Quantum Fluids of Light

Titus Franz

12.05.2015

Contents

1 Bose-Einstein condensation of non-interacting particles 2

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . 21.3 BEC of photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Microcavity exciton-polaritons 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Photons in a microcavity . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Excitons in a seminconductor quantum well . . . . . . . . . . . . . . . . 42.4 Strong coupling between photons and excitons . . . . . . . . . . . . . . 42.5 The interaction Hamiltonian and Gross-Pitaevski Equation . . . . . . . 72.6 The driven-dissipative Gross-Pitaevski equation . . . . . . . . . . . . . . 9

3 Quantum uids of light in atomic vapour 9

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Propagation of light in a non-linear medium . . . . . . . . . . . . . . . . 9

4 Superuidity 10

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Landau criterion for superuidity . . . . . . . . . . . . . . . . . . . . . . 104.3 Excitation spectrum of an interacting BEC . . . . . . . . . . . . . . . . 114.4 Vortices in Bose-Einstein-Condensates . . . . . . . . . . . . . . . . . . . 14

1

Abstract

Bose-Einstein condensation (BEC) is an exciting quantum phase of matter withoutclassical analogue. This report presents how to achieve a BEC of photons using acavity-embedded quantum well. In addition, it is shown how the propagation of light innonlinear media like atomic vapours can simulate in a relatively simple way the Gross-Pitaevski equation which describes BEC of interacting bosons. Finally, superuidityand the generation of vortices, two prominent examples of the exotic behaviour ofBose-Einstein condensates, are portrayed.

1 Bose-Einstein condensation of non-interacting

particles

1.1 Introduction

Right from the beginning of the theoretical development of quantum mechanics physi-cists realized the signicance and diversity of new quantum phases. Already in 1924 to1925 Satyendra Nath Bose and Albert Einstein discussed the statistic of Bosons leadingto the theory of Bose-Einstein condensates (BEC) [12], an entirely novel phase of mat-ter. Bosons, i.e. particles with integer spin, may occupy the same single-particle state,as they don't obey the Pauli-principle. Therefore, it is possible, that at nite tempera-ture the ground state is macroscopically populated. This is a new state of matter withexotic behaviour like superconductivity without classical analogue [3], [18], [5], [10].

1.2 The grand canonical ensemble

Neglecting the interaction between the bosons, the system may be represented in thegrand canonical ensemble with xed temperature T and chemical potential µ. Thesystem is described by the particle number equation

N =∑k

nk =∑k

1

eβ(εk−µ) − 1(1)

where nk is the number of particles with wave vector k, β = 1kBT

is the inverse temper-

ature and εk = h2k2

2m is the free boson dispersion. Note that µ < 0, otherwise nk < 0,which is unphysical.In the thermodynamical limit, the sum is replaced by an integral

∑k → lim

V→∞V

(2π)d

∫∞0ddk.

For a critical density n = NV > nc where µ = 0, the particle number equation has no

longer a solution and the single particle mode n0 with k = 0 has to be singled out:

n = n0 +

∫k 6=0

ddk

(2π)d1

eβ(εk−µ) − 1. (2)

At this point all additional bosons accumulate in the ground state and n0 becomesmacroscopic, the system is described by a Bose-Einstein-Condensate (BEC).

2

1.3 BEC of photons

BEC has already been observed in various physical systems like atomic gases or solid-state quasiparticles. However, black-body radiation, the maybe most prominent ex-ample of Bose gases, does not show this behaviour.This is explained by the vanishing chemical potential. The photon number is not con-served when the cavity is cooled down, i.e. the photons are absorbed in the cavitywalls instead of being accumulated in the ground state mode [14].

2 Microcavity exciton-polaritons

2.1 Introduction

One way to surpass the previously described problems and to observe Bose-Einsteincondensation of photons are so called cavity-conned quantum wells. The cavity mir-rors of those systems provide a conning potential and an eective mass to the photonswhich are now equivalent to massive bosons. The semiconductor helps to achieve ther-malisation by phonon-exciton interactions [18].

