CAVITY QUANTUM CAVITY QUANTUM ELECTRODYNAMICS IN PHOTONIC ELECTRODYNAMICS IN PHOTONIC

CRYSTAL STRUCTURESCRYSTAL STRUCTURES

Photonic Crystal Doctoral CoursePhotonic Crystal Doctoral CoursePO-014PO-014

Summer Semester 2009Summer Semester 2009

Konstantinos G. LagoudakisKonstantinos G. Lagoudakis

OutlineOutline

Light matter interaction Light matter interaction Normal mode splitting Normal mode splitting Trapping light and matter in small volumesTrapping light and matter in small volumes ExperimentsExperiments

How do we describe the interaction of How do we describe the interaction of light and matter?light and matter?

We have to get an expression of the total Hamiltonian We have to get an expression of the total Hamiltonian describing the system.describing the system.

It will consist of three terms , one for the unperturbed It will consist of three terms , one for the unperturbed two level system, one for the free field, and one for two level system, one for the free field, and one for the interaction.the interaction.

† † †ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ2

ototal zH g

γ

g κ

We can calculate the eigenvalues of the We can calculate the eigenvalues of the energy before and after the interactionenergy before and after the interaction

Excited atom with Excited atom with nn photons present, or photons present, or

atom in ground state with n+1 photons present. atom in ground state with n+1 photons present. Emission of photon is reversible: Exchange of energy Emission of photon is reversible: Exchange of energy The states with which we describe the system are in the The states with which we describe the system are in the

general case:general case:

,

, 1

e n

g n

Excited state with Excited state with nn photons photons

Ground state with Ground state with nn+1 photons+1 photons

Energy level diagramEnergy level diagram

EE2n2n

EEe,ne,n

EEg,n+1g,n+1ħRħRnn

E1n

Uncoupled system Coupled system

EN

ER

GY

AX

ISE

NE

RG

Y A

XIS

RRnn is the Quantum Rabi frequency is the Quantum Rabi frequency

The effect is called Normal Mode SplittingThe effect is called Normal Mode Splitting

Energy level diagramEnergy level diagram

EE2n2n

EEe,ne,n

EEg,ng,n+1+1

E1n

Uncoupled system Coupled system

EN

ER

GY

AX

ISE

NE

RG

Y A

XIS

RRnn is the Quantum Rabi frequency is the Quantum Rabi frequency

The effect is called Normal Mode SplittingThe effect is called Normal Mode Splitting

ħδħδ≈≈00

ħδħδ<<00

ħδħδ>>00

ħħ((RRnn++δδ))

Crossing and AnticrossingCrossing and Anticrossing Uncoupled system: tuning photon energy Uncoupled system: tuning photon energy →→

crossingcrossing with energy of 2level system with energy of 2level system Strongly coupled system: Strongly coupled system: AnticrossingAnticrossing

Ene

rgy

axis

0Detuning

EEe,ne,n

EEg,ng,n+1+1

E1n

EE2n2n

ħRħRnn

How would the spectrum look like?How would the spectrum look like?

We would see two delta-like function peaks We would see two delta-like function peaks corresponding to the two new eigenenergies corresponding to the two new eigenenergies

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

-3 -2 -1 0 1 2 3

Nor

mal

ised

Tra

nsm

issi

on

E1nE2n

In reality there are lossesIn reality there are losses There is a decay rate for the excited state of the atom (There is a decay rate for the excited state of the atom (γγ)) There is a decay rate for cavity photons There is a decay rate for cavity photons (κ) (κ)

γ

g κ

We define a quantity We define a quantity ξξ as as If If ξ<1 ξ<1 weak coupling regimeweak coupling regime If If ξξ≈1 ≈1 intermediate coupling regimeintermediate coupling regime For For ξξ>>1 Strong coupling regime>>1 Strong coupling regime

2 2 24g

Realistic transmission spectrumRealistic transmission spectrum The peaks become broadened into LorentziansThe peaks become broadened into Lorentzians

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

-1 0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 0

Lossless system

Realistic system

Nor

mal

ised

Tra

nsm

issi

on

E’1n E’2n

Experimental observations of the Experimental observations of the normal mode splittingnormal mode splitting

