Dynamic condensation of exciton-polaritons · A. Wannier-Mott exciton 1491 1. Exciton optical transition 1492 2. Quantum-well exciton 1492 B. Semiconductor microcavity 1492 C. Quantum-well

Chapter 10

Dynamic condensation ofexciton-polaritons

1

Exciton-polariton Bose-Einstein condensation

Hui Deng

Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

Hartmut Haug

Institut für Theoretische Physik, Goethe Universität Frankfurt, Max-von-Laue-Street 1,D-60438 Frankfurt am Main, Germany

Yoshihisa Yamamoto

Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA;National Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;and NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198,Japan

Published 12 May 2010

In the past decade, a two-dimensional matter-light system called the microcavity exciton-polariton hasemerged as a new promising candidate of Bose-Einstein condensation BEC in solids. Many pieces ofimportant evidence of polariton BEC have been established recently in GaAs and CdTe microcavitiesat the liquid helium temperature, opening a door to rich many-body physics inaccessible inexperiments before. Technological progress also made polariton BEC at room temperaturespromising. In parallel with experimental progresses, theoretical frameworks and numericalsimulations are developed, and our understanding of the system has greatly advanced. In this article,recent experiments and corresponding theoretical pictures based on the Gross-Pitaevskii equationsand the Boltzmann kinetic simulations for a finite-size BEC of polaritons are reviewed.

DOI: 10.1103/RevModPhys.82.1489 PACS numbers: 71.35.Lk, 71.36.c, 42.50.p, 78.67.n

CONTENTS

I. Introduction 1490

II. Semiconductor Microcavity Polaritons 1491

A. Wannier-Mott exciton 1491

1. Exciton optical transition 1492

2. Quantum-well exciton 1492

B. Semiconductor microcavity 1492

C. Quantum-well microcavity polariton 1493

1. The Hamiltonian 1493

2. Polariton dispersion and effective mass 1494

3. Polariton decay 1494

4. Very strong-coupling regime 1495

D. Polaritons for quantum phase research 1495

1. Critical densities of 2D quasi-BEC and

BKT phase transitions 1495

2. Quantum phases at high excitation densities 1496

3. Experimental advantages of polaritons for

BEC study 1496

E. Experimental methodology 1497

1. Typical material systems 1497

2. Structure design considerations 1498

3. In situ tuning parameters 1498

4. Measurement techniques 1498

F. Theoretical methods 1498

1. Semiclassical Boltzmann kinetics 1498

2. Quantum kinetics 1499

3. Equilibrium and steady-state properties 1499

4. System size 1499

III. Rate Equations of Polariton Kinetics 1499

A. Polariton-phonon scattering 1500

B. Polariton-polariton scattering 1500

1. Nonlinear polariton interaction coefficients 1500

2. Polariton-polariton scattering rates 1502

C. Semiclassical Boltzmann rate equations 1502

IV. Stimulated Scattering, Amplification, and Lasing of

Exciton Polaritons 1503

A. Observation of bosonic final-state stimulation 1503

B. Polariton amplifier 1504

C. Polariton lasing versus photon lasing 1505

1. Experiment 1505

2. Numerical simulation with quasistationaryand pulsed pumping 1507

V. Thermodynamics of Polariton Condensation 1508A. Time-integrated momentum distribution 1508B. Time constants of the dynamical processes 1508C. Detuning and thermalization 1509D. Time-resolved momentum distribution 1509E. Numerical simulations 1510F. Steady-state momentum distribution 1511

VI. Angular Momentum and Polarization Kinetics 1511A. Formulation of the quasispin kinetics 1512B. Time-averaged and time-resolved experiments and

simulations 1513C. Stokes vector measurement 1515

VII. Coherence Properties 1515A. The second-order coherence function 1516

1. g20 of the LP ground state 15162. Kinetic models of the g2 15173. Master equation with gain saturation 15184. Stationary solution 1519

REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL–JUNE 2010

0034-6861/2010/822/148949 ©2010 The American Physical Society1489

http://dx.doi.org/10.1103/RevModPhys.82.1489

5. Numerical solutions of the master equation 1519

6. Nonresonant two-polariton scattering

quantum depletion 1520

B. First-order temporal coherence 1521

C. First-order spatial coherence 1521

1. Long-range spatial coherence in a

condensate 1521

2. Spatial coherence among bottleneck LPs 1521

3. Spatial coherence between fragments of a

condensate 1522

4. Spatial coherence of a single-spatial mode

condensate 1522

5. Theory and simulations 1522

D. Spatial distributions 1523

1. Condensate size and critical density 1523

2. Comparison to a photon laser 1523

VIII. Polariton Superfluidity 1524

A. Gross-Pitaevskii equation 1524

B. Bogoliubov excitation spectrum 1525

C. Blueshift of the condensate mean-field energy 1525

D. Energy vs momentum dispersion relation 1526

E. Universal features of Bogoliubov excitations 1527

F. Thermal and quantum depletion 1528

G. Quantized vortices 1528

IX. Polariton Condensation in Single Traps and Periodic

Lattice Potentials 1528

A. Polariton traps 1529

1. Optical traps by metal masks 1529

2. Optical traps by fabrication 1529

3. Exciton traps by mechanical stress 1529

B. Polariton array in lattice potential 1529

1. One-dimensional periodic array of

polaritons 1530

2. Band structure, metastable state, and

stable zero-state 1531

3. Dynamics and mode competition 1532

X. Outlook 1532

Acknowledgments 1534

Appendix: Calculation of Scattering Coefficients 1534

References 1535

I. INTRODUCTION

The experimental technique of controlling spontane-ous emission of an atom by a cavity is referred to ascavity quantum electrodynamics. It has been studied fora variety of atomic systems Berman, 1994. In solids,the inhibited and enhanced spontaneous emission wasobserved for two-dimensional 2D quantum well QWexcitons in semiconductor planar microcavities Yama-moto et al., 1989; Björk et al., 1991. Due to a very strongcollective dipole interaction between QW excitons andmicrocavity photon fields, even with a relatively low-Qcavity, the planar microcavity system features a revers-ible spontaneous emission and thus normal-mode split-

ting into an upper and lower branches of polaritonsWeisbuch et al., 1992.1

The metastable state of lower polaritons at zero in-plane momentum k =0 has recently emerged as a newcandidate for observing Bose-Einstein condensationBEC in solids Imamoglu et al., 1996.2 The experimen-tal advantage of polariton BEC over conventional exci-ton BEC Hanamura and Haug, 1977 is twofold. Neark =0, a polariton has an effective mass of four orders ofmagnitude lighter than a bare exciton mass, so the criti-cal temperature for reaching polariton BEC is four or-ders of magnitude higher than that for reaching excitonBEC at the same particle density or equivalently thecritical density for reaching polariton BEC is four ordersof magnitude lower than that for reaching exciton BECat the same temperature. The second advantage of thissystem is that a polariton can easily extend a phase-coherent wave function in space through its photoniccomponent in spite of unavoidable crystal defects anddisorders, while a bare exciton is easily localized in afluctuating potential inside a crystal, which is a notoriousenemy to the exciton BEC.

Polaritons have a very short lifetime due to fast pho-ton leakage rate from a microcavity. In most cases, po-laritons have a lifetime shorter than the cooling time tothe metastable ground state, so that the system remainsin a nonequilibrium condition without a well-definedtemperature or chemical potential. However, when thecavity-photon resonance is detuned to higher than thatof the QW exciton blue detuning, the lower polaritonLP has an increased excitonic component, a longer life-time, and a shorter cooling time, facilitating thermaliza-tion. In the limit of a very large blue detuning, the dy-namical polariton laser becomes indistinguishable fromthe thermal-equilibrium exciton BEC. In a recent ex-periment, a quantum degenerate polariton gas was ob-served with a chemical potential well within the BECregime, − /kBT0.1, and at a polariton gas tempera-ture T very close to the lattice phonon reservoir tem-perature of 4 K Deng et al., 2006. The excitonic com-ponent of polaritons in this experiment is up to 80%.However, the system is still metastable and the quantumdegenerate gas exists only for several tens of picosec-onds.

The dynamical nature of a polariton condensateplaces a limitation for studying the standard BEC phys-ics, but it also provides a new experimental tool forstudying a nonequilibrium open system consisting ofhighly degenerate interacting bosons. This is not an eas-ily accessible problem by atomic BEC or superfluid 4Hesystems. One unique feature of a polariton system, com-pared to dilute atomic BEC and dense superfluid 4He, is

1Since we discuss exclusively exciton polaritons in semicon-ductors, we abbreviate exciton-polariton as polariton in thispaper.

2The metastable state of upper polaritons at k =0 in principlemay also undergo BEC. But the fast scattering from the upperto the lower branch makes such a condensation unlikely.

1490 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

the direct experimental accessibility to the quantum-statistical properties of the condensate. The main decaychannel of polaritons is the photon leakage from a cavityin which the energy and in-plane momentum are con-served between an annihilated polariton and a createdphoton. Through various quantum optical measure-ments on the leakage photon field, we can easily studythe quantum nature of the polariton condensate.

Various signatures of phase transition have been ex-perimentally established in GaAs Deng et al., 2002,2003, 2007; Balili et al., 2007; Lai et al., 2007 and CdTeRichard et al., 2005; Kasprzak et al., 2006, 2008 micro-cavities. In parallel with the experimental progress, the-oretical studies and numerical simulations are developedfor describing the thermodynamics Doan et al., 2005;Sarchi and Savona, 2007a, modified excitation spectrumSzymanska et al., 2006; Wouters and Carusotto, 2007,spin dynamics Kavokin et al., 2004; Cao et al., 2008,and particle-particle interactions Cao et al., 2008 of po-lariton condensates. Our understanding of the systemhas greatly advanced in the past few years.

In this article, we review the recent progress on ex-perimental and theoretical studies of polariton conden-sates. A comprehensive review on theoretical modelingof the polariton system can be found in Keeling et al.2007. Here we adopt the model that treats the polari-tons as weakly interacting bosons. The model is validwhen the exciton density is well below the Mott transi-tion density and is applicable to most of the experimentson polariton condensation. Since we focus on spontane-ous coherence and thermodynamic quantum phase ofpolaritons, we exclude from the current review the sub-ject of polariton parametric amplification which hasbeen reviewed by, e.g., Ciuti et al. 2003 and Keeling etal. 2007.

This article is organized as follows. Section II intro-duces microcavity polaritons, their properties relevantfor BEC research, common experimental methodolo-gies, and theoretical tools. In Sec. III, we review themicroscopic theory of nonlinear interactions among po-laritons and formulate the rate equations of isospin po-lariton kinetics. Numerical simulations based on theserate equations are used to understand the experimentsreviewed in Secs. IV, V, and VII. Section IV reviews coldcollision experiments that validate treating polaritons asa dilute Bose gas and reveal polariton-polariton scatter-ing as the microscopic driving force of polariton dynam-ics. Thermodynamics of polariton condensates, such asrelaxation and cooling of the polaritons, and temporalchanges of gas temperatures and chemical potential arepresented in Sec. V. Section VI extends the theory toinclude the spin degrees of freedom and reviews the re-lated experiments. Section VII reviews measurements ofthe coherence functions of polariton condensates, in-cluding the first- and second-order temporal coherenceand the first-order spatial coherence. Section VIII intro-duces the Gross-Pitaevskii GP equation and applies itto experiments on the ground-state energy, quantumdepletion spectra, and vortices of polariton condensates.Finally, the condensation dynamics of polaritons in a

single isolated trap and one-dimensional array of trapsare presented in Sec. IX. A minimum uncertainty wavepacket close to the Heisenberg limit and competitionbetween the s- and p-orbital superfluid states have beenobserved. We conclude with some future prospects ofthe field in Sec. X.

II. SEMICONDUCTOR MICROCAVITY POLARITONS

A “microcavity polariton” is a quasiparticle resultingfrom strong coupling between matter and light. The“matter” component is a Wannier-Mott exciton consist-ing of a bound pair of electron and hole. The “light”component is a strongly confined photon field in a semi-conductor microcavity. Half-matter, half-light microcav-ity polaritons emerged as a unique solid-state candidatefor research on phase-transition physics, as discussed inthis section.

A. Wannier-Mott exciton

A solid consists of 1023 atoms. Instead of describingthe 1023 atoms and their constituents in full detail, thecommon approaches are to treat the stable ground stateof an isolated system as a quasivacuum and to introducequasiparticles as units of elementary excitations, whichonly weakly interact with each other. An exciton is atypical example of such a quasiparticle, consisting of anelectron and a hole bound by the Coulomb interaction.The quasivacuum of a semiconductor is the state withfilled valence band and empty conduction band. Whenan electron with charge −e is excited from the valenceband into the conduction band, the vacancy it leaves inthe valence band can be described as a quasiparticlecalled a “hole.” A hole in the valence band has charge+e and an effective mass defined by −2E /p2−1, E andp denoting the energy and momentum of the hole. Ahole and an electron at p0 interact with each other viaCoulomb interaction and form a bound pair an excitonanalogous to a hydrogen atom where an electron isbound to a proton. However, due to the strong dielectricscreening in solids and a small effective-mass ratio of thehole to the electron, the binding energy of a semicon-ductor exciton is on the order of 10–100 meV and itsBohr radius is about 10–100 Å, extending over tens ofatomic sites in the crystal Hanamura and Haug, 1977.

If we define the creation operator of a conduction-band electron of momentum k as ak

† and a valence-band

hole as bk†, the exciton creation operator can be defined

as

eK,n† =

k,k

K,k+knmhk − mek

me + mhak

†bk† . 1

Here me and mh are the electron and hole mass, respec-tively; K is the center-of-mass momentum of the exciton;nk is the wave function of the relative motion of anelectron and a hole, which takes the same form as thatof an electron and a proton in a hydrogen atom; and n isthe quantum number of the relative motion. The exciton

1491Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation


operators obey the following commutation relations:

eK,n, eK,n = 0,

eK,n† , eK,n

† = 0, 2

eK,n, eK,n† = KKnn − OnexcaB

3 .

Hence excitons can be considered as bosons when theexciton interparticle spacing is much larger than its Bohrradius: nexcaB

−3.

1. Exciton optical transition

The electron and hole in an exciton form a dipole thatinteracts with electromagnetic fields of light. The inter-action strength with light at angular frequency is con-ventionally described by the exciton oscillator strength f,

f =2m*

uvr · euc 2

V

aB3 . 3

Here m*= me−1+mh

−1−1 is the effective mass of the exci-ton, uc and uv are the electron and hole Bloch func-tions, V is the quantization volume, and aB

3 is the Bohrradius of the exciton. V /aB

3 reflects the enhancement ofthe interaction due to enhanced electron-hole overlap inan exciton compared to a pair of unbound electron andhole.

2. Quantum-well exciton

A semiconductor QW is a thin layer of semiconductorwith a thickness comparable to the exciton Bohr radius,sandwiched between two barrier layers with a muchlarger band gap. The center-of-mass motion of an exci-ton in a QW is quantized along the confinement direc-tion. If only one quantized level is concerned, in mostcases the lowest-energy level, QW excitons behave astwo-dimensional quasiparticles.

The quantum confinement also modifies the valence-band structure and hence the optical transition strengthand selection rules. Most significantly, momentum con-servation in an optical transition needs to be satisfiedonly in the QW plane but not along the confinementdirection. Thus excitons in a QW couple to light with thesame in-plane wave number k and arbitrary transversewave number k. QWs are much more optically acces-sible than bulk materials.

Moreover, due to the confinement, a QW exciton hasa smaller Bohr radius and larger binding energy com-pared to a bulk exciton, leading to an enhancement ofthe oscillator strength by aB

3D/aB2D3. This enhancement

is often offset by a reduction in the overlap between thelight field and the exciton because the transverse coher-ence length of the photon field is usually much longerthan the QW thickness. Hence to achieve strongerexciton-photon coupling, it is necessary to also confinethe photon field in the z direction by introducing a mi-crocavity.

B. Semiconductor microcavity

Figure 1 shows a typical structure of a semiconductormicrocavity consisting of a c /2 cavity layer sandwichedbetween two distributed Bragg reflectors DBRs. ADBR is made of layers of alternating high and low re-fraction indices, each layer with an optical thickness of /4. Light reflected from each interface destructively in-terfere, creating a stop band for transmission. Hence theDBR acts as a high-reflectance mirror when the wave-length of the incident light is within the stop band. Whentwo such high-reflectance DBRs are attached to a layerwith an optical thickness integer times of c /2, a cavityresonance is formed at c, leading to a sharp increase ofthe transmission T at c,

T =1 − R11 − R2

1 − R1R22 + 4R1R2 sin2/2. 4

Here is the cavity round-trip phase shift of a photon atc and R1 and R2 are the reflectances of the two DBRs.The corresponding cavity quality factor Q is

Q c

cR1R21/4

1 − R1R21/2 , 5

where c is the width of the resonance. An ideal cavityhas Q= . If the cavity length is /2, Q is the averagenumber of round trips a photon travels inside the cavitybefore it escapes. Figure 2a gives an example of thereflection spectrum of a cavity with Q4000 at normalincidence. Figure 2b shows the field intensity distribu-tion Ez2 of the resonant mode. The field is concen-trated around the center of the cavity; its amplitude isenhanced by 20 times compared to the free spacevalue. Unlike in a metallic cavity, the field penetrationdepth into the DBRs is much larger. The effective cavitylength is extended in a semiconductor microcavity as

Leff = Lc + LDBR,6

LDBR c

2nc

n1n2

n1 − n2.

Here n1 and n2 are the refraction indices of the DBRlayers, as shown in Fig. 1

n1n2n1nc

n1

n2

n0=1air

substrate (top) DBR(bottom) DBR

cavity (with QW)

n1n2n1nc

n1

n2

n0=1air

substrate (top) DBR(bottom) DBR

cavity (with QW)

FIG. 1. Color online Sketch of a semiconductor /2 micro-cavity.



The planar DBR cavity confines the photon field inthe z direction but not in the x-y plane. The incidentlight from a slant angle relative to the z axis has aresonance at c / cos . As a result, the cavity has an en-ergy dispersion versus the in-plane wave number k,

Ecav =c

nc

k2 + k

2, 7

where k=nc2 /c. And there is a one-to-one corre-spondence between the incidence angle and each reso-nance mode with in-plane wave number k,

k = nc2

ctansin−1 sin

nc

2

c when k k.

8

In the region kk, we have

Ecav c

nck1 +

k2

2k2 = Ecavk = 0 +

2k2

2mcav. 9

Here the cavity-photon effective mass is

mcav =Ecavk = 0

c2/nc2 . 10

It is typically on the order of 10−5me.

C. Quantum-well microcavity polariton

1. The Hamiltonian

When the GaAs QWs are placed at the antinodes ofthe resonant field of a semiconductor microcavity, theJ=1 heavy-hole exciton doublet strongly interacts withthe confined optical field of the cavity. If the rate ofenergy exchange between the cavity field and excitonsbecomes much faster than the decay and decoherencerates of both the cavity photons and the excitons, an

excitation in the system is stored in the combined systemof photons and excitons. Thus the elementary excita-tions of the system are no longer excitons or photons buta new type of quasiparticles called the polaritons.

Using the rotating wave approximation, the linearHamiltonian of the system is written in the second quan-tization form as

Hpol = Hcav + Hexc + HI

= Ecavk,kcak

† ak+ Eexckbk

† bky

+ g0ak

† bk+ ak

bk

† . 11

Here ak

† is the photon creation operator with an in-plane wave number k and longitudinal wave number

kc=k · z determined by the cavity resonance. bk

† is theexciton creation operators with an in-plane wave num-ber k. g0 is the exciton-photon dipole interactionstrength; it is nonzero only between modes with thesame k. The above Hamiltonian can be diagonalized bythe transformations

Pk= Xk

bk+ Ck

ak, 12

Qk= − Ck

bk+ Xk

ak. 13

Then Hpol becomes

Hpol = ELPkPk

† Pk+ EUPkQk

† Qk. 14

Here Pk, Pk

† and Qk,Qk

† are the creation and anni-hilation operators of the new quasiparticles or eigen-modes of the system. They are called the lower polaritonLP and upper polariton UP, corresponding to the twobranches of lower and higher eigenenergies, respec-tively. Thus a polariton is a linear superposition of anexciton and a photon with the same in-plane wave num-ber k. Since both excitons and photons are bosons, soare the polaritons. The exciton and photon fractions ineach LP and UP are given by the amplitude squared ofXk

and Ckwhich are referred to as the Hopfield coef-

ficients Hopfield, 1958. They satisfy

Xk2 + Ck

2 = 1. 15

Let Ek=Eexck−Ecavk ,kc; Xkand Ck

are givenby

Xk2 =

121 +

EkEk2 + 4g0

2 ,

16

Ck2 =

121 −

EkEk2 + 4g0

2 .

At E=0, X2= C2= 12 , LP and UP are exactly half pho-

ton half exciton.The energies of the polaritons, which are the eigenen-

ergies of the Hamiltonian 14, are deduced from thediagonalization procedure as

700 750 800 8500

0.2

0.4

0.6

0.8

1

wavelength (nm)

reflecta

nce

0 1 2 3 42.5

3

3.5

4

z (m)

refr

action

index

n

n

|Ez|2

(a)

(b)

FIG. 2. Color online Properties of a high quality microcavity.a Reflectance of an empty /2 microcavity. b The cavitystructure and field intensity distribution Ez2 of the resonantTE mode.



ELP,UPk = 12 Eexc + Ecav ± 4g0

2 + Eexc − Ecav2 .

17

When the uncoupled exciton and photon are at reso-nance, Eexc=Ecav, LP and UP energies have the mini-mum separation EUP−ELP=2g0, which is often calledthe normal-mode splitting in analogy to the Rabi split-ting of a single-atom cavity system. Due to the couplingbetween the exciton and photon modes, the new polar-iton energies anticross when the cavity energy is tunedacross the exciton energy. This is one of the signatures of“strong coupling” Fig. 3. When Ecav−Eexcg0, polari-tons quickly become indistinguishable from a photon orexciton. Unless otherwise specified, the detuning is as-sumed to be comparable to or less than the couplingstrength in our discussions.

2. Polariton dispersion and effective mass

We define 0 as the exciton and photon energy detun-ing at k =0,

0 Ecavk = 0 − Eexck = 0 . 18

Given 0, Eq. 17 gives the polariton energy-momentum dispersions. In the region 2k

2 /2mcav2g0,the dispersions are parabolic,

ELP,UPk ELP,UP0 +2k

2

2mLP,UP. 19

The polariton effective mass is the weighted harmonicmean of the mass of its exciton and photon components,

1

mLP=

X2

mexc+

C2

mcav, 20

1

mUP=

C2

mexc+

X2

mcav, 21

where X and C are the Hopfield coefficients given byEq. 16. mexc is the effective exciton mass of its center-of-mass motion and mcav is the effective cavity-photonmass given by Eq. 10. Since mcavmexc,

mLPk 0 mcav/C2 10−4mexc,22

mUPk 0 mcav/X2.

The very small effective mass of LPs at k 0 makespossible the very high critical temperature of phase tran-sitions for the system. At large k where Ecavk−Eexckg0, dispersions of the LP and UP converge tothe exciton and photon dispersions, respectively, and LPhas an effective mass mLPmexc. Hence the LP’s effec-tive mass changes by four orders of magnitude from k

0 to large k. This peculiar shape has important impli-cations in the energy relaxation dynamics of polaritons,as discussed in Sec. IV. A few examples of the polaritondispersion with different 0 are given in Fig. 4.

3. Polariton decay

When taking into account the finite lifetime of thecavity photon and QW exciton, the eigenenergy 17 ismodified as

ELP,UPk = 12 Eexc + Ecav + icav + exc

± 4g02 + Eexc − Ecav + icav − exc2 .

23

Here cav is the out-coupling rate of a cavity photon dueto imperfect mirrors and exc is the nonradiative decayrate of an exciton. Thus the coupling strength must belarger than half of the difference in decay rates to ex-hibit anticrossing, i.e., to have polaritons as the neweigenmodes. In other words, an excitation must be able

−1 −0.5 0 0.5 1

1.58

1.585

1.59

1.595

1.6

1.605

1.61

1.615

∆/2hΩ

ener

gy(e

V)

EUP

ELP

Eexc

Ecav

2hΩ

FIG. 3. Color online Anticrossing of LP and UP energy lev-els when tuning the cavity energy across the exciton energy,calculated by the transfer-matrix method.

-5 0 5

1.62

1.64

1.66

energ

y(e

V)

dispersions

-5 0 50

0.5

1Hopfield coefficients

-5 0 5

1.62

1.64

1.66

energ

y(e

V)

-5 0 50

0.5

1

-5 0 5

1.6

1.62

1.64

energ

y(e

V)

k||

m-1

)

-5 0 50

0.5

1

|C(k||)|

2

|X(k||)|

2E

UP(k

||)

ELP

(k||)

Ecav

(k||)

Eexc

(k||)

(a)

(c)

(b)

k||

m-1

)

FIG. 4. Color online Polariton dispersions and correspondingHopfield coefficients at a =2g0, b =0, and c =−2g0.



to coherently transfer between a photon and an excitonat least once. When g0 cav−exc /2, we call the sys-tem in the strong-coupling regime. In the opposite limitwhen excitons and photons instead are the eigenmodes,the system is called to be in the weak-coupling regime,and the radiative decay rate of an exciton is given by theoptical transition matrix element. We are mostly inter-ested in microcavities with exccavg0; then Eq. 17gives a good approximation of the polariton energies.

As a linear superposition of an exciton and a photon,the lifetime of the polaritons is directly determined byexc and cav as

LP = X2exc + C2cav, 24

UP = C2exc + X2cav. 25

In typical high quality semiconductor samples availabletoday, cav=1–10 ps and exc1 ns, hence the polaritonlifetime is mainly determined by the cavity-photon life-time: LPC2cav. Polariton decays in the form of emit-ting a photon with the same k and total energy =ELP,UP. The one-to-one correspondence between theinternal polariton mode and the external out-coupledphoton mode lends convenience to experimental accessto the system see Sec. II.E.4.

