisation effects in microcavities requires itsgeneralization. Formally, kineticequations for the occupation numbers and pseudospins can bedecoupled only in thelinear regime. Thus, a theoretical description of nonlinear processes requires the selfconsistent accounting of both energy and spin relaxation processes.

Our starting point is the Hamiltonian of the system in the interaction representation.

H = Hshift +Hscat (9.32)

INTERPLAY BETWEEN SPIN AND ENERGY RELAXATION OF EXCITON-POLARITONS363

Polariton interaction with acoustic phonons and polariton-polariton scattering are takeninto account here. Only the lower polariton branch is considered, coupling with the up-per branch and dark exciton states are neglected. The “shift” term describes interactionof exciton-polaritons having the same momentum but possibly different spins:

Hshift =∑

k,σ=↑,↓

(

gBµBBLTa†σ,ka−σ,k + V

(1)k,k,0(a

†σ,kaσ,k)2 + V

(2)k,k,0(a

†σ,ka−σ,k)2

)

+∑

k,k′ 6=kσ=↑,↓

(

V(1)k,k′,0a

†σ,ka

†σ,k′aσ,kaσ,k′ + V

(2)k,k′,0a

†σ,ka

†−σ,k′aσ,ka−σ,k′

)

(9.33)

Herea↑,k, a↓,k are annihilation operators of the spin-up and spin-down polaritons,BLT,k = BLT,k,x + iBLT,k,y (BLT,k,x andBLT,k,y are thex andy projections ofthe effective magnetic field). The “scattering term”Hscat describes scattering betweenstates with different momenta:

Hscat =1

4

∑

k,k′ 6=kσ=↑,↓

exp( i

~(E(k) + E(k′) − E(k + q) − E(k′ − q))t

)

×

(

V(1)k,k′,qa

†σ,k+qa

†σ,k′−qaσ,kaσ,k′ + 2V

(2)k,k′,qa

†σ,k+qa

†−σ,k′−qaσ,ka−σ,k′ + h.c.

)

+1

2

∑

k,q 6=0σ=↑,↓

exp( i

~(E(k) + ~ωq − E(k + q))

)

Uk,qa†σ,k+qaσ,kbq + h.c. (9.34)

In Eq. (9.34),bq is an acoustic phonon operator,Uk,q is the polariton-phonon cou-pling constant,E(k) is the dispersion of the low polariton branch. The matrix ele-mentsV (1)

k,k′,q andV (2)k,k′,q describe scattering of two polaritons in the triplet and the

singlet configurations, respectively. As it has been discussed above, in real microcavi-ties the polariton-polariton interaction is strongly anisotropic: the triplet scattering isusually much stronger than the singlet one. Moreover, the matrix elementsV (1)

k,k′,q

andV (2)k,k′,q can have opposite signs, as shown by Kavokin, Renucci, Amand, Marie,

Senellart, Bloch & Sermage (2005). Indeed, the interactionbetween two polaritons withparallel spins is always repulsive, while polaritons with opposite spins are characterisedby an attractive interaction and can even form a bound state (bipolariton), as discussedby Ivanov et al. (2004).

We use the Hamiltonian (9.32) to write the Liouville equation for the total densitymatrix of the system:

i~dρ

dt= [H(t), ρ(t)] = [Hshift +Hscat(t), ρ(t)] (9.35)

We solve Eq. (9.35) within the Born-Markov approximation already used in previ-ous chapters. The Markov approximation means physically that the system is assumed


to have no phase memory. It is, in general, not true for the coherent processes describedby theHshift part of the Hamiltonian but is a reasonable approximation for the scatter-ing processes involving the momentum transfer. We apply theMarkov approximationto the scattering part of Eq. (9.35) which reduces to:

dρ

dt= − i

~[Hshift, ρ] −

1

~2

∫ t

−∞[Hscat(t), [Hscat(τ), ρ(τ)]] dτ (9.36)

The next step is to perform the Born approximationρ = ρphon ⊗ ∏k ρk. Thephonons are then traced out with their occupation numbers being treated as fixed param-eters determined by the temperature. The density matrices are given by Eq. (9.15). Theycontain information about both occupation numbers and pseudospin components of allstates in the reciprocal space. Eqs. (9.32–9.36) together with formulas (9.21) for occupa-tion numbers and pseudospins are sufficient to derive a closed set of dynamics equationsforN↑,k,N↓,k andSk,⊥ = exSk,x+eySk,y, as has been done by Shelykh et al. (2005).Maialle spin-relxation because of the TE-TM splitting and self-induced Larmor preces-sion are reduced in this model to the precession of the polariton pseudospin about aneffective magnetic fieldBeff,k that arises from the Hamiltonian termHshift. Once po-lariton populations and pseudospins are known, the intensities of the circular and linearcomponents of photoemission are given by:

I+k = N↑,k , (9.37a)

I−k = N↓,k , (9.37b)

Ixk =N↑,k +N↓,k

2+ Sx,k , (9.37c)

Iyk

=N↑,k +N↓,k

2− Sx,k . (9.37d)

Note that light polarisations parallel tox andy axes correspond to the pseudospin par-allel and antiparallel tox axis, respectively.

The general form of the kinetic equations (9.32–9.36) is complicated and their so-lution requires heavy numerical calculations. However, some particular cases can beconsidered analytically. If the pump intensity is weak, thepolariton-polariton scatteringis dominated by the scattering with acoustic phonons. The polariton-polariton interac-tion terms can be thus neglected and the system of kinetic equations becomes muchsimpler. For the occupation numbers and pseudospins we havein this case:

dNk

dt= − 1

τkNk +

∑

k′

[

(Wk→k′ −Wk′→k)(1

2NkNk′ + 2Sk · Sk′)

+ (Wk→k′Nk′ −Wk′→kNk)]

(9.38a)

dSk

dt= − 1

τskSk +

∑

k′

[

(Wk→k′ −Wk′→k)(NkSk′ +Nk′Sk)

+ (Wk→k′Sk′ −Wk′→kSk)]

+gsµB

~[Sk × BLT,k]

(9.38b)

INTERPLAY BETWEEN SPIN AND ENERGY RELAXATION OF EXCITON-POLARITONS365

where the transition rates are

Wk→k′ =

2π

~|Uk,k′−k|2nphon,k′−kδ(E(k′) − E(k) − ~ωk′−k)

2π

~|Uk,k′−k|2(nphon,k′−k + 1)δ(E(k′) − E(k) − ~ωk′−k)

(9.39)

nphon are acoustic phonon occupation numbers. The Dirac delta functions accountfor energy conservation during the scattering processes. Mathematically, they appearfrom the integration of the time-dependent exponents in thesecond term of Eq. (9.36).Writing the energy conserving delta functions in (9.39) we assume that the polaritonlongitudinal transverse splitting does not modify strongly the polariton dispersion curve.In numerical calculations, the delta functions may be replaced by resonant functions,e.g., Lorentzians, having a finite amplitude which can be estimated as an inverse en-ergy broadening of the polariton state. The polariton lifetime has been introduced inEqs. (9.38a) to take into account the radiative decay of polaritons. The pseudospin life-time is in general less thanτk, it can be estimated asτ−1

sk = τ−1k + τ−1

sl , whereτsl is thecharacteristic time of the spin-lattice relaxation (whichaccounts for all the processesof relaxation within the polariton spin doublet apart from one due to the LT splitting).The last term in Eq. (9.38b) is the same as in Eq. (9.20). It describes the pseudospinprecession about an effective in-plane magnetic field givenby the polariton LT split-ting. This term is responsible for oscillations of the circular polarisation degree of theemitted light.

Qualitatively, the spin relaxation of exciton-polaritonscan be understood from thefollowing arguments. Below the threshold, the spin system is in the collision domi-nated regime, i.e., relaxation of polaritons down to the ground state goes through a hugenumber of random passes each corresponding to the scattering process with an acousticphonon. The polarisation degree displays a monotonic decayin this case. This is easyto understand, as in each scattering event the direction of the effective magnetic fieldchanges randomly, so that on average no oscillation can be observed. The situation iscompletely analogous to one for electrons undergoing spin relaxation while moving inan effective Rashba field. Spin relaxation of spin-up and spin-down polaritons proceedswith the same rate, in general.

The situation changes dramatically above the stimulation threshold. In this case, therelaxation rates for spin-up and spin-down polaritons become different, in general. Oncethe ground state is populated preferentially by polaritonshaving a given spin orientation,the relaxation rate of polaritons having this spin orientation is enhanced. This stimulatedscattering process makes the circular polarisation degreeof the emission increase intime, as was experimentally observed. Also the polarisation degree is found to oscillatewith a period sensitive to the pumping power and the detuning. The detuning (differencebetween bare exciton and cavity mode energies) has an important effect on the polaritonspin relaxation. First of all, the LT splitting at a given value of the wave vector stronglydepends on the detuning as one can see from Fig. 9.9. Also the energy relaxation isextremely sensitive to the detuning. At positive detuning,polaritons relax to the groundstate with more random steps than at negative detuning.


Exercise 9.3 (∗∗) Polaritons linearly polarized in they-direction propagate ballisti-cally under the effect of an effective magnetic field oriented (a) in thex-direction, (b)in the y-direction, (c) along thez-axis (normal to the cavity plane). Find the time-dependent intensity of light emitted by the cavity in X,Y andcircular polarisations if thepolariton lifetime isτ .

9.9 Polarisation of Bose condensates and polaritons superfluidity

In this Section we address an important issue of polarisation of the condensates ofexciton-polaritons in microcavities. Of course, the condensate polarisation is cruciallydependent on the dynamics of its formation, i.e., pumping polarisation, polariton relax-ation mechanisms, etc. A realistic polariton system is always out of equilibrium, strictlyspeaking, just because the polaritons have a finite lifetimeand the system should bepumped from outside to compensate the losses of polaritons due to the lifetime. Thethermodynamic limit where the polariton distribution is given by an equilibrium Bose-Einstein distribution function is hardly achievable in reality.111 Nevertheless, consider-ation of this limit is very important for understanding the fundamental effects of Bosecondensation of polaritons. The characteristics of polariton condensates out of equilib-rium deviate from the parameters predicted assuming thermal equilibrium, but the mainqualitative tendencies remain valid.

Here we assume that the Bose-condensation of exciton-polaritons has already takenplace and the polariton system is fully thermalized. The condensate is in a purely co-herent state which can be characterized by a wavefunction. The lifetime of polaritons isassumed infinite and there is no pumping. Moreover, in the first part of this Section weshall assume zero temperature. We take into account the spinstructure of the conden-sate, however. An exciton polariton can have a+1 or−1 spin projection on the structureaxis, thus we have a two-component (or spinor) Bose condensate. This is an importantpeculiarity of the polariton system with respect to variousknown0-spin bosonic sys-tems (atoms, Cooper pairs, He4) and 1-spin systems (He3). We consider the heavy-holeexciton polaritons polarized in the plane of the cavity. In this case the condensate canbe described by a spinor:

φφφ =

(

φxφy

)

, (9.40)

whereφx, φy are complex functions of time and coordinate describing projections ofthe polarisation of the condensate on two corresponding in-plane axes. In this basis, thefree energy of the system can be represented as:

F = φφφ∗T (−i∇)φφφ−µ(φφφ ·φφφ∗)2 +1

4(V1 +V2)(φφφ ·φφφ∗)2 −

1

4(V1 −V2)|φφφ×φφφ∗|2 (9.41)

whereT (−i∇) is the kinetic energy operator,µ is the chemical potential, which cor-responds to the experimentally measurableblue-shiftof the photoluminescence line of

111Recent experiments of the group of Le Si Dang (Grenoble) show, however, that under sufficiently strongpumping the polariton gas quickly thermalizes, so that its distribution function becomes very close to theBose-Einstein function.

POLARISATION OF BOSE CONDENSATES AND POLARITONS SUPERFLUIDITY 367

the microcavity due to formation of the condensate,V1 andV2 are interaction constantsof polaritons with parallel and antiparallel spins, respectively. The two last terms ofEq. (9.41) describe contributions of polariton-polaritoninteractions to the free energy.It can be shown that no other terms of this order can exist in the free energy if the cavityis isotropic in thexy-plane.

At zero temperature, only the ground state is occupied, so that substitutions can bemade in Eq. (9.41):

n = φφφ · φφφ∗ , (9.42a)

Sz =i

2|φφφ×φφφ∗| , (9.42b)

wheren is the occupation number of the condensate, andSz is the normal-to-planecomponent of the pseudospin of the condensate, which characterizes the imbalance ofpopulations of spin-up,n↑, and spin-down,n↓ polaritons:

n = n↑ + n↓ , (9.43a)

Sz =1

2(n↑ − n↓) , (9.43b)

and is related to the circular polarisation degree of the emitted lightρ by:

ρ =2Szn

. (9.44)

The free energy therefore reads:

F = −µn+1

4(V1 + V2)n

2 + (V1 − V2)S2z . (9.45)

In the experimentally studied semiconductor microcavities

V1 > 0 > V2 > −V1 (9.46)

That is why, at zero magnetic field, the minimum of the free energy corresponds toSz =0 (linearly polarized condensate). In perfectly isotropic microcavities, the orientation oflinear polarisation would be chosen spontaneously by the system and would randomlychange from one experiment to another. In real structures itis generallypinnedto oneof crystallographic axes. The pinning of linear polarisation can be caused by variousfactors, including the exchange splitting of the exciton state and the photon mode split-ting atk = 0. The second scenario is the most likely one, as the slightestbirefringencein the cavity (say, in-plane variation of the refractive index by 0.01%) would yield thesplitting of the cavity mode by about 0.1meV, which is quite sufficient for pinning of thecondensate polarisation. Here and further we shall assume the condensatex-polarizedin the absence of any magnetic field.


The chemical potential of the condensate can be obtained by minimization of thefree energy overn, which yields:

µ =V1 + V2

2n . (9.47)

As we mentioned above, this value corresponds to the blue-shift of the photolumi-nescence line due to formation of the condensate. Such a blueshift, linearly dependenton the occupation number of the condensate, has been experimentally observed by thegroup of Le Si Dang (Fig. 9.12).

Fig. 9.12: Angle-dependent photoluminescence (PL) from a microcavity in the strong coupling regime atdifferent pumping intensities (in units of the threshold pumpingP0) (a, b, c) by Richard, Kasprzak, Andre,Romestain, Dang, Malpuech & Kavokin (2005). One can clearlysee formation of the condensate atk = 0point (c) as well as the blue shift of the ground state energy by about 1meV between (a) and (c).

9.10 Magnetic field effect and superfluidity

We consider an exciton-polariton condensate in a microcavity subject to a magneticfield normal to the plane (B ‖ z). We neglect the field effect on electron-hole relativemotion and resulting diamagnetic shift of the condensate and only consider the spinsplitting of exciton-polaritons resulting from the Zeemaneffect. The diamagnetic shiftwould result in a parabolic dependence of the energy of the condensate on the amplitudeof the applied field, but would not influence the superfluidityand spin structure of thepolaritons.

The free energy of the system can be represented as:

FΩ = F − iΩ|φφφ×φφφ∗| (9.48)

WhereF is the free energy in the absence of the magnetic field given byEq. (9.41)with g being the exciton-polaritong-factor,µB being Bohr magneton,B being the am-plitude of applied magnetic field.

MAGNETIC FIELD EFFECT AND SUPERFLUIDITY 369

At zero temperature, one can use Eq. (9.45) forF so that Eq. (9.48) yields

FΩ = −µn− 2ΩSz +1

4(V1 + V2)n

2 + (V1 − V2)S2z . (9.49)

If a weak magnetic field is applied to the cavity (we shall define later what “weak”means in this case), the pseudospin projection and the number of polaritonsn still canbe considered as independent variables. In this regime further referred to as theweakfield regimethe free energy minimization overSz yields

Sz =Ω

V1 − V2. (9.50)

The condensate in this case emits elliptically polarized light with

ρ =µBgB

(V1 − V2)n. (9.51)

Interestingly, in this regime the chemical potential is still given by Eq. (9.47) as imme-diately follows from minimization of the free energy. Thus,the condensate emits lightat the same energy. The red shift of the polariton energy due to Zeeman effect is exactlycompensated by increase of the blue shift due to polarisation of the condensate. Theminimum free energy of the system decreases quadratically with the field:

FΩ,min = − Ω2

V1 − V2− V1 + V2

4n2 . (9.52)

At the critical magnetic field

Bc =(V1 − V2)n

µBg, (9.53)

the condensate becomes fully circularly polarized (ρ = 1).Beyond this point,Sz andn are no longer independent parameters, andSz = n/2.

Therefore, minimization of the free energy Eq. (9.48) overn yields a different result:

µ = −Ω + V1n (9.54)

FΩ,min = −V1

2n2 . (9.55)

In this regime, further referred to as thestrong fieldregime, the emission energy ofthe condensate decreases linearly with the field, so that thenormal Zeeman effect canbe observed in photoluminescence. The magnetic susceptibility of the condensate isdiscontinuous at the critical field.

In order to reveal the superfluid properties of exciton polaritons we consider theGross-Pitaevskii equation:

i~∂φj∂t

=∂FΩ

∂φ∗j(9.56)


Landau and his group in Moscow (1956).Back row: S.S. Gershtein, L.P. Pitaevskii, L.A. Vainshtein, R.G. Arkhipov, I.E. Dzyaloshinskii.Front row: L.A. Prozorova, A.A. Abrikosov, I.M. Khalatnikov, L.D. Landau, E.M. Lifshitz.

here the free energyF is given by Eq. (9.41),j = x, y. The dynamics of in-planecomponents of the polarisation of the system is given by:

i~∂φx∂t

= Txy(−i∇)φx − µφx + iΩφy +1

2(V1 + V2)(φφφ · φφφ∗)φx

+1

2(V1 − V2)|φφφ ×φφφ∗|φy ,

(9.57a)

i~∂φy∂t

= Tyx(−i∇)φy − µφy − iΩφx +1

2(V1 + V2)(φφφ · φφφ∗)φy

− 1

2(V1 − V2)|φφφ ×φφφ∗|φx .

(9.57b)

The last terms in Eqs. (9.57) describe the self-induced Larmor precession of the polari-ton pseudospin which exactly compensates precession induced by the external magneticfield weaker than the critical field (9.53). This follows fromEqs. (9.42b, 9.50). The exactcompensation of the Larmor precession is one of the manifestations of thefull param-agnetic screening or spin Meissnereffect in polariton condensates (discussed in detailbelow).

Following Lifshitz & Pitaevskii (1980, problem of§30) one can obtain the disper-sion of excited states of the system from Eqs. (9.57) by substitution:

φφφ(r, t) =√ne + Aei(k·r−ωt) + C∗e−i(k·r−ωt) (9.58)

MAGNETIC FIELD EFFECT AND SUPERFLUIDITY 371

where

e =

x cos θ + iy sin θ , if B < Bc ,

1√2(x + iy) , if B > Bc .

(9.59)

andθ = 12 arcsin

(

Ω/[n(V1 − V2)])

.Retaining only the terms linear inA andC, and separating the terms with different

complex exponential functions, one can obtain a system of four linear algebraic equa-tions forAx,Ay,Cx andCy. The condition for existence of a non-trivial solution of thissystem allows one to obtain the spectra of excited polaritonstates. At magnetic fieldsbelowBc it reads:

ω2 = ω20 + nV1ω0 ± ω0

√

(nV2)2 + (V 21 − V 2

2 )n2Ω2

Ω2c

, (9.60)

whereω0(k) is the energy of the lower branch polariton state as a function of the in-plane wave vectork in the absence of the condensate,ω0(k = 0) = 0, Ωc = µBgBc/2.One can see that both branches start at the same point atk = 0, which corresponds tothe bottom of the lowest polariton band blue-shifted byµ (see Eq. (9.47)). This showsthat the ground state of the system remains two-fold degenerate in the presence of theexternal magnetic field belowBc, i.e., the Zeeman effect is fully suppressed (see in-set in Fig. 9.12). This suppression results from thefull paramagnetic screening of themagnetic field. The polarisation plane of light going through the microcavity does notexperience any Faraday rotation in this case. The polaritonspins orient along the field,so that the energy of the system decreases as in any paramagnetic material, but thisdecrease is exactly compensated by the increase of polariton-polariton interaction en-ergy. This effect can be considered as a spin analogue of theMeissner effectfamiliar insuperconductors.

