Chapter 10 Dynamic Condensation of Exciton-Polaritons

Chapter 10

Dynamic condensation of exciton-polaritons

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

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 2010͒

In the past decade, a two-dimensional matter-light system called the microcavity exciton-polariton has emerged as a new promising candidate of Bose-Einstein condensation ͑BEC͒ in solids. Many pieces of important evidence of polariton BEC have been established recently in GaAs and CdTe microcavities at the liquid helium temperature, opening a door to rich many-body physics inaccessible in experiments before. Technological progress also made polariton BEC at room temperatures promising. In parallel with experimental progresses, theoretical frameworks and numerical simulations 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 equations and the Boltzmann kinetic simulations for a ﬁnite-size BEC of polaritons are reviewed.

DOI: 10.1103/RevModPhys.82.1489 PACS number͑s͒: 71.35.Lk, 71.36.ϩc, 42.50.Ϫp, 78.67.Ϫn

CONTENTS A. Polariton-phonon scattering 1500 B. Polariton-polariton scattering 1500 1. Nonlinear polariton interaction coefficients 1500 I. Introduction 1490 2. Polariton-polariton scattering rates 1502 II. Semiconductor Microcavity Polaritons 1491 C. Semiclassical Boltzmann rate equations 1502 A. Wannier-Mott exciton 1491 IV. Stimulated Scattering, Amplification, and Lasing of 1. Exciton optical transition 1492 Exciton Polaritons 1503 2. Quantum-well exciton 1492 A. Observation of bosonic final-state stimulation 1503 B. Semiconductor microcavity 1492 B. Polariton amplifier 1504 C. Quantum-well microcavity polariton 1493 C. Polariton lasing versus photon lasing 1505 1. The Hamiltonian 1493 1. Experiment 1505 2. Polariton dispersion and effective mass 1494 2. Numerical simulation with quasistationary 3. Polariton decay 1494 and pulsed pumping 1507 4. Very strong-coupling regime 1495 V. Thermodynamics of Polariton Condensation 1508 D. Polaritons for quantum phase research 1495 A. Time-integrated momentum distribution 1508 1. Critical densities of 2D quasi-BEC and B. Time constants of the dynamical processes 1508 BKT phase transitions 1495 C. Detuning and thermalization 1509 2. Quantum phases at high excitation densities 1496 D. Time-resolved momentum distribution 1509 3. Experimental advantages of polaritons for E. Numerical simulations 1510 BEC study 1496 F. Steady-state momentum distribution 1511 E. Experimental methodology 1497 VI. Angular Momentum and Polarization Kinetics 1511 1. Typical material systems 1497 A. Formulation of the quasispin kinetics 1512 2. Structure design considerations 1498 B. Time-averaged and time-resolved experiments and 3. In situ tuning parameters 1498 simulations 1513 4. Measurement techniques 1498 C. Stokes vector measurement 1515 F. Theoretical methods 1498 VII. Coherence Properties 1515 1. Semiclassical Boltzmann kinetics 1498 A. The second-order coherence function 1516 2. Quantum kinetics 1499 1. g͑2͒͑0͒ of the LP ground state 1516 3. Equilibrium and steady-state properties 1499 2. Kinetic models of the g͑2͒ 1517 4. System size 1499 3. Master equation with gain saturation 1518 III. Rate Equations of Polariton Kinetics 1499 4. Stationary solution 1519

5. Numerical solutions of the master equation 1519 ting into an upper and lower branches of polaritons 1 6. Nonresonant two-polariton scattering ͑Weisbuch et al., 1992͒. ͑quantum depletion͒ 1520 The metastable state of lower polaritons at zero in- B. First-order temporal coherence 1521 plane momentum ͑kʈ =0͒ has recently emerged as a new C. First-order spatial coherence 1521 candidate for observing Bose-Einstein condensation 2 1. Long-range spatial coherence in a ͑BEC͒ in solids ͑Imamoglu et al., 1996͒. The experimen- condensate 1521 tal advantage of polariton BEC over conventional exci- ͑ ͒ 2. Spatial coherence among bottleneck LPs 1521 ton BEC Hanamura and Haug, 1977 is twofold. Near 3. Spatial coherence between fragments of a kʈ =0, a polariton has an effective mass of four orders of condensate 1522 magnitude lighter than a bare exciton mass, so the criti- 4. Spatial coherence of a single-spatial mode cal temperature for reaching polariton BEC is four orders of magnitude higher than that for reaching exciton condensate 1522 BEC at the same particle density or equivalently the 5. Theory and simulations 1522 critical density for reaching polariton BEC is four orders D. Spatial distributions 1523 of magnitude lower than that for reaching exciton BEC 1. Condensate size and critical density 1523 at the same temperature. The second advantage of this 2. Comparison to a photon laser 1523 system is that a polariton can easily extend a phase- VIII. Polariton Superfluidity 1524 coherent wave function in space through its photonic A. Gross-Pitaevskii equation 1524 component in spite of unavoidable crystal defects and B. Bogoliubov excitation spectrum 1525 disorders, while a bare exciton is easily localized in a C. Blueshift of the condensate mean-field energy 1525 fluctuating potential inside a crystal, which is a notorious D. Energy vs momentum dispersion relation 1526 enemy to the exciton BEC. E. Universal features of Bogoliubov excitations 1527 Polaritons have a very short lifetime due to fast pho- F. Thermal and quantum depletion 1528 ton leakage rate from a microcavity. In most cases, po- G. Quantized vortices 1528 laritons have a lifetime shorter than the cooling time to IX. Polariton Condensation in Single Traps and Periodic the metastable ground state, so that the system remains Lattice Potentials 1528 in a nonequilibrium condition without a well-defined A. Polariton traps 1529 temperature or chemical potential. However, when the 1. Optical traps by metal masks 1529 cavity-photon resonance is detuned to higher than that ͑ ͒ 2. Optical traps by fabrication 1529 of the QW exciton blue detuning , the lower polariton ͑ ͒ 3. Exciton traps by mechanical stress 1529 LP has an increased excitonic component, a longer life- B. Polariton array in lattice potential 1529 time, and a shorter cooling time, facilitating thermaliza- 1. One-dimensional periodic array of tion. In the limit of a very large blue detuning, the dynamical polariton laser becomes indistinguishable from polaritons 1530 the thermal-equilibrium exciton BEC. In a recent ex- 2. Band structure, metastable ␲ state, and periment, a quantum degenerate polariton gas was ob- stable zero-state 1531 served with a chemical potential ␮ well within the BEC 3. Dynamics and mode competition 1532 regime, −␮/k TϽ0.1, and at a polariton gas tempera- X. Outlook 1532 B ture T very close to the lattice ͑phonon reservoir͒ tem- Acknowledgments 1534 perature of ϳ4K͑Deng et al., 2006͒. The excitonic com- Appendix: Calculation of Scattering Coefficients 1534 ponent of polaritons in this experiment is up to 80%. References 1535 However, the system is still metastable and the quantum degenerate gas exists only for several tens of picoseconds. The dynamical nature of a polariton condensate I. INTRODUCTION places a limitation for studying the standard BEC physics, but it also provides a new experimental tool for The experimental technique of controlling spontane- studying a nonequilibrium open system consisting of ous emission of an atom by a cavity is referred to as highly degenerate interacting bosons. This is not an eas- cavity quantum electrodynamics. It has been studied for ily accessible problem by atomic BEC or superfluid 4He a variety of atomic systems ͑Berman, 1994͒. In solids, systems. One unique feature of a polariton system, com- the inhibited and enhanced spontaneous emission was pared to dilute atomic BEC and dense superfluid 4He, is observed for two-dimensional ͑2D͒ quantum well ͑QW͒ excitons in semiconductor planar microcavities ͑Yama- 1 moto et al., 1989; Björk et al., 1991͒. Due to a very strong Since we discuss exclusively exciton polaritons in semicon- collective dipole interaction between QW excitons and ductors, we abbreviate exciton-polariton as polariton in this paper. microcavity photon fields, even with a relatively low-Q 2 The metastable state of upper polaritons at kʈ =0 in principle cavity, the planar microcavity system features a revers- may also undergo BEC. But the fast scattering from the upper ible spontaneous emission and thus normal-mode split- to the lower branch makes such a condensation unlikely.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1491 the direct experimental accessibility to the quantum- single isolated trap and one-dimensional array of traps statistical properties of the condensate. The main decay are presented in Sec. IX. A minimum uncertainty wave channel of polaritons is the photon leakage from a cavity packet close to the Heisenberg limit and competition in which the energy and in-plane momentum are con- between the s- and p-orbital superfluid states have been served between an annihilated polariton and a created observed. We conclude with some future prospects of photon. Through various quantum optical measure- the field in Sec. X. ments on the leakage photon field, we can easily study the quantum nature of the polariton condensate. II. SEMICONDUCTOR MICROCAVITY POLARITONS Various signatures of phase transition have been ex- ͑ perimentally established in GaAs Deng et al., 2002, A “microcavity polariton” is a quasiparticle resulting ͒ 2003, 2007; Balili et al., 2007; Lai et al., 2007 and CdTe from strong coupling between matter and light. The ͑ ͒ Richard et al., 2005; Kasprzak et al., 2006, 2008 micro- “matter” component is a Wannier-Mott exciton consist- cavities. In parallel with the experimental progress, the- ing of a bound pair of electron and hole. The “light” oretical studies and numerical simulations are developed component is a strongly confined photon field in a semi- ͑ for describing the thermodynamics Doan et al., 2005; conductor microcavity. Half-matter, half-light microcav- ͒ Sarchi and Savona, 2007a , modified excitation spectrum ity polaritons emerged as a unique solid-state candidate ͑ ͒ Szymanska et al., 2006; Wouters and Carusotto, 2007 , for research on phase-transition physics, as discussed in ͑ ͒ spin dynamics Kavokin et al., 2004; Cao et al., 2008 , this section. and particle-particle interactions ͑Cao et al., 2008͒ of polariton condensates. Our understanding of the system has greatly advanced in the past few years. A. Wannier-Mott exciton

In this article, we review the recent progress on ex- 23 perimental and theoretical studies of polariton conden- A solid consists of 10 atoms. Instead of describing 23 sates. A comprehensive review on theoretical modeling the 10 atoms and their constituents in full detail, the of the polariton system can be found in Keeling et al. common approaches are to treat the stable ground state ͑2007͒. Here we adopt the model that treats the polari- of an isolated system as a quasivacuum and to introduce tons as weakly interacting bosons. The model is valid quasiparticles as units of elementary excitations, which when the exciton density is well below the Mott transi- only weakly interact with each other. An exciton is a tion density and is applicable to most of the experiments typical example of such a quasiparticle, consisting of an on polariton condensation. Since we focus on spontane- electron and a hole bound by the Coulomb interaction. ous coherence and thermodynamic quantum phase of The quasivacuum of a semiconductor is the state with polaritons, we exclude from the current review the sub- filled valence band and empty conduction band. When ject of polariton parametric amplification which has an electron with charge −e is excited from the valence been reviewed by, e.g., Ciuti et al. ͑2003͒ and Keeling et band into the conduction band, the vacancy it leaves in al. ͑2007͒. the valence band can be described as a quasiparticle This article is organized as follows. Section II intro- called a “hole.” A hole in the valence band has charge 2͒−1 ץ 2ץ͑ duces microcavity polaritons, their properties relevant +e and an effective mass defined by − E/ p , E and for BEC research, common experimental methodolo- p denoting the energy and momentum of the hole. A ϳ gies, and theoretical tools. In Sec. III, we review the hole and an electron at p 0 interact with each other via ͑ ͒ microscopic theory of nonlinear interactions among po- Coulomb interaction and form a bound pair an exciton laritons and formulate the rate equations of isospin po- analogous to a hydrogen atom where an electron is lariton kinetics. Numerical simulations based on these bound to a proton. However, due to the strong dielectric rate equations are used to understand the experiments screening in solids and a small effective-mass ratio of the reviewed in Secs. IV, V, and VII. Section IV reviews cold hole to the electron, the binding energy of a semicon- collision experiments that validate treating polaritons as ductor exciton is on the order of 10–100 meV and its a dilute Bose gas and reveal polariton-polariton scatter- Bohr radius is about 10–100 Å, extending over tens of ͑ ͒ ing as the microscopic driving force of polariton dynam- atomic sites in the crystal Hanamura and Haug, 1977 . ics. Thermodynamics of polariton condensates, such as If we define the creation operator of a conduction- † relaxation and cooling of the polaritons, and temporal band electron of momentum k as aˆ k and a valence-band ˆ † changes of gas temperatures and chemical potential are hole as bk, the exciton creation operator can be defined presented in Sec. V. Section VI extends the theory to as Ј include the spin degrees of freedom and reviews the re- † mhk − mek † † eˆ = ͚ ␦ Ј␾ ͩ ͪaˆ bˆ . ͑1͒ lated experiments. Section VII reviews measurements of K,n K,k+k n m + m k kЈ k,kЈ e h the coherence functions of polariton condensates, including the first- and second-order temporal coherence Here me and mh are the electron and hole mass, respec- and the first-order spatial coherence. Section VIII intro- tively; K is the center-of-mass momentum of the exciton; ␾ ͑ ͒ duces the Gross-Pitaevskii ͑GP͒ equation and applies it n k is the wave function of the relative motion of an to experiments on the ground-state energy, quantum electron and a hole, which takes the same form as that depletion spectra, and vortices of polariton condensates. of an electron and a proton in a hydrogen atom; and n is Finally, the condensation dynamics of polaritons in a the quantum number of the relative motion. The exciton

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1492 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

operators obey the following commutation relations: n1 n2 ͓ ͔ air n =1 eˆKЈ,nЈ,eˆK,n =0, 0 n n n n ͓ † † ͔ ͑ ͒ c 1 2 1 eˆKЈ,nЈ,eˆK,n =0, 2

͓ † ͔ ␦ ␦ ͑ 3 ͒ eˆKЈ,nЈ,eˆK,n = KKЈ nnЈ − O nexcaB .