4. Very strong-coupling regime

The definition of polariton operators in Eqs. 11–14follows the procedure to first diagonalize the electron-hole Hamiltonian by exciton operators and then diago-nalize the exciton-photon Hamiltonian by polariton op-erators, treating the excitons as structurelessquasiparticles. When the exciton-photon coupling is sostrong as to become comparable to exciton binding en-ergy, the question arises whether or not the above pro-cedure is still valid. A more rigorous approach is to treatthe electron, hole, and photon on equal footing, and onefinds that the LPs then consist of excitons with an evensmaller effective Bohr radius and larger binding energy,while the opposite holds for UPs. This is called the verystrong-coupling effect Khurgin, 2001; Citrin and Khur-gin, 2003. It is a consequence of sub-band mixing ofhigher-orbital excitons mixing with 1s ,2s ,2p , . . . excitonlevels.

D. Polaritons for quantum phase research

As composite bosons with not only fermionic but alsophotonic constituents, polaritons constitute a unique sys-tem for exploring both cavity QED and many-bodyphysics.

A variety of quantum phases is predicted for polari-tons, including BEC, superfluidity, Berezinskii-Kosterlitz-Thouless BKT Malpuech et al., 2003, andBardeen-Cooper-Schrieffer BCS states Keldysh andKozlov, 1968; Comte and Nozières, 1982; Malpuech etal., 2003; Littlewood et al., 2004; Sarchi and Savona,2006, 2007a. Table I compares the basic parameters ofpolaritons to semiconductor excitons and cold atoms,which consist of the major table-top Bose systems forquantum phase studies.

The parameter scales of these systems differ by manyorders of magnitude. Even for the same type of quan-tum phase, the polariton system has distinct characteris-tics and requires dedicated theoretical frameworks. Inparticular, we discuss two important fundamental issuespertaining to the ground state of a degenerate polaritongas: the possibility of BEC and BKT phases of polari-tons as two-dimensional systems and the BEC-BCScrossover or Mott transition in the high-density regimeof polaritons as a system of composite bosons. Due tothe short lifetime of polaritons compared to their ther-malization time, the excitation spectra of a polaritoncondensate differ from their atomic gas counterparts, asdiscussed in Sec. VIII.

On the experimental side, the polariton is a most ac-cessible system, especially due to the high critical tem-perature associated with its light effective mass and theone-to-one correspondence between each polariton andeach leakage cavity photon. These and some other as-pects of the experimental advantages of polaritons aresummarized later in this section.

1. Critical densities of 2D quasi-BEC and BKT phasetransitions

A uniform 2D system of bosons does not have a BECphase transition at finite temperatures in the thermody-namic limit since long-wavelength thermal fluctuationsdestroy long-range order Mermin and Wagner, 1966;Hohenberg, 1967.

If the Bose gas is confined by a spatially varying po-tential Ur Bagnato and Kleppner, 1991; Griffin et al.,

TABLE I. Parameter comparison of BEC systems.

SystemsAtomic

gases Excitons Polaritons

Effective mass m* /me 103 10−1 10−5

Bohr radius aB 10−1 Å 102 Å 102 ÅParticle spacing: n−1/d 103 Å 102 Å 1 mCritical temperature Tc 1 nK–1 K 1 mK–1 K 1–300 KThermalization time/ Lifetime 1 ms/1 s10−3 10 ps/1 ns10−2 1–10 ps / 1–10 ps=0.1–10



1995, the constant density of states DOS of the uni-form 2D system is dramatically modified by the poten-tial, and a BEC phase at finite temperature is recovered.A few experimental implementations of transverse con-finement potential in polariton systems will be discussedin Sec. IX.

Practically, any experimental system has a finite sizeand a finite number of single-particle states. With dis-crete energy levels i i=1,2 , . . . and size S=L2, thecritical condition for quasi-BEC in a finite 2D systemcan be defined as

= 1 chemical potential , 26

nc =1

S i2

1

ei/kBT − 1critical 2D density . 27

In a 2D box system of size L, the critical condition canbe fulfilled at Tc0 at a critical density,

nc =2

T2 ln L

T . 28

If the particle number N is sufficiently large and/or thetemperature is sufficiently low, the phase transitionshows similar features as a BEC phase transition definedat the thermodynamic limit Ketterle and van Druten,1996.

In addition to the quasi-BEC phase, formation of vor-tices drives another phase transition unique to the 2Dsystem—the BKT transition Berezinskii, 1971, 1972;Kosterlitz and Thouless, 1972, 1973. At low tempera-tures, thermally excited vortices may form in the systemwhich may support a local order. With further cooling ofthe system to below a critical temperature TBKT, thebound pair of oppositely circulating vortices form, whichstabilize the phase to a power-law decay versus distanceand support finite superfluid density. The critical tem-perature of the BKT transition is

kBTBKT = ns2

2m2 , 29

where ns is the superfluid density. Equation 29 can alsobe written in terms of the thermal de Broglie wavelengthT Khalatnikov, 1965,

TT 22

mkBT, 30

nsTTc2 = 4. 31

In most experiments discussed in this review, the sys-tem size is small enough such that a finite-size BECphase transition is expected before a BKT transitionKavokin et al., 2003. However, the nature of theground state and its excitation spectra, as well as thedependence of the BEC and BKT critical densities onthe system parameters, are not yet well understood atpresent. Observations of phonon modes, superfluidity,and single and bound pair of vortices is discussed in Sec.VIII. Understanding of these observations may provide

a venue to our further understanding of 2D quantumphases. For a recent review of the ground-state proper-ties of a 2D Bose gas in general, see Posazhennikova2006.

2. Quantum phases at high excitation densities

An important parameter not listed in Table I is theparticle-particle interaction strength or the scatteringlength a. For atomic gasses, a is usually much less thanthe particle spacing, satisfying the diluteness condition.In diatomic gases where each diatomic molecule consistsof two fermionic atoms, a can be defined for the inter-action between two Fermi atoms and has a hyperbolicdependence on an external magnetic field. The magneticfield tunes a from a small negative value a0, kfa1, kf is the wave vector at the Fermi surface to a verylarge negative or positive value at the unitary conditionkfa1 near the Feshbach resonance and then to asmall positive value Feshbach, 1962. Correspondingly,below a critical temperature, the system exhibits a cross-over from a degenerate Fermi gas described by the stan-dard BCS theory to a degenerate Bose gas with a BECphase.

A similar BEC to BCS crossover is also possible inexciton and polariton systems Comte and Nozières,1982; Eastham and Littlewood, 2001. Here the pairingis between an electron and a hole, and the crossover iscontrolled by the density of the electrons and holes nexc.The density nexc tunes both the interparticle spacingnexc

−1/d and, through Coulomb screening, the exciton-exciton interaction strength and exciton-photon cou-pling. The polariton gas behaves as a weakly interactingBose gas when the density is low nexcnMott, with thescattering length approximated by the exciton Bohr ra-dius aB. Increasing nexc to nMott, the BEC-BCS crossovermay take place, where degenerate Fermi gasses of elec-trons and holes are paired in the momentum space witha Bohr radius comparable to the particle spacing. Atvery high densities, we have an electron and hole plasmawith screened Coulomb interactions, which coherentlycouple to a cavity-photon field. Inhomogeneity in exci-ton energy and inelastic scattering at crystal defects,however, may suppress the BCS gap and reduce the sys-tem to an incoherent electron and hole plasma. Detailedstudies of the phase diagrams at high excitation densitiesand effects of dephasing can be found in Eastham andLittlewood 2001, Szymanska et al. 2003, and Little-wood et al. 2004.

In the current review, we focus on the BEC regime ofLPs, where the excitation density is below the excitonsaturation density and excitons and LPs can be well de-scribed as interacting bosons.

3. Experimental advantages of polaritons for BEC study

Beside as a unique system of fundamental interest,polaritons also possess many experimental advantagesfor BEC research. Most notable is the light effectivemass of polaritons in the region kkc see Sec. II.C.2..



Taking into account the excitonlike dispersion at higherk, the critical temperatures of polariton condensationrange from cryogenic temperature of a few kelvin up toroom temperature and higher.3 Not only does this sim-plify enormously the experimental setups for polaritonresearch but it also makes polaritons a natural and prac-tical candidate for quantum device applications.

A common enemy against quantum phase transitionsin solids is the unavoidable compositional and structuraldisorders. By dressing the excitons with a microcavityvacuum field, extended coherence is maintained in thecombined excitation in a microcavity polariton system,and the detrimental effects of disorders are largely sup-pressed.

Saturation and phase-space filling are another two ob-stacles against BEC of composite bosons in solids. In amicrocavity polariton system, multiple QWs can be usedto dilate the exciton density per QW for a given totalpolariton density: nexc

QW=nLP/NQW, where NQW is thenumber of QWs.

A main challenge for realizing a thermodynamicphase transition and spontaneous coherence has beenefficient cooling of hot particles. It is especially so forrelatively short-lived quasiparticles in solids. The smalleffective mass, hence small energy DOS of polaritons,comes to rescue here. Due to the small DOS, a quantumdegenerate seed of low-energy polaritons is relativelyeasy to achieve. Hence the bosonic final-state stimula-tion effect is pronounced, which accelerates energy re-laxation of the polaritons.

Finally, the cavity-photon component of a polaritoncouples out of the cavity at a fixed rate while conservingenergy and in-plane momentum. Many essential proper-ties of the polariton gas can be directly measuredthrough the leakage photon field with well-developedspectroscopic and quantum optical techniques, includingcoherence functions Deng et al., 2002, 2007; Kasprzak etal., 2006, 2008, population distributions in both coordi-nate Deng et al., 2003, 2007; Kasprzak et al., 2006 andconfiguration space Deng et al., 2003; Kasprzak et al.,2006, vortices Lagoudakis et al., 2008; Roumpos, Hoe-fling, et al., 2009, and transport properties Utsunomiyaet al., 2008; Amo et al., 2009. In comparison, compli-cated and often indirect techniques are required toprobe those properties in matter systems. Hence the po-lariton system also offers a unique opportunity forstudying new and open questions in quantum phase re-search.

E. Experimental methodology

1. Typical material systems

An optimal microcavity system for polariton BEC hasthe following: 1 high quality cavity, hence long cavityphoton and polariton lifetime; 2 large polariton-

phonon and polariton-polariton scattering cross sec-tions, hence efficient polariton thermalization; 3 smallexciton Bohr radius and large exciton binding energy,hence high exciton saturation density; and 4 strongexciton-photon coupling, hence robust strong coupling;it requires both large QW exciton oscillator strength andlarge overlap of the photon field and the QWs.

The choice of a direct band-gap semiconductor de-pends first on the fabrication technology. The best fab-rication quality of both quantum well and microcavityhas been achieved via molecular-beam epitaxy growth ofAlxGa1−xAs-based samples 0x1, thanks to theclose match of the lattice constants alat of AlAs andGaAs and a relatively large difference between theirband-gap energies Eg. At 4 K, GaAs has alat=5.64 Åand Eg=1.519 eV and AlAs has alat=5.65 Å and Eg=3.099 eV. Nearly strain and defect-free GaAs QWs arenow conventionally grown between AlxGa1−xAs. Inho-mogeneous broadening of exciton energy is limitedmainly by monolayer QW thickness fluctuation. Nearlydefect-free microcavity structures can be grown withmore than 30 pairs of AlAs/GaAs layers in the DBRsand with a cavity quality factor Q exceeding 105

Reitzenstein et al., 2007. Many signatures of polaritoncondensation were first obtained in GaAs-based systemsDeng et al., 2002, 2003, 2007; Balili et al., 2007; Lai et al.,2007.

Another popular choice is the CdTe-based II-IV sys-tem, with CdTe QWs and MgxCd1−xTe and MnxCd1−xTebarrier and DBR layers. The larger lattice mismatch iscompensated by larger binding energy and larger oscil-lator strength, as well as larger refractive index contrasthence less layers needed in the DBRs. The smallerBohr radius of CdTe excitons, on the one hand, allows alarger saturation density and, on the other hand, reducesthe polariton and acoustic phonon scattering. Hence theenergy relaxation bottleneck is more persistent in thissystem, which prevented condensation in the LP groundstate in early experiments Huang et al., 2002; Richard etal., 2005. By adjusting the detuning to facilitate ther-malization, partially localized polariton condensationinto the ground state was finally observed by Kasprzaket al. 2006.

In recent years, there has been intense interest in de-veloping certain wide-band-gap material systems withsmall lattice constants. In these materials, the excitonbinding energy and oscillator strength are large enough,so that a polariton laser may survive at 300 K, operat-ing at visible to ultraviolet wavelengths. However, fabri-cation techniques for these materials are not as matureas for GaAs- or CdTe-based systems. The structures ofwide-band-gap materials suffer a higher concentrationof impurities and crystal defects. Integration of the QWswith microcavities is challenging due to the lack of lat-tice matched DBR layers. Though the strong-couplingregime is still hard to reach or reproduce at present,there have been a few promising reports on the obser-vation of polaritons in ZnSe Pawlis et al., 2002, GaNTawara et al., 2004; Butte et al., 2006, and ZnO Shi-mada et al., 2008; Chen et al., 2009. Lasing of GaN po-

3Note that other constrains on the condensation temperatureTc apply. For example, kBTc must be smaller than the excitonbinding energy or half of the LP-UP splitting.



laritons at room temperature was reported Christopou-los et al., 2007; Christmann et al., 2008. Also activeresearch is conducted on Frenkel exciton polaritons inorganic materials Lidzey et al., 1998, 1999; Schouwink etal., 2001; Takada et al., 2003; Kena-Cohen et al., 2008.

2. Structure design considerations

To achieve maximum exciton-photon coupling, a shortcavity is generally preferred with multiple narrow QWsplaced at the antinodes of the cavity field. A /2 cavitygives the largest field amplitude at the cavity center forgiven reflectance of the DBRs. An optimal QW thick-ness, often between 0.5 and 1 Bohr radius of a bulk ex-citon, leads to smaller Bohr radius and enhanced oscil-lator strength. To increase the field-exciton overlap, onecan use multiple closely spaced narrow QWs and evenput QWs in the two antinodes next to the central one,leading to an increased total coupling strength g0

NQW. The difference in the field amplitude E at eachQW has an effect similar to inhomogeneous broadeningof the QW exciton linewidth Pau, Björk, et al., 1996.Using multiple QWs has another important benefit—anincreased saturation density of LPs, which scales withN /aB

2 . When g0 is increased to become comparable withthe exciton binding energy, the strong-coupling effectfurther reduces exciton Bohr radius in the LP branchand increases the exciton saturation density.

3. In situ tuning parameters

The microcavity system of a given composition andstructure has a few useful parameters that can be ad-justed in situ during the experiments: detuning of theuncoupled cavity resonance relative to the QW excitonresonance, density of the polaritons, and temperature ofthe host lattice.

Tapering of the cavity thickness by special samplegrowth techniques enables tuning of across thesample. Changing changes the exciton and photonfractions in the LPs, hence the dispersions and lifetimesof the LPs. It has important implications in LP dynamicswhich is discussed in Sec. V.

The density of the polaritons is directly controlled bythe excitation density. A polariton density well into theBCS limit is within reach with commercial lasers. Thedensity dependence of various properties of the systemallows systematic studies of the its phase-transition phys-ics.

The temperature of the acoustic phonon reservoir isreadily tuned by thermoelectric heaters from cryogenictemperature up to room temperature or higher. The lat-tice temperature influences the dynamics of the polari-tons. However, the short-lived LPs are often not at thesame temperature as the lattice or do not have a well-defined temperature.

4. Measurement techniques

From the measurement viewpoints, the microcavitypolariton is a most accessible BEC system. There exists

a one-to-one correspondence between an internal polar-iton in mode k and an external photon with the sameenergy and in-plane wave number Houdré et al., 1994.The internal polariton is coupled to the external photonvia its photonic component with a fixed coupling ratesee Eq. 25. Hence information on the internal polari-tons can be directly measured via the external photonfields. At the same time, due to the steep dispersion ofpolaritons, polariton modes with different k and ELPkare well resolved with conventional optics. It is mainlythrough the external photon field that we learn aboutthe polaritons.

The basic properties of a polariton system can becharacterized through reflection or transmission mea-surements using a weak light source. For instance, froma measured reflection spectrum, one extracts the reso-nances of the microcavity system, the widths of the reso-nances, as well as the cavity stop-band structure. Sincemost microcavities have a tapered cavity length, a serialof reflection measurement at various positions on thesample reveals the characteristic anticrossing crossingof the eigenenergies and thus verifies strong weak cou-pling between the cavity and exciton modes. Dispersionof the resonances may also be measured by angle-resolved reflection, although photoluminescence PL isoften more convenient for this purpose.

Polaritons are conveniently excited, resonantly ornonresonantly, by optical pumping. By changing the col-lection angle of the PL, k-dependent spectra are ob-tained, which map out the energy-momentum dispersionas well as momentum distribution of the polaritons.Complementary to momentum-resolved PL, near-fieldimaging of the sample surface maps out the spatial dis-tribution of LPs. Quantum optics measurement on thePL reveals the quantum statistics of LPs. Time-resolvedPL or imaging measurements map out the dynamics andtransport of LPs.

F. Theoretical methods

We describe the main theoretical tools considered forstudying polariton condensation. Some references forrelevant methods and concepts are given, but are by nomeans comprehensive.

1. Semiclassical Boltzmann kinetics

Since polaritons are quasiparticles with a lifetime of1–100 ps, polariton condensation in general involves in-jection, relaxation, and decay of the polaritons and thusis a transient phenomena described by the kinetics of thecondensation. The semiclassical Boltzmann rate equa-tions have been well adapted to describe the polaritonkinetics Tassone et al., 1997; Ciuti et al., 1998; Tassoneand Yamamoto, 1999; Malpuech et al., 2002; Porras et al.,2002; Kavokin et al., 2004; Doan et al., 2005. Numericalsimulations based on the Boltzmann equations have in-terwound with experimental works to understand the in-teractions in the polariton system, the mechanisms of



polariton relaxation, and the statistical properties of thesystem, which are reviewed in Secs. II.F.2–II.F.4.

Limitations of this semiclassical approach exist, how-ever, when the system is above the condensation thresh-old. The random-phase approximation of the Boltzmannequations is no longer valid when coherence emergeswithin the ground state as well as between the groundstate and its low-energy excitations. Moreover, the con-densate and the excited states see different mean-fieldenergy shifts; the spectrum of low-energy excitationschanges from a free-particle spectrum with finite effec-tive mass to a linear Bogoliubov spectrum. These prop-erties are not explained with the Boltzmann equations.The appropriate tool is quantum kinetics.

To account for the difference of a condensed statecompared to a normal state and to help understand thecoherence properties of the condensate, various modelshave been developed that go beyond the conventionalBoltzmann equations. Two common approximations inthese models are to take advantage of the peculiarenergy-momentum dispersion of the LPs and to dividethe system into two regions in the momentum space. Forstates with relative large momentum and negligiblequantum coherence, the semiclassical kinetics is used.For the ground state and a small set of states withsmaller momenta and significant photonic fractions, thecorrelation between the ground state and excitations isexamined in greater detail. For example, Laussy et al.2004 and Doan, Cao, et al. 2008 introduced a stochas-tic extension of Langevin fluctuations to the Boltzmannequations. The method enables estimation of the first-and second-order coherence functions of the conden-sate, which are reviewed in Sec. VII. Sarchi and Savona2007a extended the Boltzmann equations under theBogoliubov approximation to include virtual excitationsbetween the condensate and excited states and calcu-lated the first-order coherence function of the system.

2. Quantum kinetics

A complete and self-consistent description of the con-densation physics requires quantum kinetics. Quantumkinetics can be developed either on the basis of nonequi-librium Keldysh Green’s functions Haug and Jauho,2008 or on the basis of a truncated hierarchy of equa-tions for reduced density matrices Kuhn, 1997. Thequantum kinetics description of condensation includesthe equation of motion for the order parameter, i.e., theexpectation values of the ground-state polariton annihi-lation operator, as well as an equation for the populationof the excited states. The Dyson equation formulated inthe time regime will couple density matrices which arefunctions of only one-time argument to two-time scatter-ing self-energies and two-time Green’s functions. Onecan express these off-diagonal quantities in the time ar-guments by two-time spectral functions and one-timedensity matrices. The full set of kinetic equations thusincludes the equations for the spectral functions, i.e., theretarded and advanced Green’s functions. Through thesespectral functions, the polariton spectra—which change

during the condensation—are introduced self-consistently in the condensation kinetics. A full quan-tum kinetic simulation of polariton condensation is com-putationally intensive and presently not yet available.4

3. Equilibrium and steady-state properties

If the polariton lifetime is long compared to the relax-ation time, a condensed polariton system can be consid-ered as in quasiequilibrium, and thus the static proper-ties may be approximated by a mean-field theory basedon the Gross-Pitaevskii equation Pethick and Smith,2001; Pitaevskii and Stringari, 2003. We introduce theGross-Pitaevskii equations in Sec. VIII and apply it toexperiments on the spectral properties and superfluidityof polariton condensation.

When a short-lived polariton system is undercontinuous-wave pumping, a steady state may bereached. Properties of a steady-state polariton conden-sate have been calculated by linearizing a generalizedGross-Pitaevskii equation around the steady stateWouters and Carusotto, 2007 and also by a KeldyshGreen’s function approach Szymanska et al., 2006. Dueto constant pumping and decay of the polaritons, thelow-energy excitations are expected to be diffusive,manifested as a flat dispersion near the ground state. Avortex induced during the relaxation process at local dis-order potential was also observed Lagoudakis et al.,2008.

4. System size

The two-dimensional polariton systems typically havea transverse size of 10–100 m across. The finite sizeleads to an energy gap E32 / 2meffS between theground state and first-excited states. Given a typicalGaAs polariton mass of meff=510−5me here me is thefree-electron mass, E is about 0.2 meV. This gap re-sults in a finite-size BEC, and the condensate densityincreases with decreasing system size. However, the realtransverse extension in experiments is not known accu-rately as it is often given by the spot size of the pumpinglaser. This uncertainty in the spatial extension imposes aspecific problem in numerical simulations of the polar-iton kinetics.

III. RATE EQUATIONS OF POLARITON KINETICS

As elementary excitations of the microcavity system,polaritons are most conveniently created by a laserpump pulse, after which they relax and under appropri-ate conditions accumulate at least partly in the ground

4Currently a simplified model of quantum kinetics is given byWouters and Savona 2009.



state of the LP branch, before they completely decay. Inorder to study a spontaneous phase transition, pumpingshould be incoherent, so that there are no phase rela-tions between the pump light and the condensate. Onespeaks about a quantum statistically degenerate state ifthe number of polaritons n0 in the ground state is muchlarger than 1. In this case, stimulated relaxation pro-cesses dominate over the spontaneous ones by a factor1+n0. The main interactions which drive the relaxationand eventually the condensation stem from the exciton-phonon interactions and the Coulomb interactions be-tween excitons. The exchange Coulomb interactionsdominate over the direct Coulomb interactions betweentwo excitons and give rise to a repulsive interaction be-tween two polaritons with an identical spin. This inter-action provides the fastest scattering mechanism whichthermalizes the polariton gas, but the temperature of thegas can only be lowered by coupling to the cold acousticphonon reservoir.

Much of the physics of the polariton relaxation andcondensation can be described with the semiclassicalBoltzmann rate equations,5 which we formulate in thissection. We include the scattering due to the LP-phononand LP-LP interactions as discussed. For most experi-ments to date, the cavity-photon linewidth is on the or-der of 1 meV. The spectral change due to condensationis not extended enough to have a substantial influenceon the condensation kinetics. Hence we use only thefree-particle spectrum to calculate the scattering rates.These rate equations yield results in good agreementwith the experimental observations in Secs. IV.A andIV.B.

The rate equations calculate only the number of theLPs in the ground and excited states. However, we showin Sec. VII that a stochastic extension of the Boltzmannkinetics can calculate at least approximately first- andsecond-order coherence properties.

A. Polariton-phonon scattering

The relevant interaction between excitons andlongitudinal-acoustic phonons is provided by thedeformation-potential coupling with the electron andhole De and Dh, respectively. This interaction Hamil-tonian between the LPs and phonons is Pau, Jacobson,et al., 1996; Piermarocchi et al., 1996

HwLP-ph =

k ,q ,qz

Xk*Xk+q kHw

x-phk + q

cq ,qz− c−q ,−qz

† bk+q† bk , 32

where Xk are again the exciton Hopfield coefficients ofthe polariton. The phonons are taken to be the bulk

phonons with a wave vector qe +qzez. Only the in-planevector component is conserved. The quantum well1s-exciton envelope function is given by

r,ze,zh = 2

aB2 e−r/aBfezefhzh , 33

where aB is the 2D Bohr radius and fe,hze,h are thequantum well envelope functions. Evaluating theexciton-phonon interaction matrix element with this en-velope function, one finds

Gq ,qz = iq2 + qz21/2

2VuDeIe

qIezqz

− DhIh qIh

zqz . 34

Here is the density, u is the sound velocity, and V is themicrocavity volume. The superposition integrals for theelectron e or hole h are given by

Ie,h q = 1 + mh,e

2MqaB2−3/2

, 35

Ie,hz qz =

0

LQW

dzfe,h2 zeiqzz. 36

These integrals cut off the sums for qaB−1 and for qz

2 /LQW.With this interaction matrix element, the transition

rates for the LPs can be written as

Wk→k=k−qLP-ph =

2

qz

Xk2Xk−q

2 Gq,qz

Ek−q − Ek ± q,qz

NBq,qz +

12

±12 , 37

where NBq=1/ expq−1 is the thermal phononBose distribution function. Here the small correction ofthe phonon confinement is neglected.

B. Polariton-polariton scattering

1. Nonlinear polariton interaction coefficients

The LP-LP interaction Hamiltonian generally has theform

HI = 12Uk1,k2,qPk1

+ Pk2

+ Pk1+qPk2−q, 38

where Pk is the creation operator of a LP with the in-plane wave vector k. The dependence of Uk1 ,k2 ,q onk1, k2, and q can be neglected if the momentum ex-change q is small. In this case we replace Uk1 ,k2 ,q byUk=Uk ,k ,0.

5For an introduction of other approaches, see Sec. II.F andKeeling et al. 2007.



For a spin-polarized polariton system, the nonlinearinteraction parameter Uk can be further simplified andcalculated as

Uk =12EX +g0

2 +2

4−gn2 +

− EX2

4,

EX = EBn

nS, nS =

S

2.2aB*2X2

, 39

gn = g01 −n

nS, nS =

S

4aB*2X2

.