Both dispersion branches described by Eq. (9.58) have a linear part characteristic ofa superfluid. The peculiarity of our two-component superfluid consists in the differenceof the sound velocities for two branches:

v± =∂ω

∂k

∣

∣

∣

∣

k=0

=

√

√

√

√

V1n

2m∗ ± 1

2m∗

√

(nV2)2 + (V 21 − V 2

2 )n2Ω2

Ω2c

, (9.61)

wherem∗ is the effective mass of the lowest polariton band. Note thatat zero magneticfield the difference of two sound velocities persists, it is given by

√

n/(2m∗)(√V1 + V2−√

V1 − V2). The sound velocity of the branch polarized with the condensate increaseswith magnetic field and achieves

√

V1n/m∗ at the critical field (Fig. (9.13)). On theother hand, the sound velocity of the other branch decreaseswith increasing field andvanishes atB = Bc.

At strong fields, the ground state of the lowest polariton band is split into the right-circularly and left-circularly polarized doublet. The lowest (right-circular, ifg > 0)branch remains superfluid. Its dispersion is given by:


Fig. 9.13: Sound velocities of two branches of the excitations of a polariton condensate in a microcavity asa function of applied magnetic field, as predicted by Rubo et al. (2006). The inset shows Zeeman splitting ofthe polariton ground state,µ0 = n(V1 + V2)/2.

ω2 = ω20 + 2nV1ω0 (9.62)

The sound velocity is independent of the field and always equal√

V1n/m∗ in thisregime. On the contrary, the higher (left-circular) branchhas the same shape as thebare polariton band:

ω = ω0 + 2(Ω − Ωc) . (9.63)

The second term in the right part of Eq. (9.62) describes the Zeeman splitting of theground state.

The superfluidity of exciton-polaritons at zero temperature exists below and abovethe critical field: atB < Bc both branches of excitations have a linear dispersion, andthey touch each other atk = 0 (zero Zeeman splitting). AtB = Bc the Zeeman splittingis still zero, but one of two branches of excitations has no more linear component in thedispersion. The condensate of polaritons exists only atT = 0 in two-dimensional sys-tems with a parabolic spectrum, thus at any finite temperature the polariton superfluiditydisappears at the critical field. Interestingly, atB > Bc the superfluidity reappears againdue to the Zeeman gap which opens between the two branches of excitations: the po-lariton dispersion in the vicinity of the lowest energy state again becomes linear (see thephase diagram in Fig. 9.14). With further increase of the field the critical temperatureof the superfluid transition is expected to increase and approach the Kosterlitz-Thoulesstransition temperature, i.e., the critical temperature ofthe superfluid transition in a one-component interacting Bose gas. One can see that the two-component (spinor) natureof the polariton condensate leads to its very peculiar behaviour in magnetic field, inparticular, to the full paramagnetic screening (orspin Meissnereffect) and appearanceof a bi-critical point in the phase diagram atB = Bc.

FINITE TEMPERATURE CASE 373

Fig. 9.14: Phase diagram from Rubo et al. (2006) of the polariton superfluidity: at zero field the superfluid islinearly polarized, in the weak-field regime it is elliptically polarized and Zeeman splitting is equal to zero,at the critical field no superfluidity exists, the condensatedisappears at any finite temperature, at strong fieldsthe superfluidity exists again, the condensate is circularly polarized.

9.11 Finite temperature case

At nonzero temperature the excited polariton states are notempty. The temperatureincrease leads to depletion of the condensate, i.e., departure of the polaritons from thecondensate to the states having a non-zero in-plane wavevector) and its depolarisation(as some polaritons can flip their spins). The depletion results in the decrease ofnwith increasing temperature. Let us analyze the depolarisation effect assuming a givenoccupation number of the condensaten and finite temperatureT .

Fig. 9.15: Zeeman splitting of a polariton condensate at zero temperature (dashed lines) and finite temper-atures increasing from right to left. One can see that the spin Meissner effect is relaxed as the temperatureincreases.


We shall consider only thek = 0 state assuming that all other states are weakly oc-cupied and do not contribute to the polarisation of emission. We take into account all theexcited states of the condensate characterized by different projections of its pseudospinon the magnetic field. The chemical potential of the system can be found from:

µ =∂

∂n〈E〉 (9.64)

where the average energy of the condensate

〈E〉 = −2Ω〈Sz〉 +1

4(V1 + V2)n

2 + (V1 − V2)〈S2z 〉 (9.65)

with

〈Sz〉 =

∑n/2j=−n/2 j exp

(

− Ej/(kBT ))

∑n/2j=−n/2 exp

(

− Ej/(kBT ))

, (9.66a)

〈S2z 〉 =

∑n/2j=−n/2 j

2 exp(

− Ej/(kBT ))

∑n/2j=−n/2 exp

(

− Ej/(kBT )), (9.66b)

whereEj = −2Ωj + (V1 − V2)j2. One can see that withT → 0, Sz → Ω/(V1 −

V2) if Ω < (V1 − V2)n/2 andSz → n/2 if Ω > (V1 − V2)n/2, which yields theresults obtained above (Eqs. (9.50–9.53)). In order to obtain a compact expression forthe chemical potential we substitute the sums (9.66a) by integrals. This is perfectly validfor large enough occupation numbersn. Now the derivatives are easily calculated andthe chemical potential reads

µ =1

2(V1+V2)n−2Ω

exp(−(V1 − V2)n2/4kBT )

Σn

(

n

2sinh

Ωn

kBT− 〈Sz〉 cosh

Ωn

kBT

)

+ (V1 − V2)exp(−(V1 − V2)n

2/4kBT )

Σncosh

Ωn

kBT

(

n2

4− 〈S2

z 〉)

, (9.67)

where

Σn =

∫ n2

−n2

exp

(

−E(x)

kBT

)

dx , E(x) = −2Ωx+ (V1 − V2)x2 . (9.68)

Fig. 9.15 shows schematically the behaviour of the chemicalpotential of a conden-sate as a function of the magnetic field at different temperatures. At nonzero tempera-ture, the chemical potential decreases with the field increase even if the field is lowerthan the critical one. The condensate is never fully circularly polarized but its magneticsusceptibilityχ = ∂µ/∂B remains strongly field dependent, however. The upper setof curves shows the excitation energy of the polariton having its spin opposite to themagnetic field, i.e., the upper component of Zeeman doublet observable in reflection ortransmission experiments. One can see that at finite temperatures, the Zeeman splittingis never exactly zero. It is, however, strongly suppressed below the critical field due tothe paramagnetic screening (orspin Meissnereffect).

SPIN DYNAMICS IN PARAMETRIC OSCILLATORS 375

Exercise 9.4 (∗∗) Obtain the Bogolyubov spectrum of excitations from a Gross-Pitaevskiiequation for a scalar wavefunction.

9.12 Spin dynamics in parametric oscillators

In microcavity based optical parametric oscillators (OPOs), the redistribution of po-lariton population between the initial (“pump”) state and two final states (“signal” and“idler”) takes place:

pump ↔ signal + idler

so that both the energy and the total in-plane wavevector of the polariton system are con-served (see Section 9.7). Frequently, such OPOs are studiedin the amplification regimewhen a weak probe pulse is used to seed thesignal state and stimulate the scatteringof polaritons from the pump state. Frequently the back-scattering of polaritons to thepump state can be neglected in this regime. Experimentally,the polarisation propertiesof such an amplification process have been studied by Lagoudakis et al. (2002) (see Sec-tion 7.2.1). In terms of the classical nonlinear optics the scattering of polaritons fromthe pump to signal state stimulated by a probe pulse can be understood as a resonantfour-wave mixing process. In this Section we first consider phenomenologically the am-plification process and describe the experiment of Lagoudakis in terms of the four-wavemixing, then present the quantum theory of polarisation dependent OPO based on theLiouville equations.

9.13 Classical nonlinear optics consideration

We consider an experimental configuration of Lagoudakis et al.: a polarized pump ex-cites the cavity at the magic angle and generates a coherent polariton population at theinflection point of the lower polariton branch (see Fig. 9.16). Its polarisation is changed

Fig. 9.16: In-plane dispersion of un-coupled excitonand photon (dashed) and polariton (solid) modes ina microcavity in the strong coupling regime. Arrowsshow scattering of exciton-polaritons from the pumpstate into signal and idler states.


from right-circular to left-circular passing through linear polarisation in a series of ex-periments. A circularly polarised probe generates polaritons in the ground state (k = 0)that stimulates the resonant scattering of polaritons created by the pump pulse to theground state. The signal polarisation dependence on the pump polarisation is studied(see Fig. 9.5 (a, b, c)).

To model this experiment, one can represent an electric fieldof an electromagneticwave propagating in the structure as a Jones vector:

E =

(

ExEy

)

(9.69)

having the in-plane componentsEx, Ey . The electric field of the four-wave mixingsignal is given by:

Esigα = TαβγδPβP

∗γSδ , (9.70)

whereα, β, γ andδ take the valuesx or y, Pα are the components of the pump pulse,Sα are the components of the probe pulse.

In order to obtain the tensor let us analyse the two-component matrix

Mαδ = TαβγδPβP∗γ . (9.71)

By reasons of symmetry, it can be represented generally in the following way:

Mαβ = A(PxP∗x + PyP

∗y )δα,β +B(PαP

∗β + PβP

∗α) + C(PαP

∗β − PβP

∗α) , (9.72)

whereA, B, C are constants andδ is the Kronecker symbol. The first term in the rightpart of Eq. (9.72) describes the isotropic optical responseof the system, the second termyields the in-plane anisotropy induced by the pump pulse, and the third term describesthe pump-induced gyrotropy.

The gyrotropy comes from the spin-splitting of the exciton resonance in the caseof circular or elliptical pumping. The splitting of the exciton resonance inσ+ andσ−

polarisations influences the linear optical response of thequantum well (optically in-duced Faraday rotation results from this effect). We take this effect into account whilecalculating the linear propagation of light in the cavity. PuttingA = 1 we reduce thenumber of unknown parameters of the problem till two. The best fit to the experimentaldata (Fig. 9.5 (d, e, f ) is obtained with the following set of parameters:B = (i− 1)/2andC = i/8 for the signal,B = −1/2 andC = 2i/3 for the idler.

Note that if the pump pulse is linearly polarized, the third term in the right part ofEq. (9.72) vanishes, so that the polarisation of the signal is governed by the only param-eterB. If it would be zero (no optical anisotropy case) the signal polarisation would bethe same as the probe polarisation. In the experiment of Lagoudakis, however, the sig-nal polarisation is purely linear at circular probe. This indicates that optically inducedanisotropy (Kerr effect) takes place in microcavities: if the sample is illuminated by lin-early polarized light, the polariton ground state (signal state) splits in a doublet co- andcross-polarized with respect to the incident light. Most likely, this effect has its originsin the anisotropy of polariton-polariton interactions, aswe shall see below.

To understand the origin of constantsB andC a microscopic (quantum) considera-tion is needed.

POLARIZED OPO: QUANTUM MODEL 377

9.14 Polarized OPO: quantum modelIn Section 9.7 we have given a set of equations (9.38a) which describe polariton relax-ation assisted by acoustic phonons, polariton-polariton scattering and spin-relaxationbecause of the Maialle field. It represents a general formalism which can be applied fornumerical modelling of any particular experimental situation. To find out the qualitativetendencies analytically, this formalism can be simplified by neglecting different spin orenergy relaxation mechanisms. In OPO case, polariton-polariton scattering is the dom-inant mechanism of the polariton redistribution in the reciprocal space. If we retain inEq. (9.38a) only polariton-polariton scattering and neglect all other terms (except the ra-diative lifetime term responsible for emission of light), the system of kinetic equationsfor polariton occupation numbers and and in-plane pseudospins can be derived:

dNk,↑dt

= Tr(a†k↑ak,↑dρ

dt) = − 1

τkNk↑ +

∑

k′,q

W(1)k,k′,q

[

(Nk↑ +Nk′↑ + 1)Nk+q↑Nk′−q↑ − (Nk+q↑ +Nk′−q↑ + 1)Nk↑Nk′↑]

+W(2)k,k′,q

[

(Nk↑ +Nk′↓ + 1)Nk+q↑Nk′−q↑ +Nk+q↓Nk′−q↑ + 2S⊥,k+q · S⊥,k′−q

−(Nk↑Nk′↓+S⊥,k+q·S⊥,k′)(Nk+q↑+Nk′−q↓+Nk+q↓+Nk′−q↑+2)]

+W(12)k,k′,q

[

Nk′↑S⊥,k′ ·S⊥,k′−q+Nk′−q↑S⊥,k′ ·S⊥,k+q+Nk′↑S⊥,k·(S⊥,k′−q+S⊥,k+q)

+ S⊥,k′ · S⊥,k+q(Nk′−q↑ +Nk′−q↓ +Nk′↑ −Nk′↓)

+ S⊥,k · S⊥,k+q(Nk′−q↑ +Nk+q↓ −Nk′↑ −Nk′↓)]

, (9.73)

dS⊥,kdt

= − 1

τskS⊥,k +

∑

k′,q

W(1)k′,k′,q

2S⊥,k

[

Nk+q↑Nk′−q↑ +Nk+q↓Nk′−q↓

−Nk′↑(Nk+q↑ +Nk′−q↑ + 1) −Nk′↓(Nk+q↓ +Nk′−q↓ + 1)

+ 2S⊥,k+q(S⊥,k′ · S⊥,k′−q) + 2S⊥,k−q(S⊥,k′ · S⊥,k+q)

− 2S⊥,k′(S⊥,k′+q · S⊥,k′−q)]

+W

(2)k′,k′,q

2

[

2(S⊥,k + S⊥,k′)(Nk+q↑Nk′−q↑ +Nk+q↓Nk′−q↑ + 2S⊥,k · S⊥,k′)

− (S⊥,k(Nk′↑ +Nk′↓ +Nk↑ +Nk↓))(Nk+q↑ +Nk′−q↑ +Nk+q↓ +Nk′−q↓ + 2)

− 4S⊥,k(S⊥,k′ · S⊥,k+q + S⊥,k′ · S⊥,k′−q)]

+W

(12)k′,k′,q

2

[

S⊥,k′−q2((Nk′↑ + 1)Nk+q↑ + (Nk′↓ + 1)Nk+q↓)

+ (Nk+q↓ +Nk+q↓ −Nk′↑ −Nk′↓)(Nk↑ +Nk↓)+ S⊥,k′−q2((Nk′↑ + 1)Nk+q↑ + (Nk′↓ + 1)Nk+q↓)

+ (Nk+q↓ +Nk+q↓ −Nk′↑ −Nk′↓)(Nk↑ +Nk↓)]

(9.74)


whereez is a unitary vector in the direction of the structure growth axis. The equationfor spin-down occupation numbers can be obtained from Eq. (9.73) by changing thespin indices. The transition rates are as follows:

W(1)k′,k′,q =

2π

~|V (1)

k,k′,q|2δ(

E(k) + E(k′) − E(k + q) − E(k′ − q))

,

W(2)k′,k′,q =

2π

~|V (2)

k,k′,q|2δ(

E(k) + E(k′) − E(k + q) − E(k′ − q))

, (9.75)

W(12)k′,k′,q =

2π

~ℜ(

V(1)k,k′,qV

(2)∗k,k′,q

)

δ(

E(k) + E(k′) − E(k + q) − E(k′ − q))

.

As usual, the delta functions ensure energy conservation. The signs of the transi-tion ratesW (1)

k′,k′,q, W (2)k′,k′,q andW (12)

k′,k′,q can differ. AlthoughW (1)k′,k′,q andW (2)

k′,k′,q

are always positive,W (12)k′,k′,q can be positive or negative depending on the phase shift

between the matrix elements of the singlet and triplet scatteringV (2) andV (1), respec-tively. In particular, it is negative if these matrix elements are real and have oppositesigns, which is typically the case for microcavity polaritons.

Fig. 9.17: Scattering of two X-polarized polaritons (schematic): both linearly polarized polariton states arelinear combinations of circularly polarized (spin-up and spin-down) states. Due to different signs of the scat-tering constants in triplet (parallel spin) and singlet (antiparallel spin) configurations the pair of polaritonsafter the act of scattering will have Y-polarisation. IfV (1) = V (2) the pair of polaritons would lose theirpolarisation after the scattering process.

Modelling the polarisation dynamics in the OPO we can restrict our consideration toonly three states in reciprocal space corresponding to the pump, signal and idler (furthermarked as p, s, i states).

Let us consider the OPO pumped at the magic angle (p-state) using TE-polarizedlight. No probe pulse is sent, and the polarisation emitted by the ground state (s) is stud-ied. This corresponds to the experimental configuration used by Renucci et al. (2005b).Their experiment showed that the signal polarisation is rotated by 90 with respect topump polarisation. Let us see how this surprising observation can be interpreted withinthe quantum kinetic formalism.

In the spontaneous scattering regime, whenNs↑ = Ns↓ = 0, S⊥,s the system ofkinetic equations (9.73) and (9.74) strongly simplifies andfor the signal state we have:

CONCLUSIONS 379

dNs↑dt

= −dNs↑τs

+W (1)N2p↑ + 2W (2)(Np↑Np↓ + S⊥,p · S⊥,p) , (9.76a)

dNs↓dt

= −dNs↓τs

+W (1)N2p↓ + 2W (2)(Np↑Np↓ + S⊥,p · S⊥,p) , (9.76b)

dS⊥,sdt

= −dS⊥,sτs

+ 2W (12)S⊥p(Np↑ +Np↓) . (9.76c)

Eqs. (9.76a) show that the orientation of the in-plane pseudospin of the signal stateis governed by the sign ofW (12). For W (12) < 0 (which is fulfilled if the signsof V (1) andV (2) are different) the pseudospin of the signal is opposite to the pseu-dospin of the pump. The inversion of the pseudospin corresponds to the 90 rotationof the polarisation plane of photoemission (see the scheme in Fig. (9.17). This rota-tion has been observed experimentally by several groups, the above mentionned one butalso by Krizhanovskii et al. (2006). The linear polarisation degree of the signal in thespontaneous scattering regime can be estimated as

L =4W (12)

W (1) + 4W (2)(9.77)

As the triplet scattering amplitudeV (1) is usually about 20 times higher than the singletscattering amplitudeV (2), |W (12)| ≪ W (1) and thus L is very small (typically, a fewpercents as one can see from Fig. (9.6). However, enhancement of the pump power caninduce a considerable increase of the linear polarisation degree of the emission. Indeed,due to the polarized spontaneous scattering a seed population of polaritons is createdin the signal state. These polaritons have a weak linear polarisation perpendicular tothe pump pulse polarisation. The negative polarisation degree of this seed can be thenenhanced by bosonic stimulation if the intensity of the pumpexceeds the stimulationthreshold value (in Fig. (9.6) it achieves -70% for high pumping).

To conclude this Section, polariton-polariton scatteringis polarisation dependentand the final state polarisation does not coincide with the initial state polarisation, ingeneral. Inversion of linear polarisation between initialand final state is caused by dif-ferent signs of polariton-polariton interaction constants in singlet and triplet geome-tries. When the final state is populated by a seed of circularly polarized polaritons(Lagoudakis experiment) rotation of the polarisation plane by 45 instead of 90 degreeshas been observed, which indicates that light-induced optically anisotropy (birefrin-gence) may take place in the microcavities. In general, alsothe TE-TM splitting of po-lariton states and self-induced Faraday rotation should beaccounted for to describe thecomplicated polarisation dynamics of cavity OPOs. The kinetic equations derived us-ing the pseudospin density matrix approach and Born-Markovapproximation describeall above mentioned effects. Their simplified version retaining only polariton-polaritoninteraction allows for qualitative understanding of the experimental data in many cases.