2.2 Photons in a microcavity

The Fabry-Perot cavity used in this experiment is made out of two parallel Braggmirrors separated by λc. Each Bragg mirror consists of alternating layers of thicknessei = λ0

4ni, where λ0 is the wavelength of maximal reection and ni is the optical index

of layer i ∈ a, b (see gure 1). This cavity is excited with light incident under anangle θ with respect to the normal, such that only the z-component of the wave vectorkγ = (kγ‖ , k

γz ) satises the resonance condition

kγz =2πncλ0

(3)

Under these conditions for small incident angle θ this leads to an energy of the cavitywhich approximates to a parabolic dispersion

Eγ(kγ‖ =hc

nc

√(2πncλ0

)2

+(kγ‖

)2

(4)

≈ hc

λ0+h2k2‖

2m∗γ(5)

with an eective mass term m∗γ =n2chλ0c

.The photon lifetime of the microcavity is given by

τc = Leffncπ

F

c(6)

with an eective cavity length Leff = λc

(1 + 2 nanb

nb−na

)and a cavity nesse F .

3

Incident lightλ

| z |z | z opt ical cavity of length

λc = m λ2nc

layer awith thicknessea = m

λ4na

layer bwith thicknesseb = m

λ4nb

θ

Figure 1: Microcavity of length λ0 consisting of two Bragg mirrors. Both mirrors aremade of alternaying layers of thickness ea and eb. The cavity is excited withlight of wavelength λ and with wave vector kγ = (kγ‖ ,k

γz ) under an angle θ.

(Source: Own representation based on [24].)

2.3 Excitons in a seminconductor quantum well

In order to increase the coupling of the excitons in a semiconductor to light, theexcitons are conned in a 2D quantum well (see gure 2. As a result, along the zdirection the excitonic eigenenergies are quantized, while in the (x,y)-plane the excitonsshow a parabolic dispersion relation. Taking into account, that the ground state energyis well separated from higher lying states, the energy spectrum may be written as

E2Dexc(k

exc‖ ) = Eexc +

(hkexc‖

)2

2m∗exc(7)

where Eexc is the energy of the bottom of the exciton band, kexc‖ is the exciton wavevector in the (x,y)-plane and m∗exc is the exciton eective mass.

2.4 Strong coupling between photons and excitons

A strong coupling between photons and excitons is obtained by placing one or moresemiconductor quantum wells (see 2.3) in a Fabry-Perot microcavity (see 2.2). This isdepicted in gure 3 In dierence to bulk semiconductors only the parallel componentof the momentum is conserved when conned excitons couple to light [13]. In addition,the z component of the electromagnetic eld is quantized. This allows to excite a singleexciton by choosing the photon angle of incidence:

kexc = kγ =Enchc

sin θc, (8)

4

Figure 2: A GaAs-InGaAs quantum well as it is used in the experiment. The crystal isgrown in z direction and the thickness of the InGaAs layer is 8nm. (Right)The energy diagram as a function of the z-position. The excitons are trappedin the InGaAs-zone (red). (Source: [4])

where E is the photon energy and θc is photon angle of diraction inside the cavity.Neglecting the exciton-exciton energies this system of photons and excitons may bedescribed by the Hamiltonian

H =∑k

(Eexc(k)b†k bk + Eγ(k)a†kak +

hΩR2

(a†k bk + b†kak

)), (9)

where bk, b†k and ak, a

†k are the destruction and annihilation operators of an exciton

respectively a photon at wave vector k = |k‖|. ΩR is the the Rabi frequency andcorresponds to the frequency of the photon-exciton conversion. Note that the excitonsare fermionic pairs and therefore don't underlie denite statistics. However, at lowexciton density they may be described as Bosons [19].The Hamiltonian above can be diagonalized by the unitary transformation(

ρ(LP )k

ρ(UP )k

)=

(−Ck Xk

Xk Ck

)(akbk

). (10)

The eigenstates ρ(LP )k and ρ(UP )

k of the Hamiltonian are called upper respectively lowerpolaritons, their eigenenergies are given by:

E(UP ) =1

2

(Eexc(k) + Eγ(k) +

√δ2k + (hΩR)2

), (11)

E(LP ) =1

2

(Eexc(k) + Eγ(k)−

√δ2k + (hΩR)2

). (12)

5

InGaAs

GaAsGaAs

Incident lightλ

| z |z | z opt ical cavity of length

λc = mλ

2nc

layer awith thicknessea = m

λ4na

layer bwith thicknesseb = m

λ4nb

θ

| z | z Bragg mirror Bragg mirror

| z quantum well

Pola

rito

n

Figure 3: The quantum well from gure 2 is embedded in the microcavity from gure1. The microcavity of length λ0 consists of two Bragg mirrors. Both mirrorsare made of alternating layers of thickness ea and eb. The cavity is excitedwith light of wavelength λ and with wave vector kγ = (kγ‖ ,k

γz ) under an

angle θ. The quantum well is made of a layer of InGaAs between layers ofGaAs. (Source: Own representation based on [4]).

