Hui DengStephan Gotzinger

David PressYoshihisa

Yamamoto

Robin HuangHui Cao

Francesco Tassone

Gregor WeihsStanley Pau

(Former members)

Quantum Entanglement Project, SORST, JSTE.L. Ginzton Laboratory, Stanford University

andNational Institute of Informatics

BaCa Tec-Summer School, Würzburg, June 26 – July 01, 2005

2

Outline

• Microcavity exciton polaritons

• Polariton BEC vs. exciton BEC

• Non-equilibrium, quasi-equilibrium and thermal equilibrium BEC

• Final state stimulation in exciton-exciton scattering processes

• Amplification of exciton polaritons

• Dynamic condensation (lasing) of exciton polaritons in CdTe and GaAs MQW-microcavities

Polariton population per state N(k//)

Effective mass

Relaxation time polariton vs. lifetime 0

Momentum and real space distributions Chemical potential and polariton temperature Second order coherence function

• Transverse confinement of exciton polaritons

2

//

2

2

1

dk

Ed

Microcavity Exciton Polaritons

4

Wannier-Mott Excitons in Quantum Wells

Momentum eigenstate – A valence electron with and is excited to a conduction electron with andk ''k

Exciton state

mode index envelope function in k-space

plane wave hole electron

momentum eigenstate

5

kKkk

vkkkk

vKv

Kk

kkkkKKvC ˆˆ)

2( ˆˆ)

2

'())'(( |0,| '

',,

Exciton creation operator

)(1],[ *0,0,

dBexcvv anOCC Composite boson in the

1st order approximation

Exciton Hamiltonian

22 )(42

1)(

2

1phexkphexk

g

002g

kkkkkavCuP ˆˆˆ Diagonalize with polariton operator:

Hamiltonian of Coupled Cavity Photon-QW Exciton

Rabi-splitting: when cavity photon on resonance with bare exciton

k

kkkTPPhH ˆˆˆ

kkkkkkkkkkph

CaCaigCCexcaahH )ˆˆˆˆ(ˆˆˆˆˆ

k kk CCexchH ˆˆˆ0

QW Excitons and Microcavity Polaritons

6

1 or 2?

1 or 2?

r’A

rB

rA

r’B

e1 r’A h1 rA

e2 r’B h2 rB

~

+ e1 r’A h2 rA

e2 r’B h1 rB

e2 r’A h1 rA

e1 r’B h2 rB

e2 r’A h2 rA

e1 r’B h1 rB

ex2 rA ex1 rB

ex1 rA ex2 rB

+ 12

=

Two-Exciton State:

Spatial Correlation induced by Coulomb Interaction.

A composite particle (exciton) behaves as a “massive boson”.

Exciton as a Composite Boson

7

Polariton dispersion curves

Rabi splitting

exc0 = ph0

E

k //

QW exciton

Lower polariton

Upper plariton

Microcavity photon (mph ~ 10-5 me)

(mexc ~ 10-1 me)

(meff ~ 2 mph)

osc~ 1 THz

C. Weisbuch, et al. Phys. Rev. Lett. 69, 3314 (1992)

S. Jiang et al., Appl. Phys. Lett. 73, 3031 (1998)

Exciton Polariton Dispersion, Normal Mode Splitting and Oscillation

UP LP

OSC 1THz ～

8

Atom Cavity QED vs. Semiconductor Cavity QED

single-atom cavity QED many-atom cavity QED exciton cavity QED

single atom ensemble of atoms

QWs

d: atomic dipole momentV: optical mode volume

eigenstate of collective angular momentum

(J =N/2, N: # of atoms)

effective # of atomic oscillators:

S: cavity mode area, ~2m : Bohr radius, ~100

Non-equilibrium Polariton Laser vs. Equilibrium Polariton BEC

10

Dynamic vs. Equilibrium Condensation

Polariton decay vs. Two relaxation processes

exciton-phonon scattering

phonon phonon

k//

equilibrium is established with a lattice at lattice

exciton-exciton scattering

k//

equilibrium is established within polaritons at polariton

Non-equilibrium(multi-mode polariton laser)