Hence excitons can be considered as bosons when the substrate (bottom) DBR (top) DBR exciton interparticle spacing is much larger than its Bohr cavity (with QW) Ӷ −3 radius: nexc aB . FIG. 1. ͑Color online͒ Sketch of a semiconductor ␭/2 microcavity. 1. Exciton optical transition

The electron and hole in an exciton form a dipole that B. Semiconductor microcavity interacts with electromagnetic fields of light. The interaction strength with light at angular frequency ␻ is con- Figure 1 shows a typical structure of a semiconductor ␭ ventionally described by the exciton oscillator strength f, microcavity consisting of a c /2 cavity layer sandwiched between two distributed Bragg reflectors ͑DBRs͒.A 2m*␻ V f = ͦ͗u ͉r · e͉u ͦ͘2 . ͑3͒ DBR is made of layers of alternating high and low re- ប v c ␲ 3 aB fraction indices, each layer with an optical thickness of ␭ * ͑ −1 −1͒−1 /4. Light reflected from each interface destructively in- Here m = me +mh is the effective mass of the exciton, ͉u ͘ and ͉u ͘ are the electron and hole Bloch func- terfere, creating a stop band for transmission. Hence the c v DBR acts as a high-reflectance mirror when the wave- tions, V is the quantization volume, and a3 is the Bohr B length of the incident light is within the stop band. When radius of the exciton. V/␲a3 reflects the enhancement of B two such high-reflectance DBRs are attached to a layer the interaction due to enhanced electron-hole overlap in ␭ with an optical thickness integer times of c /2, a cavity an exciton compared to a pair of unbound electron and ␭ resonance is formed at c, leading to a sharp increase of hole. ␭ the transmission T at c, ͑1−R ͒͑1−R ͒ 2. Quantum-well exciton T = 1 2 . ͑4͒ ͓1−ͱR R ͔2 +4ͱR R sin2͑␾/2͒ A semiconductor QW is a thin layer of semiconductor 1 2 1 2 with a thickness comparable to the exciton Bohr radius, Here ␾ is the cavity round-trip phase shift of a photon at sandwiched between two barrier layers with a much ␭ c and R1 and R2 are the reflectances of the two DBRs. larger band gap. The center-of-mass motion of an exci- The corresponding cavity quality factor Q is ton in a QW is quantized along the confinement direction. If only one quantized level is concerned, in most ␭ ␲͑R R ͒1/4 Q Џ c Ӎ 1 2 , ͑5͒ cases the lowest-energy level, QW excitons behave as ⌬␭ ͑ ͒1/2 c 1− R1R2 two-dimensional quasiparticles. ⌬␭ The quantum confinement also modifies the valence- where c is the width of the resonance. An ideal cavity band structure and hence the optical transition strength has Q=ϱ. If the cavity length is ␭/2, Q is the average and selection rules. Most significantly, momentum con- number of round trips a photon travels inside the cavity servation in an optical transition needs to be satisfied before it escapes. Figure 2͑a͒ gives an example of the only in the QW plane but not along the confinement reflection spectrum of a cavity with QӍ4000 at normal direction. Thus excitons in a QW couple to light with the incidence. Figure 2͑b͒ shows the field intensity distribu- 2 same in-plane wave number kʈ and arbitrary transverse tion ͉E͑z͉͒ of the resonant mode. The field is concen- wave number kЌ. QWs are much more optically acces- trated around the center of the cavity; its amplitude is sible than bulk materials. enhanced by ϳ20 times compared to the free space Moreover, due to the confinement, a QW exciton has value. Unlike in a metallic cavity, the field penetration a smaller Bohr radius and larger binding energy com- depth into the DBRs is much larger. The effective cavity pared to a bulk exciton, leading to an enhancement of length is extended in a semiconductor microcavity as ͑ 3D 2D͒3 the oscillator strength by aB /aB . This enhancement is often offset by a reduction in the overlap between the Leff = Lc + LDBR, light field and the exciton because the transverse coher- ͑6͒ ence length of the photon field is usually much longer ␭ n n Ϸ c 1 2 than the QW thickness. Hence to achieve stronger LDBR ͉ ͉ . 2nc n1 − n2 exciton-photon coupling, it is necessary to also confine the photon field in the z direction by introducing a mi- Here n1 and n2 are the refraction indices of the DBR crocavity. layers, as shown in Fig. 1

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(a) 1 excitation in the system is stored in the combined system 0.8 of photons and excitons. Thus the elementary excitations of the system are no longer excitons or photons but 0.6 a new type of quasiparticles called the polaritons. 0.4

reflectance Using the rotating wave approximation, the linear 0.2 Hamiltonian of the system is written in the second quan- 0 tization form as 700 750 800 850 wavelength (nm) ˆ ˆ ˆ ˆ (b) Hpol = Hcav + Hexc + HI 4 |E |2 z † ˆ † ˆ = ͚ E ͑kʈ,k ͒aˆ aˆ + ͚ E ͑kʈ͒b b n cav c kʈ kʈ exc kʈ ky 3.5 ͚ ͑ˆ † ˆ ˆ ˆ † ͒ ͑ ͒ + g0 ak bkʈ + akʈbk . 11 3 ʈ ʈ †

refraction index n Here aˆ is the photon creation operator with an in- 2.5 kʈ 0 1 2 3 4 plane wave number kʈ and longitudinal wave number z(mm) k =k·zˆ determined by the cavity resonance. bˆ † is the c kʈ FIG. 2. ͑Color online͒ Properties of a high quality microcavity. exciton creation operators with an in-plane wave num- ͑ ͒ ␭ ͑ ͒ a Reﬂectance of an empty /2 microcavity. b The cavity ber kʈ. g0 is the exciton-photon dipole interaction ͉ ͉2 structure and ﬁeld intensity distribution Ez of the resonant strength; it is nonzero only between modes with the

TE mode. same kʈ. The above Hamiltonian can be diagonalized by the transformations The planar DBR cavity confines the photon field in Pˆ = X bˆ + C aˆ , ͑12͒ the z direction but not in the x-y plane. The incident kʈ kʈ kʈ kʈ kʈ light from a slant angle ␪ relative to the z axis has a ␭ ␪ ˆ ˆ ˆ ͑ ͒ resonance at c /cos . As a result, the cavity has an en- Qkʈ =−Ckʈbkʈ + Xkʈakʈ . 13 ergy dispersion versus the in-plane wave number kʈ, ˆ ប Then Hpol becomes cͱ 2 2 E = kЌ + kʈ , ͑7͒ cav ˆ ˆ † ˆ ˆ † ˆ nc H = ͚ E ͑kʈ͒P P + ͚ E ͑kʈ͒Q Q . ͑14͒ pol LP kʈ kʈ UP kʈ kʈ ͑ ␲ ␭ ͒ where kЌ =nc 2 / c . And there is a one-to-one corre- Here ͑Pˆ ,Pˆ † ͒ and ͑Qˆ ,Qˆ † ͒ are the creation and anni- spondence between the incidence angle ␪ and each reso- kʈ kʈ kʈ kʈ hilation operators of the new quasiparticles or eigen- nance mode with in-plane wave number kʈ, modes of the system. They are called the lower polariton 2␲ sin ␪ 2␲ ͑ ͒ ͑ ͒ ͫ −1ͩ ͪͬ Ϸ ␪ Ӷ LP and upper polariton UP , corresponding to the two kʈ = nc ␭ tan sin ␭ when kʈ kЌ. c nc c branches of lower and higher eigenenergies, respectively. Thus a polariton is a linear superposition of an ͑8͒ exciton and a photon with the same in-plane wave num- Ӷ In the region kʈ kЌ, we have ber kʈ. Since both excitons and photons are bosons, so 2 2 2 are the polaritons. The exciton and photon fractions in បc kʈ ប kʈ Ϸ ͩ ͪ ͑ ͒ ͑ ͒ each LP and UP are given by the amplitude squared of Ecav kЌ 1+ 2 = Ecav kʈ =0 + . 9 n 2kЌ 2m c cav Xk and Ck which are referred to as the Hopfield coef- ʈ ͑ ʈ ͒ Here the cavity-photon effective mass is ficients Hopfield, 1958 . They satisfy ͑ ͒ ͉ ͉2 ͉ ͉2 ͑ ͒ E kʈ =0 Xkʈ + Ckʈ =1. 15 m = cav . ͑10͒ cav c2/n2 c ⌬ ͑ ʈ͒ ͑ ʈ͒ ͑ ʈ ͒ Let E k =Eexc k −Ecav k ,kc ; Xkʈ and Ckʈ are given −5 by It is typically on the order of 10 me.

1 ⌬E͑kʈ͒ ͉X ͉2 = ͩ1+ ͪ, kʈ 2 ͱ⌬ ͑ ͒2 2 C. Quantum-well microcavity polariton E kʈ +4g0 ͑16͒ 1. The Hamiltonian 1 ⌬E͑kʈ͒ ͉C ͉2 = ͩ1− ͪ. When the GaAs QWs are placed at the antinodes of kʈ ͱ 2 2 2 ⌬E͑kʈ͒ +4g the resonant field of a semiconductor microcavity, the 0 ⌬ ͉ ͉2 ͉ ͉2 1 J=1 heavy-hole exciton doublet strongly interacts with At E=0, X = C = 2 , LP and UP are exactly half pho- the confined optical field of the cavity. If the rate of ton half exciton. energy exchange between the cavity field and excitons The energies of the polaritons, which are the eigenen- becomes much faster than the decay and decoherence ergies of the Hamiltonian ͑14͒, are deduced from the rates of both the cavity photons and the excitons, an diagonalization procedure as

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dispersions Hopfield coefficients 1.615 (a) 1 |X(k )|2 1.61 E (k ) || E E UP || UP cav 1.66 1.605 0.5 Ω E 1.64 E (k ) 1.6 2h exc cav || 2

energy (eV) E (k ) E (k ) LP || |C(k )| 1.595 1.62 exc || || energy (eV) 1.59 0 -5 0 5 -5 0 5 E (b) LP 1 1.585 1.66 1.58

−1 −0.5 0 0.5 1 1.64 ∆/2hΩ 0.5

FIG. 3. ͑Color online͒ Anticrossing of LP and UP energy lev- energy (eV) 1.62 els when tuning the cavity energy across the exciton energy, 0 calculated by the transfer-matrix method. -5 0 5 -5 0 5 (c) 1 1.64 ͑ ͒ 1 ͓ ͱ 2 ͑ ͒2͔ ELP,UP kʈ = 2 Eexc + Ecav ± 4g0 + Eexc − Ecav . 0.5 ͑17͒ 1.62

energy (eV) When the uncoupled exciton and photon are at reso- 1.6 0 nance, Eexc=Ecav, LP and UP energies have the mini- -5 0 5 -5 0 5 mum separation E −E =2g , which is often called k (mm -1 ) k (mm -1 ) UP LP 0 || || the normal-mode splitting in analogy to the Rabi splitting of a single-atom cavity system. Due to the coupling FIG. 4. ͑Color online͒ Polariton dispersions and corresponding ͑ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ ⌬ between the exciton and photon modes, the new polar- Hopfield coefficients at a =2g0, b =0, and c =−2g0. iton energies anticross when the cavity energy is tuned across the exciton energy. This is one of the signatures of 2 −4 ͑ ͒ ͉ ͉ӷ m ͑kʈ ϳ 0͒Ӎm /͉C͉ ϳ 10 m , “strong coupling” Fig. 3 . When Ecav−Eexc g0, polari- LP cav exc tons quickly become indistinguishable from a photon or ͑22͒ ͑ ϳ ͒Ӎ ͉ ͉2 exciton. Unless otherwise specified, the detuning is as- mUP kʈ 0 mcav/ X . sumed to be comparable to or less than the coupling The very small effective mass of LPs at kʈ ϳ0 makes strength in our discussions. possible the very high critical temperature of phase tran- ͑ ͒ sitions for the system. At large kʈ where Ecav kʈ ͑ ͒ӷ 2. Polariton dispersion and effective mass −Eexc kʈ g0, dispersions of the LP and UP converge to the exciton and photon dispersions, respectively, and LP We define ⌬ as the exciton and photon energy detun- ϳ 0 has an effective mass mLP mexc. Hence the LP’s effec- ing at kʈ =0, tive mass changes by four orders of magnitude from kʈ