6

2.5 The interaction Hamiltonian and Gross-Pitaevski Equation

The polariton-polariton interaction is due to exciton-exciton and exciton-photon inter-actions [9]. The latter arise from Coulomb electron-hole interactions and are describedby an eective Hamiltonian proportional to b†k+q bk′−q bkbk′ . The former is due to sat-uration of the coupling between photons and the pair of electrons forming the exciton.This leads to an anharmonic term proportional to a†k+q b

†k′−q bkbk′ . The inverse of the

unitary transformation (11) is applied to the interaction Hamiltonians and all termscontaining ρ(UP ) are discarded as in the experiment only the low lying polaritons areexcited with a spectrally narrow laser. The resulting Hamiltonian is given by [8,9,23]:

H = Hlin + Hint =∑k

Ekρ†kρk +1

2

∑k′,q

V pol−polk,k′,q ρ†k−qρ†k′+qρkρk′

. (13)

By taking the Fourier-Back-Transform of the annihilation operator

ρk =1

V

∫drΨ(r)e−ik·r/h (14)

and the approximation that the potential only depends on the transferred momentumq [11]

V pol−polk,k′,q = V pol−polq =

∫drV pol−pol(r)e−ik·r/h, (15)

we nd the interaction Hamiltonian of a Bose-gas:

H =

∫dr(h2

2m∇Ψ†∇Ψ + Ψ†Vext(r)Ψ

)+

1

2

∫∫drdr′

(Ψ†(r)Ψ†(r′)V pol−pol(r − r′)Ψ(r′)Ψ(r)

). (16)

From this Hamiltonian the dynamic of the system is derived using the Heisenbergequation:

ih∂Ψ(r, t)

∂t=[Ψ(r, t), H

]=

[− h

2∇2

2m+ Vext(r, t) +

∫dr′Ψ†(r′, t)V pol−pol(r′ − r)Ψ(r′, t)

]Ψ(r, t).

(17)

The idea of deriving the Gross-Pitaevski equation is similar to the Bogoliubov ansatzthat the ground state of the Bose gas remains macroscopically populated even withinteractions. Indeed, this is fullled and a Bose-Einstein condensation has been ob-served for polaritons above some critical excitation power threshold (see gure 4) [17].We write therefore:

Ψ = a0ρ0 +∑i 6=0

aiρ0 ≈ a0ρ0 ≈ Ψ0. (18)

7

Figure 4: Observation for Bose-Einstein condensation for polaritons in the same ex-perimental set-up as in gure 3. The left panels correspond to an excitationspectrum below the excitation power threshold Pthr, the middle cone is atPthr, the right one above Pthr. a: Far-eld image of the emission cone of±23. The vertical axis shows emission intensity. Above the threshold, asharp peak forms in the centre of the cone, i.e. all polaritons are in the low-est momentum state k‖ = 0. The ground state is macroscopically populatedwhich justies the approximation in equation 18. b: The same data as in a,but resolved in energy. (Source: [17])

8

In the last step the Bogoliubov approximation was realized, i.e. the annihilation/creationoperator was replaced by its mean value. The Gross-Pitaevski equation follows now bytaking a contact-point interaction between the particles V pol−pol(r′− r) = gδ(r′− r):

ddt

Ψ0(r, t) =

[− h

2∇2

2m+ Vext(r, t) + g|Ψ0(r, t)|2

]Ψ0(r, t). (19)

2.6 The driven-dissipative Gross-Pitaevski equation

One way of injecting polaritons in the microcavity is the "(quasi-)resonant injection":The microcavity is constantly pumped with a continuous laser of frequency ωL. Thespecial case of ∆ = Elas−ELP = 0 where the energy of the laser Elas = hωL equals thepolariton energy ELP , is called resonant pumping. In this case the system is describedby the driven-dissipative Gross-Pitaevski equation:

ddt

Ψ0(r, t) =

[− h

2∇2

2m+ Vext(r, t) + g|Ψ0(r, t)|2 − i hγ

2

]Ψ0(r, t) + FL(r, t). (20)

Here a pump term FL(r, t) and a relaxation term proportional to the decay of thelower polaritons γLPk ≡ γ are added to the equation.