Quasi-equilibrium(single-mode polariton laser)

Thermal equilibrium(polariton BEC)

0<polaritonlattice

polariton not defined

polariton0<lattice

polariton > Tlattice

polariton<lattice 0

polariton = Tlattice

Fragmentation of the condensate

Fock exchange term

Dynamic single-state condensation

Steady state single- state condensation

polariton decay

k//

polariton decay by leakage of photonic component at 0

leakage photon

11

Polariton BEC vs. Exciton BEC

Enemies of exciton BEC:Dissociation of excitons (screening, phase space filling)Disorder, localization and inhomogeneous broadening

Advantage of Polariton BECExtended phase coherence reinforced by a cavity field

suppressed localization, disorder and inhomogeneous broadening

Light effective mass by dressing a cavity field

mpolariton ~ 10-4 mexciton ~ 10-7 mH-atom

Enhanced binding energy/decreased Bohr radius in the very-strong-coupling regime

[J. B. Khurgin et. al., Solid State Commun. 117, 307 (2002)]

suppressed dissociation of excitonsPhotonic component out-coupling from the cavity with k conservation in contrast to spontaneous decay of an un-dressed exciton

direct experimental access to internal polariton population

higher critical temperaturelower particle density

Bosonic Final State Stimulation

13

Exciton-polariton Nonlinear Interaction

NLL HHH

))(( ))(( 21 bbbbUbbbbUH NL

h.c.bbbabbba ))((2

T ))((

2

T 21

same spins opposite spins

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

U1

T1/2 T1/2

ExcitonLP UP

smaller spitting

Blue shift

, 21, nUnUh exc

• M. Kuwata-Gonokami et al., Phys. Rev. Lett. 79, 1341 (1997)• S. Schmitt-Rink, et al., Phys. Rev. B 32, 6601 (1985)• J. Fernandez-Rossier et al., Phys. Rev. B 54, 11582 (1996)• J. Inoue, et al., Phys. Rev. B 61, 2863 (2000)

)( bababbaaH exccavL

nTnT 21

14

Measurement of Exciton Interaction – Pump-probe experiments with optical heterodyne detection

Experimental results

Probe Energy

Experimental setup

Excitons with same spins

(theory)(experiment)

(Fermionic exchange + phase space filling)

(theory)(experiment)

(Fermionic exchange)

15

Idea:UP=background free measurement window

leakage from cavity phonon scattering exciton-exciton scattering

Stimulated

scattering

0 0 0, , , (1 )UPUP k LP k k LP k LP k LP

kUP

ndn a n b n n n

dt

exc. beamexc. beam

Spontaneous

scattering

16

Observation of Bosonic Final State Stimulationin exciton-exciton scattering in a GaAs SQW-Microcavity

nexc = 1.5109 cm-2

1.2

0.54

R. Huang et al., Phys. Rev. B 61, R7854 (2000)

• Upper-polariton emission decay time ~ 95 ps

• bottle-neck polariton decay time ~ 190 ps

Amplification of Exciton Polaritons —Probing Quantum Degeneracy

18

Strong Coupling to Weak Coupling Transition

Normal mode splitting at resonance (c=exc)

(weak coupling)

polaritoncavity photon

Exciton densities:A: 1.1108 cm-2 , B: 1.1109 cm-2 , C: 5.5109 cm-2 , D: 1.11010 cm-2 , E: 2.01010 cm-2 , F: 2.71010 cm-2 G:

4.41010 cm-2 , H: 6.61010 cm-2 , I: 1.11010 cm-2S. Jiang et al., Appl. Phys. Lett. 73, 3031 (1998)

19

1. Exciton localization and inhomogeneous broadening

Dressing QW excitons with a microcavity vacuum field

Strong coupling to weak coupling transition when an exciton decoherence rate exceeds a normal mode splitting.