ϳ0 to large kʈ. This peculiar shape has important impli- ⌬ Џ E ͑kʈ =0͒ − E ͑kʈ =0͒. ͑18͒ 0 cav exc cations in the energy relaxation dynamics of polaritons, ⌬ ͑ ͒ Given 0, Eq. 17 gives the polariton energy- as discussed in Sec. IV. A few examples of the polariton ប2 2 Ӷ ⌬ momentum dispersions. In the region kʈ /2mcav 2g0, dispersion with different 0 are given in Fig. 4. the dispersions are parabolic,

2 2 ប kʈ 3. Polariton decay ͑ ͒Ӎ ͑ ͒ ͑ ͒ ELP,UP kʈ ELP,UP 0 + . 19 2mLP,UP When taking into account the finite lifetime of the ͑ ͒ The polariton effective mass is the weighted harmonic cavity photon and QW exciton, the eigenenergy 17 is mean of the mass of its exciton and photon components, modified as ͑ ͒ 1 ͕ ͑␥ ␥ ͒ 1 ͉X͉2 ͉C͉2 ELP,UP kʈ = 2 Eexc + Ecav + i cav + exc = + , ͑20͒ ͱ 2 ͓ ͑␥ ␥ ͔͒2͖ mLP mexc mcav ± 4g0 + Eexc − Ecav + i cav − exc . ͑23͒ 1 ͉C͉2 ͉X͉2 = + , ͑21͒ ␥ m m m Here cav is the out-coupling rate of a cavity photon due UP exc cav ␥ to imperfect mirrors and exc is the nonradiative decay where X and C are the Hopfield coefficients given by rate of an exciton. Thus the coupling strength must be ͑ ͒ Eq. 16 . mexc is the effective exciton mass of its center- larger than half of the difference in decay rates to ex- of-mass motion and mcav is the effective cavity-photon hibit anticrossing, i.e., to have polaritons as the new ͑ ͒ Ӷ mass given by Eq. 10 . Since mcav mexc, eigenmodes. In other words, an excitation must be able

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TABLE I. Parameter comparison of BEC systems.

Atomic Systems gases Excitons Polaritons

3 −1 −5 Effective mass m* /me 10 10 10 −1 2 2 Bohr radius aB 10 Å10Å10Å Particle spacing: n−1/d 103 Å102 Å1␮m ␮ Ͼ Critical temperature Tc 1nK–1 K 1 mK–1 K 1– 300 K Thermalization time/ Lifetime 1 ms/1 sϳ10−3 10 ps/1 nsϳ10−2 ͑1–10 ps͒/͑1–10 ps͒=0.1–10

to coherently transfer between a photon and an exciton D. Polaritons for quantum phase research ӷ͑␥ ␥ ͒ at least once. When g0 cav− exc /2, we call the system in the strong-coupling regime. In the opposite limit As composite bosons with not only fermionic but also when excitons and photons instead are the eigenmodes, photonic constituents, polaritons constitute a unique sys- the system is called to be in the weak-coupling regime, tem for exploring both cavity QED and many-body and the radiative decay rate of an exciton is given by the physics. optical transition matrix element. We are mostly inter- A variety of quantum phases is predicted for polari- ested in microcavities with ␥ Ӷ␥ Ӷg ; then Eq. ͑17͒ tons, including BEC, superfluidity, Berezinskii- exc cav 0 ͑ ͒͑ ͒ gives a good approximation of the polariton energies. Kosterlitz-Thouless BKT Malpuech et al., 2003 , and ͑ ͒ ͑ As a linear superposition of an exciton and a photon, Bardeen-Cooper-Schrieffer BCS states Keldysh and the lifetime of the polaritons is directly determined by Kozlov, 1968; Comte and Nozières, 1982; Malpuech et ␥ ␥ al., 2003; Littlewood et al., 2004; Sarchi and Savona, exc and cav as 2006, 2007a͒. Table I compares the basic parameters of polaritons to semiconductor excitons and cold atoms, ␥ = ͉X͉2␥ + ͉C͉2␥ , ͑24͒ LP exc cav which consist of the major table-top Bose systems for quantum phase studies. ␥ ͉ ͉2␥ ͉ ͉2␥ ͑ ͒ The parameter scales of these systems differ by many UP = C exc + X cav. 25 orders of magnitude. Even for the same type of quan- In typical high quality semiconductor samples available tum phase, the polariton system has distinct characteris- ␥ ␥ ϳ tics and requires dedicated theoretical frameworks. In today, cav=1–10 ps and exc 1 ns, hence the polariton lifetime is mainly determined by the cavity-photon life- particular, we discuss two important fundamental issues pertaining to the ground state of a degenerate polariton time: ␥ Ӎ͉C͉2␥ . Polariton decays in the form of emit- LP cav gas: the possibility of BEC and BKT phases of polari- ting a photon with the same kʈ and total energy ប␻ tons as two-dimensional systems and the BEC-BCS =E . The one-to-one correspondence between the LP,UP crossover or Mott transition in the high-density regime internal polariton mode and the external out-coupled of polaritons as a system of composite bosons. Due to photon mode lends convenience to experimental access the short lifetime of polaritons compared to their ther- to the system ͑see Sec. II.E.4͒. malization time, the excitation spectra of a polariton condensate differ from their atomic gas counterparts, as discussed in Sec. VIII. 4. Very strong-coupling regime On the experimental side, the polariton is a most ac- The definition of polariton operators in Eqs. ͑11͒–͑14͒ cessible system, especially due to the high critical tem- follows the procedure to first diagonalize the electron- perature associated with its light effective mass and the hole Hamiltonian by exciton operators and then diago- one-to-one correspondence between each polariton and nalize the exciton-photon Hamiltonian by polariton op- each leakage cavity photon. These and some other as- erators, treating the excitons as structureless pects of the experimental advantages of polaritons are quasiparticles. When the exciton-photon coupling is so summarized later in this section. strong as to become comparable to exciton binding energy, the question arises whether or not the above procedure is still valid. A more rigorous approach is to treat 1. Critical densities of 2D quasi-BEC and BKT phase transitions the electron, hole, and photon on equal footing, and one finds that the LPs then consist of excitons with an even A uniform 2D system of bosons does not have a BEC smaller effective Bohr radius and larger binding energy, phase transition at finite temperatures in the thermody- while the opposite holds for UPs. This is called the very namic limit since long-wavelength thermal fluctuations strong-coupling effect ͑Khurgin, 2001; Citrin and Khur- destroy long-range order ͑Mermin and Wagner, 1966; gin, 2003͒. It is a consequence of sub-band mixing of Hohenberg, 1967͒. higher-orbital excitons ͑mixing with 1s,2s,2p,... exciton If the Bose gas is confined by a spatially varying po- levels͒. tential Uϳr␩ ͑Bagnato and Kleppner, 1991; Griffin et al.,

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1496 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

1995͒, the constant density of states ͑DOS͒ of the uni- a venue to our further understanding of 2D quantum form 2D system is dramatically modified by the poten- phases. For a recent review of the ground-state proper- tial, and a BEC phase at finite temperature is recovered. ties of a 2D Bose gas in general, see Posazhennikova A few experimental implementations of transverse con- ͑2006͒. finement potential in polariton systems will be discussed in Sec. IX. 2. Quantum phases at high excitation densities Practically, any experimental system has a finite size and a finite number of single-particle states. With dis- An important parameter not listed in Table I is the ␧ ͑ ͒ 2 crete energy levels i i=1,2,... and size S=L , the particle-particle interaction strength or the scattering critical condition for quasi-BEC in a finite 2D system length a. For atomic gasses, a is usually much less than can be defined as the particle spacing, satisfying the diluteness condition. In diatomic gases where each diatomic molecule consists ␮ ␧ ͑ ͒ ͑ ͒ = 1 chemical potential , 26 of two fermionic atoms, a can be defined for the interaction between two Fermi atoms and has a hyperbolic 1 1 ͑ ͒ ͑ ͒ dependence on an external magnetic field. The magnetic nc = ͚ ␧ critical 2D density . 27 i/kBT ͑ Ͻ ͉ ͉ S iജ2 e −1 field tunes a from a small negative value a 0, kf a Ӷ ͒ 1, kf is the wave vector at the Fermi surface to a very In a 2D box system of size L, the critical condition can large negative or positive value at the unitary condition Ͼ be fulfilled at Tc 0 at a critical density, ͑ ͉ ͉ӷ ͒ kf a 1 near the Feshbach resonance and then to a ͑ ͒ 2 L small positive value Feshbach, 1962 . Correspondingly, n = lnͩ ͪ. ͑28͒ c ␭2 ␭ below a critical temperature, the system exhibits a cross- T T over from a degenerate Fermi gas described by the stan- If the particle number N is sufficiently large and/or the dard BCS theory to a degenerate Bose gas with a BEC temperature is sufficiently low, the phase transition phase. shows similar features as a BEC phase transition defined A similar BEC to BCS crossover is also possible in at the thermodynamic limit ͑Ketterle and van Druten, exciton and polariton systems ͑Comte and Nozières, 1996͒. 1982; Eastham and Littlewood, 2001͒. Here the pairing In addition to the quasi-BEC phase, formation of vor- is between an electron and a hole, and the crossover is tices drives another phase transition unique to the 2D controlled by the density of the electrons and holes nexc. system—the BKT transition ͑Berezinskii, 1971, 1972; The density nexc tunes both the interparticle spacing ͒ −1/d Kosterlitz and Thouless, 1972, 1973 . At low tempera- nexc and, through Coulomb screening, the exciton- tures, thermally excited vortices may form in the system exciton interaction strength and exciton-photon cou- which may support a local order. With further cooling of pling. The polariton gas behaves as a weakly interacting ͑ Ӷ ͒ the system to below a critical temperature TBKT, the Bose gas when the density is low nexc nMott , with the bound pair of oppositely circulating vortices form, which scattering length approximated by the exciton Bohr ra- stabilize the phase to a power-law decay versus distance dius aB. Increasing nexc to nMott, the BEC-BCS crossover and support finite superfluid density. The critical tem- may take place, where degenerate Fermi gasses of elec- perature of the BKT transition is trons and holes are paired in the momentum space with ␲ប2 a Bohr radius comparable to the particle spacing. At ͑ ͒ very high densities, we have an electron and hole plasma kBTBKT = ns 2 , 29 2m with screened Coulomb interactions, which coherently ͑ ͒ couple to a cavity-photon field. Inhomogeneity in exci- where ns is the superfluid density. Equation 29 can also be written in terms of the thermal de Broglie wavelength ton energy and inelastic scattering at crystal defects, however, may suppress the BCS gap and reduce the sys- ␭ ͑Khalatnikov, 1965͒, T tem to an incoherent electron and hole plasma. Detailed 2␲ប2 studies of the phase diagrams at high excitation densities ␭ ͑ ͒ Џ ͱ ͑ ͒ T T , 30 and effects of dephasing can be found in Eastham and mk T B Littlewood ͑2001͒, Szymanska et al. ͑2003͒, and Little- wood et al. ͑2004͒. n ␭ ͑T ͒2 =4. ͑31͒ s T c In the current review, we focus on the BEC regime of In most experiments discussed in this review, the sys- LPs, where the excitation density is below the exciton tem size is small enough such that a finite-size BEC saturation density and excitons and LPs can be well de- phase transition is expected before a BKT transition scribed as interacting bosons. ͑Kavokin et al., 2003͒. However, the nature of the ground state and its excitation spectra, as well as the 3. Experimental advantages of polaritons for BEC study dependence of the BEC and BKT critical densities on the system parameters, are not yet well understood at Beside as a unique system of fundamental interest, present. Observations of phonon modes, superfluidity, polaritons also possess many experimental advantages and single and bound pair of vortices is discussed in Sec. for BEC research. Most notable is the light effective Ӷ ͑ ͒ VIII. Understanding of these observations may provide mass of polaritons in the region kʈ kc see Sec. II.C.2. .