Here =Ecavk−Eexck is the detuning between thecavity and exciton energies at given k, EB is the excitonbinding energy, aB is the exciton Bohr radius, 2g0 is thenormal-mode splitting at =0, and S is the cross-sectional area of the system. X2= 1

2 1+ /4g02+2 is

the excitonic fraction of the polariton at k. EX repre-sents the repulsive fermionic exchange interaction be-tween QW excitons of the same spin Ciuti et al., 1998;Rochat et al., 2000. gn is the reduced normal-modesplitting due to phase-space filling and fermionic ex-change interactions Schmitt-Rink et al., 1985.

If a polariton system consists of two helicity spincomponents, the biexcitonic attractive interaction be-tween two excitons with opposite helicities spins playsa crucial role in its relaxation dynamics. A frequency-domain degenerate four-wave mixing FD-DFWMtechnique was used for such a system to evaluate rela-tive magnitudes of the phase-space filling effect and thetwo-body attractive and repulsive interaction betweenexcitons. As shown in Fig. 5a, a pump beam excitestwo counterpropagating photon fields with amplitudes1 and 2 and a test beam excites a probe field with anamplitude 3. The elastic scattering conserving energyand momentum produces a phase-conjugate signal withan amplitude 4. The nonlinear Hamiltonian of this elas-tic scattering has three terms,

Hn = H1 + H2 + H3 + H.c.,

H1 = Wb4++ b3−

+ b2+b1− + b4++ b3−

+ b2−b1+ + b4−+ b3+

+ b2+b1−

+ b4−+ b3+

+ b2−b1+ ,40

H2 = R =+,−

b4+ b3

+ b2b1,

H3 = g =+,−

b4+ b3

+ b2a1 + b4+ a3

+ b2a1 .

Here =± represents the Z component of the angular

momentum and a and b are the annihilation operators

of the cavity photon and QW excitons, respectively. H1represents attractive interactions between two excitons

with opposite spins W0. H2 represents the repulsive

interaction between two excitons with the same spin. H3

represents the reduction of the exciton-photon coupling.

In the presence of an exciton population b+b , theexciton-photon coupling strength g is reduced to g1

−b+b , here 0.Figure 5b shows the DFWM spectra at =0 for dif-

ferent polarization configurations of the pump, test, andsignal beams. When all three beams have the same lin-ear polarization in configuration A, a strong signal isobserved at the UP resonance, while the signals at UPand LP resonances are expected to be proportional toW /4+ R /2−g2 and W /4+ R /2+g2, respectively.When the pump beam has a polarization orthogonal tothat of the test and signal beams in B, a strong signalappears at the LP resonance, where the signals at UPand LP resonances should be proportional to W /4− R /2−g2 and W /4− R /2+g2. In C and D, thetest and signal beams have the same helicity, so only H2

and H3 are effective. In E the two beams have oppositehelicities so that only H1 contributes to the scattering.The data are explained by a microscopic theory for non-linear interaction coefficients based on an electron-holeHamiltonian Inoue et al., 2000.

FIG. 5. Self-pumped four-wave mixing phase conjugation ex-periment. a Schematic of the experiment. Two initial fieldsprovided by the pump beam are elastically scattered into twooutgoing fields conserving energy and momentum. b Spectraof FD-DFWM at zero detuning for various polarization con-figurations. Adapted from Kuwata-Gonokami et al., 1997.



2. Polariton-polariton scattering rates

For a nonpolarized LP gas, following Tassone andYamamoto 1999, we further simplify the scattering co-efficient as

M 2k,k

Vk−kkkk2 − kk 6E0

aB2

S, 41

where k and E0 are the screened 2D 1s-exciton wavefunction and binding energy, respectively. Vk=2 /0Skis the 2D Coulomb potential and 0 is the dielectric func-tion.

The corresponding LP transition probability is then

Wk,k;k1,k2

LP-LP =

S2

24

E2M2Xk

2Xk2 Xk1

2 Xk2

2

Ek/k2Ek1/k12Ek2/k2

2

Rk,k,k1,k2 . 42

A uniform energy grid is adopted with an energy spacingE. The terms proportional to the 2D density of statesEk /k2 stem from the conversion of integrationover momenta to summation over energies. The term Ris given by

Rk,k,k1,k2 = dq2

k + k12 − q2q2 − k − k12k + k22 − q2q2 − k − k22. 43

The integration is taken over the range in which all fourterms under the square root are non-negative. Energyconservation is built in by taking k2=kE2=Ek+Ek−Ek1

and summing over the Ek and Ek1.

C. Semiclassical Boltzmann rate equations

With these transition rates we formulate the semiclas-sical Boltzmann equations of the polariton distribution.This approach has been used before, for example, todescribe the exciton condensation kinetics by Bányai etal. 2000, Tran Thoai and Haug 2000, and Bányai andGartner 2002 and to describe the cavity polariton re-laxation by Tassone et al. 1996, 1997, Bloch and Marzin1997, Tassone and Yamamoto 1999, Cao et al. 2004,and Doan et al. 2005. In the latter two references theimportance of a finite size for the 2D condensation ki-netics has been shown.

The Boltzmann equations for the population of theexcited states nkt and the ground-state population n0tare

tnk = Pkt −

nk

k+

tnk

LP-LP+

tnk

LP-ph, 44

tn0 = −

n0

0+

tn0

LP-LP+

tn0

LP-ph. 45

Note that a separate equation is needed for the ground-state LP population n0 so as to properly account for thegap between the ground state and the lowest excited-state energies due to a finite size of the system. In thefollowing, we assume cylindrical symmetry such that nk

depends only on the amplitude of k , and we drop thevector notation for simplicity. Pkt is the time-dependent incoherent pump rate and k is the LP life-time introduced by Bloch and Marzin 1997. For small

transverse momenta 0kkcav=8 m−1 we use 1/k

=Ck2 /c given the photon Hopfield coefficient Ck. The

cavity lifetime c is only a few picoseconds for most mi-crocavities. In the range kcavkkrad=ncavEcav/c, weuse the nanosecond exciton lifetime x, i.e., 1 /k=1/x,and finally for large momenta kkrad the exciton life-time is infinite, i.e., 1 /k=0.

The LP-phonon scattering rate is given by

tnk

LP-ph= −

k

Wk→kLP-phnk1 + nk

− Wk→kLP-ph1 + nknk . 46

The LP-phonon scattering provides a dissipative coolingmechanism for the LP gas. It can reduce the excess en-ergy introduced in the LP gas with the nonresonant op-tical excitation.

The LP-LP scattering rate is given by

tnk

LP-LP= −

k,k1,k2

Wk ,k,k1,k2

LP-LP nknk1 + nk1

1 + nk2 − nk1

nk21 + nk1 + nk ,

47

where k1=k +q and k2=k−q . LP-LP scattering is ratherfast; it thermalizes the LP gas in a few picoseconds butcannot reduce the total energy of the LP gas. Thus bothscattering processes have their own distinct roles in thecondensation kinetics of the LPs.

We emphasize again that the factor 1+n0 expressesthe stimulated scattering into the ground state enhancedby the final-state occupation n0, which are the essentialdriving forces of LP condensation. Once the condensateis degenerately populated with n01, the escape of the



macroscopic coherent polariton wave is rather classicaland given by the photon leakage out of the cavity.

To study spontaneous condensation, we keep thepump energy high enough to avoid the possibility wheretwo LPs at the pumped mode can resonantly scatter intothe ground state and a higher lying state Savvidis et al.,2000. The pump pulse is assume to be of the form

Pkt = Pk e−2 ln2t/tp2

tanht/t0 48

with

Pk = P0e−2k − kp2/2mexcE 49

for pulse and stationary excitations, respectively.For GaAs QW and microcavities, the material param-

eters we used are Bockelmann, 1993; Pau et al., 1995;Bloch and Marzin, 1997; Cao et al., 2004 E0

x=1.515 eV,me=0.067m0, mh=0.45m0, aB=10 nm, the index of re-fraction ncav=3.43, the normal-mode splitting g0=5 meV, the deformation potentials De=−8.6 eV, Dh=5.7 eV, u=4.81105 cm/s, and =5.3103 kg/m3.

IV. STIMULATED SCATTERING, AMPLIFICATION, ANDLASING OF EXCITON POLARITONS

In this section we describe cold collision experimentswhich demonstrate the bosonic final-state stimulation,matter-wave amplification, and lasing of polaritons.Through those experiments we investigate the nonlinearinteractions of polaritons, which are at the heart of po-lariton condensation and the excitation spectrum dis-cussed later.

A. Observation of bosonic final-state stimulation

One distinct feature of a system of identical bosonicparticles is the final-state stimulation: the presence of Nparticles in a final state enhances the scattering rate intothat final state by a factor of 1+N. Final-state stimula-

tion is the driving mechanism behind photon lasers andamplifier and is expected to be crucial to the formationof BEC. It is of great interest to demonstrate the final-state stimulation for a massive composite particle, toconfirm its bosonic nature, and to clarify the role offinal-state stimulation in the BEC physics.

Due to the strong coupling between excitons and pho-tons, the energy DOS of the LPs decreases steeply byfour orders of magnitude from a state at large in-planewave number kkbot0.1k0 to states at k 0. Here weassume a moderate detuning and k0 is the wave numberof the cavity photon at k =0. As a result of the reducedDOS, cooling of the LPs by acoustic phonon emission isslowed down at kbot. At the same time, the lifetime ofthe LPs at kbot states is typically two orders of magni-tude longer than that of LPs at k 0. The two effectstogether cause an accumulation of incoherent excitonpopulation at kbot—the so-called energy relaxation“bottleneck” Pau et al., 1995; Tassone et al., 1996.

In an experiment on binary exciton scattering Tas-sone and Yamamoto, 1999; Huang et al., 2000, thebottleneck effect is utilized to study the bosonic final-state stimulation. As shown in Fig. 6a, two pumpbeams create incoherent excitonlike LPs at k = ±kbot.Due to the long lifetime and slow relaxation rate at kbot,the injected excitons form an incoherence reservoir.Then the LPs at k = ±kbot may elastically scatter into LPand UP at k 0. If a LP at k 0 is optically populated,the exciton-exciton scattering rate is enhanced by thebosonic stimulation factor. The other final state of thescattering, the otherwise unpopulated k 0 UPs, pro-vides a background-free measurement window.

The sample used in this experiment is a single GaAsQW embedded in a planar AlAs/AlxGa1−xAs microcav-ity. A mode-locked Ti:Al2O3 laser with a bandwidth of16 meV is spectrally filtered to obtain the pump andprobe pulses which have a bandwidth of 0.4 meV andare separated in energy by half of the LP-UP splitting of2 meV. The pump pulse is split into two beams and in-cident on the sample at ±45° from the sample normal.

FIG. 6. A cold collision experiment of polaritons. a Schematic of the experiment. The size of the solid circles depicts thepopulation in the LP states. Inset: The cross section of the microcavity structure and the experimental excitation scheme. b Theintegrated UP emission intensity as a function of probe density nLP for varying exciton density nexc. Symbols are the experimentaldata and solid lines are the theoretical prediction. The exciton densities are nexc=1.5109 squares, 1.2109 circles, and 5.4108 cm−2 triangles. c The integrated UP emission intensity as a function of nexc for a fixed nLP=1.1109 cm−2. The dotted lineindicates the acoustic phonon-scattering contribution to the UP intensity. Adapted from Huang et al., 2000.



The probe beam incidents on the sample at 3° in orderto excite the LPs at k 0, while the UP emission is de-tected in the normal direction Fig. 6a.

The time-integrated intensity of the UP emission NUPincreases linearly with the probe intensity Fig. 6b,which is the signature of the final-state stimulation. NUPalso increases quadratically with the reservoir excitondensity nexc Fig. 6c due to the binary exciton-excitonscattering. The results are described quantitatively by arate equation solid lines in Figs. 6b and 6c, wherethe dominant term for populating the UPs is 1+NLPnexc

2 ,

dNUP

dt= −

NUP

UP+ aUPnexc + bUP + b1 + NLPnexc

2 .

50

Here UP is the radiative lifetime of the UP and NLP isthe k 0 LP population determined by the probe beamintensity. The coefficient aUP represents the scatteringrate of a bottleneck exciton to an UP by acoustic pho-non absorption and has been calculated by applying Fer-mi’s golden rule to the Hamiltonian of the deformation-potential scattering. bUP is the scattering rate of twoexcitons to one UP and another exciton e.g., 1+6→UP+4. b is the scattering rate of two bottleneckexcitons at opposite in-plane momenta ±k to UP andLP both at k 01+1→UP+LP. bUP and b havebeen calculated via Fermi’s golden rule rate applied tothe Hamiltonian for exciton-exciton scattering.

In this cold collision experiment, real population ofexcitons at ±kbot are excited by the pump laser andforms incoherent reservoirs. This is evident from the ob-served exponential decay of a luminescence from thoseexciton states with a time constant of exc190 ps, whichis much longer than the pump pulse duration of 9 ps.Moreover, NUP due to the stimulated scattering decayswith a time constant of UP96 ps. The difference of afactor of 2 between exc and UP stems from the nexc

2

dependence of the stimulated scattering rate.By contrast, the polariton parametric amplification ex-

periment first performed by Savvidis et al. 2000 is acoherent optical mixing process. In this latter case, thepump beam creates virtual LP population at a magicwave number kp, which coherently scatters to the twofinal states: LPs at k 0 and excitons at 2kp. The scat-tering conserves both in-plane momentum and energybetween the initial and final states. In this coherentparametric process, the stimulated scattered signal ap-pears only when the pump beam is present, and the sig-nal intensity increases linearly with the pump intensity.In this sense, the process can be interpreted as coherentoptical four-wave mixing resonantly enhanced by the LPresonances.

B. Polariton amplifier

A standard photon laser pumped just below its oscil-lation threshold is often used as a regenerative linear

optical amplifier for an input light signal. It is a negative-conductance amplifiers based on incoherent or resis-tive gain elements. A second common type of amplifieris a nonlinear susceptance amplifier, the gain of which isbased on coherent or reactive multiwave mixing be-tween a pump, signal, and idler. A parametric amplifieris a classic example. This second type of amplifier hasbeen demonstrated with matter waves using both BECof alkali atoms Inouye et al., 1999; Kozuma et al., 1999and coherent polaritons produced by a pump laser Sav-vidis et al., 2000. Utilizing the stimulated scattering ofbottleneck polaritons, Huang et al. 2002 demonstrateda matter-wave amplifier of the first type with a polaritonsystem. The fundamental difference between a photonlaser amplifier and a polariton amplifier is that, in thelatter system, there is no population inversion betweenconduction and valance bands. Therefore the standardstimulated emission of photons cannot provide a gain.Instead, amplifications of the injected polaritons areachieved through elastic LP-LP scattering. Such a polar-iton amplifier is a matter-wave amplifier based on stimu-lated scattering of massive particles rather than a photonamplifier based on stimulated emission of photons. Anegative-conductance amplifier for matter waves is at-tractive from a practical viewpoint because electricallyinjected incoherent excitons can eventually realize anamplifier for an optical signal without requiring popula-tion inversion between conduction and valence bands.

The sample used in the experiment by Huang et al.2002 is a /2 Cd0.6Mg0.4Te cavity with two CdTe QWs.A pump laser incident from a slant angle resonantly ex-cites excitons at k 3.5 m−1. When this initial popula-tion increases to above about Np=71010 cm−2, a quan-tum degeneracy threshold is reached at a bottleneckstate with k =2 m−1. Np is nearly one order of magni-tude smaller than the saturation density of 51011 cm−2 for CdTe QW excitons. The measured LPpopulation distribution versus energy and in-plane mo-mentum is shown in Fig. 7a. When pumped abovethreshold, the maximum of the LP emission continuallyshifts toward smaller k. This behavior of bottleneckcondensation is predicted by the rate equation theoryTassone and Yamamoto, 1999. A slight blueshift of theLP energy at k 0 and 3 m−1 is due to the repulsiveinteraction between nondegenerate LPs and the quan-tum degenerate bottleneck LPs at k 2 m−1. Thesame feature is manifested in Bogoliubov excitationspectra as discussed in Sec. VIII.

When a probe pulse, with a narrow bandwidth ofabout 0.6 meV, selectively excites the bottleneck LPs,amplification of the probe beam is observed Fig. 7b.The dynamics of the bottleneck LP population NLP ismodeled by Ciuti et al. 1998,

dNLP

dt= PLP −

NLP

LP+ aLPNEP1 + NLP

+ bLPNEP2 1 + NLP . 51

Here LP is the bottleneck LP lifetime, PLP is the exter-nal injection rate of the LP by the probe-laser pulse,



aLP=47 s−1 is the scattering rate between acousticphonons and LPs, and bLP=0.45 s−1 is the LP-LP scatter-ing rate of an incoherent LP population.

The increase of the measured gain G with the reser-voir population NEP agrees quantitatively with the rateequation simulation, that is, GexpconstNEP

2 in thehigh-gain regime and G−1 constNEP

2 in the low-gain regime. The time dependence of the gain is charac-terized by a fast rise followed by a slow exponential de-cay. The decay time of 57 ps corresponds to about halfof the separately measured decay time of bottleneck LPsFig. 7c, which is consistent with the quadratic depen-dence of the gain on NEP. These results all confirm thegain mechanism by two-body scattering of incoherentreservoir LPs.

C. Polariton lasing versus photon lasing

1. Experiment

The LP-LP scattering enhanced by the bosonic final-state stimulation gives rise to large nonlinearity in themicrocavity system. This large nonlinearity has been ex-ploited to implement polariton amplifiers of incoherenttype Huang et al., 2000 and of coherent type Savvidiset al., 2000 and polariton lasers in the bottleneck regimeHuang et al., 2002. First-order coherence Richard etal., 2005 was also observed for a bottleneck polaritonlaser. However, such a system is always far from thermalequilibrium. There is a large number of degenerate las-ing modes at the bottleneck, and it is difficult to charac-terize the thermodynamic properties of the LP gas. Withthe improvement of fabrication techniques and optimi-zation of the experimental conditions, quantum degen-eracy and lasing at the k =0 LP ground state have beenobserved first with GaAs-based devices Deng et al.,2002, 2003; Balili et al., 2007 and later with CdTe-baseddevices Kasprzak et al., 2006 and GaN-based devicesChristopoulos et al., 2007.

GaAs excitons have a relatively larger Bohr radius,hence the advantage of relatively large scattering crosssections. To cope with the relatively lower saturationdensity and maintain the strong-coupling regime, Denget al. 2002 used a microcavity with 12 QWs. MultipleQWs, on the one hand, reduced the exciton density perQW for a given total LP density and, on the other hand,increased the exciton-photon coupling to reach the verystrong-coupling regime Khurgin, 2001, which leads toreduced exciton Bohr radius for the LP branch hencefurther increased exciton saturation density.

The experimental scheme is depicted in the left panelof Fig. 10. A pump laser resonantly excites hot LPs atk =5–6 m−1 so as to minimize the excitation of freeelectrons and holes, which usually increase the excitondecoherence rate. The value k =5–6 m−1 is well abovethe magic angle corresponding to k 2 m−1, so thatthe coherent LP parametric amplification Savvidis et al.,2000 is avoided.

A typical lasing behavior is observed in the input-output relationship, i.e., the pump power dependence ofthe emission intensity of the LP ground state Fig. 8. Itfeatures a sharp superlinear increase around a thresholdpumping density Pth, corresponding to an injected car-rier density of nQW3109 cm−2 per pulse per QW fora spot of D15 m in diameter. nQW is 30 times smallerthan the saturation density and two orders of magnitudesmaller than the Mott density n1/aB

2 31011 cm−2

electron-hole pairs per QW. At the threshold, the k =0LP population NLP is estimated to be of the order ofunity, which coincides with the onset of the stimulatedscattering of LPs into the k 0 LP state.

To confirm that LPs remain as the normal modes ofthe system above threshold, the energy-momentum dis-persion was measured and was found to agree with theLP dispersion instead of the cavity-photon dispersion.The LP energy remains well below the bare QW excitonenergy in spite of a small blueshift due to the Coulomb

1.62 1.63 1.640

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

4

Pump only

Counts

1.62 1.63 1.640

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

4

Probe only

Energy [eV]

1.62 1.63 1.640

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

4

Pump+Probe

(a) (b) (c)

FIG. 7. Color online A negative conductance polariton amplifier. a The LP emission intensity taken as a function of energy andin-plane wave vector. The system is above the quantum degeneracy threshold at the bottleneck. The solid line and the dashed lineindicate the theoretical dispersion of a single LP and cavity photon, respectively. The dashed line indicates the measured LPdispersion. b Observed probe emission for pump only, probe only, and simultaneous pump and probe excitation. c Gain −1 asa function of the time delay between pump and probe pulses open circles. In the same plot, the intensity of bottleneck LPemission at a low pumping intensity is shown as solid dots. Solid lines indicate exponential decay curves of 57 and 114 ps. Thedashed line indicates the convolution of the pump and the probe pulse envelopes for comparison. Adapted from Huang et al.,2002.



exchange interaction and phase-space filling effect. Thesubject will be discussed in Sec. VIII.

To distinguish the polariton laser from a coherent po-lariton parametric amplifier, the time evolution of theLP ground-state population was measured. There was aclear delay between the peak of the LP emission and thepump pulse, which should be absent in the polaritonparametric amplifier and oscillator Savvidis et al., 2000see Fig. 9.

Using the same sample and experimental setup, a di-rect comparison has been made with a conventionalphoton laser Deng et al., 2003. Since the sample’s cav-ity layer thickness is tapered by growth, detuning ofthe cavity resonance changes across the sample. To es-

tablish photon lasing, the sample is pumped at a positionwith the cavity mode bluedetuned to 15 meV above theGaAs QW band edge. The pump energy is tuned ac-cordingly to be resonant with the cavity mode energy atk =5.33104 cm−1. The experimental scheme is de-picted in the right panel of Fig. 10.

The emission in this case follows the cavity-photondispersion, and a photon lasing threshold is observed atk =0 Fig. 8. Here the injected carrier density at thresh-old is nQW 31011 cm−2 per pulse per QW, exactlyidentical to the density required for electronic popula-tion inversion at 15 meV above the band edge. This re-sult agrees with the standard photon laser mechanism.That is, when the spontaneous emission coupling con-stant is large enough, the threshold of a semiconductorlaser is solely determined by the electronic inversioncondition Björk et al., 1994. The cavity-photon numberper mode is estimated to be of the order of unity atthreshold, which coincides with the onset of the stimu-lated emission of photons into the k =0 cavity-photonmode.

A photon laser results from stimulated “emission” ofphotons into cavity modes with an occupation numberlarger than 1. The quantum degeneracy threshold of po-laritons is triggered by the same principle of bosonicfinal-state stimulation, yet the physical process isdifferent—it is stimulated “scattering” of massive quasi-particles the LPs from nonlasing states at k0 intothe lasing state at k =0. The nonlinear increase of thephoton flux is an outcome of the leakage of the LPs viacavity mirrors. Hence the observed threshold of the po-lariton system requires no electronic population inver-sion. The threshold density is of the order of 1/T

2 andthree to four orders of magnitude lower than that of aphoton laser threshold given by 1/aB

2 . It serves as aproof of principle demonstration to use polariton sys-tems as a new energy efficient source of coherent light.The difference in the operational principles of polariton

109 1010 1011 1012

10-1

100

101

102

103

inversion

PhotonsperCavityModeatk ||~0

polariton laser,ELP=1.6166 eVphoton laser,Ecav=1.6477 eV

Injected Carrier Density (cm-2)

PolaritonsperModeatk ||~0

100

no inversion

FIG. 8. Color online Number of LPs and cavity photons permode vs injected carrier density for a polariton condensatetriangles, the scheme in the left panel of Fig. 10 and a photonlaser circles, the scheme in the right panel of Fig. 10, respec-tively. The gray zone marks the population inversion densitiesfrom band edge to 15 meV above the band edge. From Denget al., 2003.

FIG. 9. Dynamic properties of the emission. a The time evo-lution of the k =0 LP emission intensity. “a.u.” in the y-axislabel denotes “arbitrary units.” b Pulse FWHM Tp vs pumpintensity P /Pth. c Turn-on delay time of the emission pulse vsP /Pth. From Deng et al., 2002.

FIG. 10. Color online Schematics of the experimental prin-ciples for a photon laser right panel compared to LP conden-sation left panel.



laser and photon laser is shown schematically in Fig. 11Imamoglu et al., 1996.

If the pump power for a polariton lasing system iscontinuously increased, the 2D carrier density eventu-ally exceeds the saturation density where the polaritonsystem undergoes a transition from strong to weak cou-pling. When this transition happens, the leaking modeenergy, which is approximately equal to the LP energyexcept for a continuous blueshift, suddenly jumps to thebare cavity resonant energy. Such behavior is shown inFig. 12 for a 12-QW GaAs microcavity Roumpos,Hoefling, et al., 2009. In this particular experiment,continuous-wave pumping is used at an energy abovethe band gap. The first threshold is that of a polaritonlaser. Whether the second threshold is that of a simplephoton laser Bajoni et al., 2008 or a new ordered stateinvolving a BCS-type correlation is unresolved at thistime.

2. Numerical simulation with quasistationary and pulsedpumping

Numerical solutions of the Boltzmann equation repro-duced the above experimental results when both LP-phonon and LP-LP scattering are considered Doan et

al., 2005. By quasistationary pumping above the bottle-neck, a quantum statistically degenerate population ofthe ground state with n01 is generated. This is shownin Fig. 13 where the ground-state population n0 is plot-ted versus the total LP density ntot taken as a measure ofthe pump rate for various values of the detuning . It isseen that above the LP lasing threshold, the groundstate is degenerate with 103–104 LPs. The total LP den-sity ntot at threshold is about 5109 cm−2 for a detuningof 6 meV, which is well below the exciton saturationdensity ns=1.51011 cm−2 Schmitt-Rink et al., 1985.The criteria for the validity of the boson approximationfor polaritons are relaxed compared to those for exci-tons by the square of the exciton Hopfield coefficient. Inexperiments, the total LP density is distributed in sev-eral QWs which have no direct interaction. The conden-sation threshold density ntot,c=510−9 cm−2 agrees withthe experimental values Deng et al., 2003.