9.15 Conclusions

Progress in polarized optics of microcavities paves the wayto realisation of “spin-optronic” devices which would transform light on a nano-meter and pico-second scale.


Nowadays, optical communication technologies are based onintensity and frequencymodulation. The light polarisation modulation is rarely used. On the other hand, po-larisation of light represents an additional degree of freedom that could be efficientlyused for encoding of information. Polarisation modulated signals can not be used forlong-range telecommunication as polarisation is quickly lost polarisation scrambling inoptical fibres. However short range optical information transfer, or even optical infor-mation transfer on a processor scale are excellent application areas for “spin-optronic”devices. For this purpose amplifiers and switches sensitiveto polarisation are needed aswell as polarisation converters, polarisation modulators, and stable sources of polarizedlight. Ideally, these several functionalities should be embedded within a single system.A spin-optronic device should be extremely low power consuming as it is supposed tobe integrated on a chip together with classical electronic functions. The device should beelectronically controlled in order to provide an efficient interface with the informationprocessing unities.

Future polariton devices based on microcavities have a potentiality to fulfil all theserequirements, that is why they seem to be well-adapted for the purposes of spin-optronics.They are expected to consume little power, have a nano-size and allow for ultrafastmanipulations with the polarisation of light. Experimentsperformed in the resonant-pumping geometry show that microcavities efficiently convert linear to circular polari-sation and vice versa. They also act as switches sensitive tothe polarisation of a refer-ence beam. The main obstacle on the way towards realisation of spin-optronic devicesbased on microcavities is linked with the need of having an electrically pumped polari-ton laser operational at room temperature. If this problem is solved, polariton deviceswould respond to the actual technological needs of spin-optronics. A new generation ofoptoelectronic devices is therefore potentially to be based on microcavities operating inthe strong coupling regime.

9.16 Further readings

Hanle (1924) provides excellent additional reading on spin-relaxation.

FURTHER READINGS 381

This glossary serves two purposes: it can first be used as anindex whichlocates where in the book the key is introduced, discussed orused. It canalso be used as adictionary which provides a succinct definition thatmight otherwise not be found elsewhere.


AAcoustic phonon SeePhonon.

Active layer In semiconductor lasers, this is the layer of a semi-conductor material, e.g., a quantum well, where the in-version of electronic population between the energy lev-els in the valence and conduction bands is achieved. Thestimulated emission of light dominates its absorption inthis layer at some frequency. See on page 4, 10, 14, 230,Fig. 6.10 on page 232, page 231, 236.

Antinode (of the light field in a cavity) is the maximum of inten-sity of the electric field of a standing light mode. Typ-ically an active element (quantum well, wire or dot) issought to be placed at the antinode of the field as thisprovides the largest exciton-light coupling strength in thecavity. See on page 55, 233.

BBandgap is the region of forbidden states in the band diagram of

a semiconductor. Thebandgap energyis the energy dif-ference between the conduction and the valence band of asemiconductor, that is, the energy required or released tobring one electron from one to the other. See on page 64,Fig. 4.4 on page 123, page 305.

Bottleneck The (phonon) bottleneck effect is a slowing down ofthe rapid polariton relaxation along the lower dispersionbranch which is rapid in the exciton-like part due to scat-tering with acoustic phonons, but then becomes slowerin the vicinity of the anticrossing point of the excitonand photon modes because of kinetic blocking of polari-ton relaxation. The main obstacle for polariton relaxationin the bottleneck region comes from the lack of acous-tic phonons that are able to scatter with polaritons of verylight effective mass. The bottleneck effect prevents polari-tons from relaxing down to their ground state atk = 0,which represents a major problem for the realisation ofpolariton lasers. The bottleneck effect exists also for bulkor quantum well exciton-polaritons. See on page 252, 253,280, 290, Fig. 8.10 on page 299, pages 303—305.

Bose-Einstein condensation (BEC), also simply “Bose condensa-tion”, is a phase transition for bosons leading to the for-mation of a coherent multi-particle quantum state char-acterized by a wave function. The Bose condensate occu-pies the lowest energy level of the system which coincideswith the chemical potential. Strictly speaking, BEC canonly take place in infinite systems with dimensionalityhigher than 2. In finite size and/or low-dimensional sys-tems one can speak of quasi-BEC and Kosterlitz-Thoulessphase transition. See on page 271 and Chapter 8.

Bra (seeketfirst.) In quantum mechanics, the dual state〈ψ| of aket|ψ〉. The product of a bra〈ψ| with any ket|φ〉 givesthebraket〈ψ|φ〉 which is the inner product of their as-sociated Hilbert space, whence the name. See note19 onpage 74.

Bragg mirror is a mirror formed by alternating layers of semicon-ductors with different refractive index. Each layer bound-ary partially reflects the incoming wave and through theeffect of constructive interferences, very high reflectiv-ity is achieved. To obtain the strongest interference, thethicknesses of these layers must be chosen equal to aquarter of the wavelength of light in the correspondingmaterial at some frequency referred to as theBragg fre-quency. The reflection spectrum of a Bragg mirror ex-hibits a plateau of very high reflectivity centered on theBragg frequency. This plateau is referred to as a stop-band, which represents a one-dimensional photonic band-gap. See on page 2, 8, 9, 44, 52, Fig. 2.8 on page 53, page55, Fig. 2.9 on page 57, Fig. 4.20 on page 147, page 232.

CCoherence is one of the basic characteristics of light. According

to the Glauber classification, different orders of the co-herence can be defined. The first order coherence is de-pendent on temporal correlations of the amplitude of thelight field, the second order coherence is dependent onthe intensity correlations etc. Fully coherent light is fullycorrelated to all orders. The coherence time and coher-ence length of light are linked to the first order coher-ence. See on page 10, Section 2.3 on page 31, page 41,Section 3.3.2 on page 98, page 103, 105. The term is alsowildly applied to other concepts, such as quantum coher-ence (e.g., page 94) or condensate coherence (Chapter 8).Often these other meanings themselves split further intomore definitions for unrelated concepts.

Collapse The “process” that a quantum state undergoes upon mea-surement to become the eigenstate|ωi0

〉 associated tothe eigenvalueωi0

measured in the course of the exper-iment. This postulate has been made to match the exper-imental fact that (immediately) repeated measures of thesame observable on a quantum system always return thesame result. This assumption is one of the pillars of theCopenhagen interpretation and is also known as there-ductionof the wavefunction orquantum jump. The exactprocess responsible for it is as yet debated but is describedby the theory of decoherence. See on pages 78—80, 82.

Copenhagen interpretation One earlyinterpretationof quantummechanics issued by the joint efforts of Bohr and Heisen-berg c. 1927 while collaborating in the capital of Den-mark (whence the name). See on page 74.

DDark Exciton Seeexciton.

Detuning refers to the difference in energy between two coupledmodes. “Changing the detuning” means bringing the twomodes in and out of resonance (equal energy). For in-stance, the detuning between photon and exciton modesin a microcavity is the difference between the eigenfre-quencies of the bare cavity mode and the exciton reso-nance at zero in-plane wavevector. One speaks ofposi-tive detuningif the cavity mode is above the exciton res-onance and ofnegative detuningif it is below. See onpage 61, 154, Fig. 4.23 on page 155, 168.

EEtching is a process to remove unwanted parts of a semiconductor

device during its fabrication. Many techniques exist, e.g.,wet etchingusing acid chemicals anddry etchingvaporiz-ing the material,selectiveor anisotropicetchings allow toshape the structure by etching different parts at differentrates. See on page 9, 11, 12, 14, 233, 349.

Exciton is a Coulomb correlated electron-hole pair in a semicon-ductor. One can distinguish between Frenkel and Wannier-Mott excitons, the former having much larger binding en-ergies and much smaller Bohr radius than the latter. Seeon page 114, Section 4.3 on page 124, Fig. 4.7 on page 126,Section 5.5.1 on page 194.Dark Exciton also referred to as anoptically inactive

excitonis an exciton which cannot be created byresonant absorption of a photon. Examples in-clude: indirect in real or reciprocal space exci-tons, excitons with a spin projection on a givenaxis equal to±2, excitons having wavevectorsexceeding the wavevector of light in vacuum attheir resonance frequency. See on page 279, 342.By analogy, a so-calledbright excitonis directlycoupled to light.


Exciton-Polariton is a quasiparticle formed by a pho-ton propagating in the crystal and an exciton res-onantly excited by light. Exciton-polaritons aretrue eigenstates of light in crystals in the vicin-ity of the resonant frequencies of excitonic tran-sitions. See on page 5, Fig. 1.3 on page 6, page114, 134, footnote 65 on page 135, page 139,143, Section 4.4.4 on page 147, Fig. 4.23 onpage 155, page 205, Section 5.8.1 on page 206,Section 5.8.2 on page 208, Chapters 7 and 8.

FFabry-Perot resonator is a kind of cavity formed by a dielectric

layer sandwiched between two mirrors. Its eigenmodesare standing waves whose wavelength is related to thesize of the resonator. If the mirrors are ideal, the integernumber of half-wave lengths of an eigenmode should beequal to the thickness of the cavity. See on page 6, 58, 62,235, 243.

Faraday rotation is a rotation of the polarisation plane of linearlypolarized light passing through a media subject to a mag-netic field parallel to the light propagation direction. Un-like natural optical activity, the Faraday effect can be ac-cumulated in optical resonators and microcavities due tothe multiple round-trips of light. See on page 44, Section9.2 on page 345, Section 9.3 on page 346, Section 9.7 onpage 360

HHanbury-Brown Twiss setup is a photon counting optical setup

which allows one to measure the intensity-intensity cor-relations in a light beam and extract from them the secondorder coherence of lightg2 . See Exercise 2.9 on page 71and Section 3.3.6 on page 103.

Hermitian operator In mathematics, an operatorΩ which is self-adjoint, i.e., such thatΩ† = Ω. As a consequence itseigenvalues are real. Such an operator is typically used todefine anobservable. See the second postulate of quan-tum mechanics on page 78.

Heterostructure The superposition of several thin layers of dif-ferent (hetero) types of semiconductors which togetherform a structurewhose bandgap varies with position. Ajunctionbetween two semiconductors is the simplest het-erostructure. In the celebrateddouble heterostructure, twosemiconductors sandwich a lower-bandgap semiconduc-tor so as to create a potential trap for both electrons andholes. Such a region is the core for semiconductor lasing.See on page 128, 230, 230, Fig. 6.10 on page 232.

Hilbert space In quantum mechanics, mainly used as a synonymfor “the set of quantum states” for a given system. Inmathematics, a separable complete vector space which isthe foundation for the mathematical formulation of thetheory. See the first postulate of quantum mechanics onpage 74 and appendix A.

KKet In quantum mechanics, a vector noted|ψ〉 (by Dirac) part

of a Hilbert space which describes the state of a quan-tum system. See “Bra” on the facing page and note19 onpage 74.

Kosterlitz-Thouless phase transition is a transition towards a su-perfluid phase in two-dimensional bosonic systems. It wasdescribed for the first time by J.M. Kosterlitz and D.J.Thouless in 1973. In infinite two-dimensional systems,Bose-condensation is impossible while a superfluid can

be formed. A superfluid is a collective bosonic state, inwhich the particles can move throughout space along aphase-coherent, dissipation-less path. In ideal infinite mi-crocavities, exciton-polaritons may undergo the Kosterlitz-Thouless-like transition and form a superfluid if a criti-cal condition linking the concentration of polaritons withtemperature is fulfilled. However, strictly speaking, theKosterlitz-Thouless theory cannot be directly applied toexciton-polaritons in microcavities as it ignores the two-component nature of a polariton superfluid coming fromthe specific spin structure of the exciton-polaritons. Seeon page 280.

LLaser is historically the acronym forLight Amplification by Stim-

ulated Emission of Radiationbut is now a generic termto refer to a device emitting a coherent output with someor all the features of the original laser, even though itsspecifics can differ (case of thepolariton laser, for in-stance, or theatom laserwhich has nothing to do withlight). In its original acceptance the laser generates light(its predecessor themasergenerates microwaves) fromstimulated emission of photons by an inverted populationof emitters (atoms, excitons. . . ) with oscillations of theradiation. The oscillation—or positive feedback—is pro-vided by the cavity. See the second half of Chapter 6.

Light cone is the cone limited by the maximal angle at which lightemitted by a spherical source inside the dielectric mediumcan go out to vacuum. This angle is dependent on the di-electric constant of the medium. See on page 137, 236.

Magic angle is the incidence angle of light which allows one to ex-cite the polariton state close to the inflection point of thelowest polariton branch so that the energy and wavevec-tor conservation conditions are fulfilled for the polariton-polariton scattering from this state into the ground state(k = 0) and some higher energy state belonging tothe lower polariton branch (called “idler”). The resonantpolariton-polartion scattering is used in microcavity-basedoptical parametric amplifiers and oscillators. Excitationatthe magic angle allows one to populate quasi-directly thepolariton ground state, thus transferring the coherence ofthe exciting laser pulse to light emitted by the cavity nor-mally to its surface. In typical GaAs-based cavities themagic angle varies between 15 and 20 degrees at detun-ings close to zero. See Fig. 7.9 on page 255, pages 256—258, page 260, 261, 351, 375, 378.

Microsphere, microdisk are semiconductor or dielectric spheres(disks) with a radius comparable to the wavelength of vis-ible light in the media. See Section 2.8.2 on page 68.

Motional narrowing is a quantum effect that consists in the nar-rowing of a distribution function of a quantum particlepropagating in a disordered medium due to averaging ofthe disorder potential on the size of the wave-function of aparticle. In other words, quantum particles that are neverlocalised at a given point of the space, but always occupysome non-zero volume, have a potential energy that is theaverage of the potential within this volume. This is why,in a random fluctuation potential, the energy distributionfunction of a quantum particle is always narrower thanthe potential distribution function. See Section 4.4.5 onpage 157, Section 7.1.3 on page 251

OOrder parameter of a phase transition is a characteristic of the

system which is zero above the critical temperature of the


transition and nonzero below. In the case of the Bose-Einstein condensation, the wavefunction of the conden-sate is an order parameter. Within the second quantiza-tion formalism, the expectation value of the boson cre-ation (annihilation) operator in the condensate plays thesame role. From the point of view of experimental obser-vation of the superfluid phase transition in the system ofspin-degenerate exciton-polaritons, the spontaneous lin-ear polarisation of the polariton condensate provides anorder parameter. See on page 270, 277, 279, 310, 311,Section 8.8.4 on page 331.

Optical parametric amplifier (OPA) is a process of resonant scat-tering of two particles (like photons or polaritons) of fre-quencyω0 into two particles of frequencyω0 + ω1andω0 − ω1 which are calledidler andsignal, respec-tively. In terms of classical optics this is a nonlinear pro-cess governed by aχ3 susceptibility. If a nonlinear me-dia generating the parametric amplification is placed ina resonator, the corresponding device can be referred toas anoptical parametric oscillator (OPO). In microcav-ities such a parametric amplification process is extremelyefficient if one pumps at themagic angle. In this case,the scattering of two pumped polaritons into a groundstate (signal) and an excited state (idler) is resonant (con-serves both energy and wave vector). The driving force ofthe scattering is the Coulomb interaction between polari-tons. The parametric amplification can be stimulated by aprobe pulse which seeds the ground state, injecting a po-lariton population larger than one. The process can alsobe strong enough to be self-stimulated. See Chapter 7.

Optical spin Hall effect is the angle-dependent conversion of lin-ear to circular polarized light in microcavities. It is basedon the resonant Rayleigh scattering of exciton-polaritonsand is governed by their longitudinal-transverse splitting.See Section 9.6 on page 358.

PPhonon quantized mode (longitudinal or transverse) of vibration

in a crystal lattice. Phonons can control thermal and elec-trical conductivities. In particular, long-wavelength phononstransport sound in a solid, whence the name (voice inGreek). There are two types of phonons,acoustic phononsandoptical phonons. See on page 122, 208, 264, Fig. 8.1on page 283.Acoustic phonon Interaction of exciton-polaritons with

acoustic phonons is one of the most importantmechanisms of the polariton energy relaxationin microcavities. See on page 135, 143, 252, 292,Section 8.6.1 on page 298.

Photonic crystal is a periodic dielectric structure characterized byphotonic bands (including allowed bands and gaps). Cav-ities in photonic crystals allow study of fully localizeddiscrete photonic states. See on page 2, 6, Section 1.7 onpage 19, Fig. 1.19 on page 19, page 64.

Pillar microcavity is a pillar etched from a planar microcavity struc-ture. Its diameter is comparable to the wavelength of lightat the frequency of the planar cavity mode. It allows oneto obtain full (three-dimensional) photonic confinement.See Section 1.5 on page 12, Fig. 1.10 on page 13, Sec-tion 2.8 on page 64, Fig. 2.14(c) on page 66.

Plasmon is a light mode propagating on a metal or a highly dopedsemiconductor. Plasmons can be longitudinal or transverse,localized at the surface or freely propagating in the bulkcrystal. In metallic microspheres or other microstructuresconfined plasmon-polaritons can be formed. See on page 20,292.

Polariton is a mixed quasiparticle formed by a photon and a crys-tal excitation (phonon, magnon, plasmon or exciton). Po-laritons can be formed in bulk crystals, at their surfaces,

in quantum confined structures and microcavities. In thisbook we mostly consider theexciton-polaritons(see itsentry in glossary).

Polariton laser A polariton laser is a coherent light source basedon Bose-Einstein condensation of exciton-polaritons. Con-trary to VCSELs polariton lasers have no threshold linkedto the population inversion. Amplification of light in po-lariton lasers is governed by the ratio between the lifetimeof exciton polaritons and their relaxation time towards thecondensate. See on page 21, 275, 280, 290, Section 8.5 onpage 292, Table 8.9 on page 296, page 305, 322, Section8.8.2 on page 324, 329, 332.

QQuantum computation Application of Quantum Information to

process and manipulate qubits to undergo usefulcomputational procedures (or algorithms) whichhave been found in some cases to outclass theirclassical equivalents. For instance the Shor al-gorithm factorises large integers in polynomialtime and Grover algorithm speeds up search inunstructured spaces. The possibility to use mi-crocavities to do quantum computation is in aprehistoric research stage. See on page 77.

quantum cryptography Application of Quantum Infor-mation to communicate a message securely, tak-ing advantage, e.g., of conjugate bases for mea-surement of a qubit or of EPR correlations. Thepossibility to use microcavities to do quantumcomputation is, like Quantum Computation, in aprehistoric research stage. See Exercise 3.24 onpage 112.

quantum dot (QD) is a semiconductor nanocrystal thatconfines excitation in all three dimensions. It isthe ultimate extension of the concept of the re-duced dimensionality of a quantum well. See be-low and Fig. 4.8 on page 129, Section 5.5.1 onpage 194.

quantum information The formulation of (classical) in-formation theory with quantum systems as thecarriers of informations, which proved to be aworthwile extension, yielding as sub-branchesquantum cryptography and quantum computa-tion.

quantum state A vector in a Hilbert space which fullydescribes a quantum mechanical system accord-ing to the postulates of quantum mechanics. Seethe first postulate on page 74.

quantum well (QW) is a semiconductor heterostructurehaving a profile of conduction and/or valenceband edges in form of a potential well where thefree carriers or excitons can be trapped in one-dimensional sheets and propagate freely in thetwo others (the so-called plane of the quantumwell.) See Fig. 4.8 on page 129.

quantum wire is an electrically conducting wire whosedimensions are so small as to impose quantumconfinement in the directions normal to the axis.It extends the concept of the reduced dimension-ality of a quantum well one step further. SeeFig. 4.8 on page 129

Quasi Bose-Einstein condensationis a term frequently employedto describe accumulation of a macroscopic quantity ofbosons in the same quantum state in a finite size quantumsystem, in obvious analogy to Bose-Enstein condensationwhich strictly speaking is a phase transition for infinitesize systems. See Chapter 8.