3 Quantum uids of light in atomic vapour

3.1 Introduction

As we have seen in the last chapter, a BEC of light can be achieved in microcavi-ties. This system can eectively be described using the Gross-Pitaevski-equation (19).However, the timescale of the system is too fast to observe the out-of-equilibrium dy-namics. In the following chapter a dierent system of light will be shown, which canbe used to simulate the time-dependent Gross-Pitaevski equation. The deduction ofthis hydrodynamical description follows the approach of [6].

3.2 Propagation of light in a non-linear medium

The Kerr χ(3) non-linearity of an optical medium can be seen as an eective interactionbetween photons. A monochromatic beam of frequency ω0 propagating in z-directionis described by the non-linear wave equation

0 = ∂2zE(r, z) +∇2

⊥E(r, t) +ω2

0

c2

(ε+ δε(r, z) + χ3)|E(r, z)|2

)E(r, z). (21)

Here E(r, z) is the amplitude of the electromagnetic eld at position (r, z) = (x, y, z),ε is the dielectric constant and δε(r, z) its variation in space which is assumed to besmall. Under this condition the paraxial approximation is justied and the second

9

derivative of E(r, z) = E(r, z)e−ik0z van be neglected. Under this approximationequation 21 takes the form of:

i∂zE(r, z) = −∇2⊥E(r, t)− k2

0

2ε

(δε(r, z) + χ3)|E(r, z)|2

)E(r, z). (22)

This equation is equivalent to the Gross-Pitaevski equation in the following sense:The time evolution in the Gross-Pitaevski equation is replaced by the propagationin space (z-direction). Therefore, a space-time mapping is needed to compare bothsystems.The spatial variation of the dielectric constant correlates to the potential term Vext(r, t)=

−k0δε(r,z)2ε

and the Kerr nonlinearity χ(3) gives rise to the photon-photon interaction constantg = −χ

(3)k0

2 .

4 Superuidity

4.1 Introduction

In the previous chapter we have seen that both systems, the exciton-polaritons in aquantum well and the uid of light in a nonlinear medium are described by the samemathematical equation and therefore equivalent. Therefore, the following discussionof properties like the excitation spectrum, superuidity or the existence of vorticesis valid for both systems. For simplicity the notation introduced in section 2 forthe microcavity is used as it resembles the standard notation of the Gross-Pitaevskiequation for Bose-gases.

4.2 Landau criterion for superuidity

Superuidity was rst observed for Helium 4 below the critical temperature of Tλ =2.17K by P. Kapitza, F.F. Allen and D. Misseneren in 1937 [1, 16]. The theoreticalexplanation was found 4 years later by Landau who gave simple criterion for whichquantum mechanics prohibits any dissipation of the moving uid [20].Landau considered a uid at low temperature moving with constant velocity v in atube. In the frame of reference moving with the uid the total energy for an excitationwith momentum p is given by E0 + ε(p). Here E0 is the non-excited uid energyand ε(p is the excitation energy. Under a Galilean transformation the energy andmomentum in the referential of the tube are given by

E′ = E0 + ε(p)− p · v +1

2Mv2. (23)

In order dissipate its energy the change of energy in the rest frame needs to be negative.Therefore we nd the condition for nite viscosity:

∆E = ε(p)− p · v < 0 (24)

⇔ v > vc =ε(p)

p. (25)

10

If the velocity is small enough that no excitation is possible, the system is superuid.This leads to the Landau criterion for superuidity:

v < vc = limp→0

ε(p)

p. (26)

4.3 Excitation spectrum of an interacting BEC

In the last chapter we have seen that it is sucient to calculate the excitation spectrumin order to determine whether the system is in the superuid or supersonic phase. Theperturbed wave function is obtained from the static wave function φ0 by a Bogoliubovtransformation:

Ψ(r, t) =[φ0(r) +Aei(l·r−ωt)B∗e−i(l·r−ωt)

]e−i

µth . (27)

Here A and B∗ are the small perturbation amplitudes. This ansatz is injected in theGross-Pitaevski equation and the terms evolving with e±ωt are isolated. The obtainedset of coupled equation is solved and the excitation spectrum is found:

hω± = ±

√(h2k2

2m

)2

+h2k2

mgn0. (28)

For small k this equation is linear and the speed of sound is given by

cs =∂ω

∂k

∣∣∣k=0

=

√gn0

m. (29)

If the velocity v = hk0

m of the condensate is slower than the speed of sound cs, thecondensate is according to the Landau criterion (equation 26) in the superuid regime.On the other hand for high velocities v > cs the condensate is able to scatter elastically,this regime is called "Cerenkov-regime". This behaviour was observed experimentally,for a wave vector gure 5.