In Fig. 13 the important influence of the detuning pa-rameter on the condensation threshold is clearly dem-

(a)

crystal ground state k

via multiple phonon emission& polariton-polariton scattering

external injection

exciton-polaritondispersion

leakage of photonsvia cavity mirror

stimulatedcooling

k = 0 LP

final bosonic mode

E

(b)

stimulated emissionof photons

external pumping

E

crystal ground state k

electron-holepair dispersion

spontaneouscooling

cavity photon

final bosonic mode

populationinversion

FIG. 11. Color online The operational principles of a polariton BEC and b photon laser. From Imamoglu et al., 1996.

FIG. 12. Color Polariton lasing and photon lasing. a Theemission energy vs pump power for a N=12 multi-QW planarmicrocavity. b The dispersion characteristics of polaritonBEC green diamond and photon lasing pink triangle as wellas the linear dispersion of UP red square and LP blue circleat low pump power. From Roumpos, Hoefling, et al., 2009.

FIG. 13. Theoretical condensate population f0=n0 vs the totalLP density ntot for quasistationary pumping and the combinedLP-LP and LP-phonon scattering for the bath temperatures ofT=4 K for various values of the detuning: =0 full line, 2.5dashed line, and 6 meV dash-dotted line. From Doan et al.,2005.



onstrated: If the cavity mode is slightly detuned in theorder of a few meV above the exciton resonance, theexciton component of the LPs increases. Because all in-teractions relevant for the relaxation kinetics are pro-vided by the exciton component, the LPs can relax moreefficiently toward the ground state, so that the conden-sation threshold is reached at lower densities.

Under a pulse excitation below threshold, a relativelyslow buildup and decay of the perpendicular emissionare also obtained numerically. Above threshold the out-put rises faster due to the stimulated scattering see Fig.14. The decay is faster too because now an increasingfraction of the particles is in the ground state where thelifetime is the shortest. These results of the Boltzmannkinetics are in good agreement with the observationshown in Fig. 9 Deng et al., 2002; Weihs et al., 2003.

V. THERMODYNAMICS OF POLARITONCONDENSATION

In Sec. IV, we show that stimulated scattering of LPsovercomes the bottleneck effect above a thresholdpumping density; the population in the k =0 LP groundstate increases nonlinearly with the pump and quicklybuilds up quantum degeneracy. Quantum degeneracy isoften associated with a transition to a macroscopic quan-tum state, such as a BEC state. To search for directmanifestation of such a phase transition, quantum statis-tics of the polariton gas has been studied by two types ofmeasurements. One is the momentum-space distributionof polaritons, which reveals the evolution and formationof a condensate and its thermodynamic properties. It isthe focus of this section. The other is coherence func-tions, specifically, first-order spatial and temporal coher-ence and second-order spatial and temporal coherence.These will be reviewed in Sec. VII. Other manifestationsof the macroscopic phase include superfluidity, vortices,and phase locking among polariton arrays, as discussedlater.

A. Time-integrated momentum distribution

As reviewed in Sec. II.E.4, a microcavity polaritonsystem has the unique advantage that the momentumdistribution of LPs can be directly measured via angle-resolved PL. Early studies revealed an energy relaxationbottleneck at relatively low excitation densities when theLP-phonon scattering was the dominant relaxationmechanism Tassone et al., 1997; Muller et al., 1999; Tar-takovskii et al., 2000. With increasing excitation density,the bottleneck migrated toward smaller k state Tartak-ovskii et al., 2000. Later, in a GaAs-based 12-QW mi-crocavity, the energy relaxation bottleneck becamelargely absent at zero or positive cavity detunings, andquantum degeneracy was achieved in the k =0 LPground state above certain excitation density Pth Denget al., 2002. Following this experiment, Deng et al.2003 measured the momentum distribution and com-pared it with Bose-Einstein BE statistics.

In the experiment of Deng et al. 2003, the microcav-ity system was pumped by a 3 ps pulsed laser on reso-nance with excitonlike states at large k. The time-integrated momentum distribution of LPs wasmeasured. For excitation densities between Pth and 2Pth,the LP ground state reached quantum degeneracy wherestimulated scattering was pronounced; the energy relax-ation time was close to but still longer than the lifetimeof the ground-state LPs. Thus the LP distribution doesnot change significantly within the LP lifetime, while atthe same time the duration of the emission pulse is com-parable to their lifetime. In this regime, the time-integrated measurement approximates the instanta-neous momentum distribution of the LPs. The measuredmomentum distribution agrees well with the quantum-mechanical BE distribution except for an extra popula-tion in the condensed k 0 state Fig. 15a. The quan-tum degeneracy parameter, the ratio of the chemicalpotential and temperature =− /kBTLP, decreases rap-idly when the excitation density increased from below toabove the threshold, reaching a minimum of 0.02 atP=1.7Pth Fig. 15b. This demonstrates that the LPsform a quantum degenerate Bose gas and establish qua-siequilibrium among themselves via efficient stimulatedLP-LP scattering.

The fitted effective LP temperature TLP, however, ishigher than the lattice temperature of 4 K, and it in-creases with increasing pump rate inset of Fig. 15b.This implies that the LP-phonon scattering does not pro-vide sufficient cooling to the injected hot excitons, andthermal equilibrium with the phonon bath is notreached.

At PPth, the LP-LP interaction becomes strong.Therefore, the condensate begins to be depleted. At thesame time, the dynamics becomes faster and the time-integrated data are no longer a good approximation ofthe instantaneous distribution.

B. Time constants of the dynamical processes

Whether or not LPs can be sufficiently cooled to reachthermal equilibrium with the phonon bath requires fur-

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

Inte

nsity

(arb

.uni

ts)

t(ps)

Pump

FIG. 14. Calculated buildup and decay of the ground-stateemission in the laser mode above and below threshold at thepump powers P /Pth=0.87 solid line, 1.87 dashed line, and3.0 dotted line, respectively. The dash-dotted line shows thepump pulse. From Doan et al., 2005.



ther investigation of the dynamical processes of the sys-tem. As discussed in Sec. III, there are mainly three dy-namical processes of the LP system: 1 LP decay via theout coupling of its photon component, 2 LP-LP scat-tering, and 3 LP-phonon scattering.

LP decay is characterized by the decay time LP

=cav/ Ck2, where Ck

2 is the photon fraction of the LPdefined in Eq. 16. The cavity lifetime is cav

=1–100 ps. For a given cavity, LP increases with de-creasing photon fraction. LP is the shortest time scalefor most microcavities at low pumping densities, leavingthe system in nonequilibrium, sometimes with a relax-ation bottleneck.

LP-LP scattering is a nonlinear process; its time scaleLP-LP shortens with increasing LP density. When LP-LP

becomes shorter than LP, LPs overcome the energy re-laxation bottleneck and reach quantum degeneracythreshold. Due to the efficient LP-LP scattering amongstates below the bottleneck, these LPs thermalize amongthemselves before they decay, establish quasiequilib-rium, and form quantum degenerate BE distribution aspresented in Sec. V.A. However, LP-LP scattering con-serves the total energy of the LPs and thus does notreduce the temperature of the LP gas.

The only mechanism to cool the LPs is the LP-phononscattering. The linear LP-phonon scattering is a ratherslow energy relaxation process with a more or less fixedtime scale phonon=10–50 ps. Fortunately, in the quan-tum degenerate regime of nk01, LP-phonon scatter-ing can be significantly enhanced by the bosonic final-state stimulation. If phonon shortens to less than LP, LPsmay have enough time to reach thermal equilibriumwith the phonon bath within their lifetime.

C. Detuning and thermalization

To facilitate LP thermalization and eventually reachthermal equilibrium with the lattice, we need fasterLP-phonon scattering and slower LP decay. For this pur-pose, there exists a simple and useful controlling param-eter for the microcavity system: the detuning =Ecav−Eexc. At positive , the excitonic fraction Xk

increases,the LP effective mass is heavier, and the energy densityof states E 1− Xk

2−1 increases. Hence LP-phonon-scattering rates to low-energy statesXk,iXk0Ei as well as LP-LP scattering rates tolow-energy states Xk,i1

Xk,i2Xk,f1

Xk0Ei1Ei2

Ef1

become larger. The flatter dispersion also re-quires less phonon-scattering events to thermalize theLPs below bottleneck. At the same time, the LP lifetimeLPcav/ 1− Xk

2 becomes longer. All these are favor-able for LPs to reach thermal equilibrium.

The compromise is a higher critical density due to aheavier effective mass, stronger exciton localization, andshorter dephasing time of the LPs. At very large positivedetunings, these detrimental effects prevent clear obser-vation of polariton condensation.

D. Time-resolved momentum distribution

Along this line, Deng et al. 2006 measured how theLP momentum distribution depends on the cavity detun-ing and obtained evidence of a thermal-equilibrium LPcondensation at moderate positive detunings. In the ex-periment, a GaAs-based microcavity with 12 QWs and a3 ps pulsed laser excitation were adopted. Time-resolvedmomentum distribution was measured by a streak cam-era with a time resolution of 4 ps. From the time evo-lution of the LP ground-state population NLPk =0, theenergy relaxation time relax of the LP system was ex-tracted based on a two-mode rate equation model. Theinstantaneous momentum distribution NLPk , t is com-pared with BE and MB distributions Fig. 16.

The relaxation time relax decreases sharply near thequantum degeneracy threshold, reflecting the onset ofstimulated scattering into the LP ground state. At posi-tive detunings and at pump levels above the quantumdegeneracy threshold, relax shortens to one-tenth of thelifetime LP0 of the LP ground state, and thermal equi-librium with the phonon bath is established for a periodof about 20 ps, with a chemical potential LP of about−0.1kBT. Near resonance 1 meV, relax is compa-rable to LP0. The stimulated LP-LP scattering is still

0 1 2 3 40

0.5

1

1.5

k||

(104cm−1)

n(k

||)(a

.u.)

P/Pth

= 1.5

1 2 30

0.5

1

k||

(104cm−1)

n(k

||)(a

.u.) P/P

th= 0.6

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.70

0.5

1

1.5

2

P/Pth

(mW)

α

1 1.2 1.4 1.60

100

P/Pth

(mW)

TLP

(meV

)

dataBE fitMB fit

a

b

FIG. 15. Momentum space distribution of LPs. a The mea-sured LP population per state vs k stars, compared with BEsolid line and MB dotted line distribution functions atpump rates P /Pth=1.5 and 0.6 inset. At P /Pth=0.6, the fittedBE and MB distribution curves almost overlap. b The quan-tum degeneracy parameter vs pump rate P /Pth and the fittedeffective LP temperature TLP vs pump rate inset. The dashedlines are guides for the eye. From Deng et al., 2003.



effective to achieve a quasiequilibrium BE distributionamong LPs in k1 m−1 at a temperature TLP signifi-cantly higher than the lattice temperature Tlat with achemical potential LP=−0.01 to −0.1kBT. Finally, atlarge negative detunings, relax is longer than LP0, sothe system never reaches equilibrium.

A quantum degenerate LP system, capable of reach-ing thermal equilibrium, is also valuable for studying theevolution from nonequilibrium to equilibrium phases,the distinction between classical and quantum systems,and the formation and dynamics of the order parameterof a quantum phase.

E. Numerical simulations

Numerical studies show that, with certain parameters,the kinetically calculated momentum distributions nkt

of LPs can indeed be fitted by thermal distributionsabout 20–40 ps after the excitation, i.e., by the Maxwell-Boltzmann or Bose-Einstein distributions below orabove threshold Doan et al., 2005; Doan, Thien Cao, etal., 2008. Figure 17 shows a typical distribution calcu-lated for 45 ps after the excitation pulse and with a lat-tice temperature of 4.2 K. Also plotted is a fit with a BEdistribution. From the fit we obtain T=5.5 K and a de-generacy parameter − /kT=4.610−3. The goodness ofthe fit shows that the LP gas has reached quasiequilib-rium with well-defined temperature and chemical poten-tial. These results agree with the experiments aboveDeng et al., 2006.

From the fits of the measured and calculated distribu-tions one also obtains the still slowly varying parametersof the thermal distributions, namely, the temperature ofthe LP gas Tt=Tpol and its chemical potential t. Us-ing these fits with a cavity lifetime of c=2 ps and apump rate P /Pth=2, the evolution of the LP tempera-ture Tpolt normalized to the lattice temperature Tlat isshown in Fig. 18 for various values of the detuning. Onesees that the temperature of LP gas reaches for a detun-ing of =4.5 meV after about 60 ps. For lager times Tpolincreases again because the low-energy LPs have a

0 2 4 6 8 10

x 10−4

10−1

100

101

NLP

(k)

0 2 4 6 8 10

x 10−4

10−1

100

101

NLP

(k)

0 2 4 6 8 10

x 10−4

101

ELP

− ELP

(k=0) (meV)

NLP

(k)

(a)

(b)

(c)

E=kB

Tlat

FIG. 16. Color online LP population per state Nt ,E vs LPkinetic energy E−E0 for different detunings at a time whenTLP reaches Tmin: a =6.7 meV, P4Pth; b =−0.9 meV,P4Pth; and c =−3.85 meV, P6Pth. Crosses are datataken from the incidence direction of the pump, open circlesare from the reflection direction, solid lines are fitting by BEdistribution, and dashed lines by MB distribution. Parametersfor the curves are the following: a TMB=4 K, TBE=4.4 K,BE=−0.04 meV; b TMB=4 K, TBE=8.1 K, BE=−0.13 meV;and c TMB=8.5 K, TBE=182 K, BE=−0.4 meV. From Denget al., 2003.

FIG. 17. Snapshot of the calculated LP energy distributioncircles at 45 ps after the pulse excitation for a detuning of4.5 meV. The full line shows the fit with a Bose-Einstein dis-tribution with the temperature of 5.5 K and a chemical poten-tial with the value =−2.810−3 meV. From Doan, ThienCao, et al., 2008.

FIG. 18. Calculated normalized temperature vs time for threevalues of the detuning. =−1 triangles, 3 squares, and4.5 meV circles. From Doan, Thien Cao, et al., 2008.



smaller lifetime. This calculated relaxation kinetics ofTpol can be compared directly to the corresponding mea-surements of Deng et al. 2006, as shown in Fig. 19.Only where full symbols are shown, the gas was in goodthermal equilibrium. The measured temperature of theLP gas also shows an improved temperature relaxationwith increasing detuning. With a detuning of 6.6 meVthe lattice temperature is reached after about 40 ps.Even with a detuning of 9 meV the lattice temperatureis reached, while the numerical studies yield for thislarge detuning a less effective relaxation. This differenceis mainly due to the larger Rabi splitting of the realsystem compared to the model system. In most otherrespects, e.g., the final increase of the temperature, themeasured and calculated transient temperatures showthe same trends.

For the same parameters the calculated degeneracyparameter −t /kTpolt is shown in Fig. 20 and themeasured degeneracy parameters are shown in Fig. 21.Although quantum degeneracy is reached for smallerdetunings at a slightly shorter delay, it stays much longerfor larger blue detunings. The temporal range in whichdegeneracy exists extends over about 60 ps. The degen-eracy is lost because the LP density decreases due to

emission from the ground state. The calculated pumpdependences of the temperature and of the degeneracyparameter have been shown to agree qualitatively wellwith the corresponding measurements of Deng et al.2006. These findings show that for favorable param-eters a finite-size thermal-equilibrium BEC of LPs ispossible.

F. Steady-state momentum distribution

Steady-state condensation of LPs was first reported ina CdTe/CdMgTe microcavity with 16 QWs, operatingalso at positive detunings Kasprzak et al., 2006. In thisexperiment, the excitation pulse duration of 1 s is fourorders of magnitude longer than the decay and energyrelaxation times of the LPs. Therefore the time-integrated measurement reflects the steady-state proper-ties of the system. Above the quantum degeneracythreshold, the high-energy tail of the time-integratedmomentum distribution decays exponentially, indicatingquasiequilibrium among LPs at an effective temperatureof 19 K the lattice temperature is 5 K if neglectingheating by the excitation laser. The low-energy LP dis-tributions are not described by a simple model, which isattributed to the relatively strong LP-LP interactions. Asimilar measurement was later performed with a GaAs-based microcavity, where an in-plane harmonic potentialwas implemented and LPs condense into the potentialminimum. An effective LP temperature of 100 K wasmeasured Balili et al., 2007.

VI. ANGULAR MOMENTUM AND POLARIZATIONKINETICS

Up to now, the spin and orbital angular momenta ofexcitons have not been considered explicitly. In this sec-tion, we formulate a pseudospin formalism and reviewexperiments with circularly and linearly polarized pump-

FIG. 19. Color Measured normalized temperature vs time forvarious values of the detuning for GaAs microcavities with aRabi splitting of 14.4 meV and a normalized pump powerP /Pth=3 for a lattice temperature of 4.2 K. =−0.9 squares,3.6 triangles, 6.6 diamonds, and 9.0 meV circles. For theopen symbols a full local equilibrium is not yet reached. FromDeng et al., 2006.

FIG. 20. Calculated degeneracy parameter vs time for threevalues of the detuning. =−1 triangles, 3 squares, and4.5 meV circles. From Doan, Thien Cao, et al., 2008.

FIG. 21. Color Measured degeneracy parameter vs time forvarious values of the detuning measured for GaAs microcavi-ties with a Rabi splitting of 14.4 meV and a normalized pumppower P /Pth=3 for a lattice temperature of 4.2 K. =−0.9squares, 3.6 triangles, 6.6 diamonds, and 9.0 meV circles.For the open symbols, a full local equilibrium is not yetreached. The dash-dotted line is the limit of quantum degen-eracy with a ground-state population of 1. From Deng et al.,2006.



ings and polarization dependent detection Deng et al.,2003; Renucci et al., 2005; Kasprzak et al., 2006.

A. Formulation of the quasispin kinetics

Shelykh et al. 2005 reviewed the theory of the exci-ton angular-momentum Boltzmann kinetics in micro-cavities. The spin kinetics was numerically evaluatedtaking into account the LP-phonon scattering, but theeven more important LP-LP scattering was omitted. Re-cently Krizhanovskii et al. 2006 and Solnyshkov et al.2007 studied spin kinetics including the LP-LP scatter-ing for the case of resonant excitations. Here we formu-late the kinetic equations for a nonresonantly pumpedpolariton system.

We consider only optically active exciton states brightexcitons with the total angular momentum m= ±1. The22 reduced density matrix k,ij can be composed con-veniently using the Pauli matrices ij

l ,

k,ijt = bk,j† tbk,it 52

=Nkt

2ij +

lSk

l ti,jl , 53

x = 0 1

1 0, y = 0 − i

i 0, z = 1 0

0 − 1 . 54

The diagonal elements of k,ij corresponds to the distri-bution functions k,iit=nk,it and we have

Nkt = i=1

2

nk,it , 55

Skzt =

12

nk,1t − nk,2t . 56

The off-diagonal elements give the x and y componentsof the spin-polarization amplitude,

Skxt =

12

k,12t + k,21t , 57

Skyt =

i

2k,12t − k,21t . 58

According to Cao et al. 2008, the Hamiltonian of thesystem is given by

H = k,s

ekbk,s† bk,s +

12

k!kbk,1

† bk,2 + H.c. + q,qz

q,qzaq,qz

† aq,qz+

k,q,qz,sGk,q,qzaq,qz

† + a−q,qzbk+q,s

† bk,s

+14

k,k,q,s

Vk + q,k − q,k,kbk,s† bk,s

† bk−q,sbk+q,s + k,k,q,ss

Uk + q,k − q,k,kbk,s† bk,s

† bk−q,sbk+q,s

+ bk,s† bk,s

† bk−q,sbk+q,s + H.c. , 59

where aq,qz

† , aq,qzare the creation and annihilation opera-

tors of acoustic phonons, respectively. ek is the disper-sion of the LP branch. !k is a complex energy, related tothe TE-TM splitting of the microcavity. q,qz

is the fre-quency of the acoustic phonon, with the in-plane wavenumber q and the wave number qz perpendicular to thequantum well layers. Gk ,q ,qz is the LP-phonon cou-pling constant. The matrix elements Vk ,k ,k−q ,k+q and Uk ,k ,k−q ,k+q describe scattering of LPsin the relative triplet and singlet configurations, respec-tively. They are given by Shelykh et al., 2005

Vk,k,k − q,k + q = 6EBaB2 Xk+qXk−qXkXk,

Uk,k,k − q,k + q

= − Vk,k,k − q,k + q, 0, 60

Tk,k,k − q,k + q = Vk,k,k − q,k + q

Uk,k,k − q,k + q ,

where EB and aB are the binding energy and Bohr radiusof the exciton in two dimensions, respectively. Using thex and y components, we define a 2D polarization vector

Sk = Skx,Sk

y , 61

2SkSk = k,21k,12 + k,12k,21. 62

Similarly we decompose the TE-TM splitting into x andy components which form an effective 2D magnetic fieldwe absorb the g factor and magnetic moment in thiseffective field,

!k =!kx + i!k

y ,63



! k = !kx,!k

y ,

where the components of the splitting field have beengiven by Kavokin et al. 2004, 2005 as

!kx =!k

kx2 − ky

2

k2 =!k cos2 ,

64

!ky =!k

2kxky

k2 =!k sin2 .

Here is the polar angle of the in-plane momentumvector, i.e., kx=k cos, ky=k sin. The momentumdependence of the splitting field 64 is due to theTE-TM splitting of the microcavity modes, together withthe momentum dependence of the optical matrix ele-ments of the exciton longitudinal-and-transverse split-ting. From the equations of the motion, we get the ki-netic equations of the distribution function nk,i and in-plane pseudospin Sk. For nk,j,

nk,j

t= −

nk,j

k+ nk,j

t

S-F+ nk,j

t

mf-s-s+ nk,j

t

scatt

LP-ph

+ nk,j

t

scatt

LP-LP

+12

pjPk,t, j = 1,2. 65

Here the changes due to the splitting field are given by

nk,j

t

S-F= −

1

− 1jezSk! k . 66

The mean-field term due to the interaction of singletLPs is given by

nk,j

t

mf-s-s=

2

− 1j

qUk,q,k,qSk Sq . 67

For Sk,

Sk

t= −

Sk

ks+ Sk

t

S-F+ Sk

t

mf-LP-LP+ Sk

t

scatt

LP-ph

+ Sk

t

scatt

LP-LP

+12

epPk,t . 68

Here the mean-field term due to the LP-LP interactionis

Sk

t

mf-LP-LP=

2

q

Vk,q,k,q − Uk,q,k,q

fq,1 − fq,2ez Sk , 69

where ez is the unit vector along the growth axis and pjand ep denote components of the pump polarization inthe z direction and x ,y plane, respectively. p1=2, p2=0 for the circular pump and p1=p2=1 for the linearpump. The changes of the in-plane polarization due tothe splitting field are given by

Sk

t

S-F=

1

fk,1 − fk,2

2ez! k. 70

The in-plane pseudospin lifetime can be estimated asks

−1=k−1+sl

−1, where sl is the characteristic time of thespin-lattice relaxation. The scattering terms in Eqs. 65and 68 in the cylindrical approximation are put to-gether in the Appendix according to Cao et al. 2008.

As a simplifying approximation, we follow Cao et al.2008 and assume a cylindrical symmetry Kavokin etal., 2004 so that the polarization has only the radialcomponent Sk=Sk

r er and the splitting field is taken to beorthogonal to the polarization vector. This assumptiondoes not take into account certain directional propertiesof the splitting field, but such effects are relatively smallwith respect to the presently investigated aspects of thespin kinetics. Using this cylindrical symmetry we obtainthe following set of equations for nk,j and the radial com-ponent Sk

r of the in-plane spin polarization:

nk,j

t= −

nk,j

k+ nk,j

t

S-F+ nk,j

t

scatt

LP-ph

+ nk,j

t

scatt

LP-LP

+12

pjPk,t, j = 1,2, 71

Skr

t= −

Skr

ks+ Sk

r

t

S-F+ Sk

r

t

scatt

LP-ph

+ Skr

t

scatt

LP-LP

+12

epPk,t , 72

where the splitting field terms reduce to

nk,j

t

S-F= −

1

− 1jSk

r!k 73

and

Skr

t

S-F=

1

nk,1 − nk,2

2!k. 74

The mean-field terms due to the LP-LP interactions 67and 69 vanish in the cylindrical approximation due tothe angle integration. For the scattering terms in Eqs.71 and 72 in the cylindrical approximation we refer tothe appendix of Cao et al. 2008.

B. Time-averaged and time-resolved experiments andsimulations

To compare with experiments, we use x=30 ps, c

=2 ps, and sl=15 ps. The smallness parameter of theinteraction between excitons in the relative singlet stateis =510−2. The bath temperature is T=4 K. The de-tuning between the cavity mode and the exciton reso-nance is assumed to be zero. The pump is centered atkp=1.7105 cm−1 with a linewidth E=0.1 meV. Thepump pulse width is as in the experiment of Deng et al.2003, tp=3 ps.

First we examine the time-averaged ground-state LPpopulation for a circularly polarized pump pulse. Herewe use the total number of injected LPs as a measure ofthe pump power. As expected, the LP population at k



=0 with cocircular polarization exhibits a clear conden-sation threshold solid line in Fig. 22a, while the popu-lation with cross-circular polarization dashed line inFig. 22a stays subcritical up to a total LP density of1011 cm−2, where it is about four orders of magnitudesmaller than the cocircular density.

Next we calculate the ground-state LP population forleft- and right-circular polarizations when the pumppulse is linearly polarized. Figure 22b shows that dueto the TE-TM splitting the ground-state population forthe left-circular LPs solid line has a lower threshold,while the right-circular LPs also condense but at about afactor of 3 higher pump rate. At the highest pump rates,the ground-state populations of both circular polariza-tions become equal. Finally, we plot in Fig. 22c thecircular degree of polarization, i.e., the normalized dif-ference between the ground-state populations of theleft- and right-circular LPs for circularly and linearly po-larized pump light. For circular pump the degree of po-larization increases rapidly at the condensation thresh-old and approaches a value close to 1. For linearlypolarized pumping the degree of polarization increasesto a value of about 0.95 but decreases rapidly when theright-circular LPs condense and approach a value closeto 0 at the highest pump rates. Comparison with theexperimental results in Fig. 22 Deng et al., 2003 showsgood qualitative agreement. In particular, the absence ofa threshold for the cross-circularly polarized LPs for cir-cular pumping is clearly shown in Figs. 22a and 23a.Similarly the successive condensations of the left- andright-circular LPs separated by about a factor of 3 in

corresponding threshold pump rates in Figs. 22b and23b agree well. Finally, the degree of polarization forcircular pumping remains high in both theory and ex-periment see Figs. 22c and 23c, while the degree ofpolarization for linearly polarized pumping decreasesabove the threshold.