Qubit A quantum two-level system. The term appears in connec-tion with quantum informationwhere it is the elementaryunit of information carried by a quantum system, and is


the support for related effects, likedense coding. In mi-crocavities, any two-level system such as the pseudospinof a polariton in principle qualifies as a qubit, providedthat the coherence time and control of the state are goodenough, which are still open questions. The term qubit hasbeen introduced by Schumacher (1995). See on page 77.

RRabi splitting is the splitting of an energy level due to the cou-

pling to a cavity mode. The term came to microcavityphysics from atomic physics where an atomic resonanceis split in energy. The appearance of Rabi splitting is a sig-nature of the strong-coupling regime in microcavities. Insemiconductor microcavities, this term is frequently usedinstead of exciton-polariton splitting. It can be detectedby anticrossing of exciton and cavity-photon resonancesin reflection spectra taken at different incidence angles. Itshould be noted, however, that two dips in reflection canbe seen even in the weak-coupling regime, if the exci-ton inhomogeneous broadening exceeds the cavity modewidth. Thus, the dip positions in reflection spectra do notcoincide, in general, with the eigenmodes of a microcav-ity. Typical values of the Rabi splitting are from a fewmeV in GaAs-based microcavities with a single QW orin bulk microcavities, to more than 100 meV in organiccavities with Frenkel excitons. See on page 149, 151, 167,Fig. 5.2 on page 177.

vacuum (field) Rabi splitting refers to the linear opti-cal regime where the interaction of a single pho-ton with a single atom is implied in the case ofatomic cavities. Nonlinear vacuum Rabi split-ting has not been evidenced so far so that thedistinction has not yet gained importance and“Rabi splitting” is often used to mean “vacuumRabi splitting”. See above.

Resonant Rayleigh scattering of light is an elastic scattering wherethe wavevector of light changes but not its frequency. It isan important tool of optical spectroscopy of semiconduc-tors. See on page 64, 146, 154, 158, 358, 359.

SSchrodinger equation is the time evolution equation for a non-

relativistic quantum state. It is the basis for other dynam-ical equations of quantum systems. See Section 3.1.2 andEq. (3.1) on page 74 for the basis equation in Schrodingerpicture.

Single photon emitter is a device which ideally emits a single pho-ton on demand. It has application in quantum cryptogra-phy. They are currently usually based on single quantumdots. Due to the Pauli exclusion principle, the dot can-not host more than one electron and one hole at the low-est energy level and in a given spin configuration. Oncesuch an electron-hole pair recombines, emitting a photon,some time is needed to recreate it again in the dot. There-fore photons are emitted one by one. The same effect canbe realized using single atoms, molecules, or defects. Seefootnote 41 on page 97.

Spin Meissner effect is the same as full paramagnetic screeningin exciton (exciton-polariton) BEC. No Zeeman splittingof the condensate takes place until some critical magneticfield dependent on the occupation number of the conden-sate and polariton-polariton interaction constants. See onpage 370.

Spontaneous symmetry breaking is a signature of any phase tran-sition according to the Landau theory. In microcavities,BEC or the superfluid phase transition of exciton polari-tons require symmetry breaking and the appearance of an

order parameter (wavefunction) in the polariton conden-sate. The signature of spontaneous symmetry breaking inisotropic planar microcavities is the buildup of linear po-larisation of light emitted by a polariton condensate. Theorientation of the polarisation plane is randomly chosenby the system. See Section 7.4.1 on page 269, page 279,312, 334.

Stimulated scattering is a scattering which is enhanced by theBose statistics. Probability of scattering becomes propor-tional to the occupation number (plus one) of the finalstate to which the bosons are scattered. The term+1 isthe term independent of the statistics (providing the scat-tering rate in absence of stimulation). If the final state ismacroscopically populated, i.e., forms a Bose condensate,scattering towards such a state is strongly amplified andbecomes extremely rapid. Exciton polaritons (which aregood bosons) are subject to stimulated scattering whichprovides the underlying action of polariton lasing. SeeSection 7.2.1 on page 253, page 255, Fig. 7.10 on page 256,Fig. 7.11 on page 257, page 263, 265, 269, 275, Sec-tion 8.6.5 on page 303, Section 8.7.2 on page 311, page340, 350, 351, 365, 375.

Strong coupling between two systems refers to a regime wherethe quantum Hamiltonian dynamics predomines over thedissipation of the system. The dynamics cannot be dealtwith perturbatively and new quantum states of the systememerge. In a cavity, the strong coupling refers to sucha coupling between exciton and light giving rise to po-laritons. It manifests itself in the appearance of a split-ting between the real parts of the eigenfrequencies of po-lariton modes, which is maximum at the resonance be-tween bare exciton and photon modes. In this regime, theimaginary parts of two polariton eigenfrequencies coin-cide at the resonance. The signature of the strong cou-pling regime is a characteristic anticrossing observed inthe reflection (transmission) spectra when the light modecrosses the exciton resonance or vica versa. It requiresdomination of the exciton-photon coupling strength overdifferent damping factors (acoustic phonon broadening,inhomogeneous broadening etc). It is the opposite ofweakcouplingregime. See Section 1.1.7 on page 5, Fig. 4.21on page 149, page 151, 207, 212, 219, Chapters 7 and 8,page 358, Fig. 9.16 on page 375

Superfluidity is a specific property of bosonic liquids at ultra-low temperatures. The liquid propagates with zero vis-cosity and has a linear dependence of the kinetic energyon wavevector. Appearance of superfluidity is a conse-quence of the repulsive interaction between bosons. Ac-cording to recent theories, exciton-polaritons in micro-cavities may become superfluid under certain conditions.See on page 266, 277, 279, 280, 281, Section 8.3.3 onpage 284, page 289, 338.

VVCSEL is an acronym forVertical Cavity Surface Emitting Laser.

It is a device based on a microcavity in the weak-couplingregime. Stimulated emission of light by an active elementinside the cavity (typically quantum wells, where the in-version of population of electron levels of the conductionand valence bands is achieved due to electrical injectionof charge carriers) pumps one of the confined light modesof the cavity. The light emitted by this laser goes out atthe right angle to the surface of the mirror, contrary to“horizontal” lasers, where the generated light propagatesin the plane of the laser cavity. See Fig. 1.11 on page 14,Fig. 6.10 on page 232, Section 6.2.3 on page 232.

W


Weak coupling regime is the regime opposed tostrong coupling(see above) where dissipation dominates over the systeminteraction so that the coupling between the modes can bedealt with pertubatively and both modes retain essentiallytheir uncoupled properties. The weak coupling betweenexciton and light manifests itself in appearance of thesplitting between imaginary parts of the eigenfrequenciesof exciton-polariton modes at the resonance between bareexciton and photon modes. In this regime the real parts oftwo polariton eigenfrequencies coincide at the resonance,and two polariton resonances in the reflection or transmis-sion spectra usually coincide (while in the case of a strongimbalance between the widths of the exciton and photonmodes the doublet structure in reflection and transmissioncan be seen even in the weak coupling regime). See Sec-tion 1.1.7 on page 5, Chapter 6.

Whispering gallery mode are standing light modes localized atthe equator of a sphere. They have been first describedby Lord Rayleigh. Contrary to the “breathing modes”,the whispering gallery modes are characterized by highorbital quantum numbers. See on page 6, Section 1.6 onpage 14, Fig. 2.16 on page 69.

ZZeeman splitting is the magnetic field induced energy splitting of

a quantum state into a couple of states characterized bydifferent spin projections onto the magnetic field direc-tion. See on page 345, 346, 348, Fig. 9.13 on page 372,Fig. 9.15 on page 373, page 374.

APPENDIX A

LINEAR ALGEBRA

This appendix reviews briefly the essential notions of linear algebra required for a goodunderstanding of the postulates and concepts of quantum mechanics discussed in Chap-ter 3. It also settles notations.

A complex vector spaceis a nonempty setH with linear stability, i.e., such that

∀ |u〉 , |v〉 ∈ H, ∀α, β ∈ C, |u〉 , |v〉 ∈ H ⇒ α |u〉 + β |v〉 ∈ H (A.1)

(we use Dirac’s notations introduced in Chapter 3). Axioms for this structure arei)associativity,|u〉 + (|v〉 + |w〉) = (|u〉 + |v〉) + |w〉, and ii) commutativity,|u〉 +|v〉 = |v〉 + |u〉, of the vector addition,iii) existence of a neutral element0 (such that|u〉+0 = |u〉), iv) existence for all|v〉 of |w〉 such that|v〉+ |w〉 = 0, v) associativity ofscalar multiplication,α(β |u〉) = (αβ) |u〉, vi) equality1 |v〉 = |v〉 with 1 the identityof C, and distributivitiesvii) over vector addition,α(|u〉 + |v〉) = α |u〉 + α |v〉, andviii) over scalar addition,(α+ β) |u〉 = α |u〉 + β |u〉.

All these axioms are trivial for a physicist and we shall passquickly over such as-pects of the theory. The mathematical structure withstanding quantum theory is put tothe front because this is largely an abstract theory which connections to physical “real-ity” or to measurements is not intrinsic to the “quantum state”, whose better definitionremains to be part of a vector space. Note that we write the null vector 0 rather than|0〉,which often will more conveniently refer to a nonzero vector.

This space is further endowed with anormwhich defines aninner productfor whichthe space iscompleteand this qualifies it as aHilbert space. A norm is an applicationNfromH to R∗

+ such that for allα ∈ C and for all|u〉, |v〉 ∈ H,

i N (|u〉) ≥ 0, positivity.ii N (α |u〉) = |α|N (|u〉), scalability.iii N (|u〉 + |v〉) ≤ N (|u〉) + N (|v〉), triangle inequality.iv N (|u〉) = 0 iff |u〉 = 0, positive definiteness.

The norm provides a notion oflengthof a vector. The inner product extends to the notionof angles and projections; it is defined as the application fromH2 to C, satisfying thefollowing axioms:

i 〈u|v + w〉 = 〈u|v〉 + 〈u|w〉, 〈u+ v|w〉 = 〈u|w〉 + 〈v|w〉, additivity.

ii 〈u|u〉 ≥ 0, nonnegativity.iii 〈u|u〉 = 0 iff |u〉 = 0, nondegeneracy.iv 〈u|v〉 = 〈v|u〉∗, conjugate symmetry.v 〈u|αv〉 = α〈u|v〉, sesquilinearity.

387

388 LINEAR ALGEBRA

We have contracted an expression which should read⟨

|u〉 , |v〉 + |w〉⟩

to 〈u|v + w〉.This demonstrates the considerable simplifications afforded by Dirac’s notation. Subse-quently,N (|u〉) will be written |〈u|u〉|2.

Completenessis a topological notion of “no-missing point” in the space, it demandsformally that all sequences of pointsun in the space which converge in the sense ofCauchy (such that whatever isǫ > 0, there existsNǫ for whichN (un, vm) < ǫ for n,m > N ), have their limit also in the space.

Although infinite-dimensional spaces of functions are alsoof prime importance,as is the case for instance to describe the harmonic oscillator, we can safely rely onintuition gained from finite-dimensional linear algebra and not study the full theory,known asspectral theory. We cannot spare a few elementary facts of linear algebra,though. We list them now for later reference (if needs be of further details, cf. Halmos,op. cit.)

Let H be a complex Hilbert space of dimensionn. There exists abasisof states|φi〉 , i ∈ [1, n], such that any|ψ〉 ∈ H can be written as:

|ψ〉 =n∑

i=1

ci |φi〉 , with ci ∈ C . (A.2)

The basis is said to be orthogonal if〈φi|φj〉 ∝ δi,j and orthonormal if each state isnormed to unity,|〈φi|φi〉|2 = 1.

Operators onH are linear applications (endomorphisms) fromH to H. A generaloperatorΩ can be decomposed along the same lines as (A.2) as

Ω =

n∑

i

n∑

j

Ωi,j |φi〉〈φj | (A.3)

whereΩi,j = 〈φi|Ω |φj〉. Especially, the unity1 defined such that1 |ψ〉 = |ψ〉 forall |ψ〉 can be decomposed on basis states as follows (this is an important propertyknown asclosure of the identitywhich we write in the case of a continuum basis also):

1 =n∑

i=1

|φi〉〈φi| , 1 =

∫

|φ〉〈φ| dφ (A.4)

from which one can recovers (A.3) by applying it on each side of Ω.The set of applications on the Hilbert space itself forms a vector space. The compo-

sition of applications, denoted with, is defined as follows: letΛ andΩ be two linearapplications onH (endomorphisms), then for every|u〉 ∈ H, the new applicationΛ Ωis given by:

(Λ Ω) |u〉 = Λ(Ω |u〉) (A.5)

whereΩ |u〉 is another vector|v〉 ∈ H on whichΛ is evaluated. Dropping the unneces-sary parentheses on the right hand side of Eq. A.5, the resultreadsΛΩ |u〉 so that thesymbol is dropped for convenience and the composition is noted as a product. When

LINEAR ALGEBRA 389

the applications in a finite dimensional space are written asmatrices, the composition iseffectively the product of their matrices.

Operators are usually not commutative, i.e.,ΛΩ 6= ΩΛ in general. Thecommutatoris the operator defined as:

[Λ,Ω] = ΛΩ − ΩΛ (A.6)

and is an important quantity in Quantum Mechanics. Sometimes the anticommutator isrequired, it is noted with curly brackets:

Λ,Ω = ΛΩ + ΩΛ . (A.7)

For a good and more detailed coverage of the mathematics involved, cf., e.g.,Finite-Dimensional Vector Spaces, P. R. Halmos, Springer (1993).

APPENDIX B

SCATTERING RATES OF POLARITONS RELAXATION

B.1 Polariton-phonon interaction

The theoretical description of carrier-phonon or of exciton-phonon interaction has re-ceived considerable attention throughout the history of semiconductor heterostructures.Here we present a simplified picture which is, however, well suited to our problem.Cavity polaritons are two-dimensional particles with onlyan in-plane dispersion. Theyare scattered by phonons which are in the QWs we consider, mainly three-dimensional(acoustic phonons) or two-dimensional (optical phonons).Scattering events should con-serve the wavevector ino the plane. We callq the phonon wavevector andq‖, qz thein-plane andz-component ofq:

q = (q‖, qz) . (B.1)

Using Fermi’s Golden Rule, the scattering rate between two discrete polariton statesof wavevectork andk′ reads:

W phonk→k′ =

2π

~

∑

q

|M(q)|2(θ± +Nq=k−k′+qz)δ(E(k′) − E(k) ± ωq) (B.2)

whereNq is the phonon distribution function andθ± is a quantity whose sign matchesthe one in the delta function corresponding to phonon emission and absorption, and isdefined asθ+ = 1 andθ− = 0. In the case of an equilibrium phonon distribution,Nq

follows the Bose distribution. The sum of Eq. (B.2) is over phonon states.M is the matrix element of interaction between phonons and polaritons. If one con-

siders polariton states with a finite energy widthγk, the function can be replaced by aLorentzian and Eq. (B.2) becomes:

W phonk→k′ =

2π

~

∑

q

|M(q)|2(θ± +Nq=k−k′+qz)γk′/π

(E(k′) − E(k) ± ωq)2 + γ′k/π2

(B.3)Wavevector conservation in the plane actually limits the sum of Eq. (B.3) to thez-

direction. In the framework of the Born approximation the matrix element of interactionreads:

|M(q)|2 =∣

∣

∣

⟨

ψpolk

∣

∣

∣Hpol−phonq

∣

∣

∣ψpolk′

⟩∣

∣

∣

2

= xkxk′

∣

∣〈ψexk |Hex−phon

q |ψexk′ 〉∣

∣

2(B.4)

where∣

∣

∣ψpol

k

⟩

is the polariton wavefunction and|ψexk 〉 the exciton wavefunction, withxk

the exciton fraction of the polariton. The exciton wavefunction reads:

390

POLARITON-PHONON INTERACTION 391

ψexk (re, rh) = fe(ze)fh(zh)

1√Seik(βere+βhrh)

√

2

π

1

a2DB

exp(

− |re − rh|a2DB

)

(B.5)

whereze, zh are electron and hole coordinates along the growth axis andfe andfh theircoordinates in the plane,fe andfh are the electron and hole wavefunctions in the growthdirection,a2D

B is the two-dimensional exciton Bohr radius,βe,h are the electron/holereduced mass,βe,h = me,h/(me +mh).

B.1.1 Interaction with longitudinal optical phonons

This interaction is mainly mediated by the Frolich interaction (Frohlich 1937). In threedimensions the exciton-LO phonon matrix element reads:

MLO(q) = −eq

√

4π~ωLO

SLε0

(

1

ε∞− 1

εs

)

=MLO

0

q√SL

(B.6)

whereωLO is the energy for creation of a LO-phonon,ε∞ is the optical dielectricconstant,εs the static dielectric constant,L is the dimension along the growth axisandS the normalisation area. In two dimensions one should consider confined opti-cal phonons with quantised wavevector in thez-direction.L becomes the QW widthandqmz = mπ/L with m an integer. Moreover, the overlap integral between excitonand phonon wave-functions quickly vanishes whilem increases. Therefore, we con-sider only the first confined phonon state and the matrix element (B.6) becomes:

MLO(q) =MLO

0√

|q‖|2 + (π/L)2√SL

(B.7)

The wavevectors exchanged in the plane are typically much smaller thanπ/L andEq. (B.7) can be approximated by:

MLO(q) =MLO

0

π

√

L

S(B.8)

Considering a dispersionless phonon dispersion for LO phonons, the LO phononcontribution to Eq. (B.3) reads:

W phon−LOk→k′ =

2L

π2Sxkxk′ |MLO

0 |2×(

θ± +1

exp(−ωLO/kBT ) − 1

)

γk′

(E(k′) − E(k) ± ωLO)2 + γ′2k(B.9)

Optical phonons interact very strongly with carriers. Theyallow fast exciton formation.Their energy of formationωLO is, however, of the order of 20 to 90meV, depending onthe nature of the semiconductor involved. An exciton with a kinetic energy smaller than20meV can no longer emit an optical phonon. The probability of absorbing an opticalphonon remains extremely small at low temperature. This implies that an exciton gas

392 SCATTERING RATES OF POLARITONS RELAXATION

cannot cool down to temperature lower than 100-200K by interacting only with opticalphonons. Optical phonons are therefore extremely efficientat relaxing a hot-carrier gas(optically or electrically created) towards an exciton gaswith a temperature 100-300K ina few picoseconds. The final cooling of this exciton gas towards the lattice temperatureshould, however, be assisted by acoustical phonons or otherscattering mechanisms.The semiconductor currently used to grow microcavities, and where optical phononsplay the largest role, is CdTe. In such a material,ωLO is only 21meV, namely largerthan the exciton binding energy in CdTe-based QWs. Moreover, the Rabi splitting is ofthe order of 10-20meV in CdTe-based cavities. This means that the direct scattering ofa reservoir exciton towards the polariton ground state is a possible process which mayplay an important role.

B.1.2 Interaction with acoustic phonons

This interaction is mainly mediated by the deformation potential. The exciton-acousticphonon matrix element reads:

Mac(q) =

√

q

µ2ρcsSLG(q‖, qz) (B.10)

whereρ is the density andcs is the speed of sound in the medium. Assuming isotropicbands,G reads

G(q‖, qz) = DeI⊥e (qz)I

‖e (q‖)DhI

⊥h (qz)I

‖h(q‖) ≈ DeI

‖e (q‖)DhI

‖h(q‖) (B.11)

De, Dh are the deformation coefficients of the conduction and valence band, respec-tively, andI⊥(‖)

e(h) are the overlap integrals between the exciton and phonon mode in thegrowth direction and in the plane, respectively:

I‖e(h)(q‖) =

(

√

2

π

1

a2DB

)2∫

exp

(

− 2r

a2DB

)

exp

(

imh(e)

me +mhq‖ · r

)

dr

=

(

1 +

(

mh(e)q‖a2DB

2Mx

))2

, (B.12a)

I⊥e(h)(qz) =

∫

|fe(h)(z)|2eiqzz dz ≈ 1 . (B.12b)

Using this matrix element and moving to the thermodynamic limit in the growthdirection, the scattering rate (B.3) becomes:

W phonk→k′ =

2π

~

L

2πxkxk′

∫

qz

q

2ρcsSL|G(k − k′|2(θ± +Nphon

k−k′+qz)

γk′/π

(E(k′) − E(k) ± ωk−k′+qz)2 + γ2

k′

(B.13)

Moving to the thermodynamic limit means that we let the system size in a given di-rection (here thez-direction) go to infinity, substituting the summation withan integral,using the formula

∑

qz→ (L/2π)

∫

dqz .