11

Figure 5: Observation of superuidity in a cavity-embedded quantum well. I − III(IV − VI): Near-eld images corresponding to position space (far-eld im-ages corresponding to momentum space) of the excitation spot around adefect. Below the excitation power threshold Pthr (I and IV) the photonsscatter elastically at the defect which leads to an interference pattern anda parabolic wavefront in position space and a Rayleigh-ring in momentumspace. Above the threshold Pthr (III and VI) the polaritons are in the super-uid regime. The emission pattern is strongly inuenced by the polariton-polariton interactions. The incident angle is chosen such that the wave vectork‖ corresponds to a velocity below the Landau critical value, therefore noscattering can occur any more. The system shows characteristics typical forthe superuid regime: In position space the interference pattern and theparabolic wavefront disappear, in momentum space the Rayleigh scatteringring is no longer observable. II and V eventually show the onset of superu-idity. (Source: [2])

12

Figure 6: Observation of the Cherenkov-regime in a cavity-embedded quantum well.I− III (IV−VI): Near-eld images corresponding to position space (far-eldimages corresponding to momentum space) of the excitation spot arounda defect. Below the excitation power threshold Pthr (I and IV) the pho-tons scatter elastically at the defect which leads to an interference patternand a parabolic wavefront in position space and a Rayleigh-ring in momen-tum space. Above the threshold Pthr (III and VI) the polaritons are inthe superuid regime. The emission pattern is strongly inuenced by thepolariton-polariton interactions. The incident angle is chosen such that thewave vector k‖ corresponds to a velocity above the Landau critical value,therefore the system shows characteristics typical for the Cherenkov-regime:In position space the wavefront becomes linear and in momentum space thescattering ring strongly deforms. II and V show the onset of the Cherenkov-regime.(Source: [2])

13

4.4 Vortices in Bose-Einstein-Condensates

The Gross-Pitaevski equation 19 can be transformed in a hydrodynamic form usingthe Madelung transformation [21]

Ψ(r, t) =√n0(r, t)eiθ(r,t). (30)

The imaginary part of the Gross-Pitaevski equation with this ansatz is given by

∂n

∂t+∇ · (n0(r, t)v(r, t)) = 0 (31)

where v(r, t) = hm∇θ(r, t) is the velocity eld of the condensate. It follows that the

condensate is irrotational as ∇× v = hm∇×∇θ = 0.

We consider now a translational invariant BEC having shaped as a disc. The wavefunction may therefore be written as

Ψ0(r = (r, φ)) =√n0(r)e.θ(φ) (32)

which leads to a velocity:

v(r) =h

mr

∂θ

∂φuφ =

h

mrluθ. (33)

In the last step it was used that the rotational invariances requires that θ is independentof the angle φ and therefore constant. Further more this constant needs to be an integerbecause the phase θ is only dened modulo 2π. Finally, this leads to the quantizationof the velocity eld around the centre:∮

Cvds =

h

ml. (34)

The divergence of 1r in (33) is hidden in a vanishing density n0(r)

r→0−→ 0. This can beobserved in experiments, one example is shown in gure ??.In a typical dilute BEC the energy is dominated by the kinetic energy. The kineticenergy needed to excite a vortex is proportional to l2 [7]:

Elkin =

∫∫1

2mv2(r)nl(r)d2r ∝ l2.. (35)

Therefore, it is more favourable to excite N vortices of charge l = 1 rather than onevortex of charge l = N . One can further show that vortices of the same sign repel eachother and vice versa:

Epair ∝∫∫

(v1(r) + v2(r))2 d2r (36)

⇒ Epair = El1kin + El2kin + const ∗ l1l2. (37)

This repel leads to the self-organization of the vortices to Abrikosov lattices in stirredBEC [15,22].

14

Figure 7: (Left) Schematic presentation of the pumping scheme. Four laser beamsarrive on the microcavity with incidence angle θ. The wave vector k‖ in-side the cavity is set by the azimuthal angle φ. (Right) The experimentallyobtained density (near-eld image) and phase (far-eld image) maps for dif-ferent azimuthal angle φ (from top to bottom: φ = 0, 5, 5, 10, 15, 21). Theincreasing azimuthal angle corresponds to an increasing angular momentumwhich manifests itself in a higher number of vortices. Each elementary vor-tex consists of a singularity in the density map. Around the singularity thephase winds from 0 to 3π. According to 35 the existence of N vortices ofcharge l = 1 is energetically more favourable than the existence of 1 vor-tex of charge l = N . Therefore, only vortices of charge l = 1 exist in theexperiment (Source: [2]).