In Fig. 24 the calculated time evolution of the degreeof circular polarization for a circular pump is shown forvarious total LP densities. The results can be comparedwith corresponding measurements of Renucci et al.

FIG. 22. LP ground-state population vs total LP density underpumping a with circularly polarized and b with linearly po-larized light at T=4 K in a GaAs microcavity. The sum of thetwo ground-state populations is given by the dotted line, thespin-up LP population by the solid line, and the spin-downpopulation by the dashed line. c Circular degree of polariza-tion of light emitted from the ground state. For linearly polar-ized pumping the light intensity is given by the solid line forcircularly polarized pumping by the dashed line. From Cao etal., 2008.

1 10

0

10

100

n(k

||~0)

Pump Power (mW)

1 10 100

Pump Power (mW)

1 10 1000

0.2

0.4

0.6

0.8

1

Circ

ular

Deg

ree

ofP

olar

isat

ion

Pump Power (mW)

cirular pumplinear pump

co−circross−cirtotal

left−cirright−cirtotal

(a) (b)

(c)

FIG. 23. The measured time-averaged polarization propertiesof a GaAs microcavity. Emitted light intensity of the LPground state a for circularly polarized and b for linearlypolarized pump light. c The circular degree of polarization isplotted for pumping with linear and circular polarizations.From Deng et al., 2003.

FIG. 24. Calculated time dependence of the circular degree ofpolarization for circularly polarized near-resonant 3 ps pumppulses which excite a total LP density of Ntot=2.01010, 2.51010, 4.61010, and 6.01010 cm−2. From Doan, Cao, et al.,2008.



2005 as shown in Fig. 25, where a resonant excitationhas been used. In both theory and experiment, the de-gree of polarization decays rapidly below threshold butquickly approaches 1 around threshold. With increasingpump rate, the degree of polarization remains close to 1for about 30–70 ps, followed by a steep decay. Particu-larly striking is the difference in time evolution belowand above the threshold, as well as the trend of a longerlasting degree of polarization at higher pump rates.

C. Stokes vector measurement

The polarization state of emission, which has a one-to-one correspondence to the quasispin state of the in-ternal polariton condensate, is characterized by theStokes vector S1 ,S2 ,S3 defined as

S1 =I0° − I90°

I0° + I90°, S2 =

I45° − I−45°

I45° + I−45°, S3 =

IL − IR

IL + IR. 75

I0°, I90°, I45°, and I−45° are the intensities of the linearlypolarized components at detection angles d=0°, 90°,45°, and −45°, respectively, as defined in Fig. 26. IL andIR are the intensities of the left- and right-circularly po-larized components, respectively. The degree of linearpolarization DOLP and the angle of the dominant lin-ear polarization " are calculated as

DOLP = S12 + S2

2, " =12

tan−1S2

S1 . 76

The degree of circular polarization is simply given by theparameter S3.

In Fig. 27a, the measured intensity for the linearlypolarized light along d=0° and 90° is shown as a func-tion of pump power in units of the injected polaritondensity per pulse QW Roumpos, Lai, et al., 2009. Thepump is horizontally polarized p=90° . The threeStokes parameters defined above are plotted in Fig.27b. Notice that the circular polarization S3 developsjust above threshold but the linear polarization S1

takes over well above threshold. For circularly polarizedpump Fig. 27c, the emission is circularly polarized upto S3=−99.4%.

If we rotate the direction p of the linearly polarizedpump Figs. 27d and 27e, the polarization directionis always rotated by 90° from the pump polarization.Also, a circularly polarized component has S3 changingsign for varying p Fig. 27f. The sign change is corre-lated with the deviation of "−p from 90° as shown inFig. 27e.

The above quasispin dynamics can be understood interms of spin-dependent Boltzmann equations Roum-pos, Lai, et al., 2009. In particular, the quantum interfer-ence effect between the repulsive scattering among isos-pins and the attractive scattering among heterospins isresponsible for the observed 90° rotation of the linearpolarization shown in Fig. 27e. This is fully compatiblewith the discussion on the LP-LP interaction in Sec.III.B.

VII. COHERENCE PROPERTIES

In BEC, a macroscopic number of particles occupy asingle-quantum state and manifest quantum correlationson the macroscopic scale. The wave function of the con-

0 50 100 150

0.0

0.5

1.0

+excitation

Pin

(W.cm-2

)

Circula

rP

ola

rization

0.3

1

2.25

4

7.5

time (ps)

FIG. 25. Measured time dependence of the circular degree ofpolarization for circularly polarized pump pulses in GaAs mi-crocavities. From Renucci et al., 2005.

FIG. 26. The polarization measurement setup. Vectors labelthe fast or polarization axes of the optical components. Thelaser pump is initially horizontally polarized p=90° and isincident at an angle of 55° with respect to the growth directionz. Luminescence is collected along the z axis. The first variableretarder VR1 and linear polarizer LP1 work as a variableattenuator. By rotating a half wave plate HWP1, and using aremovable quarter wave plate QWP1, we can implementvarious polarization states for the pump. The second variableretarder VR2 is used as a zero, half, or quarter wave plate.The combination of a quarter wave plate QWP2, variableretarder VR3, and linear polarizer LP2 is used for detectionof a particular linear polarization state, depending on the re-tardance of VR3. Inset: Definition of angles p and d corre-sponding to the polarization axes of the pump and detection,respectively Roumpos, Lai, et al., 2009.



densate serves as the order parameter; its density matrixhas finite off-diagonal elements, often referred to as off-diagonal long-range order Penrose and Onsager, 1956;Beliaev, 1958; Yang, 1962, which can be measuredthrough the first-order coherence functions. Second- andhigher-order coherence functions of the state furthercharacterize the nature of a quantum state and distin-guish it from a thermal mixture Glauber, 1963; Gar-diner and Zoller, 2000. We review in this section studiesof the coherence functions of the LP condensates anddiscuss next special properties of the macroscopic quan-tum state, such as the first-sound mode and superfluiditydue to broken gauge symmetry, appearance of quantizedvortices, and phase locking among multiple condensates.

A. The second-order coherence function

The temporal intensity-intensity correlation functionis the lowest-order coherence function which character-izes the statistical nature of a quantum state. It has beenof crucial importance in characterizing the coherence

properties of laser light. For polaritons, it was first mea-sured by Deng et al. 2002 for a GaAs microcavity with2 ps photon lifetime. Recently it was measured for aGaAs microcavity with 80 ps LP lifetime by Roumpos,Hoefling, et al. 2009 and also for a CdTe microcavity byKasprzak et al. 2008. A first attempt to explain thesecond-order coherence measurements was given byLaussy et al. 2004, 2006 in terms of a coupled Boltz-mann equation and master equation kinetics. The modelwas extended by Doan, Cao, et al. 2008 to reach betteragreement with experiments.

The time-domain second-order coherence functiong2 is defined as Glauber, 1963

g2 =E−tE−t + E+t + E+t

E−tE+t 2, 77

where E−t and E+t are the negative and positivefrequency parts of the electric-field operator at time t,respectively. g2 measures intensity correlation of the

field between time t and t+. If Et and Et+ areuncorrelated but stationary,

g2 =E−tE+t E−t + E+t +

E−tE+t 2= 1.

For a single-mode state, the maximum correlation oranticorrelation is obtained at =0, i.e., by g2=0which has the following property:

g20 = 2, thermal state,

g20 = 1 −1

n, number state n ,

g20 = 1, coherent state.

Thus defined normalized second-order coherence func-tions are independent of linear losses between thesource and detector. Moreover, while spectrally filteredthermal light can be brought into interference after aMichelson interferometer, its g20 value remains un-changed. These properties make g20 especially usefulin identifying the quantum-statistical nature of a state.

Caution is needed, however, given the finite experi-mental time resolution. When is much longer the co-herence time c of the field, g2 decays to 1. Thus adetector time resolution much larger than c will makeaccurate measurement of g20 difficult.

1. g(2)(0) of the LP ground state

Deng et al. 2002 first measured the second-ordertemporal coherence function of a condensate. A Han-bury Brown–Twiss-type setup Hanbury Brown andTwiss, 1956 is adopted, mainly consisting of a 50/50beam splitter and two single-photon counters Fig. 28.In this pulsed experiment, since the polariton dynamicsis much faster than the time resolution of the photon

FIG. 27. Color Measurement of Stokes parameters markerscompared with the theoretical model solid lines. a Horizon-tal pumping p=90° . Collected luminescence for d=0° redsquares and d=90° blue circles vs injected particle densityin m−2 per pulse per QW. A clear threshold is observed at5102 m−2. Degree of polarization measurement for b p=90° linear pumping and c left circularly polarized pumping.Blue circles: S1, green diamonds: S2, and red squares: S3 de-fined in Eq. 75. Calculated polarization parameters from themeasurement of the Stokes parameters for linear pumpingEqs. 75 and 76. d DOLP. e Angle for major axis oflinear polarization " relative to p. f Degree of circular po-larization S3 Roumpos, Lai, et al., 2009.



counters, essentially the integrated area under eachpulse was measured, which gives the numerator in thefollowing equation:

g2j =n1in2i + j i

n1 n2 , 78

where n1i and n2i+ j are the photon numbers de-tected by the two-photon counters in pulses i and i+ j,respectively. In the limit of low average count rates n0.01 per pulse in our experiment, g2j approximatestime-averaged g2 over each pulse. For j0, coinci-dence counts are recorded from adjacent pulses whichare uncorrelated and thus g2j1 for j0. Assumingthe same statistical properties of all pulses, the area ofthese peaks at j0 gives the denominator in Eq. 78.

The emission pulse of the LP ground state was firstfiltered by a monochromator with a resolution of =0.1 nm, resulting in a pulse with a coherence time ofc=8 ln 22 /c4 ps. Below threshold, the fullwidth at half maximum FWHM temporal width of thepulse is about 350 ps, much longer than the coherencetime of the pulse c4 ps. Since the measured g20 isan integration of g20 over the whole pulse as given inEq. 78 and g21 during most of the pulse, g20 isalso close to 1 even though the system is expected to bein a thermal state. Poor time resolution is a commonlimitation in Hanbury Brown–Twiss-type measurements.

Fortunately, at threshold, the pulse width shortens to8 ps and becomes comparable to c. Hence g20becomes a good approximation of g20. Here bunchingof the emitted photons was observed, with a maximumg20 of 1.77 at P /Pth1.1 Fig. 29. This bunching ef-fect, which is obscured by the time-integration effect be-low threshold, is the signature of bosonic final-statestimulation. Far above threshold, the pulse width onlydecreases further, therefore g20 remains a good esti-mate for g20. However, the g20 measured in thisregion decreases, which demonstrates the formation ofsecond-order coherence in the system or, in other words,the polariton gas starts to acquire macroscopic coher-ence.

g20 decreases only very slowly with increasing den-sity above threshold, indicating a relatively small con-densate fraction which builds up only gradually with in-creasing total density of LPs. This is typical of two-dimensional systems with relatively strong interactions,where there is large quantum depletion of the conden-sate. Theoretical and numerical studies have predictedthe same qualitative behavior Sarchi et al., 2008;Schwendimann and Quattropani, 2008.

Recently a similar measurement was performed with aCdTe-based microcavity Kasprzak et al., 2008. In thisexperiment, emission from the LP ground state, not fil-tered in energy, has a coherence time of 2 ps belowand around threshold. g20 close to 1 was observed forexcitation densities from below the threshold Pth to2Pth. At higher excitation densities, g20 increasessignificantly to 2 at 10Pth.

Kasprzak et al. 2008 also measured g20 underquasi-continuous-wave excitation with an excitationpulse duration of 1 s. In this case, the time reso-lution is limited by that of the photon counters, which is120 ps. A small bunching with gm

201.03 was mea-sured below and around Pth. If correcting for the coher-ence time and photon counter resolution, it correspondsto an actual g202.8±1. No bunching above thenoise level was observed at 1.5Pth, which indicates a de-crease in g20. gm

201 is again measured above2.3Pth, consistent with the pulsed measurement.

2. Kinetic models of the g(2)

Using a stochastic Boltzmann kinetic equation andmaster equation, we study the probability to find n LPsin the ground state Doan, Cao, et al., 2008. Thequantum-statistical correlation function g2 is given bythe correlations of the condensate operators Glauber,1963

g2 =b0

†tb0†t + b0t + b0t

b0†t + b0t + b0

†tb0t . 79

Note that the field operators are in the normal form withall creation operators to the right. In the number repre-sentation for zero delay

FIG. 28. Color online A realization Hanbury Brown–Twissmeasurement with single-photon counters. For autocorrelationmeasurement, use a single-core fiber, beam splitter, and mirrorsolid path. For cross correlation, use a double-core fiberbundle to carry the two input and output signals dashed path.

10 201.0

1.2

1.4

1.6

1.8

2.0

0 400 800

0.51.01.52.0

0 400 800

0.51.01.52.0

P ump Intens ity P/Pth

P/Pth=1

C hannel

(0)

g(2

)

P/Pth=15

Number

FIG. 29. The second-order coherence function g20 vs pumpintensity P /Pth. From Deng et al., 2002.



g20 =n0n0 − 1

n0 2 . 80

The mean condensate density n0t = b0†tb0t is

determined by a Boltzmann equation which governs thescattering kinetics in and out of the ground state,

n0 t

= Rint1 + n0t − Routtn0t . 81

The rate in is given by the scattering processes from theexcited states to the ground state by LP-LP and LP-phonon scatterings,

Rin = k,k

w0,k;k,k−kLP-LP 1 + nknknk−k

+ q,=±1

w0,qLP-phnqNq,−. 82

The rate out is given by

Rout =1

0+

k,k

w0,k;k,k−kLP-LP nk1 + nk1 + nk−k

+ q,=±1

w0,qLP-ph1 + nqNq,−. 83

Formally one has exchange rates due to the scatteringprocess k ,0→0,k . These exchange rates do not give riseto changes of the populations and thus should not beincluded in the kinetics. The mean rate equation canthen be written as

n0 t

= Rint 1 + n0t − Routt n0t . 84

To calculate the second moments of the condensate, weneed a stochastic extension of the condensate’s kineticequation. There are at least four possible formulationsof such an extension:

• One can supplement the Boltzmann equation for n0by Langevin fluctuations with shot-noise characterHaug and Haken, 1967; Lax, 1967; Haug, 1969. Thesecond moments of density are then determined bythe second moments of these shot-noise fluctuations.As it is known from laser theory, Langevin equationscan be solved approximately by linearization belowand above threshold, but there is no simple way tofind solutions in the whole density regime.

• This is different for the associate Fokker-Planckequation in which the probability for a certain den-sity value is calculated. In stationary equilibrium, onegets an analytic solution in terms of an exponentialof a generalized Ginzburg-Landau potential seeRisken 1984. The change of the potential from be-low threshold with a minimum at n=0 to a potentialwith a sharp minimum at n0 above threshold de-scribes the probabilities for all densities. In both ap-proaches, the Langevin and Fokker-Planck methods,the density is considered to be a continuous variablewhich is valid if n1. Because in a BEC the conden-

sate density varies from small values of n1 belowthreshold to large values above n1 this approxima-tion is not well justified below and at thresholdwhere n=O1.

• The third possible stochastic extension of theground-state kinetics is the master equation of theprobability to find n particles in this state. The mas-ter equation takes into account the discrete nature ofthe number of condensed particles Carmichael,2003 but disregards the phase of the condensedstate. This semiclassical approach has been studiedby Laussy et al. 2004. They calculated g20 forCdTe microcavities from numerical solutions of thetime-dependent master equation and considered LP-phonon and LP-electron scatterings. Here we use awell-converging continued fraction method for thesolution of the stationary master equation and calcu-late the results for GaAs microcavities, includingboth LP-phonon and LP-LP scattering. The resultsare compared with experimental observations Denget al., 2002.

• Still another possibility is to treat the kinetics of thesecond-order function b0

†tb0†tb0tb0t in addi-

tion to the kinetics Sarchi et al., 2008; Schwendi-mann and Quattropani, 2008.

3. Master equation with gain saturation

We now formulate the master equation for the prob-ability Wnt to find n particles in the condensate at timet, dropping the index 0 to ease the notation. With thegain saturation rate Gn=Rn

inn+1 and decay rate Dn

=Rnout, we have the master equation

dWn

dt= − Gn + DnWn + Dn+1Wn+1 + Gn−1Wn−1

= m=n−1

m=n+1

Mn,mWm. 85

Mn,m is a tridiagonal matrix Risken, 1984. Its elementsare

Mn,n = − Gn − Dn, Mn,n+1 = Dn+1, Mn,n−1 = Gn−1.

86

The rate equation does not provide the full informa-tion for constructing the corresponding master equationbecause the transition rates are only known for themean n but not for n see, e.g., Carmichael 2003.Systematically, one would have to determine the popu-lations of the excited states as a function of the pumprate and the number of LPs in the ground state. It can bedone analytically for a homogeneously broaden three-level laser but only numerically in our case from thesolution of the Boltzmann equation. Simple proceduresto include this correction Laussy et al., 2004 are to ex-pand the distributions of the excited states linearlyaround n as nkn= nk + nk /nn− n and estimate



nk nn

n

=nk

n − N, 87

where N is the total number of polaritons. This yields

nk n = nk 1 +n − n n − N

. 88

These correction terms of the scattering rates in themaster equation vanish approximately if one calculateswith the modified master equation and with n =nnWn. We use these corrections of the scattering ratesin the linear approximation. Because nk 1 for all ex-cited states, stimulated scattering into these states is neg-ligible and 1+ nk n . Thus the LP-phonon-scatteringrate Rin

LP-ph has a correction term Fn= n− n / n −N, while the LP-LP scattering rate Rin

LP-LP has a cor-rection factor 2Fn. In the rate out, only the LP-LPscattering rate Rout

LP-LP is corrected by a factor Fn, whileother rates remain unchanged.

4. Stationary solution

Under stationary pumping, if a stationary state isreached, the rates in and out of the ground state nolonger change with time. The stationary solution ismaintained if the rates between two successive statesbalance. From Fig. 30 we see that the following detailedbalance relation holds:

DnWn = Gn−1Wn−1, 89

which yields the detailed balance solution

Wn =Gn−1

DnWn−1 =

k=1

k=nGk−1

DkW0. 90

W0 is obtained from the normalization nWn=1.In general, the stationary solutions are obtained from

m=n−1

m=n+1

Mn,mWm = 0. 91

Following Risken 1984, we introduce the ratio Sn=Wn+1 /Wn and rewrite Eq. 91 as

Mn,n + Mn,n+1Sn +Mn,n−1

Sn−1= 0. 92

By changing n→n+1, we obtain

Sn = −Mn+1,n

Mn+1,n+1 + Mn+1,n+2Sn+1. 93

Equation 93 is well suited for numerical iteration. Byanalytical iteration one obtains continued fractions forthe ratio Sn and Wn is then given by

Wn = k=0

k=n−1

SkW0. 94

The value of W0 follows again from the normalizationcondition. One can start the iteration at a value N n when WN+1=WN+2= ¯ =0, so that SN=0. This iterativeprocedure can also be generalized for the solution of thetime-dependent master equation Risken, 1984 wheredetailed balance is no longer fulfilled.

5. Numerical solutions of the master equation

The stationary probability distributions obtained byiteration are shown in Fig. 31 for three pump powers.Below threshold, the distribution peaks at n=0 and de-cays monotonically. Slightly above threshold, the distri-bution peaks at a value n 1 but is still rather broad.Well above threshold for P /Pth=4.1, the distributionpeaks sharply around a larger density value of aboutn 104. We checked that the solutions obtained itera-tively agree indeed with those obtained from the de-tailed balance condition.

Once all Wn are calculated including the normaliza-tion W0, one obtains g20= nn−1 / n 2. We see inFig. 32 how the thermal limit with g20=2 is smoothlyconnected with g20=1. Surprisingly, the transitiontakes place not in the threshold region but around apump power of P2Pth. This delayed transition fromthe thermal to the coherent limit is due to the semiclas-sical approach which assumes an n-diagonal density ma-trix.

As discussed, the fast decay of the correlations abovethreshold is not observed in the experiments Deng etal., 2002. This can be understood qualitatively, takinginto account that g2 varies as g2=2eRin−Rout asone can show, e.g., in a Langevin approach. An averageover the delay times yields g20g20 / Rout−Rin. Asthe pump power decreases away from threshold Rout

W

W

W

D

D G

n

n+1

n−1

n n−1

n+1Gn

FIG. 30. Level diagram for master equation.

FIG. 31. Calculated distribution functions Wn for various nor-malized pump powers P /Pth. From Doan, Cao, et al., 2008.



−Rin increases. Thus the averaged g20 decreases rap-idly below threshold, as seen in the experiment with thestrongly fluctuating 2 ps GaAs microcavities. In a recentexperiment with improved GaAs microcavities Roum-pos, Hoefling, et al., 2009, the minimum of g2 as a func-tion of pump power seems to occur at threshold, and therange of correlations above the Poisson limit extendsover a much wider pump range. Similar observationshave been made in CdTe microcavities Kasprzak et al.,2008. Further many-body effects have to be found inorder to account for the difference between these ex-perimental findings and the theory.

6. Nonresonant two-polariton scattering (quantum depletion)

One way to improve the g2 calculations is to includequantum depletion processes. Following the idea ofSchwendimann and Quattropani 2008, one can con-sider a quadratic correction term in the form of a scat-tering of two condensate LPs to the excited states withthe momenta q and −q. This process is nonresonant andonly takes place when the linewidth is finite. The corre-sponding transition rate is

q

w0,0;q,−qLP-LP nn − 11 + nq1 + n−q

− 1 + n2 + nnqn−q , 95

with n1. The energy-conserving delta function has tobe replaced, e.g., by a Gaussian resonance

22e0 − 2eq ⇒2

e−4e0 − eq2/2

. 96

This yields two-particle transition rates of the form

Gn2 = Rin,2n + 1n + 2 ,

97Rin,2 =

qw0,0;q,−q

LP-LP nqn−q

and

Dn2 = Rout,2nn − 1 ,

98

Rout,2 = q

w0,0;q,−qLP-LP 1 + nq1 + n−q .

The last finite decay rates are D12=D0

2=0 and D22

=2Rout,2 because at least two LPs are needed in theground state for a quadratic scattering process out of theground state.

The two quadratic terms extend the three-diagonalmaster equation to a five-diagonal one,

dWn

dt= − Gn + Dn + Gn

2 + Dn2Wn + Dn+1Wn+1

+ Dn+22 Wn+2 + Gn−1Wn−1 + Gn−2

2 Wn−2

= m=n−2

m=n+2

Mn,mWm. 99

Solving this pentadiagonal master equation iterativelyfor details see Doan, Thien Cao, et al. 2008 one findsfor realistic values of the collision broadening a rathersmall increase of g2 above threshold. In contrast,Schwendimann and Quattropani 2008 found additionalcorrelations above the coherent limit and a minimum ofg2 around threshold, as has been observed, under theassumption of a thermal exciton reservoir. If the kineticsof the excited states is included Sarchi and Savona,2008, the minimum in the threshold region is nearlyeliminated but the extra correlations remain. The differ-ences between the results of Doan, Thien Cao, et al.2008 and Sarchi and Savona 2008 could be partly dueto the fact that the quadratic corrections have been ne-glected in the Boltzmann equation.

In order to give an alternative explanation, Doan,Thien Cao, et al. 2008 considered a phenomenologicalkinetic saturation model. It is known that the 2D exci-tons ionize once their density is so high that the excitonwave functions start to overlap Schmitt-Rink et al.,1985. The critical saturation density due to phase-spacefilling has been evaluated from the reduction of the ex-citon oscillator strength to be 1/ns= 32

7 a2D2 . This will

also result in a reduction of the exciton density n ton / 1+ n /ns2. In order to incorporate the saturation ef-fect kinetically in the master equation, we introducephenomenologically a two-polariton ionization rate.Due to the overlap of the wave functions in a collision oftwo polaritons, the two polaritons are assumed to byionized and thus lost for the condensate,

Dn2 =

1

0nn − 1

rs

ns. 100

We have included a small numerical factor rs in order toavoid an unphysically large ionization rate far belowsaturation. We used the lifetime 0 to set the scale forthe ionization rate. Any direct backscattering from theionization continuum to the condensate can be ne-glected.

In Fig. 33 the results are plotted for the saturationmodel. We see that with increasing interaction strengthrs the range where g21 extends to larger pumping

FIG. 32. Calculated second-order coherence function g20 vsnormalized pump power P /Pth. The full line is obtained byiteration and the circles by the detailed balance relation. FromDoan, Cao, et al., 2008.



rates as has been observed in the experiment Deng etal., 2002. This simple model brings the results closer tothose of the experiment.

B. First-order temporal coherence

The first-order temporal coherence is characterized bythe energy linewidth of the LP emission. Interestingly,the linewidth above threshold does not decrease accord-ing to the Shallow-Townes formula but increases againPorras and Tejedor, 2003 with increasing density of theLPs in the ground state. This observed effect can becalculated by the Langevin equations for the Bogoliubovmodel of weakly interacting bosons. The LP populationin the ground state gives rise to a Kerr-effect-like behav-ior which in turn causes the linewidth increase Tassoneand Yamamoto, 2000.

C. First-order spatial coherence

First-order spatial coherence or the so-called off-diagonal long-range order ODLRO is studied by Pen-rose and Onsager 1956, Beliaev 1958, and Yang1962.

1. Long-range spatial coherence in a condensate

The Bose field operator #r of an ideal gas is definedas

#r =1

V

iaie

ipir =a0eip0r

V+

1V

i0

aieipir

=#0r +#Tr , 101

where r is the coordinate vector and pi is the momentumvector of the state i. #Tr consists of terms of the ex-cited states. In the thermodynamic limit of V→ , #Trvanishes for r0. #0r is the ground-state term. In acondensed phase, a0 can be approximated by a c-numberamplitude and #0r has a finite and constant amplitudegiven by the square root of the condensate fraction.Thus #r also serves as the order parameter of BEC.