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1497

Taking into account the excitonlike dispersion at higher phonon and polariton-polariton scattering cross sec- kʈ, the critical temperatures of polariton condensation tions, hence efficient polariton thermalization; ͑3͒ small range from cryogenic temperature of a few kelvin up to exciton Bohr radius and large exciton binding energy, room temperature and higher.3 Not only does this sim- hence high exciton saturation density; and ͑4͒ strong plify enormously the experimental setups for polariton exciton-photon coupling, hence robust strong coupling; research but it also makes polaritons a natural and prac- it requires both large QW exciton oscillator strength and tical candidate for quantum device applications. large overlap of the photon field and the QWs. A common enemy against quantum phase transitions The choice of a direct band-gap semiconductor de- in solids is the unavoidable compositional and structural pends first on the fabrication technology. The best fab- disorders. By dressing the excitons with a microcavity rication quality of both quantum well and microcavity vacuum field, extended coherence is maintained in the has been achieved via molecular-beam epitaxy growth of ͑ ഛ ഛ ͒ combined excitation in a microcavity polariton system, AlxGa1−xAs-based samples 0 x 1 , thanks to the and the detrimental effects of disorders are largely sup- close match of the lattice constants alat of AlAs and pressed. GaAs and a relatively large difference between their Saturation and phase-space filling are another two ob- band-gap energies Eg. At 4 K, GaAs has alat=5.64 Å stacles against BEC of composite bosons in solids. In a and Eg =1.519 eV and AlAs has alat=5.65 Å and Eg microcavity polariton system, multiple QWs can be used =3.099 eV. Nearly strain and defect-free GaAs QWs are to dilate the exciton density per QW for a given total now conventionally grown between Al Ga As. Inho- QW x 1−x polariton density: nexc =nLP/NQW, where NQW is the mogeneous broadening of exciton energy is limited number of QWs. mainly by monolayer QW thickness fluctuation. Nearly A main challenge for realizing a thermodynamic defect-free microcavity structures can be grown with phase transition and spontaneous coherence has been more than 30 pairs of AlAs/GaAs layers in the DBRs efficient cooling of hot particles. It is especially so for and with a cavity quality factor Q exceeding 105 relatively short-lived quasiparticles in solids. The small ͑Reitzenstein et al., 2007͒. Many signatures of polariton effective mass, hence small energy DOS of polaritons, condensation were first obtained in GaAs-based systems comes to rescue here. Due to the small DOS, a quantum ͑Deng et al., 2002, 2003, 2007; Balili et al., 2007; Lai et al., degenerate seed of low-energy polaritons is relatively 2007͒. easy to achieve. Hence the bosonic final-state stimula- Another popular choice is the CdTe-based II-IV sys- tion effect is pronounced, which accelerates energy re- tem, with CdTe QWs and MgxCd1−xTe and MnxCd1−xTe laxation of the polaritons. barrier and DBR layers. The larger lattice mismatch is Finally, the cavity-photon component of a polariton compensated by larger binding energy and larger oscil- couples out of the cavity at a fixed rate while conserving lator strength, as well as larger refractive index contrast energy and in-plane momentum. Many essential proper- ͑hence less layers needed in the DBRs͒. The smaller ties of the polariton gas can be directly measured Bohr radius of CdTe excitons, on the one hand, allows a through the leakage photon field with well-developed larger saturation density and, on the other hand, reduces spectroscopic and quantum optical techniques, including the polariton and acoustic phonon scattering. Hence the ͑ coherence functions Deng et al., 2002, 2007; Kasprzak et energy relaxation bottleneck is more persistent in this ͒ al., 2006, 2008 , population distributions in both coordi- system, which prevented condensation in the LP ground ͑ ͒ nate Deng et al., 2003, 2007; Kasprzak et al., 2006 and state in early experiments ͑Huang et al., 2002; Richard et ͑ configuration space Deng et al., 2003; Kasprzak et al., al., 2005͒. By adjusting the detuning to facilitate ther- ͒ ͑ 2006 , vortices Lagoudakis et al., 2008; Roumpos, Hoe- malization, partially localized polariton condensation ͒ ͑ fling, et al., 2009 , and transport properties Utsunomiya into the ground state was finally observed by Kasprzak ͒ et al., 2008; Amo et al., 2009 . In comparison, compli- et al. ͑2006͒. cated and often indirect techniques are required to In recent years, there has been intense interest in de- probe those properties in matter systems. Hence the po- veloping certain wide-band-gap material systems with lariton system also offers a unique opportunity for small lattice constants. In these materials, the exciton studying new and open questions in quantum phase re- binding energy and oscillator strength are large enough, search. so that a polariton laser may survive at Ͼ300 K, operating at visible to ultraviolet wavelengths. However, fabri- E. Experimental methodology cation techniques for these materials are not as mature as for GaAs- or CdTe-based systems. The structures of 1. Typical material systems wide-band-gap materials suffer a higher concentration An optimal microcavity system for polariton BEC has of impurities and crystal defects. Integration of the QWs the following: ͑1͒ high quality cavity, hence long cavity with microcavities is challenging due to the lack of lat- photon and polariton lifetime; ͑2͒ large polariton- tice matched DBR layers. Though the strong-coupling regime is still hard to reach or reproduce at present, there have been a few promising reports on the obser- 3Note that other constrains on the condensation temperature vation of polaritons in ZnSe ͑Pawlis et al., 2002͒, GaN ͑ ͒ ͑ Tc apply. For example, kBTc must be smaller than the exciton Tawara et al., 2004; Butte et al., 2006 , and ZnO Shi- binding energy or half of the LP-UP splitting. mada et al., 2008; Chen et al., 2009͒. Lasing of GaN po-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1498 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation laritons at room temperature was reported ͑Christopou- a one-to-one correspondence between an internal polar- los et al., 2007; Christmann et al., 2008͒. Also active iton in mode kʈ and an external photon with the same research is conducted on Frenkel exciton polaritons in energy and in-plane wave number ͑Houdré et al., 1994͒. organic materials ͑Lidzey et al., 1998, 1999; Schouwink et The internal polariton is coupled to the external photon al., 2001; Takada et al., 2003; Kena-Cohen et al., 2008͒. via its photonic component with a fixed coupling rate ͓see Eq. ͑25͔͒. Hence information on the internal polari- 2. Structure design considerations tons can be directly measured via the external photon fields. At the same time, due to the steep dispersion of To achieve maximum exciton-photon coupling, a short polaritons, polariton modes with different kʈ and E ͑kʈ͒ ͑ ͒ LP cavity is generally preferred with multiple narrow QWs are well resolved with conventional optics. It is mainly ͑ ͒ ␭ placed at the antinode s of the cavity field. A /2 cavity through the external photon field that we learn about gives the largest field amplitude at the cavity center for the polaritons. given reflectance of the DBRs. An optimal QW thick- The basic properties of a polariton system can be ness, often between 0.5 and 1 Bohr radius of a bulk ex- characterized through reflection or transmission mea- citon, leads to smaller Bohr radius and enhanced oscil- surements using a weak light source. For instance, from lator strength. To increase the field-exciton overlap, one a measured reflection spectrum, one extracts the reso- can use multiple closely spaced narrow QWs and even nances of the microcavity system, the widths of the reso- put QWs in the two antinodes next to the central one, nances, as well as the cavity stop-band structure. Since leading to an increased total coupling strength g0 most microcavities have a tapered cavity length, a serial ϰͱ NQW. The difference in the field amplitude E at each of reflection measurement at various positions on the QW has an effect similar to inhomogeneous broadening sample reveals the characteristic anticrossing ͑crossing͒ of the QW exciton linewidth ͑Pau, Björk, et al., 1996͒. of the eigenenergies and thus verifies strong ͑weak͒ cou- Using multiple QWs has another important benefit—an pling between the cavity and exciton modes. Dispersion increased saturation density of LPs, which scales with of the resonances may also be measured by angle- 2 ͑ ͒ N/aB. When g0 is increased to become comparable with resolved reflection, although photoluminescence PL is the exciton binding energy, the strong-coupling effect often more convenient for this purpose. further reduces exciton Bohr radius in the LP branch Polaritons are conveniently excited, resonantly or and increases the exciton saturation density. nonresonantly, by optical pumping. By changing the col-

lection angle of the PL, kʈ-dependent spectra are ob- 3. In situ tuning parameters tained, which map out the energy-momentum dispersion The microcavity system of a given composition and as well as momentum distribution of the polaritons. structure has a few useful parameters that can be ad- Complementary to momentum-resolved PL, near-field imaging of the sample surface maps out the spatial dis- justed in situ during the experiments: detuning ⌬ of the tribution of LPs. Quantum optics measurement on the uncoupled cavity resonance relative to the QW exciton PL reveals the quantum statistics of LPs. Time-resolved resonance, density of the polaritons, and temperature of PL or imaging measurements map out the dynamics and the host lattice. transport of LPs. Tapering of the cavity thickness by special sample growth techniques enables tuning of ⌬ across the sample. Changing ⌬ changes the exciton and photon fractions in the LPs, hence the dispersions and lifetimes F. Theoretical methods of the LPs. It has important implications in LP dynamics We describe the main theoretical tools considered for which is discussed in Sec. V. studying polariton condensation. Some references for The density of the polaritons is directly controlled by relevant methods and concepts are given, but are by no the excitation density. A polariton density well into the means comprehensive. BCS limit is within reach with commercial lasers. The density dependence of various properties of the system allows systematic studies of the its phase-transition phys- 1. Semiclassical Boltzmann kinetics ics. The temperature of the acoustic phonon reservoir is Since polaritons are quasiparticles with a lifetime of readily tuned by thermoelectric heaters from cryogenic 1–100 ps, polariton condensation in general involves in- temperature up to room temperature or higher. The lat- jection, relaxation, and decay of the polaritons and thus tice temperature influences the dynamics of the polari- is a transient phenomena described by the kinetics of the tons. However, the short-lived LPs are often not at the condensation. The semiclassical Boltzmann rate equa- same temperature as the lattice or do not have a well- tions have been well adapted to describe the polariton ͑ defined temperature. kinetics Tassone et al., 1997; Ciuti et al., 1998; Tassone and Yamamoto, 1999; Malpuech et al., 2002; Porras et al., 2002; Kavokin et al., 2004; Doan et al., 2005͒. Numerical 4. Measurement techniques simulations based on the Boltzmann equations have in- From the measurement viewpoints, the microcavity terwound with experimental works to understand the in- polariton is a most accessible BEC system. There exists teractions in the polariton system, the mechanisms of

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1499 polariton relaxation, and the statistical properties of the during the condensation—are introduced self- system, which are reviewed in Secs. II.F.2–II.F.4. consistently in the condensation kinetics. A full quan- Limitations of this semiclassical approach exist, how- tum kinetic simulation of polariton condensation is com- ever, when the system is above the condensation thresh- putationally intensive and presently not yet available.4 old. The random-phase approximation of the Boltzmann equations is no longer valid when coherence emerges within the ground state as well as between the ground state and its low-energy excitations. Moreover, the con- 3. Equilibrium and steady-state properties densate and the excited states see different mean-field If the polariton lifetime is long compared to the relax- energy shifts; the spectrum of low-energy excitations ation time, a condensed polariton system can be consid- changes from a free-particle spectrum with finite effec- ered as in quasiequilibrium, and thus the static proper- tive mass to a linear Bogoliubov spectrum. These prop- ties may be approximated by a mean-field theory based erties are not explained with the Boltzmann equations. on the Gross-Pitaevskii equation ͑Pethick and Smith, The appropriate tool is quantum kinetics. 2001; Pitaevskii and Stringari, 2003͒. We introduce the To account for the difference of a condensed state Gross-Pitaevskii equations in Sec. VIII and apply it to compared to a normal state and to help understand the experiments on the spectral properties and superfluidity coherence properties of the condensate, various models of polariton condensation. have been developed that go beyond the conventional When a short-lived polariton system is under Boltzmann equations. Two common approximations in continuous-wave pumping, a steady state may be these models are to take advantage of the peculiar reached. Properties of a steady-state polariton conden- energy-momentum dispersion of the LPs and to divide sate have been calculated by linearizing a generalized the system into two regions in the momentum space. For Gross-Pitaevskii equation around the steady state states with relative large momentum and negligible ͑Wouters and Carusotto, 2007͒ and also by a Keldysh quantum coherence, the semiclassical kinetics is used. Green’s function approach ͑Szymanska et al., 2006͒. Due For the ground state and a small set of states with to constant pumping and decay of the polaritons, the smaller momenta and significant photonic fractions, the low-energy excitations are expected to be diffusive, correlation between the ground state and excitations is manifested as a flat dispersion near the ground state. A examined in greater detail. For example, Laussy et al. vortex induced during the relaxation process at local dis- ͑ ͒ ͑ ͒ 2004 and Doan, Cao, et al. 2008 introduced a stochas- order potential was also observed ͑Lagoudakis et al., tic extension of Langevin fluctuations to the Boltzmann 2008͒. equations. The method enables estimation of the first- and second-order coherence functions of the condensate, which are reviewed in Sec. VII. Sarchi and Savona ͑2007a͒ extended the Boltzmann equations under the 4. System size Bogoliubov approximation to include virtual excitations between the condensate and excited states and calcu- The two-dimensional polariton systems typically have ͑ ͒ ␮ lated the first-order coherence function of the system. a transverse size of 10–100 m across. The finite size ␦ Ӎ ប2 ͑ ͒ leads to an energy gap E 3 / 2meffS between the ground state and first-excited states. Given a typical ϫ −5 ͑ 2. Quantum kinetics GaAs polariton mass of meff=5 10 me here me is the free-electron mass͒, ⌬E is about 0.2 meV. This gap re- A complete and self-consistent description of the con- sults in a finite-size BEC, and the condensate density densation physics requires quantum kinetics. Quantum increases with decreasing system size. However, the real kinetics can be developed either on the basis of nonequi- transverse extension in experiments is not known accu- ͑ librium Keldysh Green’s functions Haug and Jauho, rately as it is often given by the spot size of the pumping ͒ 2008 or on the basis of a truncated hierarchy of equa- laser. This uncertainty in the spatial extension imposes a ͑ ͒ tions for reduced density matrices Kuhn, 1997 . The specific problem in numerical simulations of the polar- quantum kinetics description of condensation includes iton kinetics. the equation of motion for the order parameter, i.e., the expectation values of the ground-state polariton annihilation operator, as well as an equation for the population of the excited states. The Dyson equation formulated in III. RATE EQUATIONS OF POLARITON KINETICS the time regime will couple density matrices which are functions of only one-time argument to two-time scatter- As elementary excitations of the microcavity system, ing self-energies and two-time Green’s functions. One polaritons are most conveniently created by a laser can express these off-diagonal quantities in the time ar- pump pulse, after which they relax and under appropri- guments by two-time spectral functions and one-time ate conditions accumulate at least partly in the ground density matrices. The full set of kinetic equations thus includes the equations for the spectral functions, i.e., the retarded and advanced Green’s functions. Through these 4Currently a simplified model of quantum kinetics is given by spectral functions, the polariton spectra—which change Wouters and Savona ͑2009͒.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1500 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

