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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