The off-diagonal element of the first reduced densitymatrix of #r is

1r,r = #†r#r

= #0†r#0r + #T

† r#Tr

=N0

Veip0r−r +

1

V i0

ai+ai eipir−r. 102

1r ,r describes the first-order coherence between thefields at positions r and r. We again can separate it intoterms of the excited states and a term from the groundstate. When r−r→ , the phase pir−r of theexcited-state terms becomes random, and the summa-tion of these terms vanishes when V→ . The ground-state term, however, has a finite and constant amplitudeof 0n0n in the BEC phase regardless of the separa-tion r−r. Hence 1r ,r=n0= #02 is finite and uni-form even over macroscopic distances. In other words,1 has an ODLRO. This is an essential character ofBEC.

In two dimensions, long-wavelength thermal fluctua-tions of the phase destroy a genuine long-range order inthe thermodynamic limit Mermin and Wagner, 1966;Hohenberg, 1967. But quasi-BEC is established whencoherence is established throughout the system size inan interacting Bose gas Lauwers et al., 2003, such asmanifested in extended spatial coherence.

2. Spatial coherence among bottleneck LPs

Spatial coherence of LPs was first studied by Richardet al. 2005. In this experiment, a CdTe-based microcav-ity was used. Above a threshold excitation density Pth,LP lasing was observed in bottleneck states with in-plane wave number k =kbot and energy Ekbot. The 2DFourier plane image of the LPs was uniform at PPth.At PPth, speckles appeared in the images, which wasrelated to an increase in spatial coherence among the LPemitters. At the same time, the emission linewidth atEkbot was broadened in k, and interference in k wasobserved by a Billet interferometer setup. It again indi-cated increased spatial coherence Fig. 34.

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5

g2 (0)

P/Pth

rs=10-1

rs=10-2

rs=10-6

rs=0

FIG. 33. Calculated second-order coherence function g20 vsnormalized pump power P /Pth for the kinetic saturationmodel.

FIG. 34. Color online Interference fringes between two frag-ments 6 m apart, measured below open triangles andabove open circles the nonlinear threshold, with contrasts of5% and 46%, respectively. Horizontal axis is the relativephase. Adapted from Kasprzak et al., 2006.



3. Spatial coherence between fragments of a condensate

Later Kasprzak et al. 2006 measured directly the spa-tial coherence of a fragmented condensate. LPs con-densed into the k 0 state but at multiple localizedsmall areas of 4 m in diameter within the 20–30 mexcitation spot. PL from different localized condensateswas brought into interference after a Michelson interfer-ometer. The fringe visibility gives the first-order coher-ence function g1r,

g1r =E−r0E+r0 + r

E−r0E+r0 E−r0 + rE+r0 + r

=1r0,r0 + r

1r0,r01r0 + r,r0 + r. 103

Interestingly, between two fragments of condensate r6 m apart, g1r increased sharply across the thresh-old from 5% to 8% below threshold to a maximumvalue of 46% at 2Pth Fig. 34. This verifies that al-though local disorder caused spatial fragmentation ofthe LP gas, different fragments are phase locked witheach other.

4. Spatial coherence of a single-spatial mode condensate

A more systematic study of spatial coherence was car-ried out by Deng et al. 2007. A GaAs-based microcav-ity was used, where the LP condensate forms a single-spatial mode. A double slit was placed in the near-fieldimage of the LPs, which passes PL from different spatialregions of the condensate. Far-field interference be-tween light through the two slits was then recorded,which consists of a cosine oscillation imposed on a sincfunction envelope due to the finite slit width. The ampli-tude of the cosine oscillation corresponds to g1r, rgiven by the slit separation.

g1r increases sharply across the threshold and satu-rates at P3Pth. The maximum g1r of 0.8 was mea-

sured for r=1.3 m, which is approximately the intrinsiccoherence length of a single LP. At r=8 m, which is thesize of the excitation spot, g1r remains above e−.From r=1.3 to 8 m, decay of g1r is well modeled bythe Fourier transform of the momentum distribution ofLPs. The e− decay length of g1r was about eighttimes the thermal de Broglie wavelength above thresh-old Fig. 35. This demonstrates that coherence extendsto a much longer range than both the average LP-LPspacing and LP’s intrinsic coherence length. Finite co-herence is maintained throughout the condensate. Atthe same time, there is a significant thermal excitationfrom the condensate.

5. Theory and simulations

An early attempt to calculate the first-order spatialcorrelation function is due to Sarchi and Savona 2007a,2007b, 2008. With the condensation kinetics they ob-tained relatively large condensate densities which re-sulted in a strong off-diagonal long-range order. Theyincorporated many-body effects using the Popov ap-proximation Shi and Griffin, 1998 in order to calculatethe depletion of the condensate. Here we calculate g1

using nkt and n0t obtained through theBoltzmann equations.

We write the first-order spatial coherence function interms of the LP field operator "r as

g1r1,r2 ="†r1"r2

"†r1 "r2 . 104

Inserting a plane-wave expansion

"r =1Sb0 +

kbkeik ·r , 105

we find in the free-particle approximation

g1r1,r2 =

n0 + k

nkeik ·r1−r2

n0. 106

With the calculated populations we can now evaluatethe pump power dependence of g1r= r1−r2 for vari-ous values of the distance r. We use a positive detuningof 4 meV and a system size of 100 m in cross section.The pump power P is normalized to the threshold valuePth. Figure 36 shows how the range of the spatial coher-ence increases above threshold. These results are in ex-cellent agreement with the double-slit experiment byDeng et al. 2007 as shown in Fig. 37.

As a next test we plot the distance dependence ofg1r in Fig. 38 for various values of the pump power.We see a more or less exponential decay of the coher-ence function. The coherence length increases withpump power, i.e., with the degeneracy of the condensedLPs. The corresponding measurements shown in Fig. 35are again in quantitative agreement with the calculatedvalues of the coherence function.

0 2 4 6 810

−2

10−1

100

r (µm)

g(1) (r

)

6.72.01.71.3

P/Pth

1/e

FIG. 35. Color online g1r vs r for different pump ratesP /Pth as given in the legend. The symbols are measured g1r.The solid lines are fits by the modified Bessel function of thefirst kind, corresponding to the 2D Fourier transform of themomentum distribution of kBT / Ek−. The dashed lineat P /Pth=7 is fit by the 2D Fourier transform of a Maxwell-Boltzmann distribution. The dash-dotted line marks whereg1r=1/e Deng et al., 2007.



D. Spatial distributions

The presence of a condensate with a long coherencelength changes the spatial distribution of LPs in charac-teristic ways. If there is a harmonic potential, the spatialdistribution features a Thomas-Fermi profile, and thewidth of the distribution scales inversely proportional tothe width in the momentum space. In the absence ofin-plane confinement potential, the spatial propertiesare governed by two factors: the spatial distribution dueto the profile of the excitation laser and the finite exten-sion of the macroscopic coherence as a 2D system.

1. Condensate size and critical density

If the excitation laser has a nonuniform profile, suchas a Gaussian with a FWHM spot size of p, the centerof the spot reaches the degeneracy threshold first andforms a quasicondensate with a spatial extension cp. With increasing excitation density, c also in-creases. If the critical density is constant independent ofthe system size, c will expand as

c = p1 − log21 +Pth

P . 107

In two dimensions, however, a higher critical densityof condensation ncc is required to maintain coherenceover a larger area Ketterle and van Druten, 1996. Thiseffect was first observed by Deng et al. 2003. A planar

microcavity was used without in-plane confinement po-tential. The excitation laser has a Gaussian profile with aFWHM spot size p15 m. While the LP emissionfollows the same profile below threshold, the emissionspot size PL shrank steeply when the center of the spotreached the critical density locally. At PPth, the in-crease of PL is much slower than predicted by Eq.107.

Taking into account the size dependence of the criticaldensity, Eq. 107 is modified as Deng et al., 2007

P/Pth = p1 − log21 +1

P/Pth

ncP/Pthncc

.

108

Here cP /Pth=1 is the FWHM of the condensatewhen it first appears and corresponds to the smallestmeasured size. As a simplified model, consider nc for a2D boson gas confined in a finite size L=2 Ketterleand van Druten, 1996: nc= 2/$T

2 ln2 /$T. Oneobtains

P/Pth = p1 − log21 +Pth

P

ln2/$Tln2c/$T . 109

Here $T is the LP thermal de Broglie wavelength.The measured increase of the LP spot size was well de-scribed by Eq. 109, as shown in Fig. 39 Deng et al.,2007.

2. Comparison to a photon laser

In the case of a photon laser Deng et al., 2003, areduction in the spot size was also observed at the lasingthreshold. In sharp contrast to the LP condensation, theincrease of the spot size above threshold is much fasterand was explained well by the classical local oscillatormodel of Eq. 107 Fig. 39c. This confirms that in aconventional photon laser gain is determined by localdensity of the electron-hole pairs with a threshold den-sity independent of the system size.

Moreover, there are multiple transverse modes in thespatial profile above threshold Fig. 39b, as is typicalin large-area vertical cavity surface emitting lasers. Co-herence in this case is present only in the photon field

FIG. 36. Calculated first-order coherence function g1r vsnormalized pump power P /Pth for various distances r. FromDoan, Cao, et al., 2008.

FIG. 37. Color online Measured first-order coherence func-tion g1r vs normalized pump power P /Pth for various dis-tances r. From Deng et al., 2007.

FIG. 38. Calculated first-order coherence function g1r vsdistances r for various normalized pump powers P /Pth. FromDoan, Cao, et al., 2008.



but not in the electronic media. In the case of polaritoncondensation, however, a uniform Gaussian profile ismaintained up to very high pump rates without obviousmultiple transverse modes.

VIII. POLARITON SUPERFLUIDITY

The occurrence of BEC was originally predicted foran ideal gas of noninteracting bosonic particles Ein-stein, 1925. However, particle-particle interaction andresulting peculiar excitation spectra are keys for under-standing BEC and superfluidity physics Tilley andTilley, 1990; Pitaevskii and Stringari, 2003. A quantumfield-theoretical formulation for a weakly interactingBose condensed system was first developed by Bogoliu-bov, which predicted the phononlike linear excitationspectrum at low-momentum regime Bogoliubov, 1947.The experimental verification of the Bogoliubov theoryon a quantitative level was first performed for atomic

BEC using two-photon Bragg scattering techniqueStamper-Kurn et al., 1999. The particle-particle interac-tion and the Bogoliubov excitation spectrum have beenstudied theoretically for exciton polaritons Shelykh etal., 2005; Sarchi and Savona, 2008. Due to a short po-lariton lifetime, it is expected that the mode becomesdiffusive and the dispersion is flat in a small momentumregime Szymanska et al., 2006; Wouters and Carusotto,2007. In this section, we describe the observation ofinteraction effects on the condensate mean-field energyand the excitation spectra, which are in quantitativeagreement with the Bogoliubov theory. The slope of thelinear dispersion at the small momentum regime corre-sponds to a superfluid critical velocity according to Lan-dau’s criterion Landau and Lifshitz, 1987. It is eightorders of magnitude faster than that for atomic BECStamper-Kurn et al., 1999. More direct evidence of po-lariton superfluidity came from recent observations of aquantized vortex pinned to a potential defect La-goudakis et al., 2008 and a vortex-antivortex bound pairin a finite-size polariton BEC Roumpos, Hoefling, et al.,2009. Different from superfluidity arising from LP-LPinteractions in a condensate, a superfluid LP was alsocreated by coherent parametric pumping Amo et al.,2009.

A. Gross-Pitaevskii equation

The Hamiltonian of the exciton-polariton condensateexpressed in terms of the LP field operator " is

H = 2

2m "+ "dr

+12 "+"+Vr − r""drdr , 110

where Vr is the two-body interaction potential. The

Heisenberg equation of motion of " is

i

t"r,t = "r,t,H

= −22

2m+ "+r,t

Vr − r"r,tdr"r,t . 111

If the field operator is written in the form

"r =1

Vp

apeip·r/, 112

where V is a quantization volume and ap is the annihi-lation operator for the single-particle state with a mo-mentum p, Eq. 110 can be rewritten as

FIG. 39. Color Spatial distributions and spot sizes of a polar-iton laser vs a photon laser. a and b Spatial profiles of LPsand lasing cavity mode at 1.4 times the threshold pump pow-ers, respectively. c Comparison of the expansion of the spotsize vs pump rate for the LP condensate circles and the pho-ton laser stars. The red dashed line is fit by Eq. 108, withp=16.5 m, and c=2.8 m. The green dotted lines are fit byEq. 107 assuming a pump spot size of p=9 m for the LPcondensate and p=23 m for the photon laser. From Deng etal., 2003.



H = p

p2

2map

+ap +1

2Vp1

p2

q

Vqap1+q+ ap2−q

+ ap1ap2

,

113

where Vq=Vrexp−iq ·r /dr is the Fourier transformof Vr.

As far as the macroscopic properties of the conden-sate system are concerned, only small momenta are in-volved so that we are allowed to consider the q=0 valueof Vq,

V0 = g = Vrdr , 114

and to rewrite the Hamiltonian in the form

H = p

p2

2map

+ap +g

2Vp1

p2

q

ap1+q+ ap2−q

+ ap1ap2

.

115

According to the Bogoliubov prescription, we replacethe operator a0 for the condensate with a c number,

a0 , 116

in Eq. 115. The corresponding c-number field ampli-tude order parameter "0r , t varies slowly over dis-tances comparable to the range of the interparticleforce, thus we substitute r for r in Eq. 111 and obtainthe Gross-Pitaevskii equation,

i

t"0r,t = −

22

2m+ g"0r,t2"0r,t . 117

B. Bogoliubov excitation spectrum

In a dilute and low-temperature limit, the occupationnumber for excited states with p0 is finite but small. Inthis first-order approximation, we neglect all terms inEq. 115 containing ap and ap

+ with p0. Then theground-state energy is

E0 =g

2VN2. 118

The corresponding chemical potential or mean-field en-ergy shift is given by

Un E0

N= gn , 119

where n=N /V is the particle density.In order to obtain the excitation spectrum and study

the quantum fluctuation, however, this level of approxi-mation is insufficient. We have to work with higher ac-curacy using the normalization relation

a0+a0 +

p0ap

+ap = 2 + p0

ap+ap = N . 120

Substituting Eq. 120 into 115, we obtain the newHamiltonian

H =g

2VN2 +

p0

p2

2map

+ap

+12

gn p0

2ap+ap + ap

+a−p+ + apa−p . 121

The above Hamiltonian can be diagonalized by the lin-ear transformation Bogoliubov, 1947

ap = upbp + v−p* b−p

+ , ap+ = up

*bp+ + v−pb−p. 122

If the two coefficients up and v−p satisfy

up2 + v−p2 = 1, 123

a new set of operators bp and bp+ satisfies the bosonic

commutation relation bp , bp+ =pp. In order to make

the non-number conserving terms bp+b−p

+ and bpb−p inEq. 121 vanish, the coefficients up and v−p must satisfy

Un2

up2 + v−p2 + p2

2m+ Unupv−p = 0. 124

From Eqs. 123 and 124, we can solve up and v−p asfollows:

up =p2/2m + Unp

+12

, 125

v−p = −p2/2m + Unp

−12

, 126

where

p =Unm

p2 + p2

2m2

. 127

With this linear transformation, the Hamiltonian 121 isreduced to the diagonal form

H = E0 + p0pbp

+bp. 128

p is the well-known Bogoliubov dispersion law for theelementary excitations from the condensate system.

C. Blueshift of the condensate mean-field energy

The k=0 LP mean-field energy is blueshifted with in-creasing number of LPs as shown in Fig. 40. This is adirect manifestation of the repulsive interaction amongLPs. The k=0 LP energy shift Un is calculated by Eq.39. If we assume a constant size of the condensate, thetheoretical energy shift is shown by the light blue line inFig. 40. We can numerically solve the Gross-PitaevskiiGP equation Eq. 117 to incorporate the condensateexpansion due to repulsive interaction. The results areshown by red dots in Fig. 40. These two theoretical pre-dictions are compared to the experimental results bluediamonds. Note that the above mentioned nonlinearmodel based on weakly interacting bosons Schmitt-Rink et al., 1985; Ciuti et al., 1998; Rochat et al., 2000can reproduce the experimental data only in low LP



density regimes. Recently the Popov approximation isapplied to the LP system and quantitative agreementwith the experimental data on Un is obtained as shownby the pink solid line Sarchi and Savona, 2008. We usethe experimental values not theoretical values for Unas the interaction energy in subsequent discussions forthe universal feature of the Bogoliubov excitations.

D. Energy vs momentum dispersion relation

Using a microcavity with a very long cavity-photonlifetime of 80 ps, Roumpos, Hoefling, et al. 2009 mea-sured again the excited-state energy E dispersion versusin-plane wave number k, as shown in Fig. 41 for a pumprate P /Pth=3. The pump spot size is 30 m in diam-eter. The LP condensate is formed in an area deter-mined by the pump spot size and the pump rate due tothe limited lateral diffusion and varying spatial densityof the LPs Deng et al., 2007; Lai et al., 2007. Figure41a represents a linear plot of the luminescence inten-sity, while Figs. 41b–41d employ logarithmic plots ofthe intensity to magnify the excitation spectra. Abovethreshold, two drastic changes are noticed compared tothe standard quadratic dispersion observed belowthreshold. One is the blueshift of the k=0 LP energy asdiscussed in Sec. VIII.C and the other is the phononlikelinear dispersion relation at low-momentum regimek%1, where %= /2mUn is the healing length.White and black lines in Fig. 41b represent the twoquadratic dispersion relations, ELP=−Un+ k2 /2mand ELP = k2 /2m, where m is the effective mass of thek=0 LP. Here we choose the zero energy as the blue-shifted condensate mean-field energy for convenience.Neither of the two theoretical curves can explain themeasurement result. A solid pink line in Fig. 41b isBogoliubov excitation energy 127, which can be rewrit-ten as Pitaevskii and Stringari, 2003; Ozeri et al., 2005

EB = ELP ELP + 2Un = Unk%2k%2 + 2 .

129

The measured dispersion relation for the excitation en-ergy versus in-plane wave number is in good agreement

with the Bogoliubov excitation spectrum without any fit-ting parameter.

A circularly polarized pump-laser beam was used toinject spin-polarized LPs in this experiment. In this case,the LP condensate preserves the original spin polariza-tion of optically injected LPs as described in Sec. VIII.CDeng et al., 2003; Roumpos, Hoefling, et al., 2009. Theresult shown in Fig. 41b was taken for detecting theleakage photons with the same circular polarization asthat of the pump cocircular detection. A small amountof cross-circularly polarized photons was also detectedbecause of spin relaxation during the cooling processDeng et al., 2003; Roumpos, Hoefling, et al., 2009.However, the standard quadratic dispersion was ob-served but with slightly lower energy, compared to thecondensate energy, as shown in Fig. 41c. With increas-ing pump rate, the number of the condensed LPs is in-creased so that the weak attractive interaction betweenthe condensate and excited states with the opposite spinresults in a slight redshift compared to ELP black linein Fig. 41c. In Fig. 41d we simultaneously detected asmall amount of the cocircular polarized photons withthe cross-circular polarized photons. The difference be-tween the Bogoliubov excitations with cocircular polar-ization and the standard but redshifted quadratic excita-tion with cross-circular polarization is clearly seen.

Popov theoryBogolubov theory Experiment

Numerical modelPopov theoryBogolubov theory Experiment

Numerical model

FIG. 40. Color The measured mean-field energy shift vs totalnumber of polaritons dark blue diamond. The analyticalblue solid line and numerical red dot results based on Bo-goliubov approximation Utsunomiya et al., 2008. The analyti-cal result based on Popov approximation pink solid line isprovided by Savona et al. private communication.

FIG. 41. Color Polarization dependence of the excitationspectrum for an untrapped condensate system. a A linearplot of the intensity, while b–d employ three-dimensionallogarithmic plots of the intensity to magnify the excitationspectra. A circularly polarized pump beam was incident withan angle of 60°. Three detection schemes: a and b detectionof the leakage photons with the cocircular polarization as thepump beam, c detection of the cross-circular polarization,and d detection of a small amount of mixture of the cocircu-lar polarization with the cross-circular polarization. The theo-retical curves represent the Bogoliubov excitation energy EBpink line, the quadratic dispersion relations ELP black linewhich starts from the condensate energy, and the noninteract-ing free LP dispersion relation ELP white line which is experi-mentally determined by the data taken far below the thresholdP=0.001Pth. From Utsunomiya et al., 2008.



As shown in Fig. 41b, the condensate at an energyELP stretches to a finite wave number k0, which is notexpected from the Bogoliubov dispersion law Eqs. 127and 128. This experimental result has two possible ex-planations. One explanation is the Heisenberg uncer-tainty relationship between the position and the momen-tum of condensate particles. The spatial extent of thepolariton condensate is determined by the pump spotsize and the size-dependent critical density Deng et al.,2003, as discussed in Sec. VII. It is expected that thecondensate has a finite momentum uncertainly p whenit is confined in a finite space region. Indeed, the mea-sured product of the spot size x and the spread in mo-mentum p is close to the minimum allowable value,xp= /2, within a factor of 2–4 above the thresholdUtsunomiya et al., 2008. It is also shown by the spa-tially inhomogeneous Gross-Pitaevskii equation that thedensity of states for the excitation spectrum disappearsin a small momentum regime p /2x as shown inFig. 42a, for which the condensate at ELP fills in thevacant momentum regime Utsunomiya et al., 2008.

The other possible explanation is the finite lifetime ofcondensate particles. Using a generalized Gross-Pitaevskii equation with polariton loss and gain terms, itcan be found that the dispersion of elementary excita-tions around the condensate becomes diffusive, i.e., isflat in a small momentum regime and then rises up toapproach the Bogoliubov spectrum superfluidity-diffusion crossover, as shown in Fig. 42b Szymanskaet al., 2006; Wouters and Carusotto, 2007. It is not wellunderstood at present which of the above two factors, afinite condensate size or a finite condensate lifetime, ismainly responsible for the observed flat dispersion neark=0.

E. Universal features of Bogoliubov excitations

As indicated by Eq. 129, the Bogoliubov excitationenergy normalized by the mean-field energy shiftEB /Un is a universal function of the wave number nor-malized by the healing length k%. In Fig. 43a, this uni-versal relation green solid line is compared to the ex-perimental results for four different condensate systemswith varying detuning parameters. In both the phonon-like regime at k%1 and the free-particle regime atk%1, the experimental results agree well with the uni-

versal curve except for the region of small momentak%1 as mentioned above. On the other hand, at apump rate far below threshold, the measured dispersionrelation is completely described by the single LP disper-sion ELP red solid line.

The sound velocity deduced from the phononlike lin-ear dispersion spectrum is of the order of 108 cm/sand is proportional to the square root of the condensatepopulation n0 at reasonable pump rates. This valueis eight orders of magnitude larger than that of atomicBEC and four orders of magnitude larger than that ofsuperfluid 4He. This enormous difference comes fromthe fact that the LP mass is nine to ten orders of magni-tude smaller than the atomic mass and the LP interac-tion energy is six to seven orders of magnitude larger

FIG. 42. Color online Excitations of a condensate. a Falsecolor contour plot of the density of states for elementary exci-tations of a finite-size polariton condensate Utsunomiya et al.,2008. b The dispersion of elementary excitations of a dissi-pative polariton condensate Wouters and Carusotto, 2007.

FIG. 43. Color Universal features of Bogolinbor excitations.a Excitation energy normalized by the interaction energyE /Un as a function of normalized wave number k% for fourdifferent untrapped condensate systems. A blue dot: =1.41 meV, P=4Pth Pth=6.3 mW; B light blue dot: =0.82 meV, P=8Pth Pth=8.2 mW; C red dot: =4.2 meV,P=4Pth Pth=6.4 mW; and D pink dot: =−0.23 meV, P=24Pth Pth=8.2 mW. The experimental data far belowthreshold are also plotted by blue crosses for system A. Threetheoretical dispersion curves normalized by the interaction en-ergy are plotted; the Bogoliubov excitation energy EB /Ungreen solid line, the quadratic dispersion curves ELP /Ungray solid line, and the free LP dispersion ELP/Un redsolid line. b and c The energy shift EB−ELP in the free-particle regime k%=1 plotted as a function of the interactionenergy Un for the same four different untrapped systems asin a and b and four different trapped systems c, wheretrapped condensate systems are labeled as below: E bluecircle: ddiameter=7 m, =3.3 meV; F light blue circle:d=7 m, =2.9 meV; G red circle: d=8 m, =1.6 meV;and H pink circle: d=8 m, =2.5 meV. The dashed linerepresents the theoretical prediction EB−ELP=2Un. d LPpopulation distribution normalized by the value at k%=0.3 forthe trapped system G in c at the pump rate; P=2Pth Pth=4 mW. Theoretical 1 /k2 dependence for the thermal deple-tion is shown by the blue line and 1/k dependence for thequantum depletion is shown by the blue dotted line. FromUtsunomiya et al., 2008.



than the atomic interaction energy. According to Land-au’s criterion Landau and Lifshitz, 1987, the observa-tion of this linear dispersion at low-momentum regime isan indication of superfluidity in the exciton-polaritonsystem. It has recently been reported that the scatteringrate for the moving exciton-polariton condensate is dra-matically suppressed when the initial velocity given tothe polariton system is lower than this critical velocityAmo et al., 2009. It was also observed that the Bogo-liubov theory does not fit to the measured dispersion ofelementary excitations at very high pump rates. This isan expected result because the Bogoliubov theory isbased on the weakly interacting Bose gas and this ap-proximation is violated at a pump rate far above thresh-old.

In the free-particle regime k%1, the excitation en-ergy associated with the condensate is larger by 2Unthan that of a single LP energy for the same wave num-ber. A factor of 2 difference between the mean-field en-ergy shifts of the condensate and excited states origi-nates from a doubled exchange interaction energyamong different modes. The fact was already manifestedin the kink observed in Fig. 40. In Figs. 43b and 43c,this important prediction of the Bogoliubov theory iscompared to the experimental results for four differentuntrapped and trapped condensate systems, respectively.The experimental data were determined as the differ-ence between the measured excitation energy with thepresence of the condensate and the standard quadraticdispersion for a single LP state, which is determined bythe experimental data obtained for a pump rate far be-low threshold P /Pth1. The experimental data are ingood agreement with the theoretical curve gray dashedline for both untrapped and trapped cases.