state of the LP branch, before they completely decay. In phonons with a wave vector qeជʈ +qzeជz. Only the in-plane order to study a spontaneous phase transition, pumping vector component is conserved. The quantum well should be incoherent, so that there are no phase rela- 1s-exciton envelope function is given by tions between the pump light and the condensate. One speaks about a quantum statistically degenerate state if 2 ␹͑ ͒ ͱ −rʈ/aB ͑ ͒ ͑ ͒ ͑ ͒ rជʈ,z ,z = e f z f z , 33 the number of polaritons n0 in the ground state is much e h ␲a2 e e h h larger than 1. In this case, stimulated relaxation pro- B cesses dominate over the spontaneous ones by a factor ͑ ͒ where aB is the 2D Bohr radius and fe,h ze,h are the 1+n0. The main interactions which drive the relaxation quantum well envelope functions. Evaluating the and eventually the condensation stem from the exciton- exciton-phonon interaction matrix element with this en- phonon interactions and the Coulomb interactions be- velope function, one finds tween excitons. The exchange Coulomb interactions dominate over the direct Coulomb interactions between ប͑q2 + q2͒1/2 two excitons and give rise to a repulsive interaction be- ͑ជ ͒ ͱ z ͓ ʈ ͑ ͒ z͑ ͒ G q,qz = i ␳ DeIe q Ie qz tween two polaritons with an identical spin. This inter- 2 Vu action provides the fastest scattering mechanism which ʈ ͑ ͒ z͑ ͔͒ ͑ ͒ − DhIh q Ih qz . 34 thermalizes the polariton gas, but the temperature of the gas can only be lowered by coupling to the cold acoustic Here ␳ is the density, u is the sound velocity, and V is the phonon reservoir. microcavity volume. The superposition integrals for the Much of the physics of the polariton relaxation and electron ͑e͒ or hole ͑h͒ are given by condensation can be described with the semiclassical 5 2 −3/2 Boltzmann rate equations, which we formulate in this ʈ m I ͑q͒ = ͫ1+ͩ h,e qa ͪ ͬ , ͑35͒ section. We include the scattering due to the LP-phonon e,h 2M B and LP-LP interactions as discussed. For most experiments to date, the cavity-photon linewidth is on the or- L der of 1 meV. The spectral change due to condensation QW Iz ͑q ͒ = ͵ dzf2 ͑z͒eiqzz. ͑36͒ is not extended enough to have a substantial influence e,h z e,h 0 on the condensation kinetics. Hence we use only the free-particle spectrum to calculate the scattering rates. Ͼ −1 These integrals cut off the sums for q aB and for qz These rate equations yield results in good agreement Ͼ ␲ 2 /LQW. with the experimental observations in Secs. IV.A and With this interaction matrix element, the transition IV.B. rates for the LPs can be written as The rate equations calculate only the number of the LPs in the ground and excited states. However, we show ␲ LP-ph 2 2 2 Wជ ជ ជ = ͚ X X G͑q,q ͒ in Sec. VII that a stochastic extension of the Boltzmann k→kЈ=k−qជ ប k k−q z kinetics can calculate at least approximately first- and qz second-order coherence properties. ϫ␦͑E − E ± ប␻ ͒ k−q k q,qz 1 1 ϫͩN ͑ប␻ ͒ + ± ͪ, ͑37͒ B q,qz 2 2 A. Polariton-phonon scattering ͑ប␻ ͒ ͓ ͑ប␻ ͒ ͔ where NB q =1/ exp q −1 is the thermal phonon The relevant interaction between excitons and Bose distribution function. Here the small correction of longitudinal-acoustic phonons is provided by the the phonon confinement is neglected. deformation-potential coupling with the electron and hole De and Dh, respectively. This interaction Hamil- ͑ tonian between the LPs and phonons is Pau, Jacobson, B. Polariton-polariton scattering et al., 1996; Piermarocchi et al., 1996͒ 1. Nonlinear polariton interaction coefficients

LP-ph ͚ * ជ ͗ ͉ x-ph͉ ͘ The LP-LP interaction Hamiltonian generally has the Hw = XkX͉k+qជ͉ k Hw k + q ជ k,qជ,qz form † ϫ͑ † ͒ ជ ͑ ͒ cqជ,q − c ជ bជ ជbk, 32 z −q,−qz k+q Hˆ = 1 U͑k ,k ,q͒Pˆ + Pˆ + Pˆ Pˆ , ͑38͒ I 2 1 2 k1 k2 k1+q k2−q where Xk are again the exciton Hopﬁeld coefﬁcients of ˆ where Pk is the creation operator of a LP with the in- the polariton. The phonons are taken to be the bulk ͑ ͒ plane wave vector k. The dependence of U k1 ,k2 ,q on k1, k2, and q can be neglected if the momentum ex- 5 ͑ ͒ For an introduction of other approaches, see Sec. II.F and change q is small. In this case we replace U k1 ,k2 ,q by ͑ ͒ ͑ ͒ Keeling et al. 2007 . Uk =U k,k,0 .

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1501

For a spin-polarized polariton system, the nonlinear interaction parameter Uk can be further simpliﬁed and calculated as 1 ⌬2 ͑⌬ − ␦E ͒2 U = ␦E + ͱg2 + − ͱg͑n͒2 + X , k 2 X 0 4 4

n ␦ S ͑ ͒ EX = EB , nS = , 39 *2 2 nS ␲ ͉ ͉ 2.2 aB X

͑ ͒ ͩ n ͪ Ј S g n = g0 1− , n = . nЈ S ␲ *2͉ ͉2 S 4 aB X ⌬ ͑ ͒ ͑ ͒ Here =Ecav k −Eexc k is the detuning between the cavity and exciton energies at given k, EB is the exciton binding energy, aB is the exciton Bohr radius, 2g0 is the normal-mode splitting at ⌬=0, and S is the cross- ͉ ͉2 1 ͑ ⌬ ͱ 2 ⌬2͒ sectional area of the system. X = 2 1+ / 4g0 + is ␦ the excitonic fraction of the polariton at k. EX represents the repulsive fermionic exchange interaction between QW excitons of the same spin ͑Ciuti et al., 1998; Rochat et al., 2000͒. g͑n͒ is the reduced normal-mode splitting due to phase-space filling and fermionic exchange interactions ͑Schmitt-Rink et al., 1985͒. If a polariton system consists of two helicity ͑spin͒ components, the biexcitonic attractive interaction between two excitons with opposite helicities ͑spins͒ plays a crucial role in its relaxation dynamics. A frequency- FIG. 5. Self-pumped four-wave mixing ͑phase conjugation͒ ex- domain degenerate four-wave mixing ͑FD-DFWM͒ periment. ͑a͒ Schematic of the experiment. Two initial fields technique was used for such a system to evaluate rela- provided by the pump beam are elastically scattered into two tive magnitudes of the phase-space filling effect and the outgoing fields conserving energy and momentum. ͑b͒ Spectra two-body attractive and repulsive interaction between of FD-DFWM at zero detuning for various polarization con- excitons. As shown in Fig. 5͑a͒, a pump beam excites figurations. Adapted from Kuwata-Gonokami et al., 1997. two counterpropagating photon fields with amplitudes ␣ ␣ 1 and 2 and a test beam excites a probe field with an ␣ represents the reduction of the exciton-photon coupling. amplitude 3. The elastic scattering conserving energy and momentum produces a phase-conjugate signal with In the presence of an exciton population ͗bˆ +bˆ ͘, the ␣ an amplitude 4. The nonlinear Hamiltonian of this elas- exciton-photon coupling strength g is reduced to g͑1 tic scattering has three terms, −␯͗bˆ +bˆ ͒͘, here ␯Ͼ0. ˆ ˆ ˆ ˆ ͑ ͒ ⌬ Hn = H1 + H2 + H3 + H.c., Figure 5 b shows the DFWM spectra at =0 for different polarization configurations of the pump, test, and ˆ ͑ ˆ + ˆ + ˆ ˆ ˆ + ˆ + ˆ ˆ ˆ + ˆ + ˆ ˆ signal beams. When all three beams have the same lin- H1 = W b4+b3−b2+b1− + b4+b3−b2−b1+ + b4−b3+b2+b1− ear polarization ͓in configuration ͑A͔͒, a strong signal is ˆ + ˆ + ˆ ˆ ͒ + b4−b3+b2−b1+ , observed at the UP resonance, while the signals at UP ͑40͒ and LP resonances are expected to be proportional to ˆ ˆ + ˆ + ˆ ˆ ͉W/4+͑R/2−g␯͉͒2 and ͉W/4+͑R/2+g␯͉͒2, respectively. H2 = R ͚ b4␴b3␴b2␶b1␴, ␴=+,− When the pump beam has a polarization orthogonal to that of the test and signal beams ͓in ͑B͔͒, a strong signal ˆ ␯ ͓ ˆ + ˆ + ˆ ˆ + + ˆ ͔ appears at the LP resonance, where the signals at UP H3 = g ͚ b4␴b3␴b2␴aˆ 1␴ + b4␴aˆ 3␴b2␴aˆ 1␴ . ␴=+,− and LP resonances should be proportional to ͉W/4 ͑ ␯͉͒2 ͉ ͑ ␯͉͒2 ͑ ͒ ͑ ͒ Here ␴=± represents the Z component of the angular − R/2−g and W/4− R/2+g .In C and D , the test and signal beams have the same helicity, so only H ˆ ˆ 2 momentum and a␴ and b␴ are the annihilation operators ͑ ͒ ˆ and H3 are effective. In E the two beams have opposite of the cavity photon and QW excitons, respectively. H1 helicities so that only H contributes to the scattering. represents attractive interactions between two excitons 1 The data are explained by a microscopic theory for non- ͑ Ͻ ͒ ˆ with opposite spins W 0 . H2 represents the repulsive linear interaction coefficients based on an electron-hole ˆ ͑ ͒ interaction between two excitons with the same spin. H3 Hamiltonian Inoue et al., 2000 .

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1502 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

2. Polariton-polariton scattering rates ␲ 2 LP-LP S W Ј = For a nonpolarized LP gas, following Tassone and k,k ;k1,k2 ប ͑2␲͒4 ͑ ͒ Yamamoto 1999 , we further simplify the scattering co- ⌬ 2͉ ͉2 2 2 2 2 E M XkX ЈXk Xk efﬁcient as ϫ k 1 2 2͔ ץ ͒ ͑ ץ2͔͓ ץ ͒ ͑ ץЈ2͔͓ ץ Ј͒ ͑ ץ͓ 2 E k / k E k1 / k1 E k2 / k2 aB Ӎ ជ ជ ␸ ␸ ͑␸2 ␸ ␸ ͒Ӎ ͑ ͒ M 2 ͚ Vk−kЈ k kЈ k − k kЈ 6E0 , 41 ϫ ͑ Ј ͒ ͑ ͒ S R k,k ,k1,k2 . 42 k,kЈ ␸ where k and E0 are the screened 2D 1s-exciton wave A uniform energy grid is adopted with an energy spacing ␲ ⑀ ⌬ function and binding energy, respectively. Vk =2 / 0Sk E. The terms proportional to the 2D density of states Ј2 ץ Ј͒ ͑ ץ ⑀ is the 2D Coulomb potential and 0 is the dielectric func- E k / k stem from the conversion of integration tion. over momenta to summation over energies. The term R The corresponding LP transition probability is then is given by

dq2 R͑k,kЈ,k ,k ͒ = ͵ . ͑43͒ 1 2 ͱ͓͑ ͒2 2͔͓ 2 ͑ ͒2͔͓͑ ͒2 2͔͓ 2 ͑ ͒2͔ k + k1 − q q − k − k1 kЈ + k2 − q q − kЈ − k2