POLARITON ELECTRON INTERACTION 393

Eq. (B.13) can be easily simplified:

W phonk→k′ =

|G(k − k′|22πZρcs

xkxk′

∫

qz

|k − k′ + qz |(θ± +Nphonk−k′+qz

)

γk′/π

(E(k′) − E(k) ± ωk−k′+qz)2 + γ2

k′

. (B.14)

B.2 Polariton electron interaction

The polariton-electron scattering rate is calculated using Fermi’s Golden Rule as

W elk→k′ =

2π

~

∑

q

|M elq,k,k′|2xkxk′N e

q(1 −N eq+k′−k)

γk′

(

E(k′) − E(k) + ~2

2me(q2 − |q + k− k′|2)

)2

+ γ2k′

. (B.15)

whereN eq is the electron distribution function andme the electron mass. If one considers

electrons at thermal equilibrium, it is given by the Fermi-Dirac electron distributionfunction with a chemical potential

µe = kBT ln(

exp( ~2ne

πkBTme

)

− 1)

(B.16)

wherene is the electron concentration.M el is the matrix element of interaction betweenan electron and an exciton. A detailed calculation of the electron-exciton matrix elementhas been given by Ramon et al. (2003).M el is composed of a direct contribution and ofan exchange contribution:

M el = M eldir ±M el

exc , (B.17)

The + sign corresponds to a triplet configuration (parallel electron spins) and the−to a singlet configuration (antiparallel electron spins). If both electrons have the samespin, the total exciton spin is conserved through the exchange process. However, if bothelectron spins are opposite, an active exciton state of spin+1, for example, will bescattered towards a dark state of spin+2 through the exchange process. Here and inwhat follows we shall consider only the triplet configuration for simplicity.

In order to calculateM el we adopt the Born approximation and obtain:

M eldir =

∫∫∫

ψ∗k(re, rh)f∗

q(r′e)[

V(

|re − r′e|)

− V(

|rh − r′e|)

]

ψk′(re, rh)f∗q+k−k′(r′e)dredrhdre′

(B.18a)

M elexc =

∫∫∫

ψ∗k(re, rh)f∗

q(r′e)[

V(

|re − r′e|)

− V(

|rh − re|)

− V(

|rh − r′e|)

]

ψk′(re, rh)f∗q+k−k′(r′e)dredrhdre′

(B.18b)


with Coulomb potentialV (r) = e2/(4πε0εr) with ε the dielectric susceptibility of theQW.

The free-electron wavefunctionf is given by:

fq(r′e) =1√Seiq·r

′e (B.19)

Integrals (B.18) can be calculated analytically. One finds:

M eldir =

e2

2Sε0ε|k − k′|[

(1 + ξ2h)−3/2 − (1 + ξ2e )−3/2]

(B.20)

Mdirdir =

2e2

Sε0ε

[

(1 + ξ2c )−3/2 − (1 + 4ξ2h)−3/2

(

a−2 + |q − βek′|2)1/2

− (1 + 4ξ2h)−3/2

(

a−2 + |k′ − k − q − βek′|2)1/2

] (B.21)

where

ξe,h =1

2βe,h|k′ − k|a2D

B and ξc = |βek + k′ − k − q|a2DB . (B.22)

Passing to the thermodynamic limit, Eq. (B.15) becomes

W elk→k′ =

S

2π

∫

q

|M eldir +M el

exc|2xkxk′N eq(1 −N e

q+k′−k)×γk′

(

E(k′) − E(k) + ~2

2me(q2 − |q + k− k′|2)

)2

+ γ2k′

. (B.23)

The polariton-electron interaction is a dipole-charge interaction which takes placeon a picosecond time scale. An equilibrium electron gas can thermalise a polaritongas quite efficiently. A more complex effect may, however, take place such as trionformation or exciton dephasing.

B.3 Polariton-polariton interaction

The polariton-polariton scattering rate reads:

W polk→k′ =

2π

~

∑

q

|M ex|2xkxk′xqxq+k′−kNpolq (1 +Npol

q+k′−k)×

γk′

(

E(k′) − E(k) + E(q + k′ − k) − E(q))2

+ γ2k′

. (B.24)

The exciton-exciton matrix element of interaction is also composed of a direct andan exchange term. It has been investigated and calculated byCiuti et al. (1998). Here

POLARITON-POLARITON INTERACTION 395

and in what follows we use a numerical estimate provided by Tassone & Yamamoto(1999) that we further assume constant over the whole reciprocal space:

Mex ≈ 6(a2D

B )2

SEb =

1

SM0

exc (B.25)

whereEb is the exciton binding energy. Passing to the thermodynamiclimit in the plane,Eq. (B.24) becomes:

W polk→k′ =

1

2πS

∫

|M0ex|2xkxk′xqxq+k′−kN

polq (1 +Npol

q+k′−k)×γk′

(

E(k′) − E(k) + E(q + k′ − k) − E(q))2

+ γ2k′

. (B.26)

As one can see, thea priori unknown polariton distribution function is needed to cal-culate scattering rates. This means that in any simulation these scattering rates shouldbe updated dynamically throughout the simulation time, which can be extremely timeconsuming.

Polariton-polariton scattering has been shown to be extremely efficient when a mi-crocavity is resonantly excited. It also plays a fundamental role in the case of non-resonant excitation. Depending on the excitation condition and on the nature of the semi-conductor used, the exciton-exciton interaction may be strong enough to self-thermalisethe exciton reservoir at a given temperature.

B.3.1 Polariton decay

There are three different regions in reciprocal space:[0, ksc], ]ksc, kL] and]kL,∞[.

• [0, ksc] is the region where the exciton-photon anticrossing takes place. Cavitymirrors reflect the light only within a finite angular cone, which corresponds toan in-plane wavevectorksc which depends on the detuning. In this central regionthe polariton decay is mainly due to the finite cavity photon lifetimeΓk = ck/τcwhereck is the photon fraction of the polariton (Hopfield coefficient) andτc isthe cavity photon lifetime.ksc values are typically of the order of 4 to8×106m−1

andτc is in the range between 1 and 10ps.

• ]ksc, kL] wherekL is the wavevector of light in the medium. In this region excitonsare only weakly coupled to the light and polariton decay isΓk = Γ0 which is theradiative decay rate of QW excitons.

• ]kL,∞[. BeyondkL excitons are no longer coupled to light. They only decaynon-radiatively with a decay rateΓnr. We do not wish to enter into the detailsof the mechanism involved in this decay, which we consider asconstant in thewhole reciprocal space. This quantity is given by the decay time measured intime-resolved luminescence experiments, and is typicallyin the range between100ps and 1ns.


B.4 Polariton-structural disorder interaction

This scattering process is mainly associated with the excitonic part of polaritons. Struc-tural disorder induces coherent elastic (Rayleigh) scattering with a typical time scale ofabout 1ps. It couples very efficiently all polaritons situated on the same “elastic circle”in reciprocal space (see the results of Freixanet et al. (1999) and Langbein & Hvam(2002)). This allows us to simplify the description of polariton relaxation by assum-ing cylindrical symmetry of the polariton distribution function. Also, disorder induces abroadening of the polariton states, which should be accounted for when scattering ratesare calculated.

APPENDIX C

DERIVATION OF THE LANDAU CRITERION OFSUPERFLUIDITY AND LANDAU FORMULA

Let us consider a uniform fluid at zero temperature flowing along a capillary at a con-stant velocityv. The only dissipative process assumed is the creation of elementaryexcitations due to the interaction between the fluid and the boundaries of the capillary.If this process is allowed by the conservation laws and Galilean invariance, the flowwill demonstrate viscosity; otherwise it will be superfluid, i.e., dissipativeless. The ba-sic idea of the derivation is to calculate energy and momentum in the reference framemoving with the fluid and in the static frame, then making the link between the twoframes by a Galilean transformation. If a single excitationwith momentump = ~k ap-pears, the total energy in the moving frame isE = E0 + ǫ(k), whereE0 is the energyof the ground state andǫ(k) is the dispersion of the fluid excitations. In the static framehowever, the energy and momentum of the fluid read:

E′ = E0 + ǫ(k) + ~k · v +1

2Mv2 (C.1a)

P′ = p +Mv (C.1b)

whereM is the total mass of the fluid.Eq. (C.1a) shows that the energy of the elementary excitations in the static system

is ǫ(k) + ~k · v). Dissipation is possible, only if the creation of elementary excitationsis profitable energetically, which means:

ǫ(k) + ~k · v < 0 . (C.2)

Dissipation can therefore take place only ifv > ǫ(k)/(~k). On the other hand the fluxis stable if the velocity is smaller than:

vc = min

(

ǫ(k)

~k

)

(C.3)

Formula (C.3) is the Landau criterion of superfluidity. In the case of a parabolic disper-sion,vc is zero and there is no superfluid motion. In the opposite caseof a Bogoliubovdispersion (Eq. 8.16),vc is nothing but the speed of sound which means that the fluidcan move without dissipation with any velocity smaller thanthe speed of sound.

We now consider the finite temperature case. In such a situation it is natural toassume that part of the thermally excited particles do not remain superfluid. We there-fore consider the coexistence of a superfluid component of velocity vs and a normalcomponent of velocityvn. In the frame moving with the normal fluid, the energy of an

397

398 LANDAU CRITERION OF SUPERFLUIDITY AND FORMULA

elementary excitation readsǫ(k)+~k·(vs−vn). The occupation number of elementaryexcitations is

fB(ǫ(k) + ~k · (vs − vn) (C.4)

wherefB is the Bose distribution function, Eq. (8.1). The total massdensity of the fluidcan be written asρ = ρn +ρs whereρn andρs are the mass densities of the normal fluidand superfluid. At this stage, we will assume that all particles of the fluid have the samemassm, which is not correct, in principle, for polaritons. In the static frame the masscurrent of the liquid reads:

mj = ρsvs + ρnvn (C.5)

Following Eq. (C.1b), the total momentum of the fluid in the static frame can alsobe written asP = Mvs +

∑

i ~ki where the sum is taken over all the excitations. Themass current therefore reads:

mj = ρvs +1

S

∑

i

~ki = ρvs +~

(2π)2

∫

fB

(

ǫ(k) + ~k · (vs − vn

)

k dk . (C.6)

Comparing Eqs. (C.5) and (C.6), one gets:

ρs(vs − vn) =~

(2π)2

∫

fB

(

ǫ(k) + ~k · (vs − vn

)

k dk . (C.7)

We are now going to assume that|vs − vn| is small with respect to the speed ofsound and develop Eq. (C.4) in power series ofvs − vn which gives:

fB(ǫ(k) + ~k · (vs − vn) ≈ fB(ǫ(k)) + ~k · (vs − vn)dfB(ǫ(k))

dǫ. (C.8)

We can now insert Eq. (C.8) in Eq. (C.7) to get:

ρn = − ~2

(2π)2

∫

dNp(ǫ, 0)

dǫk2dk . (C.9)

This formula is known as theLandau formula of superfluidity. It is here written fora 2D system. It is valid only for the case of a parabolic dispersion. In Chapter 8, weuse this result for polaritons, replacing~2k2/(2m) by the polariton dispersion. The for-mula used remains however fundamentally inexact since the hypothesis of the constantparticle mass is needed in order to derive the Eq. (C.9).

APPENDIX D

LANDAU QUANTISATION AND RENORMALISATION OF RABISPLITTING

This appendix complements Chapter 9 with an effect recentlyproposed by two of theauthors.

Consider an exciton confined in a QW and subject to a magnetic field normal tothe QW plane. Separating the exciton centre of mass motion and relative electron-holemotion in the QW plane, and assuming that the exciton does notmove as a whole in theQW plane, we obtain the following exciton Hamiltonian:

H = He + Hh + Hex , (D.1)

where

Hν = − ~2

2mν

∂2

∂z2ν

+ Vν(zν) − µBgνsνB + (lν + 1/2)~ωνc , ν = e, h,

Hex = − ~2

2µ

[1

ρ

∂

∂ρ

(

ρ∂

∂ρ

)

− ρ2

4L4

]

− e2

4πε0ǫ√

ρ2 + (ze − zh)2, (D.2)

ρ is the coordinate of electron-hole relative motion in the QWplane,L =√

~c/(eB) isthe so-called magnetic length,B is the magnetic field,se(h) is the electron (hole) spin,me(h) is the electron (hole) effective mass in normal to the plane direction,Ve(h) is

the QW potential for an electron (hole),ge(h) andωe(h)c are the electron (hole)g-factor

and cyclotron frequency, respectively,l = 0, 1, 2, · · · andǫ is the dielectric constant.Hereafter we neglect the heavy-light hole mixing.

The excitonic Hamiltonian (D.2) was first derived by Russiantheorists Gor’kov &Dzialoshinskii (1967). It contains a parabolic term dependent on the magnetic field. Ifthe field increases, the magnetic lengthL decreases, which leads to the shrinkage ofthe wavefunction of the electron-hole relative motion. Thus, the probability of findingthe electron and hole at the same point increases, leading toan increase of the excitonoscillator strength. In order to estimate this effect, let us solve the Schrodinger equa-tion (3.1) variationally for the wavefunctionΨexc(ze, zh, ρ) = 〈ze, zh, ρ|Ψ〉 chosen asan approximate (ortrial ) expression equal to

Ψexc(ze, zh, ρ) = Ue(ze)Uh(zh)f(ρ); (D.3)

whereze(h) is the electron (hole) coordinate in the direction normal tothe plane andρ isthe coordinate of electron-hole in-plane relative motion.If the conduction band offsetsare large in comparison to the exciton binding energy, whichis the case in conventional

399

400 LANDAU QUANTISATION AND RENORMALISATION OF RABI SPLITTING

GaAs/AlGaAs QWs, we findUe,h(ze,h) as a solution of single-particle problems ina rectangular QW. We separate variables in the excitonic Schrodinger equation, andchoose

f(ρ) =

√

2

π

1

a⊥e−ρ/a⊥ , (D.4)

wherea⊥ is a variational parameter. Substituting this trial function into the Schrodingerequation for electron-hole relative motion with Hamiltonian (D.1), we obtain the excitonbinding energy

EB = − 3

16

~2a2⊥

µL4− ~2

2µa2⊥

+4

a2⊥

∫ ∞

0

ρdρe−2ρ/a⊥V (ρ)−(le+1/2)~ωec−(lh+1/2)~ωh

c ,

(D.5)where

V (ρ) =e2

4πε0ǫ

∫ ∞

−∞

∫ ∞

−∞dzedzh

U2e (ze)U

2h(zh)

√

ρ2 + (ze − zh)2. (D.6)

The parametera⊥ should maximize the binding energy. Differentiating Eq. (D.5)with respect toa⊥ we obtain:

Fig. D.1: Typical “fan-diagram” of an InGaAs/GaAs QW. Circles show the resonances in transmission spectraof the sample associated with the heavy-hole exciton transition. In the limit of strong fields Landau quanti-zation dominates over the Coulomb interaction of electron and hole, and the energies of excitonic transitionsincrease linearly with field. Square and diamond correspondto the light-hole exciton transitions. From Seisyanet al. (2001).

LANDAU QUANTISATION AND RENORMALISATION OF RABI SPLITTING 401

~2

8µ

[

1 − a⊥2L

4]

=

∫ ∞

0

ρdρ(

1 − ρ

a⊥

)

e−2ρ/a⊥V (ρ) . (D.7)

The exciton radiative damping rateΓ0, defined in Chapter 3, can be expressed interms of exciton parameters as

Γ0 =ω0

cωLT

√ǫa2

Ba−2⊥ J2

eh . (D.8)

Here Jeh =∫

Ue(z)Uh(z) dz, ω0 is the exciton resonance frequency andωLT

andaB are the longitudinal-transverse splitting and Bohr radiusof the bulk exciton,respectively112 The vacuum-field Rabi splitting in a microcavity is

Ω√

Γ01

a⊥. (D.9)

Shrinkage of the wavefunction of electron-hole relative motion in the magnetic fieldbecomes essential if the magnetic lengthL is comparable to the exciton Bohr radiusaB,i.e., for magnetic fields of about 3T and more in the case of GaAs QWs. Taking intoaccount the fact thatL ≈ 70A atB = 10T, the exciton Bohr radius can be realisticallyreduced by a factor of two.

112Typical parameters for GaAs areωLT = 0.08meV andaB = 14nm.

REFERENCES

Abrikosov, A. A., Gorkov, L. P. & Dzyaloshinski, I. E. (1963). Methods of Quantum Field Theoryin Statistical Physics, Dover.

Agranovich, V., Benisty, H. & Weisbuch, C. (1997). Organic and inorganic quantum wells in amicrocavity: Frenkel-wannier-mott excitons hybridization and energy transformation,SolidState Commun.102: 631.

Agranovich, V. M. (1957). On the influence of reabsorption onthe decay of fluorescence in molec-ular crystals,Optika i Spectr.3: 84.

Agranovich, V. M. & Bassani, G. F. (eds) (2003).Electronic Excitations in Organic Based Nanos-tructures, Vol. 31 of Thin Films and Nanostructures, Academic Press, Amsterdam.

Agranovich, V. M. & Dubovskii, O. A. (1966). Effect of retarded interaction on the exciton spectrumin one-dimensional and two-dimensional crystals,JETP Lett.3: 233.

Agranovich, V. M., Litinskaia, M. & Lidzey, D. G. (2003). Cavity polaritons in microcavities con-taining disordered organic semiconductors,Phys. Rev. B67: 85311.

Ajiki, H., Tsuji, T., Kawano, K. & Cho, K. (2002). Optical spectra and exciton-light coupled modesof a spherical semiconductor nanocrystal,Phys. Rev. B66: 245322.

Akahane, Y., Asano, T., Song, B. & Noda, S. (2005). Fine-tuned high-Q photonic-crystal nanocavity,Opt. Exp.13: 1202.

Alferov, Z. I. (2001). Nobel lecture: The double heterostructure concept and its applications inphysics, electronics, and technology,Rev. Mod. Phys.73: 767.

Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E.& Cornell, E. A. (1995). Observa-tion of Bose-Einstein condensation in a dilute atomic vapor, Science269: 198.

Anderson, P. W. (1966). Considerations on the flow of superfluid helium,Rev. Mod. Phys.38: 298.Andreani, L. C., Panzarini, G., Kavokin, A. V. & Vladimirova, M. R. (1998). Effect of inhomoge-

neous broadening on optical properties of excitons in quantum wells,Phys. Rev. B57: 4670.Andreani, L. C., Tassone, F. & Bassani, F. (1991). Radiativelifetime of free excitons in quantum

wells,Solid State Commun.77: 641.Apker, L. & Taft, E. (1950). Photoelectric emission from f-centers in KI,Phys. Rev.79: 964.Armani, D. K., Kippenberg, T. J., Spillane, S. M. & Vahala, K.J. (2003). Ultra-high-q toroid

microcavity on a chip,Nature421: 925.Baas, A., Karr, J. P., Eleuch, H. & Giacobino, E. (2004). Optical bistability in semiconductor

microcavities,Phys. Rev. A69: 023809.Bajoni, D., Perrin, M., Senellart, P., Lemaıtre, A., Sermage, B. & Bloch, J. (2006). Dynamics of

microcavity polaritons in the presence of an electron gas,Phys. Rev. B73: 205344.Balili, R. B., Snoke, D. W., Pfeiffer, L. & West, K. (2006). Actively tuned and spatially trapped

polaritons,Appl. Phys. Lett.88: 031110.Basov, N. (1964). Semiconductor lasers,Nobel Lecture.Bastard, G. (1988).Wave mechanics applied to semiconductor heterostructures, Les editions de

physique, Paris.Baumberg, J. J., Heberle, A. P., Kavokin, A. V., Vladimirova, M. R. & Kohler, K. (1998). Polariton

motional narrowing in semiconductor multiple quantum wells,Phys. Rev. Lett.80: 3567.Baumberg, J. J., Savvidis, P. G., Stevenson, R. M., Tartakovskii, A. I., Skolnick, M. S., Whittaker,

D. M. & Roberts, J. S. (2000). Parametric oscillation in a vertical microcavity: A polaritoncondensate or micro-optical parametric oscillation,Phys. Rev. B62: R16247.