F. Thermal and quantum depletion

Figure 43d shows the normalized excitation popula-tion nk /nk

0 versus the normalized wave number k% for atrapped condensate, where nk

0 is evaluated at k%=0.3for convenience. The LP occupation number nk in theexcitation spectrum can be calculated by applying theBose-Einstein distribution for the Bogoliubov quasipar-ticles and subsequently taking the inverse Bogoliubovtransformation Pitaevskii and Stringari, 2003,

nk = v−k2 +uk2 + v−k2

exp&EB − 1, 130

where

uk,v−k = ± k2/2m + Un2EB

±211/2

and &=1/kBT. The first and second terms on the right-hand side of Eq. 130 represent the real particles LPscreated by the quantum depletion and the thermaldepletion, respectively. In the present LP condensatesystem, the thermal depletion is much stronger than thequantum depletion so that the second term on the right-hand side of Eq. 130 dominates over the first term. In

this case, the LP population is given by nk

mkBT / k2 at k%1 regime, while nk1/22 /k%if the quantum depletion is dominant Pitaevskii andStringari, 2003. The theoretical prediction of the 1/k2

dependence of nk for thermal depletion is compared tothe experimental data in Fig. 43d and reasonableagreement was obtained.

G. Quantized vortices

A hallmark of superfluidity is quantized vortices. In aBEC phase, vortices appear when finite angular momen-tum is transferred to the condensate. A uniform 2D sys-tem of bosons does not undergo BEC at finite tempera-tures. However, such a system turns superfluid in theBKT phase see discussion in Sec. II.D.1. Elementaryexcitations of this superfluid phase are bound pairs ofvortices with opposite winding numbers. The BKT tonormal-fluid-phase transition is identified by the disso-ciation of the vortex pair into free vortices Hadzibabicet al., 2006; Posazhennikova, 2006.

A quantized vortex of LPs was first observed in aCdTe microcavity. The vortex seemed to be created dur-ing the thermalization process and was pinned by a crys-tal defect induced potential fluctuation Lagoudakis etal., 2008. A fixed winding direction of the observed vor-tex was attributed to a specific potential landscape.

Recently vortex-antivortex pairs were observed in afinite-size GaAs microcavity Roumpos, Hoefling, et al.,2009. According to the numerical analysis of the Gross-Pitaevskii equation, a vortex-antivortex pair in the BKTtopological order moves in parallel with a velocity closeto the sound velocity of the Bogoliubov excitation spec-trum if the 2D system is uniform. If the 2D system istrapped to a finite size, the vortex-antivortex pair is alsotrapped with a residual micromotion inside a trapFraser et al., 2009. The experiment of Roumpos,Hoefling, et al. 2009 observed this behavior. In the ex-periment, the ensemble-averaged 2D distributions of thepopulation nr, phase r, and first-order spatial coher-ence function g1r are measured simultaneously by aMichelson interferometer. The existence of a vortex-antivortex pair is identified by the population dip, phase shift, and reduced spatial coherence at the bound-ary of the pair. These experimental results are success-fully reproduced by the numerical analysis of the Gross-Pitaevskii equation with a gain-loss term.

IX. POLARITON CONDENSATION IN SINGLE TRAPSAND PERIODIC LATTICE POTENTIALS

Most of the experiments we discussed in this reviewused planar microcavities without an intentional in-plane trapping potential. The size of the system islargely determined by the spot size of the pump laser. Incomparison, a trapped polariton system may offer sev-eral advantages. With a trapping potential, genuine BECbecomes possible in two dimensions, and spatial conden-sation accompanies the configuration-space condensa-



tion. A strong trapping potential may lead to discretepolariton modes, hence single-mode BEC. Signatures ofBEC should be more pronounced in all these cases. Thetrap can also serve as a tuning knob for the system. Witha lattice potential, coherence and interactions amongmultiple condensates can be studied. In this section, wereview the efforts to implement single and lattice poten-tials for polaritons.

A. Polariton traps

A single isolated trap for polaritons is implementedby modifying either the cavity photon or the QW exci-ton energies.

1. Optical traps by metal masks

A rather versatile type of polariton trap is based onmodulating the cavity layer thickness and transmissionwith a thin metal mask deposited on the surface of themicrocavity Lai et al., 2007; Kim et al., 2008; Ut-sunomiya et al., 2008.

As shown in Fig. 44a, a trap potential of 200 eVis provided by a hole surrounded by a thin metalTi/Au film Lai et al., 2007; Utsunomiya et al., 2008.The cavity resonant field has normally an antinode atthe AlGaAs-air interface. However, the antinode posi-tion is shifted inside the AlGaAs layer with the metalfilm as shown in Fig. 44a, which results in the blueshiftof the cavity photon and subsequently LP resonances.The LPs were confined in circular holes of varying diam-eters from 5 to 100 m. In a trap with 5–10 m diam-eter, a single fundamental transverse mode dominatesthe condensation dynamics over higher-order transversemodes due to the relatively weak confining potential. Aweak confining potential helps to cool polaritons effi-ciently because high-energy polaritons created by two-body scattering are quickly expelled from the trap re-gion.

The top panels in Fig. 44b show the near-field-emission patterns from a trap with 8 m diameter atpump rates below, just above, and well above condensa-tion threshold. The measured standard deviations for

the LP position x and wave number k are plotted as afunction of P /Pth in Fig. 44b. The sudden decreases inx and k were clearly observed at threshold PPth.Just above threshold, the measured uncertainty productxk is 0.98, which is compared to the Heisenberglimit xk0.5 for a minimum uncertainty wavepacket. Here 103 LPs condense into a nearly minimumuncertainty wave packet at threshold. The monotonicincrease in x and xk at higher pump rates stemsfrom the repulsive interaction among LPs in a conden-sate and is well reproduced by the numerical analysisusing the spatially inhomogeneous GP equation asshown by solid lines in Fig. 44b.

2. Optical traps by fabrication

Strong optical confinement, hence discrete polaritonmodes, has been realized by modifying the microcavitystructure. Earlier efforts focused on etching small pillarstructures out of a planar microcavity to achieve strongtransverse confinement of the photon modes. In pillarsof a few microns across, discrete polariton modes Blochet al., 1998; Gutbrod et al., 1998; Obert et al., 2004 andpolariton parametric scattering Dasbach et al., 2001were observed. In very small pillars, leakage of thecavity-photon fields and loss of excitons at the side wallsprevent strong coupling.

Alternatively, in a similar spirit as the metal masks,Daif et al. 2006 devised a way to grow small regionswith thicker cavity layers and lower photon and thuspolariton energies. Compared to using metal masks, theenergy difference between inside and outside the trapregion is large enough to discretize the polariton modes.Compared to the etching method, the QW excitons areuntouched and will not suffer surface recombination;transport and interactions are possible between un-trapped and trapped polaritons. Discrete polariton spec-tra were observed in such a structure, in agreement withcalculations Kaitouni et al., 2006.

3. Exciton traps by mechanical stress

A polariton trap with a relatively larger area has beenformed by mechanically pressing a planar microcavitywafer Balili et al., 2006. The mechanical stress intro-duces a strain field which acts as a harmonic potential onthe QW excitons Negoita et al., 1999 and thus on thepolaritons as well. A potential up to about 5 meV deepwas created by a stressing tip of about 50 m in radius.LPs excited away from the trap center drifted over morethan 50 m into the trap Balili et al., 2006. Condensa-tion of LPs in the trap was also observed, with a thresh-old density lower than that required when without thetrap and with a contracted spatial profile Balili et al.,2007.

B. Polariton array in lattice potential

Extending a single trap potential to a lattice potential,the long-range spatial coherence leads to the existence

FIG. 44. Color online Polariton condensation in an opticaltrap by metal masks. a A schematic of a polariton trap. bThe measured standard deviations of LP distribution in coor-dinate x circles and in wave number k crosses are plot-ted as a function of P /Pth. Theoretical values for x and kobtained by the GP equation are shown by the solid lines Ut-sunomiya et al., 2008.



of phase-locked multiple condensates, such as in an ar-ray of superfluid helium Davis and Packard, 2002, su-perconducting Josephson junctions Hansen and Linde-lof, 1984; Hadley et al., 1988; Cataliotti et al., 2001, andatomic BECs Anderson and Kasevich, 1998; Cataliottiet al., 2001; Orzel et al., 2001; Greiner et al., 2002. Undercertain circumstances, a quantum phase difference of is predicted to develop among weakly coupled Joseph-son junctions Bulaevskii et al., 1977. Such a metastable state was discovered in a weak link of superfluid 3He,which is characterized by a p-wave order parameterBackhaus et al., 1998. Possible existence of such a state in weakly coupled atomic BECs has also been pro-posed Smerzi et al., 1997 but remains undiscovered inatomic systems. Recently Lai et al. 2007 observedspontaneous buildup of in-phase zero-state and an-tiphase -state coherent states in a polariton conden-sate array connected by weak periodic potential barri-ers. These states reflect the band structure of the one-dimensional periodic potential for polaritons andfeature the dynamical competition between metastableand stable polariton condensates.

1. One-dimensional periodic array of polaritons

Employing the same metal deposition technique de-scribed in Sec. IX.B, an array of one-dimensional cigar-shaped polaritons is created by the deposition of peri-odic strips of a metallic thin film on the top surface of amicrocavity structure, as shown in Fig. 45a. Under the

metallic layer, the cavity resonance energy Ecav for k

=0 is expected to increase by 400 eV according tothe transfer-matrix analytical method Yeh, 2005. Whenthe cavity photon Ecav is near resonance with the QWexciton Eexc at k =0, the shift of the LP energy inducedby the metallic layer is 200 eV, approximately one-half of the cavity-photon resonance shift Fig. 45b.Therefore, the LPs are expected to be trapped in the gapregion where the LP energy is lower. The measured spa-tial modulation of LP energy is 100 eV, which is lessthan the theoretical prediction due to the diffraction-limited spatial resolution 2 m of our optical detec-tion system Fig. 45c.

The microcavity is excited by a mode-locked Ti:sap-phire laser with an 2.5 ps pulse near the QW excitonresonance at an incident angle of 60°, corresponding toan in-plane wave number k 7104 cm−1 in air. Thelarge k of the pump assures that the coherence of thepump laser is lost by multiple phonon emissions beforethe LPs scatter into k 0 states.

Considering a periodic and coherent array of conden-sates aligned along the x axis, we can approximate theorder parameter in momentum space as Pedri et al.,2001

"kx = "0kx n=0,±1,. . .,±nM

einkxa+

= "0kxsinNkxa/2sinkxa/2

if = 0 , 131

where n labels the condensate array element, N=2nM+1 is the total number of periodic array elements, a isthe pitch distance between two neighboring elements,and "0kx is the momentum-space wave function of anindividual element. The overall momentum distributionkx= "kx2 reflects the coherence properties and na-ture of the condensate arrays through distinctive inter-ference patterns. In the presence of the periodic lattice,the momentum distribution displays narrow peaks atkx=m2 /a m integer with an envelope functiongiven by "0kx2. If all the elements are locked in phase=0, at a LP wavelength =780 nm and a gratingpitch distance a=2.8 m, a Fraunhofer diffraction pat-tern has a central peak at =0° and two side lobes at =sin−1 /a ±16° k = ±2 /a, m= ±1 first-order diffrac-tion. The LP emissions from a polariton condensate ar-ray are expected to display such an interference patternin momentum space, just like in an atomic BEC experi-ment Anderson and Kasevich, 1998; Cataliotti et al.,2001; Orzel et al., 2001; Greiner et al., 2002.

The observed LP distributions in both coordinate andmomentum space of the condensate array are shown inFigs. 46a and 46b. Below the condensation threshold,the near-field image reveals the cigar-shaped LP emis-sions through the 1.4-m-wide gaps between the metal-lic strips. The corresponding LP momentum distributionis broad and isotropic, independent of the periodicmodulation of LP population and the elliptically shapedpumping spot. The result indicates that there is negli-

FIG. 45. Color online Formation of a polariton array. a Aschematic of a polariton array formed by depositing periodicthin metallic strips Au/Ti on top of a microcavity structure.b Dispersion curves of the cavity-photon mode and LP. cSpatial LP energy modulation is detected from the position-dependent central energy of LP emissions. From Lai et al.,2007.



gible phase coherence among different cigar-shapedLPs. With an increasing pumping rate, two strong andtwo weak side lobes emerge at ±8° and ±24°, whichcorrespond to k = ± /a and ±3 /2a, respectively. Theparticular diffraction pattern suggests that the phase dif-ference of adjacent condensates in the array is lockedexactly to i.e., = in Eq. 131. Moreover, the cor-responding near-field image Fig. 46a reveals thatstrong LP emission is generated from under the metallicstrips rather than from the gaps. The dramatic contrastis better observed near the boundary of the central con-densate and the outside thermal LP emissions, whichcome through the gaps. The dominant LP emissionthrough the metallic layers despite the lower transmis-sion indicates that LP condensates are strongly localizedunder the metallic strips. With a further increasingpumping rate, the intensity of the central peak near =0°, with weak side lobes at ±16°, gradually sur-passes the peaks at ±8° and ±24° Fig. 46b. Simul-taneously, the dominant LP emission is observedthrough the gaps.

These two distinct interference patterns suggest thetransition from an antiphase state localized under themetallic strips to an in-phase state localized in the gaps.Both the zero and state can be observed for a polar-iton condensate array consisting of more than 30 stripsof condensates total array dimension 100 m. Thesecollective states are manifestations of the long-rangespatial coherence as well as phase locking across the ar-ray.

2. Band structure, metastable state, and stable zero-state

Figure 47a shows the energy versus in-plane momen-tum dispersion relation for a one-dimensional LP arraynear resonance EcavEexc and below threshold. Fromthe dispersion curve, we deduce the LP kinetic energy at

k =G0 /2= /a one-half of the primitive reciprocal-lattice vector, Ek=2G0 /22 /2m*500 eV, where theLP effective mass m*=910−5me. We deduce the bandstructure of the polariton array assuming a “nearly freepolariton” in the presence of the periodic square-wellpotential. Given a one-dimensional periodic potentialUX with a lattice constant a, the band structure can beobtained using the standard Bloch wave-function for-malism. Only those LPs with in-plane momentum closeto the Bragg condition k =mG0 /2, m integer are sub-ject to backscattering due to the periodic potential, andgaps appear at these Bragg planes where standing wavesare formed. Similar to the standard extended-zonescheme Madelung, 1996, the band structure can beconstructed by starting with original and displaced pa-rabolas ELPk ±mG0 as shown in Fig. 47b. Here thesize of from thermal population of LPs. This is well re-produced in the observed LP energy versus in-plane mo-mentum with a below threshold pumping rate Fig.47a. The first band gap is on the order of the barrierpotential, U0200 eV, between the first and secondbands near the zone boundaries Bragg planes. Thissmall gap is masked in the measured spectra by the largeinhomogeneous broadening of the LP emission lineslinewidth 500 eV.

In Fig. 47c, the energy versus in-plane momentum ofa polariton condensate array above threshold and Ecav−Eexc ±6 meV is shown. Above threshold, LP emis-sions occur in two states with an energy difference ofabout 1 meV. These two states with emission peaks at

FIG. 46. Color Near-field and far-field images of polaritons ina one-dimensional array. a Near-field images showing the LPdistribution across a polariton array in coordinate space underpumping powers of 20, 70, and 200 mW left to right. Thethreshold pumping power is about 45 mW. b Correspondingfar-field images showing the LP distribution in momentumspace. From Lai et al., 2007.

FIG. 47. Color online Energy-momentum dispersions andreal space wave functions of polaritons in a one-dimensionalarray. a Time-integrated energy vs in-plane momentum of apolariton array at near resonance EcavEexc. The pumpingpower is P=10 mW below threshold. b Extended-zonescheme of the band structure for the polariton array under aweak periodic potential with a lattice constant a. c Energy vsin-plane momentum of a polariton condensate array at bluedetuning Ecav−Eexc6 meV above threshold P=40 mW. dSchematic of the Bloch wave functions for states labeled asA–C in b. From Lai et al., 2007.



k =0, ±G0 and at k = ±G0 /2 , ±3G0 /2 correspond to theaforementioned zero and state, respectively.

On referring to the multivalley band structure Fig.47b, the dynamic condensation process of the polar-iton array becomes transparent. Without the spatialmodulation, the quasistationary state of the LP conden-sate is the lowest-energy state at k =0. Here the bottomof LP dispersion parabola serves as a trap of LPs inmomentum space. When the spatial potential modula-tion is introduced, a metastable dynamic condensate canoccur near the bottom of the second band point A inFig. 47b when the pumping rate is just above thresh-old. In these points, LPs experience a relaxation bottle-neck, resulting in metastable condensates. Eventually,the LP system will relax to the lowest-energy state pointC in Fig. 47b as the pumping rate is increased.

To understand the spatial distributions of LPs in thearray near-field images shown in Fig. 46a, the Blochwave functions must be considered for the above twostates. The Bloch wave functions for these eigenstatesexhibit not only opposite relative phase between adja-cent elements in the array but also different orbital wavefunctions. Schematic Bloch wave functions for selectedeigenstates labeled as A–C in Fig. 47b are shown inFig. 47d. The zero state at point C carries to the s-likewave with maximal amplitude in potential wells andidentical phase between adjacent elements across the ar-ray. The wave functions of A and B at k =G0 /2 bothexhibit a relative phase difference between adjacentelements. The lower energy unstable state at point Bcorresponds to antibonding of s waves, while the higherenergy metastable state at point A corresponds to anti-bonding of p waves. As the atomic p wave, the LP den-sity vanishes at the center of the potential well for this state. This explains the near-field imaging shown in Fig.46a. Under the weak potential modulation, the Blochwave functions are superpositions of forward-and-backward traveling plane waves at the zone boundaryand have a relatively broad distribution in space com-pared to the standard tight-binding model of a normalcondensed-matter system.

The metastability of the state antibonding of pwaves is a unique property for dynamic polariton con-densates. Under strong pumping rates, the zero statebonding of s waves dominates eventually due to en-hanced cooling by stimulated LP-LP scattering at highdensities. A typical evolution from the metastable state to the stable zero state is illustrated by the LPmomentum distribution profiles for various pumpingrates in Fig. 48. At a high pumping rate, the LP emissionfrom the zero state surpasses that from the state bymore than one order of magnitude.

3. Dynamics and mode competition

The state consisting of local p-type wave functionsis a metastable state with a higher energy than the zerostate consisting of local s-type wave functions. Bothstates decay to the crystal ground state by emitting pho-tons with a finite lifetime. After high-energy excitonlike

LPs with large in-plane wave number k are opticallyinjected into the system, they cool down to the meta-stable state first by continuously emitting phonons orvia LP-LP scattering. If the decay rate of the state isslower than the loading rate into the state, the popu-lation of the state exceeds 1 quantum degeneracycondition and induces the bosonic final-state stimulatedscattering into this mode. This is similar to a so-calledbottleneck condensation effect Tartakovskii et al., 2000;Huang et al., 2002. Here the LPs with finite k dynami-cally condense due to the slow relaxation rate into thek0 region, where the density of states is reduced.

When the pump rate is further increased, the zerostate eventually acquires the population greater than 1.The onset of stimulated scattering into the zero statefrom the metastable state favors the condensation atthe zero state. The reverse stimulated process from thezero to the state requires the absorption of phononsand thus is energetically unfavorable. This mode compe-tition behavior is well described by the coupled Boltz-mann rate equation analysis Lai et al., 2007.

Assuming reasonable numerical parameters, thequalitative agreement with the experimental results isobtained. With increasing pumping, the state appearsfirst and then is surpassed by the dominant zero state,consistent with experimental observation. The time-resolved LP emissions from the zero and state areshown in Fig. 49. The simple rate equations with param-eters fixed for all pumping rates can describe the ob-served mode competition.

X. OUTLOOK

With regard to fundamental physics of polariton BEC,it is important to understand the ground state as well asthe excitations of the quantum phases to establish theexperimental phase diagram of the polariton system andto compare experiments with theoretical analysis takinginto account both thermal and quantum depletions of acondensate. The dynamical nature of the polariton con-densate due to the short polariton lifetime imposes chal-

FIG. 48. Color online Momentum distribution profiles of the phase connected p state and zero phase connected s state ofpolaritons in a one-dimensional array. a Profiles of the “state” as a function of in-plane momentum taken at a crosssection in the dispersion curve inset for a polariton conden-sate array for various pumping rates. b Profiles of the “zerostate.” The intensity of the zero state surpasses that of the state at around P=45 mW. From Lai et al., 2007.



lenges on this kind of study. Fortunately, the Q value ofa GaAs microcavity has been dramatically improved re-cently Reitzenstein et al., 2007, so one may expect aquantitative comparison between theory and experimentin the near future.

The quantum-statistical properties of the BEC groundstate are expected to be influenced by phase locking be-tween the condensed particles at k =0 and excitations at±k0 Nozieres, 1995. The current polariton BEC isstill limited by thermal depletion, but with high-Q mi-crocavities and more efficient cooling schemes the quan-tum nature of the BEC ground state should be eluci-dated soon.

Superfluidity associated with polariton BEC is an-other urgent issue. A first sign has been reported on thesound velocity Utsunomiya et al., 2008, and prelimi-nary results on quantized vortices have been reportedrecently Lagoudakis et al., 2008. A more quantitativestudy on quantized vortices, in particular, dynamical for-mation process of vortices in a rotating polariton con-densate, is needed.

The crossover from polariton BEC in a relativelysmall 2D polariton system to a BKT phase in a relativelylarge 2D system is an interesting topic in studying thepolariton phase diagram. The decay of the first-orderspatial coherence, the recombination of thermally ex-cited vortices, as well as the temperature dependence ofa fractional condensate will be decisive experimental sig-natures for the BEC-BKT crossover. The crossover from

polariton BEC to electron-hole BCS phase is anotherintriguing topic. With increasing exciton density toabove the Mott density, excitons overlap with each otherand both the electron and hole liquids are normal exceptat the Fermi surface. The remaining binding energy isexpected to create a gap at the Fermi energy. A mea-surement of the eigenenergies will help elucidate thecrossover behavior.

The complex phase diagram and rich physics of thepolariton system make it a unique candidate for thequantum simulation of many-body Hamiltonians in oneand two dimensions. The superfluid to Mott insulator orBose glass phase transition in two dimensions andTanks gas in one-dimensional polariton systems havebeen addressed theoretically so far, while intense experi-mental research is underway.

With regard to applications of polariton BEC, mostpromising is a polariton BEC as a low-threshold sourceof coherent light. A fundamental limit on the thresholdof a standard semiconductor laser is required to createan electronic population inversion in the gain medium:na

B*−2, where a

B* is the exciton Bohr radius. On the

other hand, the polariton BEC threshold is given by nT

2 , where T is the thermal de Broglie wavelength.Since a

B* /T is on the order of 10−2, the fundamental

limit of the polariton BEC threshold density is 10−4 ofthat of a counterpart semiconductor laser. A proof ofprinciple comparison between the two types of lasers

0 20 40 60 80 1000

1

2

3

4

5

PLIntensity(arb.units)

0 20 40 60 80 1000

2

4

6

8

10

PLIntensity(arb.units)

Time (ps)0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

zero-state

Intensity(arb.units)

Time (ps)

-state

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

zero-state

Intensity(arb.units)

-state Initial Pump n3=2

Initial Pump n3=20

SimulationExperiment

P=80mW

P=160mW

a

b

c

d

FIG. 49. Color online Dynamics of the zero and state. a LP dynamics measured by a streak camera system. The black dashedcurve in a is the system response measured with scattered laser pulse from the sample. b Simulation. From Lai et al., 2007.



was made by Deng et al. 2003 with a GaAs microcavityat liquid helium temperature. Recently Tsintzos et al.2008 demonstrated electronically pumped polaritonlight-emitting diode operating at 235 K. An electricallypumped BEC with a p-n junction diode at high tempera-tures may soon be within experimental reach.

ACKNOWLEDGMENTS

We wish to acknowledge and thank present andformer members of the Stanford group for their contri-butions to the polariton research reported herein. Theseresearchers include H. Cao, R. Huang, N. Y. Kim, C. W.Lai, S. Pau, G. Roumpos, F. Tassone, L. Tian, S. Ut-sunomiya, and G. Weihs. We also wish to acknowledgeand thank T. D. Doan, H. Thien Cao, and D. B. TranThoai at the Vietnam Centre for Natural Science andTechnology for their contributions to the theoretical andnumerical works reported herein. The research at Stan-

ford at present is supported by the JST/SORST programand Special Coordination Funds for Promoting Scienceand Technology from University of Tokyo, Japan.

APPENDIX: CALCULATION OF SCATTERINGCOEFFICIENTS

In this appendix we list the LP-phonon and LP-LPscattering rates for the quasispin kinetics Eqs. 65 and68 of Sec. VI. In the Markov approximation, the LP-phonon-scattering rate of the polariton population nk,i isgiven by

nk,i

t

scatt

p-ph

= −2

k

Wk,knk,i1 + nk,i + SkSk

− Wk,knk,i1 + nk,i + SkSk A1

and the LP-phonon-scattering rate of the pseudospin Skis given by

Sk

t

scatt

p-ph

= −2

kWk,kSk +

12

i=1,2nk,iSk + nk,iSk − Wk,kSk +

12

i=1,2nk,iSk + nk,iSk . A2

Here the LP-phonon transition rate Wk,k is given in Eq. 37. Similarly, the LP-LP scattering rates are given by

nk,i

t

scatt

p-p

= −2

k,q

ek−q + ek+q − ek − ekVk,k,k − q,k + q2nk,ink,i1 + nk−q,i + nk+q,i

− nk−q,ink+q,i1 + nk,i + nk,i + Uk,k,k − q,k + q2nk,ink,i + SkSk2 + j=1,2

nk−q,j + nk+q,j− 1 + nk,i + nk,i2Sk−qSk+q +

j=1,2nk−q,jnk+q,j + Tk,k,k − q,k + q

j=1,2nk,j − nk−q,jSk+qSk

+ nk,j − nk+q,jSk−qSk + 2nk,iSkS

k−q + SkS

k+q − nk−q,iSkS

k+q − nk+q,iSkS

k−q , A3

Sk

t

scatt

p-p

= −

k,q

ek+q + ek−q − ek − ekVk,k,k − q,k + q2Sk i=1,2

nk,i1 + nk−q,i + nk+q,i − nk−q,ink+q,i

− 2Sk+qSkS

k−q + Sk−qSkS

k+q − SkSk−qSk+q+ Uk,k,k − q,k + q2

i=1,2Sknk,i + Sknk,i2 +

i=1,2nk−q,i + nk+q,i

− 2Sk + Sk i=1,2

nk−q,ink+q,i + 2Sk−qSk+q + Tk,k,k − q,k + q4SkSkS

k−q + SkS

k+q

+ Sk−q i=1,2

nk,i i=1,2

nk,i − nk+q,i− 2

i=1,2nk+q,i1 + nk,iSk+q

i=1,2nk,i

i=1,2nk,i − nk−q,i − 2

i=1,2nk−q,i1 + nk,i . A4

The matrix elements for the relative triplet and singlet configurations are given in Eq. 60.