Ͻ Ͻ ␮ −1 ␶ The integration is taken over the range in which all four transverse momenta 0 k kcav=8 m we use 1/ k terms under the square root are non-negative. Energy 2 ␶ =Ck / c given the photon Hopfield coefficient Ck. The ͑ ␶ conservation is built in by taking k2 =k E2 =Ek +EkЈ cavity lifetime c is only a few picoseconds for most mi- −E ͒ and summing over the E and E . Ͻ Ͻ ប k1 kЈ k1 crocavities. In the range kcav k krad=ncavEcav/ c,we ␶x ␶ ␶ use the nanosecond exciton lifetime , i.e., 1/ k =1/ x, Ͼ C. Semiclassical Boltzmann rate equations and finally for large momenta k krad the exciton life- ␶ time is infinite, i.e., 1/ k =0. With these transition rates we formulate the semiclas- The LP-phonon scattering rate is given by sical Boltzmann equations of the polariton distribution. ץ This approach has been used before, for example, to LP-ph ͯ ជͯ ͓ ជ͑ ជ ͒ n =−͚ Wជ ជ n 1+n k k→kЈ k kЈ ץ describe the exciton condensation kinetics by Bányai et t LP-ph ជЈ al. ͑2000͒, Tran Thoai and Haug ͑2000͒, and Bányai and k LP-ph Gartner ͑2002͒ and to describe the cavity polariton re- ͑ ͒ ជ ͔ ͑ ͒ − WជЈ→ជ 1+nkជ nkЈ . 46 laxation by Tassone et al. ͑1996, 1997͒, Bloch and Marzin k k ͑ ͒ ͑ ͒ ͑ ͒ 1997 , Tassone and Yamamoto 1999 , Cao et al. 2004 , The LP-phonon scattering provides a dissipative cooling ͑ ͒ and Doan et al. 2005 . In the latter two references the mechanism for the LP gas. It can reduce the excess en- importance of a finite size for the 2D condensation ki- ergy introduced in the LP gas with the nonresonant op- netics has been shown. tical excitation. The Boltzmann equations for the population of the The LP-LP scattering rate is given by ͑ ͒ ͑ ͒ excited states nk t and the ground-state population n0 t ץ are LP-LP ͯ ជͯ ͓ ជ ជ ͑ ជ ͒ n =− ͚ Wជ ជ ជ ជ n n 1+n k k,kЈ,k ,k k kЈ k1 ץ t LP-LP ជЈ ជ ជ 1 2 ץ ץ nkជ ץ ជ ជ͑ ͒ ͯ ជͯ ͯ ជͯ ͑ ͒ k ,k1,k2 nk = Pk t − + nk + nk , 44 tץ tץ t ␶ជץ k LP-LP LP-ph ϫ͑1+nជ ͒ − n ជ nជ ͑1+nជ͒͑1+nជ ͔͒, k2 k1 k2 k kЈ 47͒͑ ץ ץ ץ n0 ͯ ͯ ͯ ͯ ͑ ͒ n0 . 45 ץ + n0 ץ + n0 =− ␶ ץ t 0 t LP-LP t LP-ph ជ ជ ជ ជ where k1 =k+qជ and k2 =kЈ−qជ. LP-LP scattering is rather Note that a separate equation is needed for the ground- fast; it thermalizes the LP gas in a few picoseconds but state LP population n0 so as to properly account for the cannot reduce the total energy of the LP gas. Thus both gap between the ground state and the lowest excited- scattering processes have their own distinct roles in the state energies due to a finite size of the system. In the condensation kinetics of the LPs. following, we assume cylindrical symmetry such that n k We emphasize again that the factor 1+n0 expresses depends only on the amplitude of kជ, and we drop the the stimulated scattering into the ground state enhanced ͑ ͒ vector notation for simplicity. Pkជ t is the time- by the final-state occupation n0, which are the essential ␶ dependent incoherent pump rate and kជ is the LP life- driving forces of LP condensation. Once the condensate ͑ ͒ ӷ time introduced by Bloch and Marzin 1997 . For small is degenerately populated with n0 1, the escape of the

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1503

FIG. 6. A cold collision experiment of polaritons. ͑a͒ Schematic of the experiment. The size of the solid circles depicts the population in the LP states. Inset: The cross section of the microcavity structure and the experimental excitation scheme. ͑b͒ The integrated UP emission intensity as a function of probe density nLP for varying exciton density nexc. Symbols are the experimental ϫ 9 ͑ ͒ ϫ 9 ͑ ͒ data and solid lines are the theoretical prediction. The exciton densities are nexc=1.5 10 squares , 1.2 10 circles , and 5.4 ϫ 8 −2 ͑ ͒ ͑ ͒ ϫ 9 −2 10 cm triangles . c The integrated UP emission intensity as a function of nexc for a fixed nLP=1.1 10 cm . The dotted line indicates the acoustic phonon-scattering contribution to the UP intensity. Adapted from Huang et al., 2000. macroscopic coherent polariton wave is rather classical tion is the driving mechanism behind photon lasers and and given by the photon leakage out of the cavity. amplifier and is expected to be crucial to the formation To study spontaneous condensation, we keep the of BEC. It is of great interest to demonstrate the final- pump energy high enough to avoid the possibility where state stimulation for a massive composite particle, to two LPs at the pumped mode can resonantly scatter into confirm its bosonic nature, and to clarify the role of the ground state and a higher lying state ͑Savvidis et al., final-state stimulation in the BEC physics. 2000͒. The pump pulse is assume to be of the form Due to the strong coupling between excitons and pho- ͓͑ ͒2͔ tons, the energy DOS of the LPs decreases steeply by e−2 ln 2t/tp P ͑t͒ = P ϫ ͭ ͮ ͑48͒ four orders of magnitude from a state at large in-plane k k ͑ ͒ Ͼ ϳ ϳ tanh t/t0 wave number k kbot 0.1k0 to states at kʈ 0. Here we with assume a moderate detuning and k0 is the wave number of the cavity photon at kʈ =0. As a result of the reduced ប2͑ ͒2 ⌬ − k − kp /2mexc E ͑ ͒ Pk = P0e 49 DOS, cooling of the LPs by acoustic phonon emission is for pulse and stationary excitations, respectively. slowed down at kbot. At the same time, the lifetime of For GaAs QW and microcavities, the material param- the LPs at kbot states is typically two orders of magni- ϳ eters we used are ͑Bockelmann, 1993; Pau et al., 1995; tude longer than that of LPs at kʈ 0. The two effects ͒ x together cause an accumulation of incoherent exciton Bloch and Marzin, 1997; Cao et al., 2004 E0 =1.515 eV, population at kbot—the so-called energy relaxation me =0.067m0, mh =0.45m0, aB =10 nm, the index of re- ͑ ͒ fraction n =3.43, the normal-mode splitting g “bottleneck” Pau et al., 1995; Tassone et al., 1996 . cav 0 In an experiment on binary exciton scattering ͑Tas- =5 meV, the deformation potentials D =−8.6 eV, D e h sone and Yamamoto, 1999; Huang et al., 2000͒, the =5.7 eV, u=4.81ϫ105 cm/s, and ␳=5.3ϫ103 kg/m3. bottleneck effect is utilized to study the bosonic final- state stimulation. As shown in Fig. 6͑a͒, two pump IV. STIMULATED SCATTERING, AMPLIFICATION, AND beams create incoherent excitonlike LPs at kʈ =±kbot. LASING OF EXCITON POLARITONS Due to the long lifetime and slow relaxation rate at kbot, the injected excitons form an incoherence reservoir. In this section we describe cold collision experiments Then the LPs at kʈ =±kbot may elastically scatter into LP which demonstrate the bosonic final-state stimulation, and UP at kʈ ϳ0.IfaLPatkʈ ϳ0 is optically populated, matter-wave amplification, and lasing of polaritons. the exciton-exciton scattering rate is enhanced by the Through those experiments we investigate the nonlinear bosonic stimulation factor. The other final state of the interactions of polaritons, which are at the heart of po- scattering, the otherwise unpopulated kʈ ϳ0 UPs, pro- lariton condensation and the excitation spectrum dis- vides a background-free measurement window. cussed later. The sample used in this experiment is a single GaAs QW embedded in a planar AlAs/AlxGa1−xAs microcav- A. Observation of bosonic final-state stimulation ity. A mode-locked Ti:Al2O3 laser with a bandwidth of 16 meV is spectrally filtered to obtain the pump and One distinct feature of a system of identical bosonic probe pulses which have a bandwidth of 0.4 meV and particles is the final-state stimulation: the presence of N are separated in energy by half of the LP-UP splitting of particles in a final state enhances the scattering rate into 2 meV. The pump pulse is split into two beams and in- that final state by a factor of 1+N. Final-state stimula- cident on the sample at ±45° from the sample normal.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1504 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

The probe beam incidents on the sample at 3° in order optical ampliﬁer for an input light signal. It is a negative-

to excite the LPs at kʈ ϳ0, while the UP emission is de- conductance amplifiers based on incoherent ͑or resis- tected in the normal direction ͓Fig. 6͑a͔͒. tive͒ gain elements. A second common type of amplifier The time-integrated intensity of the UP emission NUP is a nonlinear susceptance amplifier, the gain of which is increases linearly with the probe intensity ͓Fig. 6͑b͔͒, based on coherent ͑or reactive͒ multiwave mixing be- which is the signature of the final-state stimulation. NUP tween a pump, signal, and idler. A parametric amplifier also increases quadratically with the reservoir exciton is a classic example. This second type of amplifier has ͓ ͑ ͔͒ density nexc Fig. 6 c due to the binary exciton-exciton been demonstrated with matter waves using both BEC scattering. The results are described quantitatively by a of alkali atoms ͑Inouye et al., 1999; Kozuma et al., 1999͒ rate equation ͓solid lines in Figs. 6͑b͒ and 6͑c͔͒, where and coherent polaritons produced by a pump laser ͑Sav- the dominant term for populating the UPs is ϰ͑1 vidis et al., 2000͒. Utilizing the stimulated scattering of ͒ 2 bottleneck polaritons, Huang et al. ͑2002͒ demonstrated +NLP n , exc a matter-wave amplifier of the first type with a polariton dN N UP UP ͓ Ј͑ ͔͒ 2 system. The fundamental difference between a photon =− ␶ + aUPnexc + bUP + b 1+NLP nexc. dt UP laser amplifier and a polariton amplifier is that, in the ͑50͒ latter system, there is no population inversion between conduction and valance bands. Therefore the standard ␶ Here UP is the radiative lifetime of the UP and NLP is stimulated emission of photons cannot provide a gain. the kʈ ϳ0 LP population determined by the probe beam Instead, amplifications of the injected polaritons are intensity. The coefficient aUP represents the scattering achieved through elastic LP-LP scattering. Such a polar- rate of a bottleneck exciton to an UP by acoustic pho- iton amplifier is a matter-wave amplifier based on stimu- non absorption and has been calculated by applying Fer- lated scattering of massive particles rather than a photon mi’s golden rule to the Hamiltonian of the deformation- amplifier based on stimulated emission of photons. A potential scattering. bUP is the scattering rate of two negative-conductance amplifier for matter waves is at- excitons to one UP and another exciton ͑e.g., 1+6Ј tractive from a practical viewpoint because electrically →UP+4Ј͒. bЈ is the scattering rate of two bottleneck injected incoherent excitons can eventually realize an

excitons at opposite in-plane momenta ±kʈ to UP and amplifier for an optical signal without requiring popula- Ϸ ͑ → ͒ tion inversion between conduction and valence bands. LP both at kʈ 0 1+1Ј UP+LP . bUP and bЈ have been calculated via Fermi’s golden rule rate applied to The sample used in the experiment by Huang et al. ͑ ͒ ␭ the Hamiltonian for exciton-exciton scattering. 2002 is a /2 Cd0.6Mg0.4Te cavity with two CdTe QWs. In this cold collision experiment, real population of A pump laser incident from a slant angle resonantly ex- ϳ ␮ −1 excitons at ±kជ are excited by the pump laser and cites excitons at kʈ 3.5 m . When this initial popula- bot ϫ 10 −2 forms incoherent reservoirs. This is evident from the ob- tion increases to above about Np =7 10 cm , a quan- served exponential decay of a luminescence from those tum degeneracy threshold is reached at a bottleneck −1 ␶ ϳ state with kʈ =2 ␮m . N is nearly one order of magni- exciton states with a time constant of exc 190 ps, which p is much longer than the pump pulse duration of ϳ9 ps. tude smaller than the saturation density of 5 ϫ1011 cm−2 for CdTe QW excitons. The measured LP Moreover, NUP due to the stimulated scattering decays ␶ ϳ population distribution versus energy and in-plane mo- with a time constant of UP 96 ps. The difference of a ␶ ␶ 2 mentum is shown in Fig. 7͑a͒. When pumped above factor of 2 between exc and UP stems from the nexc dependence of the stimulated scattering rate. threshold, the maximum of the LP emission continually By contrast, the polariton parametric amplification ex- shifts toward smaller kʈ. This behavior of bottleneck periment first performed by Savvidis et al. ͑2000͒ is a condensation is predicted by the rate equation theory ͑ ͒ coherent optical mixing process. In this latter case, the Tassone and Yamamoto, 1999 . A slight blueshift of the Ӎ տ ␮ −1 pump beam creates virtual LP population at a magic LP energy at kʈ 0 and 3 m is due to the repulsive wave number k , which coherently scatters to the two interaction between nondegenerate LPs and the quan- p Ӎ ␮ −1 ϳ ϳ tum degenerate bottleneck LPs at kʈ 2 m . The final states: LPs at kʈ 0 and excitons at 2kp. The scattering conserves both in-plane momentum and energy same feature is manifested in Bogoliubov excitation between the initial and final states. In this coherent spectra as discussed in Sec. VIII. parametric process, the stimulated scattered signal ap- When a probe pulse, with a narrow bandwidth of pears only when the pump beam is present, and the sig- about 0.6 meV, selectively excites the bottleneck LPs, ͓ ͑ ͔͒ nal intensity increases linearly with the pump intensity. amplification of the probe beam is observed Fig. 7 b . In this sense, the process can be interpreted as coherent The dynamics of the bottleneck LP population NLP is ͑ ͒ optical four-wave mixing resonantly enhanced by the LP modeled by Ciuti et al. 1998 , resonances. dNLP NLP ͑ ͒ = PLP − ␶ + aLPNEP 1+NLP dt LP B. Polariton amplifier 2 ͑ ͒ ͑ ͒ + bLPNEP 1+NLP . 51 ␶ A standard photon laser pumped just below its oscil- Here LP is the bottleneck LP lifetime, PLP is the exter- lation threshold is often used as a regenerative linear nal injection rate of the LP by the probe-laser pulse,