402

REFERENCES 403

Bayer, M., Reinecke, T. L., Weidner, F., Larionov, A., McDonald, A. & Forchel, A. (2001). Inhibi-tion and enhancement of the spontaneous emission of quantumdots in structured microres-onators,Phys. Rev. Lett.86: 3168.

Bellessa, J., Bonnand, C., Plenet, J. C. & Mugnier, J. (2004). Strong coupling between surfaceplasmons and excitons in an organic semiconductor,Phys. Rev. Lett.93: 036404.

Berger, J., Lyngnes, O., Gibbs, H., Khitrova, G., Nelson, T., Lindmark, E., Kavokin, A., Kaliteevski,M. & Zapasskii, V. (1996). Magnetic-field enhancement of theexciton-polariton splittingin a semiconductor quantum-well microcavity: The strong coupling threshold,Phys. Rev. B54: 1975.

Berman, O., Lozovik, Y., Snoke, D. & Coalson, R. (2004). Collective properties of indirect excitonsin coupled quantum wells in a random field,Phys. Rev. B70: 235310.

Bertet, P., Osnaghi, S., Milman, P., Auffeves, A., Maioli, P., Brune, M., Raimond, J. M. & Haroche,S. (2002). Generating and probing a two-photon Fock state with a single atom in a cavity,Phys. Rev. Lett.88: 143601.

Binder, R. & Koch, S. W. (1995). Nonequilibrium semiconductor dynamics,Prog. in QuantumElectron.19: 307.

Bjork, G. & Yamamoto, Y. (1991). Analysis of semiconductormicrocavity laers using rate equa-tions,IEEE J. Quant. Elec.27: 2386.

Bjork, G., Heitmann, H. & Yamamoto, Y. (1993). Spontaneous-emission coupling factor and modecharacteristics of planar dielectric microcavity lasers,Phys. Rev. A47: 4451.

Blatt, I. M., Boer, K. W. & Brandt, W. (1962). Bose-Einstein condensation of excitons,Phys. Rev.126: 1691.

Bogoliubov, N. N. (1947). On the theory of superfluidity,J. Phys. USSR11: 23.Bogoliubov, N. N. (1970).Lectures on Quantum Statistics, Vol. 1: Quantum Statistics, Gordon and

Breach Science Publisher, New York, London, Paris.Bohm, D. & Aharonov, Y. (1957). Discussion of experimental proof for the paradox of einstein,

rosen, and podolsky,Phys. Rev.108: 1070.Born, M. & Wolf, E. (1970).Principles of optics, Pergamon, London.Bose, S. N. (1924). Plancks gesetz und lichtquantenhypothese,Z. Phys.26: 178.Brune, M., Schmidt-Kaler, F., Maali, A., Dreyer, J., Hagley, E., Raimond, J. M. & Haroche, S.

(1996). Quantum Rabi oscillation: A direct test of field quantization in a cavity,Phys. Rev.Lett.76: 1800.

Butov, L. V., Ivanov, A. L., Imamoglu, A., Littlewood, P. B.,Shashkin, A. A., Dolgopolov, V. T.,Campman, K. L. & Gossard, A. C. (2001). Stimulated scattering of indirect excitons in cou-pled quantum wells: Signature of a degenerate Bose-gas of excitons,Phys. Rev. Lett.86: 5608.

Butov, L. V., Lai, C. W., Ivanov, A. L., Gossard, A. C. & Chemla, D. S. (2002). Towards Bose-Einstein condensation of excitons in potential traps,Nature417: 47.

Butte, R., Skolnick, S., Whittaker, D. M., Bajoni, D., & Roberts, J. S. (2003). Dependence ofstimulated scattering in semiconductor microcavities on pump power, angle, and energy,Phys.Rev. B68: 115325.

Butte, R., Delalleau, G., Tartakovskii, A. I., Skolnick, M. S., Astratov, V. N., Baumberg, J. J.,Malpuech, G., Carlo, A. D., Kavokin, A. V. & Roberts, J. S. (2002). Transition fromstrong to weak coupling and the onset of lasing in semiconductor microcavities,Phys. Rev. B65: 205310.

Carmichael, H. J. (2002).Statistical Methods in Quantum Optics 1, 2nd edn, Springer.Chang-Hasnain, C. (2000). Tunable VCSEL,IEEE J. Sel. Top. Quant. Elect.6: 978.Chew, W. C. (1995).Waves and Fields in Inhomogeneous Media, IEEE Press New York.Ciuti, C. (2004). Branch-entangled polariton pairs in planar microcavities and photonic wires,Phys.

404 REFERENCES

Rev. B69: 245304.Ciuti, C., Savona, V., Piermarocchi, C., Quattropani, A. & Schwendimann, P. (1998). Role of

the exchange of carriers in elastic exciton-exciton scattering in quantum wells,Phys. Rev.B 58: 7926.

Ciuti, C., Schwendimann, P., Deveaud, B. & Quattropani, A. (2000). Theory of the angle-resonantpolariton amplifier,Phys. Rev. B62: R4825.

Ciuti, C., Schwendimann, P. & Quattropani, A. (2001). Parametric luminescence of microcavitypolaritons,Phys. Rev. B63: 041303.

Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G. (2001).Photons et atomes, EDP Sciences.Coleman, S. (1973). There are no goldstone bosons in two dimensions,Comm. Math. Phys.31: 259.Collot, L., Lefevre-Seguin, V., Brune, M., Raimond, J. M. & Haroche, S. (1993). Very high-Q

whispering-gallery mode resonances observed on fused silica microspheres,Europhys. Lett.23: 327.

Comte, C. & Nozieres, P. (1982). Exciton Bose condensation- the ground state of an electron holegas: Mean field description of a simplified model,J. Physique43: 1069.

Cui, G., Hannigan, J. M., Loeckenhoff, R., Matinaga, F. M., Raymer, M. G., Bhongale, S., Holland,M., Mosor, S., Chatterjee, S., Gibbs, H. M. & Khitrova, G. (2006). A hemispherical, high-solid-angle optical micro-cavity for cavity-qed studies,Opt. Exp.14: 2289.

Dasbach, G., Dremin, A. A., Bayer, M., Kulakovskii, V. D., Gippius, N. A. & Forchel, A. (2003).Oscillations in the differential transmission of a semiconductor microcavity with reducedsymmetry,Phys. Rev. B65: 245316.

de Martini, F., Innocenti, G., Jacobovitz, G. R. & Mataloni,P. (1987). Anomalous spontaneousemission time in a microscopic optical cavity,Phys. Rev. Lett.59: 2955.

Deng, H., Press, D., Gotzinger, S., Solomon, G. S., Hey, R.,Ploog, K. H. & Yamamoto, Y. (2006).Quantum degenerate exciton-polaritons in thermal equilibrium, Phys. Rev. Lett.97: 146402.

Deng, H., Weihs, G., Santori, C., Bloch, J. & Yamamoto, Y. (2002). Condensation of semiconductormicrocavity exciton polaritons,Science298: 199.

Deng, H., Weihs, G., Snoke, D., Bloch, J. & Yamamoto, Y. (2003). Polariton lasing vs. photonlasing in a semiconductor microcavity,Proc. Nat. Acad. of Sciences100: 15318.

Deveaud, B., Clerot, F., Roy, N., Satzke, K., Sermage, B. & Katzern, D. S. (1991). Enhancedradiative recombination of free excitons in GaAs quantum wells, Phys. Rev. Lett.67: 2355.

Di Carlo, A., Pescetelli, S., Kavokin, A., Vladimirova, M. &Lugli, P. (1998). Off-resonanceγ-Xmixing in semiconductor quantum wires,Phys. Rev. B57: 9770.

Dicke, R. H. (1954). Coherence in spontaneous radiation processes,Phys. Rev.93: 99.Diederichs, C. & Tignon, J. (2005). Design for a triply resonant vertical-emitting micro-optical

parametric oscillator,Appl. Phys. Lett.87: 251107.Dirac, P. A. M. (1927). The quantum theory of the emission andabsorption of radiation,Proc. Roy.

Soc. London A114: 243.Dirac, P. A. M. (1930).The Principles of Quantum Mechanics, Oxford Science Publications.D’yakonov, M. & Perel, V. (1971a). Current-induced spin orientation of electrons in semiconduc-

tors,Phys. Lett. (a)35A: 459.D’yakonov, M. & Perel, V. (1971b). Spin relaxation of conduction electrons in noncentrosymmetric

semiconductors,Fiz. Tverd. Tela13: 3851. Sov. Phys. Solid State 13, 3023 (1972).Eastham, P. R. & Littlewood, P. B. (2001). Bose condensationof cavity polaritons beyond the linear

regime: The thermal equilibrium of a model microcavity,Phys. Rev. B64: 235101.Einstein, A. (1925). Quantentheorie des einatomigen idealen gases,Zweite Abhandlung, Sitzber.

Kgl. Preuss. Akad. Wiss.1: 3.Ell, C., Prineas, J., T. R. Nelson, J., Park, S., Gibbs, H. M.,Khitrova, G., Koch, S. W. & Houdre,

REFERENCES 405

R. (1998). Influence of structural disorder and light coupling on the excitonic response ofsemiconductor microcavities,Phys. Rev. Lett.21: 4795.

Elliot, R. J. (1954). Spin-orbit coupling in band theory-character tables for some ”double” spacegroups,Phys. Rev.96: 280.

Emam-Ismail, M., Stevenson, R. M., Skolnick, M. S., Astratov, V. N., Whittaker, D. M., Baumberg,J. J. & Roberts, J. S. (2000). Relaxation bottleneck and its suppression in semiconductormicrocavities,Phys. Rev. B62: R2283.

Fermi, E. (1932). Quantum theory of radiation,Rev. Mod. Phys.4: 87.Fernandez, S., Naranjo, F. B., Calle, F., Sanchez-Garcıa, M. A., Calleja, E., Vennegues, P., Trampert,

A. & Ploog, K. H. (2001). Mbe-grown high-quality (al,ga)n/gan distributed Bragg reflectorsfor resonant cavity leds,Semicond. Sci. Technol.16: 913.

Fetter, A. L. & Walecka, J. D. (2003).Quantum Theory of Many-Particle Systems, Dover.Fisher, D. S. & Hohenberg, P. C. (1988). Dilute Bose gas in twodimensions,Phys. Rev. B37: 4936.Fisher, M., Weichman, P., Grinstein, G. & Fisher, D. (1989).Boson localization and the superfluid-

insulator transition,Phys. Rev. B40: 546.Fisher, T. A., Afshar, A. M., Skolnick, M. S., Whittaker, D. M. & Roberts, J. S. (1996). Vacuum rabi

coupling enhancement and zeeman splitting in semiconductor quantum microcavity structuresin a high magnetic field,Phys. Rev. B53: R10469.

Fisher, T. A., Afshar, A. M., Whittaker, D. M., Roberts, J. S., Hill, G. & Pate, M. A. (1995). Electric-field and temperature tuning of exciton-photon coupling in quantum microcavity structures,Phys. Rev. B51: 2600.

Frateschi, N. C., Kanjamala, A. P., Levi, A. F. J. & Tanbun-Ek, T. (1995). Polarization of lasingemission in microdisk laser diodes,Appl. Phys. Lett.66: 1859.

Freixanet, T., B.Sermage, Bloch, J., Marzin, J. Y. & Planel,R. (1999). Annular resonant rayleighscattering in the picosecond dynamics of cavity polaritons, Phys. Rev. B60: R8509.

Frenkel, J. (1931). On the transformation of light into heatin solids. I,Phys. Rev.37: 17.Frenkel, Y. I. (1936). On the solid body model of heavy nuclei, Phys. Z. Soviet Union9: 158.Frohlich, H. (1937). Theory of electrical breakdown in ionic crystals,Proc. Roy. Soc. A160: 230.Fujita, M. & Baba, T. (2002). Microgear laser,Appl. Phys. Lett.80: 2051.Gahl, A., Balle, S. & Miguel, M. (1999). Polarization dynamics of optically pumped VCSELs,IEEE

Journal of Quantum Electronics35: 342.Gardiner, C. W. & Zoller, P. (1997). Quantum kinetic theory:A quantum kinetic master equation

for condensation of a weakly interacting Bose gas without a trapping potential,Phys. Rev. A55: 2902.

Gerace, D., Galli, M., Bajoni, D., Guizzetti, G., Andreani,L. C., Riboli, F., Melchiorri, M., Dal-dosso, N., Pavesi, L., Pucker, G., Cabrini, S., Businaro, L.& Fabrizio, E. D. (2005). Wide-band transmittance of one-dimensional photonic crystals carved in si3n4/sio2 channel waveg-uides,Appl. Phys. Lett.87: 211116.

Gibbs, H. M. (1985). Optical bistability: Controlling light with light, Academic, Orlando, Fla..Gil, B. & Kavokin, A. V. (2002). Giant exciton-light coupling in zno quantum dots,Appl. Phys.

Lett.81: 748.Gippius, N. A., Tikhodeev, S. G., Kulakovskii, V. D., Krizhanovskii, D. N. & Tartakovskii, A. I.

(2004). Nonlinear dynamics of polariton scattering in semiconductor microcavity: Bistabilityvs. stimulated scattering,Europhys. Lett.67: 997.

Glauber, R. J. (1963). Photon correlations,Phys. Rev. Lett.10: 84.Goldstone, J. (1961). Spontaneous symmetry breaking,Nuovo Cimento19: 154.Goldstone, J., Salam, A. & Weinberg, S. (1962). Broken symetries,Phys. Rev.127: 965.Gor’kov, L. & Dzialoshinskii, I. (1967). On the theory of mott excitons in high magnetic fields,

406 REFERENCES

Journal of Experimental Theoretical Physics53: 717.Goy, P., Raimond, J. M., Gross, M. & Haroche, S. (1983). Observation of cavity-enhanced single-

atom spontaneous emission,Phys. Rev. Lett.50: 1903.Griffin, A., Snoke, D. & Stringari, S. (eds) (1995).Bose Einstein Condensation, Cambridge Univer-

sity Press.Gripp, J., Mielke, S. L. & Orozco, L. A. (1997). Evolution of the vacuum rabi peaks in a detuned

atom-cavity system,Phys. Rev. A56: 3262.Gross, E. F., Zakharchenia, B. P. & Reinov, N. M. (1956). Linear and quadratic Zeeman effect and

the diamagnetism of excitons in CuO2 crystals,Doklady of the Academy of Sciences of USSR111: 564.

Gurioli, M., Borri, P., Colocci, M., Gulia, M., Rossi, F., Molinari, E., Selbmann, P. E. & Lugli, P.(1998). Exciton formation and relaxation in GaAs epilayers, Phys. Rev. B58: R13403.

Gerard, J. M., Barrier, D., Marzin, J. Y., Kuszelewicz, R.,Manin, L., Costard, E., Thierry-Mieg, V.& Rivera, T. (1996). Quantum boxes as active probes for photonic microstructures: The pillarmicrocavity case,Appl. Phys. Lett.69: 449.

Gerard, J.-M., Sermage, B., Gayral, B., Legrand, B., Costard, E. & Thierry-Mieg, V. (1998). En-hanced spontaneous emission by quantum boxes in a monolithic optical microcavity,Phys.Rev. Lett.81: 1110.

Hachair, X., Barland, S., Furfaro, L., Giudici, M., Balle, S., Tredicce, J. R., Brambilla, M., Mag-gipinto, T., Perrini, I. M., Tissoni, G. & Lugiato, L. (2004). Cavity solitons in broad-areavertical-cavity surface-emitting lasers below threshold, Phys. Rev. A69: 43817.

Hall, R. N., Fenner, G. E., Kingsley, J. D., Soltys, T. J. & Carlson, R. O. (1962). Coherent lightemission from GaAs junctions,Phys. Rev. Lett..

Hanbury Brown, R. & Twiss, R. (1958). Interferometry of the intensity fluctuations in light. III.Applications to astronomy,Proc. Roy. Soc.248: 199.

Hanbury Brown, R. & Twiss, R. Q. (1956). A test of a new type of stellar interferometer on sirius,nature,Nature178: 1046.

Hanle, W. (1924). Uber magnetische beeinflussung der polarisation der resonanzfluoreszenz,Z.Physik30: 93.

Harel, R., Cohen, E., Linder, E., Ron, A. & Pfeiffer, L. N. (1996). Absolute transmission, reflection,and absorption studies in GaAs/AlAs quantum wells containing a photoexcited electron gas,Phys. Rev. B53: 7868.

Haug, H. & Hanamura, E. (1975). Derivation of the two-fluid model for Bose-condensed excitons,Phys. Rev. B11: 3317.

Haug, H. & Koch, S. W. (1990).Quantum Theory of the Optical and Electronic Properties ofSemiconductors, World Scientific.

Heinzen, D. J., Childs, J. J., Thomas, J. E. & Feld, M. S. (1987). Enhanced and inhibited visiblespontaneous emission by atoms in a confocal resonator,Phys. Rev. Lett.58: 1320.

Hennrich, M., Kuhn, A. & Rempe, G. (2005). Transition from antibunching to bunching in cavityqed,Phys. Rev. Lett.94: 053604.

Hohenberg, P. (1967). Existence of long-range order in one and two dimensions,Phys. Rev.158: 383.

Holonyak, N. & Bevacqua, S. F. (1962). Coherent (visible) light emission from Ga(As1−xPx)junctions,Appl. Phys. Lett.1: 80.

Hoogland, S., Baumberg, J. J., Coyle, S., Baggett, J., Coles, M. J. & Coles, H. J. (2002).Self-organized patterns and spatial solitons in liquid-crystal microcavities,Phys. Rev. A66: 055801.

Hopfield, J. J. (1958). Theory of the contribution of excitons to the complex dielectric constant of

REFERENCES 407

crystals,Phys. Rev.112: 1555.Houdre, R., Weisbuch, C., Stanley, R. P., Oesterle, U., Pellandin, P. & Ilegems, M. (1994). Mea-

surement of cavity-polariton dispersion curve from angle-resolved photoluminescence exper-iments,Phys. Rev. Lett.73: 2043.

Hubner, M., Kuhl, J., Stroucken, T., Knorr, A., S.W. Koch, S., Hey, R. & Ploog, K. (1996). Collectiveeffects of excitons in multiple-quantum-well Bragg and anti-Bragg structures,Phys. Rev. Lett.76: 4199.