REFERENCES

Amo, A., D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle,M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M.S. Skolnick, C. Tejedor, and L. Viña, 2009, Nature London457, 291.

Anderson, B. P., and M. A. Kasevich, 1998, Science 282, 1686.Backhaus, S., S. Pereverzev, R. W. Simmonds, A. Loshak, J. C.

Davis, and R. E. Packard, 1998, Nature London 392, 687.Bagnato, V., and D. Kleppner, 1991, Phys. Rev. A 44, 7439.Bajoni, D., P. Senellant, E. Wertz, I. Sagnes, A. Miard, A. Le-

maitre, and J. Bloch, 2008, Phys. Rev. Lett. 100, 047401.Balili, R., V. Hartwell, D. Snoke, L. Pfeiffer, and K. West,

2007, Science 316, 1007.Balili, R. B., D. W. Snoke, L. Pfeiffer, and K. West, 2006, Appl.

Phys. Lett. 88, 031110.Bányai, L., and P. Gartner, 2002, Phys. Rev. Lett. 88, 210404.Bányai, L., P. Gartner, O. M. Schmitt, and H. Haug, 2000,

Phys. Rev. B 61, 8823.Beliaev, S. T., 1958, Sov. Phys. JETP 34, 417.Berezinskii, V. L., 1971, Sov. Phys. JETP 32, 493.Berezinskii, V. L., 1972, Sov. Phys. JETP 34, 610.Berman, P. R., 1994, Cavity Quantum Electrodynamics (Ad-

vances in Atomic, Molecular and Optical Physics) Academic,Boston.

Björk, G., A. Karlsson, and Y. Yamamoto, 1994, Phys. Rev. A50, 1675.

Björk, G., S. Machida, Y. Yamamoto, and K. Igeta, 1991, Phys.Rev. A 44, 669.

Bloch, J., F. Boeuf, J. M. Grard, B. Legrand, J. Y. Marzin, R.Planel, V. Thierry-Mieg, and E. Costard, 1998, Physica EAmsterdam 2, 915.

Bloch, J., and J. Y. Marzin, 1997, Phys. Rev. B 56, 2103.Bockelmann, U., 1993, Phys. Rev. B 48, 17637.Bogoliubov, N. N., 1947, J. Phys. USSR 11, 23.Bulaevskii, L. N., V. V. Kuzii, and A. A. Sobyanin, 1977, JETP

Lett. 25, 290.Butte, R., G. Christmann, E. Feltin, J. F. Carlin, M. Mosca, M.

Ilegems, and N. Grandjean, 2006, Phys. Rev. B 73, 033315.Cao, H. T., T. D. Doan, D. Thoai, and H. Haug, 2004, Phys.

Rev. B 69, 245325.Cao, H. T., T. D. Doan, D. B. T. Thoai, and H. Haug, 2008,

Phys. Rev. B 77, 075320.Carmichael, H. J., 2003, Statistical Methods in Quantum Optics

1: Master Equations and Fokker-Planck Equations, 2nd ed.Springer, Berlin.

Cataliotti, F. S., S. Burger, C. Fort, P. Maddaloni, F. Minardi,A. Trombettoni, A. Smerzi, and M. Inguscio, 2001, Science293, 843.

Chen, J. R., T. C. Lu, Y. C. Wu, S. C. Lin, W. R. Liu, W. F.Hsieh, C. C. Kuo, and C. C. Lee, 2009, Appl. Phys. Lett. 94,061103.

Christmann, G., R. Butte, E. Feltin, J.-F. Carlin, and N. Grand-jean, 2008, Appl. Phys. Lett. 93, 051102.

Christopoulos, S., G. B. H. von Hogersthal, A. J. D. Grundy, P.G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christ-mann, R. Butte, E. Feltin, J. F. Carlin, and N. Grandjean,2007, Phys. Rev. Lett. 98, 126405.

Citrin, D. S., and J. B. Khurgin, 2003, Phys. Rev. B 68, 205325.Ciuti, C., V. Savona, C. Piermarocchi, A. Quattropani, and P.

Schwendimann, 1998, Phys. Rev. B 58, 7926.Ciuti, C., P. Schwendimann, and A. Quattropani, 2003, Semi-

cond. Sci. Technol. 18, S279.

Comte, C., and P. Nozières, 1982, J. Phys. France 43, 1069.Dasbach, G., M. Schwab, M. Bayer, and A. Forchel, 2001,

Phys. Rev. B 64, 201309R.Davis, J. C., and R. E. Packard, 2002, Rev. Mod. Phys. 74, 741.Deng, H., D. Press, S. Gotzinger, G. S. Solomon, R. Hey, K. H.

Ploog, and Y. Yamamoto, 2006, Phys. Rev. Lett. 97, 146402.Deng, H., G. Solomon, R. Hey, K. H. Ploog, and Y. Yama-

moto, 2007, Phys. Rev. Lett. 99, 126403.Deng, H., G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto,

2002, Science 298, 199.Deng, H., G. Weihs, D. Snoke, J. Bloch, and Y. Yamamoto,

2003, Proc. Natl. Acad. Sci. U.S.A. 100, 15318.Doan, T. D., H. T. Cao, D. B. T. Thoai, and H. Haug, 2008,

Phys. Rev. B 78, 205306.Doan, T. D., C. Huy Thien, D. Thoai, and H. Haug, 2005, Phys.

Rev. B 72, 085301.Doan, T. D., H. Thien Cao, D. B. Tran Thoai, and H. Haug,

2008, Solid State Commun. 145, 48.Eastham, P. R., and P. B. Littlewood, 2001, Phys. Rev. B 64,

235101.Einstein, A., 1925, Sitzungsber. Preuss. Akad. Wiss., Phys.

Math. Kl. 1925, 3.El Daif, O., A. Baas, T. Guillet, J. P. Brantut, R. I. Kaitouni, J.

L. Staehli, F. Morier-Genoud, and B. Deveaud, 2006, Appl.Phys. Lett. 88, 061105.

Feshbach, H., 1962, Sugaku 19, 287.Fraser, M. D., G. Roumpos, and Y. Yamamoto, 2009, New J.

Phys. 11, 113048.Gardiner, C. W., and P. Zoller, 2000, Quantum Noise Springer-

Verlag, New York.Glauber, R. J., 1963, Phys. Rev. 130, 2529.Greiner, M., O. Mandel, T. Esslinger, T. W. Hänsch, and I.

Bloch, 2002, Nature London 415, 39.Griffin, A., D. W. Snoke, and S. Stringari, 1995, Bose-Einstein

Condensation Cambridge University Press, Cambridge.Gutbrod, T., M. Bayer, A. Forchel, J. P. Reithmaier, T. L.

Reinecke, S. Rudin, and P. A. Knipp, 1998, Phys. Rev. B 57,9950.

Hadley, P., M. R. Beasley, and K. Wiesenfeld, 1988, Phys. Rev.B 38, 8712.

Hadzibabic, Z., P. Krger, M. Cheneau, B. Battelier, and J. Dali-bard, 2006, Nature London 441, 1118.

Hanamura, E., and H. Haug, 1977, Phys. Rep. 33, 209.Hanbury Brown, R., and R. Q. Twiss, 1956, Nature London

178, 1046.Hansen, J. B., and P. E. Lindelof, 1984, Rev. Mod. Phys. 56,

431.Haug, H., 1969, Phys. Rev. 184, 338.Haug, H., and H. Haken, 1967, Z. Phys. A: Hadrons Nucl. 204,

262.Haug, H., and A. Jauho, 2008, Quantum Kinetics in Transport

and Optics of Semiconductors, 2nd ed. Springer, Berlin.Hohenberg, P., 1967, Phys. Rev. 158, 383.Hopfield, J. J., 1958, Phys. Rev. 112, 1555.Houdré, R., C. Weisbuch, R. P. Stanley, U. Oesterle, P. Pellan-

dini, and M. Ilegems, 1994, Phys. Rev. Lett. 73, 2043.Huang, R., F. Tassone, and Y. Yamamoto, 2000, Phys. Rev. B

61, R7854.Huang, R., Y. Yamamoto, R. André, J. Bleuse, M. Muller, and

H. Ulmer-Tuffigo, 2002, Phys. Rev. B 65, 165314.Imamoglu, A., R. J. Ram, S. Pau, and Y. Yamamoto, 1996,

Phys. Rev. A 53, 4250.Inoue, J.-i., T. Brandes, and A. Shimizu, 2000, Phys. Rev. B 61,



http://dx.doi.org/10.1038/nature07640


http://dx.doi.org/10.1126/science.282.5394.1686

http://dx.doi.org/10.1038/33629

http://dx.doi.org/10.1103/PhysRevA.44.7439

http://dx.doi.org/10.1103/PhysRevLett.100.047401

http://dx.doi.org/10.1126/science.1140990

http://dx.doi.org/10.1063/1.2164431

http://dx.doi.org/10.1063/1.2164431


http://dx.doi.org/10.1103/PhysRevB.61.8823





http://dx.doi.org/10.1013/S1386-9477(98)00186-6

http://dx.doi.org/10.1013/S1386-9477(98)00186-6









http://dx.doi.org/10.1063/1.3079398

http://dx.doi.org/10.1063/1.3079398

http://dx.doi.org/10.1063/1.2966369




http://dx.doi.org/10.1088/0268-1242/18/10/301

http://dx.doi.org/10.1088/0268-1242/18/10/301

http://dx.doi.org/10.1051/jphys:019820043070106900






http://dx.doi.org/10.1073/pnas.2634328100




http://dx.doi.org/10.1016/j.ssc.2007.09.034



http://dx.doi.org/10.1063/1.2172409

http://dx.doi.org/10.1063/1.2172409

http://dx.doi.org/10.1088/1367-2630/11/11/113048

http://dx.doi.org/10.1088/1367-2630/11/11/113048

http://dx.doi.org/10.1103/PhysRev.130.2529

http://dx.doi.org/10.1038/415039a






http://dx.doi.org/10.1016/0370-1573(77)90012-6

http://dx.doi.org/10.1038/1781046a0

http://dx.doi.org/10.1038/1781046a0







http://dx.doi.org/10.1103/PhysRevB.61.R7854





2863.Inouye, S., T. Pfau, S. Gupta, A. P. Chikkatur, A. Görlitz, D. E.

Pritchard, and W. Ketterle, 1999, Nature London 402, 641.Kaitouni, R. I., O. El Daif, A. Baas, M. Richard, T. Paraiso, P.

Lugan, T. Guillet, F. Morier-Genoud, J. D. Ganiere, J. L.Staehli, V. Savona, and B. Deveaud, 2006, Phys. Rev. B 74,155311.

Kasprzak, J., M. Richard, A. Baas, B. Deveaud, R. André, J. P.Poizat, and L. S. Dang, 2008, Phys. Rev. Lett. 100, 067402.

Kasprzak, J., et al., 2006, Nature London 443, 409.Kavokin, A., G. Malpuech, and M. Glazov, 2005, Phys. Rev.

Lett. 95, 136601.Kavokin, A., G. Malpuech, P. G. Lagoudakis, J. J. Baumberg,

and K. Kavokin, 2003, Phys. Status Solidi A 195, 579.Kavokin, K. V., I. A. Shelykh, A. V. Kavokin, G. Malpuech,

and P. Bigenwald, 2004, Phys. Rev. Lett. 92, 017401.Keeling, J., F. Marchetti, M. Szymanska, and P. Littlewood,

2007, Semicond. Sci. Technol. 22, R1.Keldysh, L., and A. N. Kozlov, 1968, Sov. Phys. JETP 27, 521.Kena-Cohen, S., M. Davanco, and S. R. Forrest, 2008, Phys.

Rev. Lett. 101, 116401.Ketterle, W., and N. J. van Druten, 1996, Phys. Rev. A 54, 656.Khalatnikov, I. M., 1965, An Introduction to the Theory of Su-

perfluidity Benjamin, New York.Khurgin, J. B., 2001, Solid State Commun. 117, 307.Kim, N. Y., C.-W. Lai, S. Utsunomiya, G. Roumpos, M. Fraser,

H. Deng, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, andY. Yamamoto, 2008, Phys. Status Solidi B 245, 1076.

Kosterlitz, J., and D. Thouless, 1972, J. Phys. C 5, L124.Kosterlitz, J., and D. Thouless, 1973, J. Phys. C 6, 1181.Kozuma, M., Y. Suzuki, Y. Torii, T. Sugiura, T. Kuga, E. W.

Hagley, and L. Deng, 1999, Science 286, 2309.Krizhanovskii, D. N., D. Sanvitto, I. A. Shelykh, M. M. Glazov,

G. Malpuech, D. D. Solnyshkov, A. Kavokin, S. Ceccarelli,M. S. Skolnick, and J. S. Roberts, 2006, Phys. Rev. B 73,073303.

Kuhn, T., 1997, in Theory of Transport Properties of Semicon-ductor Nanostructures, edited by E. Scholl Springer, NewYork, p. 113.

Kuwata-Gonokami, M., S. Inouye, H. Suzuura, M. Shirane, R.Shimano, T. Someya, and H. Sakaki, 1997, Phys. Rev. Lett.79, 1341.

Lagoudakis, K. G., M. Wouters, M. Richard, A. Baas, I. Caru-sotto, R. André, L. S. Dang, and B. Deveaud-Pledran, 2008,Nat. Phys. 4, 706.

Lai, C., N. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M.Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, and Y.Yamamoto, 2007, Nature London 450, 529.

Landau, L. D., and E. M. Lifshitz, 1987, Fluid Mechanics Per-gamon, Oxford.

Laussy, F. P., G. Malpuech, A. Kavokin, and P. Bigenwald,2004, Phys. Rev. Lett. 93, 016402.

Laussy, F. P., I. A. Shelykh, G. Malpuech, and A. Kavokin,2006, Phys. Rev. B 73, 035315.

Lauwers, J., A. Verbeure, and V. A. Zagrebnov, 2003, J. Phys.A 36, L169.

Lax, M., 1967, IEEE J. Quantum Electron. 3, 37.Lidzey, D. G., D. D. C. Bradley, M. S. Skolnick, T. Virgili, S.

Walker, and D. M. Whittaker, 1998, Nature London 395, 53.Lidzey, D. G., D. D. C. Bradley, T. Virgili, A. Armitage, M. S.

Skolnick, and S. Walker, 1999, Phys. Rev. Lett. 82, 3316.Littlewood, P. B., P. R. Eastham, J. M. J. Keeling, F. M. Mar-

chetti, B. D. Simon, and M. H. Szymanska, 2004, J. Phys.:

Condens. Matter 16, S3597, and references therein.Madelung, O., 1996, Introduction to Solid-State Theory

Springer, Berlin.Malpuech, G., A. Kavokin, A. Di Carlo, and J. J. Baumberg,

2002, Phys. Rev. B 65, 153310.Malpuech, G., Y. G. Rubo, F. P. Laussy, P. Bigenwald, and A.

V. Kavokin, 2003, Semicond. Sci. Technol. 18, S395.Mermin, N. D., and H. Wagner, 1966, Phys. Rev. Lett. 17, 1133;

17, 1307E 1966.Muller, M., J. Bleuse, R. André, and H. Ulmer-Tuffigo, 1999,

Physica B 272, 476.Negoita, V., D. W. Snoke, and K. Eberl, 1999, Appl. Phys. Lett.

75, 2059.Nozieres, P., 1995, in Bose-Einstein Condensation, edited by A.

Griffin, D. W. Snoke, and S. Stringari Cambridge UniversityPress, Cambridge, Vol. 15A.

Obert, M., J. Renner, A. Forchel, G. Bacher, R. Andre, and D.L. S. Dang, 2004, Appl. Phys. Lett. 84, 1435.

Orzel, C., A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M.A. Kasevich, 2001, Science 291, 2386.

Ozeri, R., N. Katz, J. Steinhauer, and N. Davidson, 2005, Rev.Mod. Phys. 77, 187.

Pau, S., G. Björk, H. Cao, E. Hanamura, and Y. Yamamoto,1996, Solid State Commun. 98, 781.

Pau, S., G. Björk, J. Jacobson, H. Cao, and Y. Yamamoto,1995, Phys. Rev. B 51, 7090.

Pau, S., J. Jacobson, G. Bjrk, and Y. Yamamoto, 1996, J. Opt.Soc. Am. B 13, 1078.

Pawlis, A., A. Khartchenko, O. Husberg, D. J. As, K. Lischka,and D. Schikora, 2002, Solid State Commun. 123, 235.

Pedri, P., L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S.Cataliotti, P. Maddaloni, F. Minardi, and M. Inguscio, 2001,Phys. Rev. Lett. 87, 220401.

Penrose, O., and L. Onsager, 1956, Phys. Rev. 104, 576.Pethick, C. J., and H. Smith, 2001, Bose-Einstein Condensation

in Dilute Gases Cambridge University Press, Cambridge.Piermarocchi, C., F. Tassone, V. Savona, A. Quattropani, and P.

Schwendimann, 1996, Phys. Rev. B 53, 15834.Pitaevskii, L. P., and S. Stringari, 2003, Bose Einstein Conden-

sation Clarendon, Oxford.Porras, D., C. Ciuti, J. J. Baumberg, and C. Tejedor, 2002,

Phys. Rev. B 66, 085304.Porras, D., and C. Tejedor, 2003, Phys. Rev. B 67, 161310.Posazhennikova, A., 2006, Rev. Mod. Phys. 78, 1111.Reitzenstein, S., C. Hofmann, A. Gorbunov, M. Strauss, S. H.

Kwon, C. Schneider, A. Loffler, S. Hofling, M. Kamp, and A.Forchel, 2007, Appl. Phys. Lett. 90, 251109.

Renucci, P., T. Amand, X. Marie, P. Senellart, J. Bloch, B. Ser-mage, and K. V. Kavokin, 2005, Phys. Rev. B 72, 075317.

Richard, M., J. Kasprzak, R. Romestain, R. Andre, and L. S.Dang, 2005, Phys. Rev. Lett. 94, 187401.

Risken, H., 1984, The Fokker-Planck Equation: Methods of So-lutions and Applications Springer, Berlin.

Rochat, G., C. Ciuti, V. Savona, C. Piermarocchi, A. Quattro-pani, and P. Schwendimann, 2000, Phys. Rev. B 61, 13856.

Roumpos, G., S. Hoefling, A. Forchel, and Y. Yamamoto,2009, APS March Meeting unpublished, Paper No.H1600010.

Roumpos, G., C. W. Lai, T. C. H. Liew, Y. G. Rubo, A. V.Kavokin, and Y. Yamamoto, 2009, Phys. Rev. B 79, 195310.

Sarchi, D., and V. Savona, 2006, e-print arXiv:cond-mat/0411084.

Sarchi, D., and V. Savona, 2007a, Phys. Rev. B 75, 115326.




http://dx.doi.org/10.1038/45194







http://dx.doi.org/10.1002/pssa.200306155


http://dx.doi.org/10.1088/0268-1242/22/5/R01




http://dx.doi.org/10.1016/S0038-1098(00)00469-5

http://dx.doi.org/10.1002/pssb.200777610

http://dx.doi.org/10.1088/0022-3719/5/11/002

http://dx.doi.org/10.1088/0022-3719/6/7/010

http://dx.doi.org/10.1126/science.286.5448.2309





http://dx.doi.org/10.1038/nphys1051




http://dx.doi.org/10.1088/0305-4470/36/11/102

http://dx.doi.org/10.1088/0305-4470/36/11/102

http://dx.doi.org/10.1109/JQE.1967.1074444

http://dx.doi.org/10.1038/25692


http://dx.doi.org/10.1088/0953-8984/16/35/003

http://dx.doi.org/10.1088/0953-8984/16/35/003


http://dx.doi.org/10.1088/0268-1242/18/10/314


http://dx.doi.org/10.1016/S0921-4526(99)00324-5

http://dx.doi.org/10.1063/1.124915

http://dx.doi.org/10.1063/1.124915

http://dx.doi.org/10.1063/1.1651646




http://dx.doi.org/10.1016/0038-1098(96)00204-9


http://dx.doi.org/10.1364/JOSAB.13.001078

http://dx.doi.org/10.1364/JOSAB.13.001078

http://dx.doi.org/10.1016/S0038-1098(02)00251-X







http://dx.doi.org/10.1063/1.2749862





http://arXiv.org/abs/arXiv:cond-mat/0411084

http://arXiv.org/abs/arXiv:cond-mat/0411084


Sarchi, D., and V. Savona, 2007b, Solid State Commun. 144,371.

Sarchi, D., and V. Savona, 2008, Phys. Rev. B 77, 045304.Sarchi, D., P. Schwendimann, and A. Quattropani, 2008, Phys.

Rev. B 78, 073404.Savvidis, P. G., J. J. Baumberg, R. M. Stevenson, M. S.

Skolnick, D. M. Whittaker, and J. S. Roberts, 2000, Phys.Rev. Lett. 84, 1547.

Schmitt-Rink, S., D. S. Chemla, and D. A. B. Miller, 1985,Phys. Rev. B 32, 6601.

Schouwink, P., H. V. Berlepsch, L. Dhne, and R. F. Mahrt,2001, Chem. Phys. Lett. 344, 352.

Schwendimann, P., and A. Quattropani, 2008, Phys. Rev. B 77,085317.

Shelykh, I. A., G. Malpuech, and A. V. Kavokin, 2005, Phys.Status Solidi A 202, 2614.

Shi, H., and A. Griffin, 1998, Phys. Rep. 304, 1.Shimada, R., J. Xie, V. Avrutin, U. Ozgur, and H. Morkoc,

2008, Appl. Phys. Lett. 92, 011127.Smerzi, A., S. Fantoni, S. Giovanazzi, and S. R. Shenoy, 1997,

Phys. Rev. Lett. 79, 4950.Solnyshkov, D., I. Shelykh, M. Glazov, G. Malpuech, T.

Amand, P. Renucci, X. Marie, and A. Kavokin, 2007, Semi-conductors 41, 1080.

Stamper-Kurn, D. M., A. P. Chikkatur, A. Görlitz, S. Inouye,S. Gupta, D. E. Pritchard, and W. Ketterle, 1999, Phys. Rev.Lett. 83, 2876.

Szymanska, M. H., J. Keeling, and P. B. Littlewood, 2006, Phys.Rev. Lett. 96, 230602.

Szymanska, M. H., P. B. Littlewood, and B. D. Simons, 2003,Phys. Rev. A 68, 013818.

Takada, N., T. Kamata, and D. D. C. Bradley, 2003, Appl.Phys. Lett. 82, 1812.

Tartakovskii, A. I., M. Emam-Ismail, R. M. Stevenson, M. S.

Skolnick, V. N. Astratov, D. M. Whittaker, J. J. Baumberg,and J. S. Roberts, 2000, Phys. Rev. B 62, R2283.

Tassone, F., C. Piermarocchi, V. Savona, A. Quattropani, and P.Schwendimann, 1996, Phys. Rev. B 53, R7642.

Tassone, F., C. Piermarocchi, V. Savona, A. Quattropani, and P.Schwendimann, 1997, Phys. Rev. B 56, 7554.

Tassone, F., and Y. Yamamoto, 1999, Phys. Rev. B 59, 10830.Tassone, F., and Y. Yamamoto, 2000, Phys. Rev. A 62, 063809.Tawara, T., H. Gotoh, T. Akasaka, N. Kobayashi, and T. Sai-

toh, 2004, Phys. Rev. Lett. 92, 256402.Tilley, D. R., and J. Tilley, 1990, Superfluidity and Supercon-

ductivity Hilger, New York.Tran Thoai, D. B., and H. Haug, 2000, Solid State Commun.

115, 379.Tsintzos, S. I., N. T. Pelekanos, G. Konstantinidis, Z. Hat-

zopoulos, and P. G. Savvidis, 2008, Nature London 453, 372.Utsunomiya, S., L. Tian, G. Roumpos, C. W. Lai, N. Kumada,

T. Fujisawa, M. Kuwata-Gonokami, A. Loffler, S. Hofling, A.Forchel, and Y. Yamamoto, 2008, Nat. Phys. 4, 700.

Weihs, G., H. Deng, R. Huang, M. Sugita, F. Tassone, and Y.Yamamoto, 2003, Semicond. Sci. Technol. 18, S386.

Weisbuch, C., M. Nishioka, A. Ishikawa, and Y. Arakawa,1992, Phys. Rev. Lett. 69, 3314.

Wouters, M., and I. Carusotto, 2007, Phys. Rev. Lett. 99,140402.

Wouters, M., and V. Savona, 2009, Phys. Rev. B 79, 165302.Yamamoto, Y., G. Bjök, K. Igeta, and Y. Horikoshi, 1989, in

Coherence and Quantum Optics VI, edited by R. Loudon, J.H. Eberly, L. Mandel, and E. Wolf Plenum, New York, p.1249.

Yang, C. N., 1962, Rev. Mod. Phys. 34, 694.Yeh, P., 2005, Optical Waves in Layered Media Wiley, Hobo-

ken.











http://dx.doi.org/10.1016/S0009-2614(01)00808-9





http://dx.doi.org/10.1016/S0370-1573(98)00015-5

http://dx.doi.org/10.1063/1.2830022


http://dx.doi.org/10.1134/S1063782607090138

http://dx.doi.org/10.1134/S1063782607090138






http://dx.doi.org/10.1063/1.1559950

http://dx.doi.org/10.1063/1.1559950







http://dx.doi.org/10.1016/S0038-1098(00)00200-3

http://dx.doi.org/10.1016/S0038-1098(00)00200-3


http://dx.doi.org/10.1038/nphys1034

http://dx.doi.org/10.1088/0268-1242/18/10/313






Documents

Dynamic condensation of exciton-polaritons · A. Wannier-Mott exciton 1491 1. Exciton optical transition 1492 2. Quantum-well exciton 1492 B. Semiconductor microcavity 1492 C. Quantum-well