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1505

4 4 4 x10 x10 x10 (a) (b) 5.5 5.5 5.5 (c) Pump only Probe only Pump+Probe 5 5 5

4.5 4.5 4.5

4 4 4

3.5 3.5 3.5

3 3 3

Counts 2.5 2.5 2.5

2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0 1.62 1.63 1.64 1.62 1.63 1.64 1.62 1.63 1.64 Energy [eV]

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

−1 aLP=47 s is the scattering rate between acoustic GaAs excitons have a relatively larger Bohr radius, −1 phonons and LPs, and bLP=0.45 s is the LP-LP scatter- hence the advantage of relatively large scattering cross ing rate of an incoherent LP population. sections. To cope with the relatively lower saturation The increase of the measured gain G with the reser- density and maintain the strong-coupling regime, Deng ͑ ͒ voir population NEP agrees quantitatively with the rate et al. 2002 used a microcavity with 12 QWs. Multiple ϰ ͑ ϫ 2 ͒ equation simulation, that is, G exp const NEP in the QWs, on the one hand, reduced the exciton density per ϰ͑ ϫ 2 ͒ QW for a given total LP density and, on the other hand, high-gain regime and G−1 const NEP in the low- gain regime. The time dependence of the gain is charac- increased the exciton-photon coupling to reach the very terized by a fast rise followed by a slow exponential de- strong-coupling regime ͑Khurgin, 2001͒, which leads to cay. The decay time of ϳ57 ps corresponds to about half reduced exciton Bohr radius for the LP branch hence of the separately measured decay time of bottleneck LPs further increased exciton saturation density. ͓Fig. 7͑c͔͒, which is consistent with the quadratic depen- The experimental scheme is depicted in the left panel dence of the gain on NEP. These results all confirm the of Fig. 10. A pump laser resonantly excites hot LPs at −1 gain mechanism by two-body scattering of incoherent kʈ =5–6 ␮m so as to minimize the excitation of free reservoir LPs. electrons and holes, which usually increase the exciton −1 decoherence rate. The value kʈ =5–6 ␮m is well above −1 the magic angle corresponding to kʈ ϳ2 ␮m , so that C. Polariton lasing versus photon lasing the coherent LP parametric amplification ͑Savvidis et al., ͒ 1. Experiment 2000 is avoided. A typical lasing behavior is observed in the input- The LP-LP scattering enhanced by the bosonic final- output relationship, i.e., the pump power dependence of state stimulation gives rise to large nonlinearity in the the emission intensity of the LP ground state ͑Fig. 8͒.It microcavity system. This large nonlinearity has been ex- features a sharp superlinear increase around a threshold ploited to implement polariton amplifiers of incoherent pumping density P , corresponding to an injected car- type ͑Huang et al., 2000͒ and of coherent type ͑Savvidis th rier density of n Ϸ3ϫ109 cm−2 per pulse per QW for et al., 2000͒ and polariton lasers in the bottleneck regime QW a spot of DϷ15 ␮m in diameter. n is 30 times smaller ͑Huang et al., 2002͒. First-order coherence ͑Richard et QW al., 2005͒ was also observed for a bottleneck polariton than the saturation density and two orders of magnitude ϳ ␲ 2 ϳ ϫ 11 −2 laser. However, such a system is always far from thermal smaller than the Mott density n 1/ aB 3 10 cm equilibrium. There is a large number of degenerate las- electron-hole pairs per QW. At the threshold, the kʈ =0 ing modes at the bottleneck, and it is difficult to charac- LP population NLP is estimated to be of the order of terize the thermodynamic properties of the LP gas. With unity, which coincides with the onset of the stimulated the improvement of fabrication techniques and optimi- scattering of LPs into the kʈ ϳ0 LP state. zation of the experimental conditions, quantum degen- To confirm that LPs remain as the normal modes of eracy and lasing at the kʈ =0 LP ground state have been the system above threshold, the energy-momentum dis- observed first with GaAs-based devices ͑Deng et al., persion was measured and was found to agree with the 2002, 2003; Balili et al., 2007͒ and later with CdTe-based LP dispersion instead of the cavity-photon dispersion. devices ͑Kasprzak et al., 2006͒ and GaN-based devices The LP energy remains well below the bare QW exciton ͑Christopoulos et al., 2007͒. energy in spite of a small blueshift due to the Coulomb

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1506 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

polariton laser,

3 E =1.6166 eV 0 10 LP ~0 || ~

|| photon laser, k E =1.6477 eV 102 cav eat d o

M 101

100 100 tons per i ar l o 10-1 P no inversion inversion Photons per Cavity Mode at k 109 1010 1011 1012 Injected Carrier Density(cm-2)

FIG. 8. ͑Color online͒ Number of LPs and cavity photons per mode vs injected carrier density for a polariton condensate ͑triangles, the scheme in the left panel of Fig. 10͒ and a photon FIG. 10. ͑Color online͒ Schematics of the experimental prin- laser ͑circles, the scheme in the right panel of Fig. 10͒, respec- ciples for a photon laser ͑right panel͒ compared to LP conden- tively. The gray zone marks the population inversion densities sation ͑left panel͒. from band edge to 15 meV above the band edge. From Deng et al., 2003. tablish photon lasing, the sample is pumped at a position with the cavity mode bluedetuned to 15 meV above the exchange interaction and phase-space filling effect. The GaAs QW band edge. The pump energy is tuned ac- subject will be discussed in Sec. VIII. cordingly to be resonant with the cavity mode energy at 4 −1 To distinguish the polariton laser from a coherent po- kʈ =5.33ϫ10 cm . The experimental scheme is de- lariton parametric amplifier, the time evolution of the picted in the right panel of Fig. 10. LP ground-state population was measured. There was a The emission in this case follows the cavity-photon clear delay between the peak of the LP emission and the dispersion, and a photon lasing threshold is observed at pump pulse, which should be absent in the polariton kʈ =0 ͑Fig. 8͒. Here the injected carrier density at thresh- parametric amplifier and oscillator ͑Savvidis et al., 2000͒ old is nЈ Ϸ3ϫ1011 cm−2 per pulse per QW, exactly see Fig. 9. QW Using the same sample and experimental setup, a di- identical to the density required for electronic popula- rect comparison has been made with a conventional tion inversion at 15 meV above the band edge. This re- photon laser ͑Deng et al., 2003͒. Since the sample’s cav- sult agrees with the standard photon laser mechanism. ity layer thickness is tapered by growth, detuning ⌬ of That is, when the spontaneous emission coupling con- the cavity resonance changes across the sample. To es- stant is large enough, the threshold of a semiconductor laser is solely determined by the electronic inversion condition ͑Björk et al., 1994͒. The cavity-photon number per mode is estimated to be of the order of unity at threshold, which coincides with the onset of the stimu-

lated emission of photons into the kʈ =0 cavity-photon mode. A photon laser results from stimulated “emission” of photons into cavity modes with an occupation number larger than 1. The quantum degeneracy threshold of polaritons is triggered by the same principle of bosonic ﬁnal-state stimulation, yet the physical process is different—it is stimulated “scattering” of massive quasi-

particles ͑the LPs͒ from nonlasing states at kʈ Ͼ0 into

the lasing state at kʈ =0. The nonlinear increase of the photon ﬂux is an outcome of the leakage of the LPs via cavity mirrors. Hence the observed threshold of the polariton system requires no electronic population inver- ϳ ␭2 sion. The threshold density is of the order of 1/ T and three to four orders of magnitude lower than that of a FIG. 9. Dynamic properties of the emission. ͑a͒ The time evo- photon laser threshold given by ϳ1/a2 . It serves as a lution of the kʈ =0 LP emission intensity. “a.u.” in the y-axis B ͑ ͒ ⌬ label denotes “arbitrary units.” b Pulse FWHM Tp vs pump proof of principle demonstration to use polariton sys- ͑ ͒ intensity P/Pth. c Turn-on delay time of the emission pulse vs tems as a new energy efﬁcient source of coherent light. P/Pth. From Deng et al., 2002. The difference in the operational principles of polariton

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1507

(a) (b) E E electron-hole exciton-polariton pair dispersion dispersion external injection spontaneous stimulated cooling cooling

via multiple phonon emission external pumping & polariton-polariton scattering population k =0LP inversion stimulated emission of photons

final bosonic mode cavity photon leakage of photons via cavity mirror final bosonic mode

k crystal ground state k crystal ground state

FIG. 11. ͑Color online͒ The operational principles of ͑a͒ polariton BEC and ͑b͒ photon laser. From Imamoglu et al., 1996. laser and photon laser is shown schematically in Fig. 11 al., 2005͒. By quasistationary pumping above the bottle- ͑Imamoglu et al., 1996͒. neck, a quantum statistically degenerate population of ӷ If the pump power for a polariton lasing system is the ground state with n0 1 is generated. This is shown continuously increased, the 2D carrier density eventu- in Fig. 13 where the ground-state population n0 is plot- ally exceeds the saturation density where the polariton ted versus the total LP density ntot taken as a measure of system undergoes a transition from strong to weak cou- the pump rate for various values of the detuning ⌬.Itis pling. When this transition happens, the leaking mode seen that above the LP lasing threshold, the ground energy, which is approximately equal to the LP energy state is degenerate with 103 –104 LPs. The total LP den- except for a continuous blueshift, suddenly jumps to the ϫ 9 −2 sity ntot at threshold is about 5 10 cm for a detuning bare cavity resonant energy. Such behavior is shown in of 6 meV, which is well below the exciton saturation Fig. 12 for a 12-QW GaAs microcavity ͑Roumpos, density n =1.5ϫ1011 cm−2 ͑Schmitt-Rink et al., 1985͒. Hoefling, et al., 2009͒. In this particular experiment, s The criteria for the validity of the boson approximation continuous-wave pumping is used at an energy above for polaritons are relaxed compared to those for exci- the band gap. The first threshold is that of a polariton tons by the square of the exciton Hopfield coefficient. In laser. Whether the second threshold is that of a simple experiments, the total LP density is distributed in sev- photon laser ͑Bajoni et al., 2008͒ or a new ordered state eral QWs which have no direct interaction. The conden- involving a BCS-type correlation is unresolved at this sation threshold density n =5ϫ10−9 cm−2 agrees with time. tot,c the experimental values ͑Deng et al., 2003͒. In Fig. 13 the important influence of the detuning pa- 2. Numerical simulation with quasistationary and pulsed rameter ⌬ on the condensation threshold is clearly dem- pumping Numerical solutions of the Boltzmann equation reproduced the above experimental results when both LP- phonon and LP-LP scattering are considered ͑Doan et

͑ ͒ ͑ ͒ FIG. 12. Color Polariton lasing and photon lasing. a The FIG. 13. Theoretical condensate population f0 =n0 vs the total emission energy vs pump power for a N=12 multi-QW planar LP density ntot for quasistationary pumping and the combined microcavity. ͑b͒ The dispersion characteristics of polariton LP-LP and LP-phonon scattering for the bath temperatures of BEC ͑green diamond͒ and photon lasing ͑pink triangle͒ as well T=4 K for various values of the detuning: ␦=0 ͑full line͒, 2.5 as the linear dispersion of UP ͑red square͒ and LP ͑blue circle͒ ͑dashed line͒, and 6 meV ͑dash-dotted line͒. From Doan et al., at low pump power. From Roumpos, Hoeﬂing, et al., 2009. 2005.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1508 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

Pump A. Time-integrated momentum distribution 1 As reviewed in Sec. II.E.4, a microcavity polariton 0.8 system has the unique advantage that the momentum distribution of LPs can be directly measured via angle- 0.6 resolved PL. Early studies revealed an energy relaxation bottleneck at relatively low excitation densities when the 0.4 LP-phonon scattering was the dominant relaxation