Use of multiple QWs

Use of excitons with small Bohr radius

populationpolaritongivenafor1 QWexc

QWRabi

Nn

N

MQWsGaAs12

2*sat 1 Ban

DQWCdTe

2. Exciton saturation

QW excitons are easily trapped by a local minimum of a QW potential fluctuation.

Comparison of Exciton Properties

2510Binding Energy (meV)

2890Bohr Radius in QW(A)

504Saturation Density

(1010 cm-2)

CdTeGaAs

Obstacles and Tricks for Polariton Lasing

Rabiousinhomogene extended phase coherence

20

Gain=15

BareExcitonk// = 0:LP

Bottleneckeffect

Bottleneck Exciton decay rate = 120 ps

Gain decay rate = 60 ps

A CdTe QW exciton survives at higher densities due to small Bohr radius.

R. Huang et al., Phys. Rev. B 65, 165314 (2002)

Observation of Stimulated Scattering Gainin a CdTe DQW-Microcavity

A gain is provided by two-body exciton-exciton scattering.

21

Nexc=3.4x106 Gain =23

Nexc=1.6x106

Gain =5.4

Nexc = 0.41x106

Gain = 0.34

)exp( 2excNconstg

Probe (mW/cm2) Circles: 2104 Squares: 900

Rate equationsolutions

21exc

Nconstg Low Gain Regime High Gain Regime

R. Huang et al., Phys. Rev. B 65, 165314 (2002)

)1()1(2

lpexlplpexlpnnbnna

exciton-phononscattering

exciton-excitonscattering

lp

lp

lplp τ

nPndt

d

A. Imamoglu et al.Phys. Rev. A 53, 4250 (1996)

F. Tassone et al.,Phys. Rev. B 59, 10830 (1999)

Amplification of Polaritonsin a CdTe DQW-Microcavity

22

Spontaneous build-up of ground state populationPolariton effective massSpontaneous spin polarizationSecond order coherenceReal space distribution (spontaneous localization)Momentum space distribution (BE) (chemical potential and temperature)

Experimental evidence:

Dynamic Condensation (Lasing) of Exciton Polaritons

23

Polariton Lasing vs. Photon Lasing

109 1010 1011 1012

10-1

100

101

102

103

with inversion

Fig. 2

ph

oto

ns

pe

r ca

vity

mo

de

at

k||~

0

polariton, Elp=1.6166 eV

cavity mode, Ecav

=1.6477 eV

injected carrier density (cm-2)

po

lari

ton

s p

er

mo

de

at

k||~

0

100

no electronic inversion

lasing threshold observed without electronic population inversion

H. Deng et al., Proc. Natl. Acad. Sci., 100, 15318 (2003)

polariton laser

standard photonlaser

24

Effective Mass Measurement: Polariton and Photon Dispersions

-2 0 2

1.612

1.614

1.616

1.618

1.62

k|| (10 4 cm -1)

Ene

rgy

(eV

)

P/Pth

=7.6-2 0 2

1.646

1.648

1.65

1.652

1.654

k|| (10 4 cm -1)

Ene

rgy

(eV

)

photon laser P/P'

th=3

-2 0 2

1.612

1.614

1.616

1.618

1.62

k|| (10 4 cm -1)

Ene

rgy

(eV

)

P/Pth

=0.5

photon

polariton

Polariton Laser

Photon Laser

polariton mass measured to be ~ twice the photon mass

strong-coupling preserved above threshold

H. Deng et al., Proc. Natl. Acad. Sci., 100, 15318 (2003)

25

Spontaneous Spin Polarization

26

Second Order Coherence (Hanbury Brown-Twiss experiment)

21

21

2)()(

)()()()(

)2()()(

)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ)(ˆ)(

nn

jinin

tEtE

tEtEtEtEg i

single-mode coherent state

Poissonian light

single-mode thermal state

102 g

102 g

202 g

H. Deng et al., Science 298, 199 (2002)

The on-set of bosonic final state stimulation manifests itself by increased g (2)(0). A gradual decrease in g(2)(0) suggests non-standard macroscopic coherence.