Source: H.J.Kimble “Observation of the normal-mode splitting for atoms in optical cavity” P.R.L. 68:8 1132, (1992)

TRANSMISSION

SPECTROMETER SIDELIGHTEMISSION

Source: M. S. Feld “Normal Mode Line Shapes for Atoms in Standing-Wave Optical Resonators ” P.R.L. 77:14 2901, (1996)

Source: M. S. Feld “Normal Mode Line Shapes for Atoms in Standing-Wave Optical Resonators ” P.R.L. 77:14 2901, (1996)

Up to now we investigated the effects Up to now we investigated the effects in in atomic cavity QEDatomic cavity QED

How can we manage this by means of How can we manage this by means of solid state photonic crystals??solid state photonic crystals??

Replace atoms by QDs Replace atoms by QDs (atomic like spectra)(atomic like spectra)

Replace simple mirror cavities with PC Replace simple mirror cavities with PC cavitiescavities High Q factors and tiny mode volumesHigh Q factors and tiny mode volumes

Cavity QED in PC structuresCavity QED in PC structures

Cavity Cavity constructionconstruction

placing QDplacing QD

Source: K. Hennesy “Quantum Nature of a strongly coupled single quantum dot-cavity system ” Nature 445 896 , (2007)

Tuning exciton resonance or cavity?Tuning exciton resonance or cavity?

Two available options :Two available options :Cavity tuning by condensation of innert Cavity tuning by condensation of innert

gases on surface of PCgases on surface of PCExciton resonance tuning by varying a Exciton resonance tuning by varying a

gate voltage (when applicable)gate voltage (when applicable)

Source: K. Hennesy “Quantum Nature of a strongly coupled single quantum dot-cavity system ” Nature 445 896 , (2007)

Here the first method was appliedHere the first method was applied

Tuning exciton resonance or cavity?Tuning exciton resonance or cavity?

When tuning cavity resonant to QD exciton:When tuning cavity resonant to QD exciton:

Source: K. Hennesy “Quantum Nature of a strongly coupled single quantum dot-cavity system ” Nature 445 896 , (2007)

Anticrossing is Anticrossing is evidenced → evidenced → Signature of Signature of strong coupling strong coupling

Note the existence Note the existence of central peakof central peak

Cavity QED in PC structuresCavity QED in PC structures Complementary second order autocorrelation Complementary second order autocorrelation

measurements For the ‘trio’ of peaksmeasurements For the ‘trio’ of peaks

Source: K. Hennesy “Quantum Nature of a strongly coupled single quantum dot-cavity system ” Nature 445 896 , (2007)

Antibunching of Antibunching of emitted photons emitted photons (one photon at a (one photon at a

time)time) Reduction of X Reduction of X

lifetime lifetime

Alternate method :Tuning exciton Alternate method :Tuning exciton resonanceresonance

Changing Bias voltage Changing Bias voltage Use of quantum confined stark effectUse of quantum confined stark effect Changes exciton resonanceChanges exciton resonance

A. Laucht "Electrical control of spontaneous emission and strong coupling for a single quantum dot" NJPh 11 023034, (2009)

Alternate method :Tuning exciton Alternate method :Tuning exciton resonanceresonance

Strong couplingStrong coupling No empty cavity peak?No empty cavity peak?

A. Laucht "Electrical control of spontaneous emission and strong coupling for a single quantum dot" NJPh 11 023034, (2009)

Cavity QED in PC structuresCavity QED in PC structures

Advantages: Monolithic structuresAdvantages: Monolithic structuresPossibility of devices “photon on demand”Possibility of devices “photon on demand”Single photon gunSingle photon gunCavity QED on a chipCavity QED on a chip

SummarySummary

cavity QED suggests the appearance of effects cavity QED suggests the appearance of effects that cannot be described classicallythat cannot be described classically

they are experimentally observable in two they are experimentally observable in two fundamentally different communitiesfundamentally different communities

these effects are of great interest because they these effects are of great interest because they are direct evidence of the quantised nature of are direct evidence of the quantised nature of field in cavitiesfield in cavities