15

References

[1] J. F. Allen and A. D. Misner. Flow Phenomena in Liquid Helium II. Nature,142(3597):643644, 1938.

[2] Alberto Amo, Jérôme Lefrère, Simon Pigeon, Claire Adrados, Cristiano Ciuti,Iacopo Carusotto, Romuald Houdré, Elisabeth Giacobino, and Alberto Bra-mati. Superuidity of polaritons in semiconductor microcavities. Nature Physics,5(11):805810, 2009.

[3] M H Anderson, J R Ensher, M R Matthews, C E Wieman, and E A Cornell.Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science,269(5221):198201, 1995.

[4] Mauro Antezza, Pavlos Savvidis, Raaele Colombelli, Maxime Richard, ValiaVoliotis, and Alberto Bramati. Controlled vortex lattices and non-classical lightwith microcavity polaritons. 2014.

[5] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-Einstein Condensation ofLithium: Observation of Limited Condensate Number. Physical Review Letters,78(6):985989, feb 1997.

[6] Iacopo Carusotto. Superuid light in bulk nonlinear media. Proceedings of the

Royal Society of London A: Mathematical, Physical and Engineering Sciences,470(2169), 2014.

[7] F. Chevy, K. W. Madison, and J. Dalibard. Measurement of the AngularMomentum of a Rotating Bose-Einstein Condensate. Physical Review Letters,85(11):22232227, sep 2000.

[8] C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann.Role of the exchange of carriers in elastic exciton-exciton scattering in quantumwells. Physical Review B, 58(12):79267933, sep 1998.

[9] C Ciuti, P Schwendimann, and Quattropani. Theory of polariton parametric inter-actions in semiconductor microcavities. Semiconductor Science and Technology,18(10):S279S293, oct 2003.

[10] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M.Kurn, and W. Ketterle. Bose-Einstein Condensation in a Gas of Sodium Atoms.Physical Review Letters, 75(22):39693973, nov 1995.

[11] Hui Deng, Hartmut Haug, and Yoshihisa Yamamoto. Exciton-polariton Bose-Einstein condensation. Reviews of Modern Physics, 82(2):14891537, 2010.

[12] Albert Einstein. Quantentheorie des einatomigen idealen Gases, 1924.

[13] J. J. Hopeld. Theory of the Contribution of Excitons to the Complex DielectricConstant of Crystals. Physical Review, 112(5):15551567, dec 1958.

16

[14] Kerson Huang. Statistical Mechanics, 2nd Edition, 1987.

[15] A. Imamog¯lu, R. J. Ram, S. Pau, and Y. Yamamoto. Nonequilibrium conden-sates and lasers without inversion: Exciton-polariton lasers. Physical Review A,53(6):42504253, jun 1996.

[16] P. Kapitza. Viscosity of liquid helium below the l-point, 1938.

[17] J Kasprzak, M Richard, S Kundermann, A Baas, P Jeambrun, J M J Keeling,F M Marchetti, J L Staehli, V Savona, P B Littlewood, B Deveaud, Le Si Dang,R Andre, and M H Szyman. BoseâEinstein condensation of exciton polaritons.443(September):409414, 2006.

[18] Jan Klaers, Julian Schmitt, Frank Vewinger, and Martin Weitz. Bose-Einsteincondensation of photons in an optical microcavity. Nature, 468(7323):545548,2010.

[19] B Laikhtman. Are excitons really bosons? Journal of Physics: Condensed Matter,19(29):295214, jul 2007.

[20] J Landau. de Physique, UR S. S, 5:71, 1941.

[21] E. Madelung. Quantentheorie in hydrodynamischer Form. Zeitschrift f�r Physik,40(3-4):322326, mar 1927.

[22] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard. Vortex Formation ina Stirred Bose-Einstein Condensate. Physical Review Letters, 84(5):806809, jan2000.

[23] G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, andP. Schwendimann. Excitonic Bloch equations for a two-dimensional system ofinteracting excitons. Physical Review B, 61(20):1385613862, may 2000.

[24] Vera Giulia Sala. Coherence, dynamics and polarization properties of polariton

condensates in single and coupled micropillars. PhD thesis, 2013.

17