Imamo glu, A. (2002). Cavity-QED using quantum dots,Optics & Photonics Newsp. 25.Imamo glu, A. & Ram, R. J. (1996). Quantum dynamics of exciton lasers,Physics Letter A214: 193.Imamo glu, A., Ram, R. J., Pau, S. & Yamamoto, Y. (1996). Nonequilibrium condensates and lasers

without inversion: Exciton-polariton lasers,Phys. Rev. A53: 4250.Inguscio, M., Stringari, S. & Wieman, C. E. (eds) (1999).Bose-Einstein Condensation in Atomic

Gases, Ios Pr Inc.Ivanov, A. L., Borri, P., Langbein, W. & Woggon, U. (2004). Radiative corrections to the excitonic

molecule state in GaAs microcavities,Phys. Rev. B69: 075312.Ivchenko, E. L. (1992). Exchange interaction and scattering of light with reversal of the hole angular

momentum at an acceptor in quantum-well structures,Sov. Phys. Solid State34: 254.Ivchenko, E. L. (2005).Optical Spectroscopy of Semiconductor Nanostructures, Alpha Science.Ivchenko, E. L., Nesvizhskii, A. I. & Jorda, S. (1994). Resonant Bragg reflection from quantum-well

structures,Superlatt. & Microstruct.16: 17.Ivchenko, E. L. & Pikus, G. E. (1997).Superlattices and other heterostructures. Symmetry and

Optical Phenomena, Springer, Berlin.Jackson, J. D. (1975).Classical Electrodynamics, 2nd edn, Wiley.Jahnke, F., Kira, M., Koch, S. W., Khitrova, G., Lindmark, E.K., Nelson, T. R., Wick, J. D. V.,

Berger, J. D., Lyngnes, O., Gibbs, H. M. & Tai, K. (1996). Excitonic nonlinearities of semi-conductor microcavities in the nonperturbative regime,Phys. Rev. Lett.77: 5257.

Jaksch, D., Gardiner, C. W. & Zoller, P. (1997). Quantum kinetic theory. II. Simulation of thequantum boltzmann master equation,Phys. Rev. A56: 575.

Jaynes, E. & Cummings, F. (1963). Comparison of quantum and semiclassical radiation theory withapplication to the beam maser,Proc. IEEE51: 89.

Jhe, W., Hinds, E. A., Meschede, D., Moi, L. & Haroche, S. (1987). Suppression of spontaneousdecay at optical frequencies: Test of vacuum-field anisotropy in confined space,Phys. Rev.Lett.58: 666.

Joannopoulos, J. D., Meade, R. D. & Winn, J. N. (1995).Photonic Crystals: Molding the Flow ofLight, Princeton University Press.

Kaliteevski, M. A., Brand, S., Abram, R. A., Nikolaev, V. V.,Maximov, M. V., Torres, C. M. S. &Kavokin, A. V. (2001). Electromagnetic theory of the coupling of zero-dimensional excitonand photon states: A quantum dot in a spherical microcavity,Phys. Rev. B64.

Kasprzak, J., Richard, M., Kundermann, S., Baas, A., Jeambrun, P., Keeling, J. M. J., Marchetti,F. M., Szymanska, M. H., Andre, R., Staehli, J. L., Savona, V., Littlewood, P. B., Deveaud, B.& Dang, L. S. (2006). Bose-Einstein condensation of excitonpolaritons,Nature443: 409.

Kato, Y., Myers, R., Gossard, A. & Awschalom, D. (2004). Observation of the spin hall effect insemiconductors,Science306: 1910.

Kavokin, A. & Malpuech, G. (2003).Cavity Polaritons, Vol. 32 of Thin Films and Nanostructures,Elsevier.

Kavokin, A., Malpuech, G. & Glazov, M. (2005). Optical spin hall effect, Phys. Rev. Lett.95: 136601.

408 REFERENCES

Kavokin, A., P.G.Lagoudakis, Malpuech, G. & J.J.Baumberg (2003). Polarization rotation in para-metric scattering of polaritons in semiconductor microcavities,Phys. Rev. B67: 195321.

Kavokin, A. V. (1997). Motional narrowing of inhomogeneously broadened excitons in a semicon-ductor microcavity: Semiclassical treatment,Phys. Rev. B57: 3757.

Kavokin, A. V. & Kaliteevski, M. A. (1995). Excitonic light reflection and absorption in semicon-ductor microcavities at oblique incidence,Solid State Commun.95: 859.

Kavokin, A. V., Malpuech, G. & Langbein, W. (2001). Theory ofpropagation and scattering ofexciton-polaritons in quantum wells,Solid State Commun.120: 259.

Kavokin, K., Renucci, P., Amand, T., Marie, X., Senellart, P., Bloch, J. & Sermage, B. (2005). Linearpolarisation inversion: A signature of coulomb scatteringof cavity polaritons with oppositespins,Phys. Stat. Sol. C2: 763.

Kavokin, K., Shelykh, I., Kavokin, A., Malpuech, G. & Bigenwald, P. (2004a). Polarization controlof the nonlinear emission of semiconductor microcavities,Phys. Rev. Lett.89: 077402.

Kavokin, K. V., Shelykh, I. A., Kavokin, A. V., Malpuech, G. &Bigenwald, P. (2004b). Quantumtheory of spin dynamics of exciton-polaritons in microcavities,Phys. Rev. Lett.92: 017401.

Keeling, J. (2006). Response functions and superfluid density in a weakly interacting bose gas withnonquadratic dispersion,Phys. Rev. B74: 155325.

Keeling, J., Eastham, P. R., Szymanska, M. H. & Littlewood, P. B. (2004). Polariton condensationwith localized excitons and propagating photons,Phys. Rev. Lett.93: 226403.

Keeling, J., Eastham, P. R., Szymanska, M. H. & Littlewood, P. B. (2005). Bcs-bec crossover in asystem of microcavity polaritons,Phys. Rev. B72: 115320.

Keldysh, L. V. & Kozlov (1968). Collective properties of excitons in semiconductors,Sov. Phys.Sol. JETP36: 1193.

Khalatnikov, K. M. (1965).An Introduction to the Theory of Superfluidity, New York: Benjamin.Khitrova, G., Gibbs, H. M., Jahnke, F., Kira, M. & Koch, S. W. (1999). Nonlinear optics of normal-

mode-coupling semiconductor microcavities,Rev. Mod. Phys71: 1591.Kimble, H. J., Dagenais, M. & Mandel, L. (1977). Photon antibunching in resonance fluorescence,

Phys. Rev. Lett.39: 691.Kira, M., Jahnke, F., Koch, S. W., Berger, J., Wick, D., Nelson Jr., T. R., Khitrova, G. & Gibbs, H. M.

(1997). Quantum theory of nonlinear semiconductor microcavity luminescence explaining“Boser” experiments,Phys. Rev. Lett.79: 5170.

Kiselev, V. A., Uraltsev, I. N. & Razbirin, B. S. (1973). (in russian),JETP Lett.18: 296.Kleppner, D. (1981). Inhibited spontaneous emission,Phys. Rev. Lett.47: 233.Koinov, Z. G. (2000). Bose-Einstein condensation of excitons in a single quantum well,Phys. Rev.

B 61: 8411.Kosterlitz, J. M. & Thouless, D. J. (1973). Ordering, metastability and phase transition in twodi-

mensional systems,J. Phys. C6: 1181.Krizhanovskii, D. N., Sanvitto, D., Shelykh, I. A., Glazov,M. M., Malpuech, G., Solnyshkov, D. D.,

Kavokin, A., Ceccarelli, S., Skolnick, M. S. & Roberts, J. S.(2006). Rotation of the plane ofpolarization of light in a semiconductor microcavity,Phys. Rev. B73: 073303.

Kroemer, H. (1963). A proposed class of hetero-junction injection lasers,Proc. IEEE51: 1782.Kronenberger, A. & Pringsheim, P. (1926).Uber das Absorptionsspektrum des festen Benzols bei -

180, Z. Phys. B40: 75.Kundermann, S., Saba, M., Ciuti, C., Guillet, T., Oesterle,U., Staehli, J. L. & Deveaud, B. (2003).

Coherent control of polariton parametric scattering in semiconductor microcavities,Phys.Rev. Lett.91: 107402.

Lagoudakis, P. G., Martin, M. D., Baumberg, J. J., Qarry, A.,Cohen, E. & Pfeiffer, L. N. (2003).Electron-polariton scattering in semiconductor microcavities,Phys. Rev. Lett.90: 206401.

REFERENCES 409

Lagoudakis, P. G., Savvidis, P. G., Baumberg, J. J., Whittaker, D. M., Eastham, P. R., Skolnick,M. S. & Roberts, J. S. (2002). Stimulated spin dynamics of polaritons in semiconductormicrocavities,Phys. Rev. B65: 161310R.

Lamb Jr., W. E. (1964). Theory of an optical maser,Phys. Rev.134: A1429.Landau, L. D. (1941). The theory of superfluidity of helium II, J. Phys. USSR5: 71.Landau, L. D. (1947). The theory of superfluidity of helium II, J. Phys. USSR11: 87.Langbein, W. & Hvam, J. M. (2002). Elastic scattering dynamics of cavity polaritons: Evidence for

time-energy uncertainty and polariton localization,Phys. Rev. Lett.88: 47401.Laussy, F. P., Glazov, M. M., Kavokin, A. V. & Malpuech, G. (2005). Multiplets in the optical

emission spectra of Dicke states of quantum dots excitons coupled to microcavity photons,Phys. Stat. Sol. C2: 3819.

Laussy, F. P., Malpuech, G. & Kavokin, A. (2004). Spontaneous coherence buildup in a polaritonlaser,Phys. Stat. Sol. C1: 1339.

Laussy, F. P., Malpuech, G., Kavokin, A. & Bigenwald, P. (2004a). Spontaneous coherence buildupin a polariton laser,Phys. Rev. Lett.93: 016402.

Laussy, F. P., Malpuech, G., Kavokin, A. V. & Bigenwald, P. (2004b). Coherence dynamics inmicrocavities and polariton lasers,J. Phys.: Condens. Matter16: S3665.

Laussy, F. P., Shelykh, I. A., Malpuech, G. & Kavokin, A. (2006). Effects of Bose-Einstein conden-sation of exciton polaritons in microcavities on the polarization of emitted light,Phys. Rev. B73: 035315.

Le Si Dang, D., Heger, D., Andre, R., Boeuf, F. & Romestain, R. (1998). Stimulation of polaritonphotoluminescence in semiconductor microcavity,Phys. Rev. Lett.81: 3920.

Lidzey, D., Fox, A. M., Rahn, M. D., Skolnick, M. S., Agranovich, V. M. & Walker, S. (2002).Experimental study of light emission from strongly coupledorganic semiconductor micro-cavities following nonresonant laser excitation,Phys. Rev. B65: 195312.

Lidzey, D. G., Bradley, D. D. C., Skolnick, M. S., Virgili, T., Walker, S. & Whittaker, D. M. (1998).Strong exciton-photon coupling in an organic semiconductor microcavity,Nature395: 53.

Lifshitz, L. & Pitaevskii, L. (1980).Statistical Physics II, Pergamon Press.London, F. (1938). The l-phenomenon of liquid helium and theBose-Einstein degeneracy,Nature ,

74 (1938)141: 74.Loudon, R. (2000).The Quantum Theory of Light, 3rd edn, Oxford Science Publications.Louisell, W. H., Yariv, A. & Siegman, A. E. (1961). Quantum fluctuations and noise in parametric

processes. i.,Phys. Rev.124: 1646.Lozovik, Y. E., Bergmann, O. L. & Panfinov, A. A. (1998). The excitation spectra and superfluidity

of the electron-hole system in coupled quantum wells,Phys. Stat. Sol. B209: 287.Lozovik, Y. E. & Ovchinnikov, I. V. (2002). Many-photon coherence of Bose-condensed excitons:

Luminescence and related nonlinear optical phenomena,Phys. Rev. B66: 075124.Lozovik, Y. E. & Yudson, V. (1975). On the possibility of superfluidity of spatially separated elec-

trons and holes due to their coupling: new mechanism of superfluidity, Pis’ma Zh.Eksp. Teor.Fiz. 22: 26. [JETP Lett. 22, 26 (1975)].

Lozovik, Y. E. & Yudson, V. (1976a). New mechanism of superconductivity: coupling betweenspatially separated electrons and holes,Zh.Eksp. Teor. Fiz.71: 738. [Sov. Phys. JETP 44, 389(1976)].

Lozovik, Y. E. & Yudson, V. (1976b). Superconductivity at dielectric pairing of spatially separatedquasiparticles,Solid State Commun.19: 391.

Lu, Q.-Y., Chen, X.-H., Guo, W.-H., Yu, L.-J., Huang, Y.-Z.,Wang, J. & Luo, Y. (2004). Mode char-acteristics of semiconductor equilateral triangle microcavities with side length of 5-20µm,IEEE Phot. Tech. Lett.16: 359.

410 REFERENCES

Maialle, M. Z., de Andrada de Silva, E. A. & Sham, L. J. (1993).Exciton spin dynamics in quantumwells,Phys. Rev. B47: 15776.

Malpuech, G., Di Carlo, A., Kavokin, A., Baumberg, J. J., Zamfirescu, M. & Lugli, P. (2002). Room-temperature polariton lasers based on gan microcavities,Appl. Phys. Lett.81: 412.

Malpuech, G. & Kavokin, A. (2001). Picosecond beats in coherent optical spectra of semiconductorheterostructures: photonic bloch and exciton-polariton oscillations,Semicond. Sci. Technol.16: R1.

Malpuech, G., Kavokin, A., Di Carlo, A. & Baumberg, J. J. (2002). Polariton lasing by exciton-electron scattering in semiconductor microcavities,Phys. Rev. B65: 153310.

Mandel, L., Sudarshan, E. C. G. & Wolf, E. (1964). Theory of photoelectric detection of lightfluctuations,Proc. Phys. Soc.84: 435.

Mandel, L. & Wolf, E. (1961). Correlation in the fluctuating outputs from two square-law detectorsilluminated by light of any state of coherence and polarization, Phys. Rev.124: 1696.

Mandel, L. & Wolf, E. (1995).Optical coherence and quantum optics, Cambridge University Press.Marchetti, F. M., Keeling, J., Szymanska, M. H. & Littlewood, P. B. (2006). Thermodynamics and

excitations of condensed polaritons in disordered microcavities,Phys. Rev. Lett.96: 066405.Marchetti, F. M., Simons, B. D. & Littlewood, P. B. (2004). Condensation of cavity polaritons in a

disordered environment,Phys. Rev. B70: 155327.Martin, M., Aichmayr, G., Vina, L. & Andre, R. (2002). Polarization control of the nonlinear

emission of semiconductor microcavities,Phys. Rev. Lett.89: 77402.Maxwell, J. C. (1865). A dynamical theory of the electromagnetic field,Philosophical Transactions

of the Royal Society of London155: 459.McKeever, J., Boca, A., Boozer, A. D., Buck, J. R. & Kimble, H.J. (2003). Experimental realization

of a one-atom laser in the regime of strong coupling,Nature425: 268.Meier, B. & Zakharchenia, B. P. (eds) (1984).Optical orientation, Elsevier Science, North Holland,

Amsterdam.Mermin, N. & Wagner, H. (1966). Absence of ferromagnetism orantiferromagnetism in one or

two-dimensional isotropic heisenberg models,Phys. Rev. Lett.22: 1133.Merzbacher, E. (1998).Quantum Mechanics, 3rd edn, John Wiley & sons, Inc.Messiah, A. M. L. & Greenberg, O. W. (1964). Symmetrization postulate and its experimental

foundation,Phys. Rev.136: B248.Messin, G., Karr, J. P., Baas, A., Khitrova, G., Houdre, R.,Stanley, R. P., Oesterle, U. & Giacobino,

E. (2001). Parametric polariton amplification in semiconductor microcavities,Phys. Rev. Lett.87: 127403.

Mollow, B. R. (1969). Power spectrum of light scattered by two-level systems,Phys. Rev.188: 1969.Moskalenko, S. A. (1962). Reversible opto-hydrodynamicalphenomena in a non-ideal exciton gas,

Fiz. Tverd. Tela4: 276.Moskalenko, S. & Snoke, D. (2000).Bose Einstein Condensation of Excitons and Biexcitons, Cam-

bridge University Press.Mott, N. F. (1938). Conduction in polar crystals. II. The conduction band and ultra-violet absorption

of alkali-halide crystals,Trans. Faraday Soc.34: 500.Muller, M., Bleuse, J., Andre, R. & Ulmer-Tuffigo, H. (1999).Observation of bottleneck effects on

the photoluminescence from polaritons in ii-vi microcavities,Physica B272: 467.Nathan, M., Dumke, W. P., Burns, G., F. H. Dill, J. & Lasher, G.(1962). Stimulated emission of

radiation from GaAsp − n junctions,apl 1: 62.Negele, J. W. & Orland, H. (1998).Quantum Many-Particle Systems, Perseus.Nelson, D. R. & Kosterlitz, J. M. (1977). Universal jump in the superfluid density of two dimen-

sional superfluid,Phys. Rev. Lett.39.

REFERENCES 411

Nockel, J. U. (2001). Optical feedback and the coupling problem in semiconductor microdisk lasers,Phys. Stat. Sol. A188: 921.

Noda, S., Tomoda, K., Yamamoto, N. & Chutinan, A. (2000). Full three-dimensional photonicbandgap crystals at near-infrared wavelengths,Science289: 604.

Panzarini, G. & Andreani, L. C. (1997). Bulk polariton beatings and two-dimensional radiativedecay: Analysis of time-resolved transmission through a dispersive film,Solid State Commun.102: 505.

Panzarini, G., Andreani, L. C., Armitage, A., Baxter, D., Skolnick, M. S., Astratov, V. N., Roberts,J. S., Kavokin, A. V., Vladimirova, M. R. & Kaliteevski, M. A.(1999a). Cavity-polariton dis-persion and polarization splitting in single and coupled semiconductor microcavities,Physicsof the Solid State41: 1223.

Panzarini, G., Andreani, L. C., Armitage, A., Baxter, D., Skolnick, M. S., Astratov, V. N., Roberts,J. S., Kavokin, A. V., Vladimirova, M. R. & Kaliteevski, M. A.(1999b). Exciton-light cou-pling in single and coupled semiconductor microcavities: Polariton dispersion and polariza-tion splitting,Phys. Rev. B59: 5082.

Parascandolo, G. & Savona, V. (2005). Long-range radiativeinteraction between semiconductorquantum dots,Phys. Rev. B71: 045335.

Patrini, M., Galli, M., Marabelli, F., Agio, M., Andreani, L. C., Peyrade, D. & Chen, Y. (2002).Photonic bands in patterned silicon-on-insulator waveguides,IEEE Journal of Quantum Elec-tronics38: 885.

Pau, S., Cao, H., Jacobson, J., Bjork, G., Yamamoto, Y. &Imamo glu, A. (1996). Observation of alaserlike transition in a microcavity exciton polariton system,Phys. Rev. A54: R1789.

Pauli, W. (1940). The connection between spin and statistics,Phys. Rev.58: 716.Pekar, S. I. (1957). Theory of electromagnetic waves in a crystal in which excitons arise,J. Exp.

Teor. Fiz. USSR33: 1022.Penrose, O. & Onsager, L. (1954). Bose-Einstein condensation and liquid helium,Phys. Rev.

104: 576.Perrin, M., Senellart, P., Lemaıtre, A. & Bloch, J. (2005).Polariton relaxation in semiconductor

microcavities: Efficiency of electron-polariton scattering,Phys. Rev. B72: 075340.Peter, E., Senellart, P., Martrou, D., Lemaıtre, A., Hours, J., Gerard, J. M. & Bloch, J. (2005).

Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,Phys. Rev. Lett.95: 067401.

Pethick, C. J. & Smith, H. (2001).Bose-Einstein Condensation in Dilute Gases, Cambridge Uni-versity Press.

Pikus, G. & Bir, G. (1971). Exchange interaction in excitonsin semiconductors,Zh. Eksp. Teor. Fiz.60: 195. [Sov. Phys. JETP 33, 108 (1971)].

Pitaevskii, L. & Stringari, S. (2003).Bose-Einstein Condensation, Oxford University Press.Porras, D., Ciuti, C., Baumberg, J. J. & Tejedor, C. (2002). Polariton dynamics and Bose-Einstein

condensation in semiconductor microcavities,Phys. Rev. B66: 085304.Porras, D. & Tejedor, C. (2003). Linewidth of a polariton laser: Theoretical analysis of self-

interaction effects,Phys. Rev. B67: 161310R.Prakash, G. V., Besombes, L., Kelf, T., Baumberg, J. J., Bartlett, P. N. & Abdelsalam, M. E. (2004).