27

Real Space Distribution

photon laserfitted spot size: 26 m

polariton lasersuppressed ‘expansion’

P/Pth = 1.5

polariton photon

below thresholdbroad Gaussian

above thresholdsteep central peak

28

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

k|| (104cm-1)

n (

k ||)

(a.u

.)

Momentum Space Distribution

Exp. Data

BE Fit

MB Fit

P/Pth =1.5

0 1 2 3 40

0.5

1

k|| (104cm-1)

n (k

||)

(a.u

.)

P/Pth =0.6

resolution

H. Deng et al., Proc. Natl. Acad. Sci., 100, 15318 (2003)

J ． Keeling et al., Phys. Rev. Lett. 93, 226403 (2004)

The k-distribution is in agreement with BE distribution, except for the population at k11=0.

29

Chemical Potential and Effective Temperature

• chemical potential ~ -kBT at threshold• chemical potential zero above threshold

fitted polariton temperaturefitted (normalized) chemical potential

1 1.2 1.4 1.60

20

40

60

80

100

120

140

160

P/Pth

TL

P (K

)

1 1.2 1.4 1.60

0.5

1

1.5

2

P/Pth

=

- /

(kBT

)

1 1.2 1.4 1.6

-6

-4

-2

(me

V)

Tpolariton >> Tlattice

BEC threshold

H. Deng et al., Proc. Natl. Acad. Sci., 100, 15318 (2003)

From Non-equilibrium Polariton Laser To Equilibrium Polariton BEC

31

Relaxation Rate vs. Decay Rate of Polaritons

Non-equilibrium

Equilibrium

0 0

0

R R R

R relax

R

relax

cavity photon

dN N NP t

dt

dN N N

dt

E E

32

Future Prospects

Room temperature polariton laser (GaN （ＮＴＴ） , ZnSe （Ｐａｄｅｒｂｏｒｎ） , ·····)

Practical issues:Practical issues:

Very low-threshold and very fast (~psec) coherent light source

Theoretical issues:Theoretical issues:Bogoliubov theory predicting a squeezed ground state

F. Tassone et al., Phys.Rev.B59,10830(1999)F.P. Laussy et al., Phys. Rev. Lett. 93, 016402

(2004)

0 0exp vack k kb b b

phase-locked

Transverse confinement and long polariton lifetime by a 2D photonic crystal or microdisk cavity

BEC BCS phase transition Impurity bound exciton in homogeneous bulk 　(F. M. Marchetti, et al., arXiv:cond-mat/0405295)

Acknowledgement

Atac Imamoglu, David Snoke, Jacqueline Bloch, Regis Andres, Hiromi Ezaki

Transverse confinement by a cavity trap (V~10meV for 12 GaAs MQW)

33

-3 -2 -1 0 1 2 3

-10

-5

0QW exciton energy

cavity field amplitudeat MQWs

LP energy

en

erg

y (

me

V)

x (m)

pillar size, 5 m

Optical Trapping of Microcavity Polaritons

-4 -2 0 2 4 6 8

764

766

768

770

772

774

position on sample (mm), dy~7/3 dx

wav

ele

ng

th (

nm

)

detuning by reflection measuremnt

2 4 6 81010152025304050

exc

=768.8nm

pillar size (m)

LP

UP ~ 6.8 nm

A. Forchel (Würzburg)

34

UP

LP

condensate

normalstate

thick lines: quasi-particle excitations in the condensed phase upper line: creating a quasi-particle lower line: absorbing a quasi-particle

gap4g|

Coherent light

BEC-BCS Phase Transition

M.H. Szymanska et al., Solid State Comm. 124, 103 (2002)