Tunable resonant optical microcavities by self-assembledtemplating,Opt. Lett.29: 1500.Prineas, J. P., Zhou, J. Y., Kuhl, J., Gibbs, H. M., Khitrova,G., Koch, S. W. & Knorr, A. (2002).

Ultrafast ac stark effect switching of the active photonic band gap from Bragg-periodic semi-conductor quantum wells,Appl. Phys. Lett.81: 4332.

Prokhorov, A. M. (1964). Quantum electronics,Nobel Lecture.Purcell, E. (1956). The question of correlation between photons in coherent light rays,Nature

412 REFERENCES

178: 1449.Purcell, E. M. (1946). Spontaneous emission probabilitiesat radio frequencies,Phys. Rev.69: 681.Purcell, E. M., Torrey, H. C. & Pound, R. V. (1946). Resonanceabsorption by nuclear magnetic

moments in a solid,Phys. Rev.69: 37.Qarry, A., Ramon, G., Rapaport, R., Cohen, E., Ron, A., Mann,A., Linder, E. & Pfeiffer, L. N.

(2003). Nonlinear emission due to electron-polariton scattering in a semiconductor micro-cavity,Phys. Rev. B67: 115320.

Que, W. (1992). Excitons in quantum dots with parabolic confinement,Phys. Rev. B45: 11036.Quist, T. M., Rediker, R. H., Keyes, R. J., Krag, W. E., Lax, B., McWhorter, A. L. & Zeigler, H. J.

(1962). Semiconductor MASER of GaAs,Appl. Phys. Lett.1: 91.Raizen, M. G., Thompson, R. J., Brecha, R. J., Kimble, H. J. & Carmichael, H. J. (1989). Normal-

mode splitting and linewidth averaging for two-state atomsin an optical cavity,Phys. Rev.Lett.63: 240.

Ramon, G., Mann, A. & Cohen, E. (2003). Theory of neutral and charged exciton scattering withelectrons in semiconductor quantum wells,Phys. Rev. B67: 045323.

Rapaport, R., Harel, R., Cohen, E., Ron, A. & Linder, E. (2000). Negatively charged quantumwell polaritons in a GaAs/AlAs microcavity: An analog of atoms in a cavity,Phys. Rev. Lett.84: 1607.

Rashba, E. I. & Gurgenishvili, G. E. (1962). Edge absorptiontheory in semiconductors,Sov. Phys.Solid State4: 759.

Rashba, E. I. & Portnoi, M. E. (1993). Anyon excitons,Phys. Rev. B70: 3315.Reithmaier, J. P., Sek, G., Loffler, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L. V., Ku-

lakovskii, V. D., Reinecker, T. L. & Forchel, A. (2004). Strong coupling in a single quantumdot–semiconductor microcavity system,Nature432: 197.

Rempe, G., Thompson, R. J., Kimble, H. J. & Lalezari, R. (1992). Measurement of ultralow lossesin an optical interferometer,Opt. Lett.17: 363.

Renucci, P., Amand, T., Marie, X., Senellart, P., Bloch, J.,Sermage, B. & Kavokin, K. (2005a). Mi-crocavity polariton spin quantum beats without a magnetic field: A manifestation of coulombexchange in dense and polarized polariton systems,Phys. Rev. B72: 075317.

Renucci, P., Amand, T., Marie, X., Senellart, P., Bloch, J.,Sermage, B. & Kavokin, K. V. (2005b).Microcavity polariton spin quantum beats without a magnetic field: A manifestation ofcoulomb exchange in dense and polarized polariton systems,Phys. Rev. B72: 075317.

Richard, M., Kasprzak, J., Andre, R., Romestain, R., Dang,L. S., Malpuech, G. & Kavokin, A.(2005). Experimental evidence for nonequilibrium bose condensation of exciton polaritons,Phys. Rev. B72: 201301R.

Richard, M., Kasprzak, J., Romestain, R., Andre, R. & Le Si Dang (2005). Spontaneous coherentphase transition of polaritons in cdte microcavities,Phys. Rev. Lett.94: 187401.

Romanelli, M., Leyder, C., Karr, J.-P., Giacobino, E. & Bramati, A. (2005). Generation of correlatedpolariton modes in a semiconductor microcavity,Phys. Stat. Sol. C2: 3924.

Rubo, Y. G., Kavokin, A. & Shelykh, I. (2006). Suppression ofsuperfluidity of exciton-polaritonsby magnetic field,Phys. Lett. A358: 227.

Rubo, Y. G., Laussy, F. P., Malpuech, G., Kavokin, A. & Bigenwald, P. (2003). Dynamical theoryof polariton amplifiers,Phys. Rev. Lett.91: 156403.

Saba, M., Ciuti, C., Bloch, J., Thierry-Mieg, V., Andre, R.,Dang, L. S., Kundermann, S., Mura, A.,Bongiovanni, G., Staehli, J. L. & Deveaud, B. (2003). High temperature ultrafast polaritonamplification in semiconductor microcavities,Nature414: 731.

Sanchez-Mondragon, J. J., Narozhny, N. B. & Eberly, J. H. (1983). Theory of spontaneous-emissionline shape in an ideal cavity,Phys. Rev. Lett.51: 550.

REFERENCES 413

Sanvitto, D., Daraei, A., Tahraoui, A., Hopkinson, M., Fry,P. W., Whittaker, D. M. & Skolnick,M. S. (2005). Observation of ultrahigh quality factor in a semiconductor microcavity,Appl.Phys. Lett.86: 191109.

Sanvitto, D., Krizhanovskii, D. N., Whittaker, D. M., Ceccarelli, S., Skolnick, M. S. & Roberts, J. S.(2006). Spatial structure and stability of the macroscopically occupied polariton state in themicrocavity optical parametric oscillator,Phys. Rev. B73: 241308(R).

Savasta, S., Stefano, O. D., Savona, V. & Langbein, W. (2005). Quantum complementarity of mi-crocavity polaritons,Phys. Rev. Lett.94: 246401.

Savona, V., Piermarocchi, C., Quattropani, A., Schwendimann, P. & Tassone, F. (1998). Opticalproperties of microcavity polaritons,Phase Transitions68: 169.

Savona, V., Quattropani, A., Andreani, L. C. & Schwendimann, P. (1995). Quantum well excitons insemiconductor microcavities: Unified treatment of weak andstrong coupling regimes,SolidState Commun.93: 733.

Savvidis, P. G., Baumberg, J. J., Porras, D., Whittaker, D. M., Skolnick, M. S. & Roberts, J. S.(2002). Ring emission and exciton-pair scattering in semiconductor microcavities,Phys. Rev.B 65: 073309.

Savvidis, P. G., Baumberg, J. J., Stevenson, R. M., Skolnick, M. S., Whittaker, D. M. & Roberts,J. S. (2000). Angle-resonant stimulated polariton amplifier, Phys. Rev. Lett.84: 1547.

Savvidis, P. G., Ciuti, C., Baumberg, J. J., Whittaker, D. M., Skolnick, M. S. & Roberts, J. S. (2001).Off-branch polaritons and multiple scattering in semiconductor microcavities,Phys. Rev. B64: 075311.

Savvidis, P. G., Connolly, L. G., Skolnick, M. S., Lidzey, D.G. & Baumberg, J. J. (2006). Ultrafastpolariton dynamics in strongly coupled zinc porphyrin microcavities at room temperature,Phys. Rev. B74: 113312.

Schumacher, B. (1995). Quantum coding,Phys. Rev. A51: 2738.Scully, M. O. (1999). Condensation of n bosons and the laser phase transition analogy,Phys. Rev.

Lett.82: 3927.Scully, M. O. & Lamb Jr., W. E. (1967). Quantum theory of an optical maser. i. general theory,

Phys. Rev.159: 208.Scully, M. O. & Zubairy, M. S. (2002).Quantum optics, Cambridge University Press.Seisyan, R., Moumanis, K., Kokhanovskii, S., Sasin, M., Kavokin, A., Gibbs, H. & Khitrova, G.

(2001). Fine structure of excitons in high quality thin quantum wells,Recent Res. Devel.Physics 2, Transworld Research Networkp. 289. Kerala, India.

Selbmann, P. E., Gulia, M., Rossi, F., Molinari, E. & Lugli, P. (1996). Coupled free-carrier andexciton relaxation in optically excited semiconductors,Phys. Rev. B54: 4660.

Senellart, P. & Bloch, J. (1999). Nonlinear emission of microcavity polaritons in the low densityregime,Phys. Rev. Lett.82: 1233.

Senellart, P., Bloch, J., Sermage, B. & Marzin, J. Y. (2000).Microcavity polariton depopulation asevidence for stimulated scattering,Phys. Rev. B62: R16263.

Shelykh, I. A., Glazov, M. M., Solnyshkov, D. D., Galkin, N. G., Kavokin, A. V. & Malpuech, G.(2005). Spin dynamics of polariton parametric amplifiers,Phys. Stat. Sol. C2: 768.

Shelykh, I., Kavokin, K., Kavokin, A., Malpuech, G., Bigenwald, P., Deng, H., Weihs, G. & Ya-mamoto, Y. (2004). Semiconductor microcavity as a spin-dependent optoelectronic device,Phys. Rev. B70: 035320.

Shen, Y. R. (1967). Quantum statistics of nonlinear optics,Phys. Rev.155: 921.Silinish, E. A. (1980).Organic molecular crystals: Their electronic states, Springer-Verlag, Berlin.Snoke, D. (2002). Spontaneous bose coherence of excitons and polaritons,Science298: 5597.Snoke, D., Denev, S., Liu, Y., Pfeiffer, L. & West, K. (2002).Long-range transport in excitonic dark

414 REFERENCES

states in coupled quantum wells,Nature418: 754.Song, J.-H., He, Y., Nurmikko, A. V., Tischler, J. & Bulovic,V. (2004). Exciton-polariton dynamics

in a transparent organic semiconductor microcavity,Phys. Rev. B69: 235330.Stevenson, R. M., Astratov, V. N., Skolnick, M. S., Whittaker, D. M., Emam-Ismail, M., Tar-

takovskii, A. I., Savvidis, P. G., Baumberg, J. J. & Roberts,J. S. (2000). Continuous waveobservation of massive polariton redistribution by stimulated scattering in semiconductor mi-crocavities,Phys. Rev. Lett.85: 3680.

Stoyanov, N. S., Ward, D. W., Feurer, T. & Nelson, K. A. (2002). Terahertz polariton propagationin patterned materials,Nature Materials1: 95.

Sudarshan, E. C. G. (1963). Equivalence of semiclassical and quantum mechanical descriptions ofstatistical light beams,Phys. Rev. Lett.10: 277.

Tartakovskii, A. I., Emam-Ismail, M., Stevenson, R. M., Skolnick, M. S., Astratov, V. N., Whittaker,D. M., Baumberg, J. J. & Roberts, J. S. (2000). Relaxation bottleneck and its suppression insemiconductor microcavities,Phys. Rev. B62: R2283.

Tartakovskii, A. I., Krizhanovskii, D. N., Malpuech, G., Emam-Ismail, M., Chernenko, A. V., Ka-vokin, A. V., Kulakovskii, V. D., Skolnick, M. S. & Roberts, J. S. (2003). Giant enhancementof polariton relaxation in semiconductor microcavities bypolariton-free carrier interaction:Experimental evidence and theory,Phys. Rev. B67: 16.

Tartakovskii, A. I., Kulakovskii, V. D., Krizhanovskii, D.N., Skolnick, M. S., Astratov, V. N., Ar-mitage, A. & Roberts, J. S. (1999). Nonlinearities in emission from the lower polariton branchof semiconductor microcavities,Phys. Rev. B60: R11293.

Tassone, F., Bassani, F. & Andreani, L. (1992). Quantum-well reflectivity and exciton-polaritondispersion,Phys. Rev. B45: 6023.

Tassone, F., Piermarocchi, C., Savona, V., Quattropani, A.& Schwendimann, P. (1997). Bottle-neck effects in the relaxation and photoluminescence of microcavity polaritons,Phys. Rev. B56: 7554.

Tassone, F. & Yamamoto, Y. (1999). Exciton-exciton scattering dynamics in a semiconductor mi-crocavity and stimulated scattering into polaritons,Phys. Rev. B59: 10830.

Tignon, J., Voisin, P., Delalande, C., Voos, M., Houdre, R., Oesterle, U. & Stanley, R. P. (1995).From fermi’s golden rule to the vacuum rabi splitting: Magnetopolaritons in a semiconductoroptical microcavity,Phys. Rev. Lett.74: 3967.

Tikhodeev, S. G. (2000). Comment on ”critical velocities inexciton superfluidity”,Phys. Rev. Lett.84: 352.

Tischler, J. R., Bradley, M. S., Bulovic?, V., Song, J. H. & Nurmikko, A. (2005). Strong coupling ina microcavity led,Phys. Rev. Lett.95: 036401.

Townes, C. H. (1964). Production of coherent radiation by atoms—and molecules,Nobel lecture.Tredicucci, A., Chen, Y., Pellegrini, V., Borger, M. & Bassani, F. (1996). Optical bistability of

semiconductor microcavities in the strong-coupling regime,Phys. Rev. A54: 3493.Tredicucci, A., Chen, Y., Pellegrini, V., Borger, M., Sorba, L., Beltram, F. & Bassani, F. (1995).

Controlled exciton-photon interaction in semiconductor bulk microcavities,Phys. Rev. Lett.75: 3906.

Torok, P., Higdon, P. D., Juskaitis, R. & Wilson, T. (1998). Optimising the image contrast of con-ventional and confocal optical microscopes imaging finite sized spherical gold scatterers,Opt.Commun.155: 335.

Uhlenbeck, G. E. & Gropper, L. (1932). The equation of state of a non-ideal einstein-bose or fermi-dirac gas,Phys. Rev.41: 79.

Unitt, D. C., Bennett, A. J., Atkinson, P., Ritchie, D. A. & Shields, A. J. (2005). Polarization controlof quantum dot single-photon sources via a dipole-dependent purcell effect,Phys. Rev. B

REFERENCES 415

72: 033318.Vahala, K. J. (2003). Optical microcavities,Nature424: 839.Vladimirova, M. R., Kavokin, A. V. & Kaliteevski, M. A. (1996). Dispersion of bulk exciton polari-

tons in a semiconductor microcavity,Phys. Rev. B54: 14566.von Neumann, J. (1932).Mathematical Foundations of Quantum Mechanics, Princeton University

Press.Vurgaftman, I., Meyer, J. R. & Ram-Mohan, L. R. (2001). Band parameters for III–V compound

semiconductors and their alloys,Journ. Appl. Phys.89: 5815.Wannier, G. H. (1937). The structure of electronic excitation levels in insulating crystals,Phys. Rev.

52: 191.Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. & Zeilinger, A. (1998). Violation of bell’s

inequality under strict einstein locality conditions,Phys. Rev. Lett.81: 5039.Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. (1992). Observation of the coupled

exciton-photon mode splitting in a semiconductor quantum microcavity, Phys. Rev. Lett.69: 3314.

Whittaker, D. M. (2001). Classical treatment of parametricprocesses in a strong-coupling planarmicrocavity,Phys. Rev. B63: 193305.

Whittaker, D. M. (2005). Effects of polariton-energy renormalization in the microcavity opticalparametric oscillator,Phys. Rev. B71: 115301.

Whittaker, D. M., Kinsler, P., Fisher, T. A., Skolnick, M. S., Armitage, A., Afshar, A. M., Sturge,M. D. & Roberts, J. S. (1996). Motional narrowing in semiconductor microcavities,Phys.Rev. Lett.77: 4792.

Widmann, F., Daudin, B., Feuillet, G., Samson, Y., Arlery, M. & Rouviere, J. L. (1997). Evidenceof 2D-3D transition during the first stages of GaN growth on AlN, MIJ-NSR2: 20.

Wood, R. & Ellett, A. (1924). Polarized resonance radiationin weak magnetic fields,Phys. Rev.24: 243.

Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. (2005). Micrometre-scale silicon electro-optic mod-ulator,Nature435: 325.

Yablonovitch, E. (1987). Inhibited spontaneous emission in solid-state physics and electronics,Phys. Rev. Lett.58: 2059.

Yablonovitch, E. (2001). Photonic crystals: Semiconductors of light,Scientific American285.Yang, C. N. (1962). Concept of off-diagonal long-range order and the quantum phases of liquid he

and of superconductors,Rev. Mod. Phys.34(694).Yariv, A. & Yeh, P. (2002).Optical Waves in Crystals: Propagation and Control of LaserRadiation,

reprint edn, Wiley-Interscience.Yoshie, T., Scherer, A., Heindrickson, J., Khitrova, G., Gibbs, H. M., Rupper, G., Ell, C., Shchekin,

O. B. & Deppe, D. G. (2004). Vacuum Rabi splitting with a single quantum dot in a photoniccrystal nanocavity,Nature432: 200.

Zagrebnov, V. A. & Bru, J. B. (2001). The bogoliubov model of weakly imperfect Bose gas,Phys.Rep.250.

Zamfirescu, M., Kavokin, A., Gil, B., Malpuech, G. & Kaliteevski, M. (2002). ZnO as a mate-rial mostly adapted for the realization of room-temperature polariton lasers,Phys. Rev. B65: R161205.

Zel’dovich, B. Y., Perelomov, A. M. & Popov, V. S. (1968). Relaxation of a quantum oscillator,Zh.Esper. Teor. Fiz55: 589.

Zou, Z., Huffaker, D. L. & Deppe, D. G. (2000). Ultralow-threshold cryogenic vertical-cavitysurface-emitting laser,IEEE Photonics Technol. Lett.12.

AUTHORS

Prof. Alexey Kavokin

Chair of Nanophysics and Photonics at the university of Southampton and Marie-CurieChair of Excellence at the University of Rome II. Coordinator of the EU Research-Training network “Physics of microcavities.” One of the world-leading theoreticiansspecialized in the optics of semiconductors and spintronics. Ioffe institute prizes forthe Best scientific work of the year in 1995 and 1999. Previously worked in Russia,France, Italy. Married with two children, his hobbies include writing (published fairytale “Saladine the Cat”, in Russian), chess, drawing.

Prof. Jeremy Baumberg

Director of NanoScience and NanoTechnology at the University of Southampton andProfessor in the Schools of Physics and of Electronics & Computer Science. Estab-lished innovator in NanoPhotonics, opening new areas for exploitation. 2004 Royal So-ciety Mullard Prize, 2004 Mott Lectureship of the Instituteof Physics and the CharlesVernon Boys Medal in 2000. Previously worked for Hitachi andIBM and recently spun-out his research into a company, Mesophotonics Ltd. Wide range of research interestsincluding ultrafast coherent control, magnetic semiconductors, ultrafast phonon prop-agation, photonic crystals, single semiconductor quantumdots, semiconductor micro-cavities, and self-assembled photonic and plasmonic nano-structures. Married with twochildren, his hobbies include playing the piano, tennis andmaking kinetic sculptures.

Dr. Guillaume Malpuech

Researcher at the Centre Nacional de la Recherche Scientifique (CNRS) in Clermont-Ferrand, France. Author of about 80 research papers on optical effects in semiconduc-tors. Among other contributions, he proposed the concept ofspinoptronic devices. Au-thor of the monograph “Cavity polaritons” (with Alexey Kavokin). Hobbies includeskiing, alpinism and hiking.

Dr. Fabrice P. Laussy

Postdoctoral researcher at the Universidad Autonoma de Madrid (Spain), known for sig-nificant contributions to the quantum theory of polariton lasers and Bose-condensationof exciton-polaritons. PhD of Universite Blaise Pascal, Clermont-Ferrand, 2005, re-search associate at the University of Sheffield (United Kingdom), 2006. Hobbies includesmoking pipe, theatre, poetry.

416

AUTHORS 417

The Authors (2006)Alexey KAVOKIN , Jeremy BAUMBERG, Guillaume MALPUECH and Fabrice LAUSSY

PREFACE - University of California, San...

Documents

Transcript of PREFACE - University of California, San...