Intensity (arb. units) mechanism ͑Tassone et al., 1997; Muller et al., 1999; Tar- 0.2 takovskii et al., 2000͒. With increasing excitation density, ͑ 0 the bottleneck migrated toward smaller kʈ state Tartak- 0 10 20 30 40 50 60 70 80 ovskii et al., 2000͒. Later, in a GaAs-based 12-QW mi- t(ps) crocavity, the energy relaxation bottleneck became largely absent at zero or positive cavity detunings, and FIG. 14. Calculated buildup and decay of the ground-state quantum degeneracy was achieved in the kʈ =0 LP emission in the laser mode above and below threshold at the ͑ ͑ ͒ ͑ ͒ ground state above certain excitation density Pth Deng pump powers P/Pth=0.87 solid line , 1.87 dashed line , and ͒ ͑ ͒ et al., 2002 . Following this experiment, Deng et al. 3.0 dotted line , respectively. The dash-dotted line shows the ͑ ͒ pump pulse. From Doan et al., 2005. 2003 measured the momentum distribution and compared it with Bose-Einstein ͑BE͒ statistics. In the experiment of Deng et al. ͑2003͒, the microcav- onstrated: If the cavity mode is slightly detuned in the ity system was pumped by a 3 ps pulsed laser on reso- order of a few meV above the exciton resonance, the nance with excitonlike states at large kʈ. The time- exciton component of the LPs increases. Because all in- integrated momentum distribution of LPs was teractions relevant for the relaxation kinetics are pro- measured. For excitation densities between Pth and 2Pth, vided by the exciton component, the LPs can relax more the LP ground state reached quantum degeneracy where efficiently toward the ground state, so that the conden- stimulated scattering was pronounced; the energy relax- sation threshold is reached at lower densities. ation time was close to but still longer than the lifetime Under a pulse excitation below threshold, a relatively of the ground-state LPs. Thus the LP distribution does slow buildup and decay of the perpendicular emission not change significantly within the LP lifetime, while at are also obtained numerically. Above threshold the out- the same time the duration of the emission pulse is com- put rises faster due to the stimulated scattering ͑see Fig. parable to their lifetime. In this regime, the time- 14͒. The decay is faster too because now an increasing integrated measurement approximates the instanta- fraction of the particles is in the ground state where the neous momentum distribution of the LPs. The measured lifetime is the shortest. These results of the Boltzmann momentum distribution agrees well with the quantum- kinetics are in good agreement with the observation mechanical BE distribution except for an extra popula- shown in Fig. 9 ͑Deng et al., 2002; Weihs et al., 2003͒. tion in the condensed kʈ ϳ0 state ͓Fig. 15͑a͔͒. The quantum degeneracy parameter, the ratio of the chemical ␣ ␮ potential and temperature =− /kBTLP, decreases rapidly when the excitation density increased from below to V. THERMODYNAMICS OF POLARITON above the threshold, reaching a minimum of ␣ϳ0.02 at CONDENSATION ͓ ͑ ͔͒ P=1.7Pth Fig. 15 b . This demonstrates that the LPs In Sec. IV, we show that stimulated scattering of LPs form a quantum degenerate Bose gas and establish qua- overcomes the bottleneck effect above a threshold siequilibrium among themselves via efficient stimulated LP-LP scattering. pumping density; the population in the kʈ =0 LP ground state increases nonlinearly with the pump and quickly The fitted effective LP temperature TLP, however, is builds up quantum degeneracy. Quantum degeneracy is higher than the lattice temperature of 4 K, and it in- ͓ ͑ ͔͒ often associated with a transition to a macroscopic quan- creases with increasing pump rate inset of Fig. 15 b . tum state, such as a BEC state. To search for direct This implies that the LP-phonon scattering does not pro- manifestation of such a phase transition, quantum statis- vide sufficient cooling to the injected hot excitons, and tics of the polariton gas has been studied by two types of thermal equilibrium with the phonon bath is not measurements. One is the momentum-space distribution reached. ӷ of polaritons, which reveals the evolution and formation At P Pth, the LP-LP interaction becomes strong. of a condensate and its thermodynamic properties. It is Therefore, the condensate begins to be depleted. At the the focus of this section. The other is coherence func- same time, the dynamics becomes faster and the time- tions, specifically, first-order spatial and temporal coher- integrated data are no longer a good approximation of ence and second-order spatial and temporal coherence. the instantaneous distribution. These will be reviewed in Sec. VII. Other manifestations B. Time constants of the dynamical processes of the macroscopic phase include superfluidity, vortices, and phase locking among polariton arrays, as discussed Whether or not LPs can be sufficiently cooled to reach later. thermal equilibrium with the phonon bath requires fur-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1509

P/P = 1.5 a th The only mechanism to cool the LPs is the LP-phonon 1.5 scattering. The linear LP-phonon scattering is a rather P/P = 0.6 th slow energy relaxation process with a more or less ﬁxed 1 time scale ␶ =10–50 ps. Fortunately, in the quan-

) (a.u.) phonon ||

) 1 Ͼ 0.5 tum degenerate regime of nkϳ0 1, LP-phonon scatter- n(k a.u.

( ing can be signiﬁcantly enhanced by the bosonic ﬁnal-

) 0 || ␶ ␶ k state stimulation. If shortens to less than , LPs ( 1 2 3 phonon LP

n 0.5 k (104cm−1) may have enough time to reach thermal equilibrium || data with the phonon bath within their lifetime. BE fit 0 MB fit 0 1 2 3 4 C. Detuning and thermalization k (104cm−1) || To facilitate LP thermalization and eventually reach b thermal equilibrium with the lattice, we need faster 2 LP-phonon scattering and slower LP decay. For this purpose, there exists a simple and useful controlling param- 1.5 100 (meV) ⌬ eter for the microcavity system: the detuning =Ecav LP

T ⌬ −Eexc. At positive , the excitonic fraction Xkʈ increases,

α 1 0 the LP effective mass is heavier, and the energy density 1 1.2 1.4 1.6 of states ␳͑E͒ϰ͑1−͉X ͉2͒−1 increases. Hence LP- 0.5 P/P (mW) kʈ th phonon-scattering rates to low-energy states ͓ϰX X ϳ ␳͑E ͔͒ as well as LP-LP scattering rates to 0 kʈ,i kʈ 0 i 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 low-energy states ͓ϰX X X X ϳ ␳͑E ͒␳͑E ͒ kʈ,i1 kʈ,i2 kʈ,f1 kʈ 0 i1 i2 P/P (mW) ϫ␳͑E ͔͒ become larger. The flatter dispersion also re- th f1 quires less phonon-scattering events to thermalize the FIG. 15. Momentum space distribution of LPs. ͑a͒ The mea- LPs below bottleneck. At the same time, the LP lifetime ͑ ͒ sured LP population per state vs kʈ stars , compared with BE ␶ Ϸ␶ /͑1−͉X ͉2͒ becomes longer. All these are favor- ͑ ͒ ͑ ͒ LP cav kʈ solid line and MB dotted line distribution functions at able for LPs to reach thermal equilibrium. ͑ ͒ pump rates P/Pth=1.5 and 0.6 inset .AtP/Pth=0.6, the fitted The compromise is a higher critical density due to a BE and MB distribution curves almost overlap. ͑b͒ The quan- ␣ heavier effective mass, stronger exciton localization, and tum degeneracy parameter vs pump rate P/Pth and the fitted shorter dephasing time of the LPs. At very large positive effective LP temperature T vs pump rate ͑inset͒. The dashed LP detunings, these detrimental effects prevent clear obser- lines are guides for the eye. From Deng et al., 2003. vation of polariton condensation.

ther investigation of the dynamical processes of the sys- D. Time-resolved momentum distribution tem. As discussed in Sec. III, there are mainly three dynamical processes of the LP system: ͑1͒ LP decay via the Along this line, Deng et al. ͑2006͒ measured how the out coupling of its photon component, ͑2͒ LP-LP scat- LP momentum distribution depends on the cavity detun- tering, and ͑3͒ LP-phonon scattering. ing and obtained evidence of a thermal-equilibrium LP ␶ condensation at moderate positive detunings. In the ex- LP decay is characterized by the decay time LP =␶ /͉C ͉2, where ͉C ͉2 is the photon fraction of the LP periment, a GaAs-based microcavity with 12 QWs and a cav kʈ kʈ 3 ps pulsed laser excitation were adopted. Time-resolved defined in Eq. ͑16͒. The cavity lifetime is ␶ cav momentum distribution was measured by a streak cam- =1–100 ps. For a given cavity, ␶ increases with de- LP era with a time resolution of ϳ4 ps. From the time evo- ␶ creasing photon fraction. LP is the shortest time scale ͑ ͒ lution of the LP ground-state population NLP kʈ =0 , the for most microcavities at low pumping densities, leaving ␶ energy relaxation time relax of the LP system was ex- the system in nonequilibrium, sometimes with a relax- tracted based on a two-mode rate equation model. The ation bottleneck. ͑ ͒ instantaneous momentum distribution NLP k,t is com- LP-LP scattering is a nonlinear process; its time scale pared with BE and MB distributions ͑Fig. 16͒. ␶ ␶ LP-LP shortens with increasing LP density. When LP-LP The relaxation time ␶ decreases sharply near the ␶ relax becomes shorter than LP, LPs overcome the energy re- quantum degeneracy threshold, reflecting the onset of laxation bottleneck and reach quantum degeneracy stimulated scattering into the LP ground state. At posi- threshold. Due to the efficient LP-LP scattering among tive detunings and at pump levels above the quantum ␶ states below the bottleneck, these LPs thermalize among degeneracy threshold, relax shortens to one-tenth of the ␶ ͑ ͒ themselves before they decay, establish quasiequilib- lifetime LP 0 of the LP ground state, and thermal equi- rium, and form quantum degenerate BE distribution as librium with the phonon bath is established for a period ␮ presented in Sec. V. A . However, LP-LP scattering con- of about 20 ps, with a chemical potential LP of about ͉͑⌬͉Ͻ ͒ ␶ serves the total energy of the LPs and thus does not −0.1kBT. Near resonance 1 meV , relax is compa- ␶ ͑ ͒ reduce the temperature of the LP gas. rable to LP 0 . The stimulated LP-LP scattering is still

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1510 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

E=k T (a) B lat 1 10

0 (k) 10 LP N

−1 10

0 2 4 6 8 10 −4 (b) x10 1 10 FIG. 17. Snapshot of the calculated LP energy distribution ͑circles͒ at 45 ps after the pulse excitation for a detuning of 0 (k) 10 4.5 meV. The full line shows the ﬁt with a Bose-Einstein dis- LP

N tribution with the temperature of 5.5 K and a chemical potential with the value ␮=−2.8ϫ10−3 meV. From Doan, Thien −1 10 Cao, et al., 2008.

0 2 4 6 8 10 −4 x10 of LPs can indeed be ﬁtted by thermal distributions (c) about 20–40 ps after the excitation, i.e., by the Maxwell- Boltzmann or Bose-Einstein distributions below or above threshold ͑Doan et al., 2005; Doan, Thien Cao, et al., 2008͒. Figure 17 shows a typical distribution calcu- (k)

LP lated for 45 ps after the excitation pulse and with a lat- N tice temperature of 4.2 K. Also plotted is a fit with a BE 1 distribution. From the fit we obtain T=5.5 K and a de- 10 generacy parameter −␮/kT=4.6ϫ10−3. The goodness of 0 2 4 6 8 10 the fit shows that the LP gas has reached quasiequilib- E −E (k=0) (meV) −4 LP LP x10 rium with well-defined temperature and chemical potential. These results agree with the experiments above FIG. 16. ͑Color online͒ LP population per state N͑t,E͒ vs LP ͑Deng et al., 2006͒. kinetic energy E−E0 for different detunings at a time when From the fits of the measured and calculated distribu- ͑ ͒ ⌬ Ϸ ͑ ͒ ⌬ TLP reaches Tmin: a =6.7 meV, P 4Pth; b =−0.9 meV, tions one also obtains the still slowly varying parameters Ϸ ͑ ͒ ⌬ Ϸ P 4Pth; and c =−3.85 meV, P 6Pth. Crosses are data of the thermal distributions, namely, the temperature of taken from the incidence direction of the pump, open circles the LP gas T͑t͒=T and its chemical potential ␮͑t͒. Us- are from the reflection direction, solid lines are fitting by BE pol ing these fits with a cavity lifetime of ␶ =2 ps and a distribution, and dashed lines by MB distribution. Parameters c ͑ ͒ pump rate P/Pth=2, the evolution of the LP tempera- for the curves are the following: a TMB=4 K, TBE=4.4 K, ␮ ͑ ͒ ␮ ture T ͑t͒ normalized to the lattice temperature T is BE=−0.04 meV; b TMB=4 K, TBE=8.1 K, BE=−0.13 meV; pol lat ͑ ͒ ␮ shown in Fig. 18 for various values of the detuning. One and c TMB=8.5 K, TBE=182 K, BE=−0.4 meV. From Deng et al., 2003. sees that the temperature of LP gas reaches for a detun- ␦ ing of =4.5 meV after about 60 ps. For lager times Tpol effective to achieve a quasiequilibrium BE distribution increases again because the low-energy LPs have a Ͻ ␮ −1 among LPs in kʈ 1 m at a temperature TLP significantly higher than the lattice temperature Tlat with a ␮ chemical potential LP=−0.01 to −0.1kBT. Finally, at ␶ ␶ ͑ ͒ large negative detunings, relax is longer than LP 0 ,so the system never reaches equilibrium. A quantum degenerate LP system, capable of reaching thermal equilibrium, is also valuable for studying the evolution from nonequilibrium to equilibrium phases, the distinction between classical and quantum systems, and the formation and dynamics of the order parameter of a quantum phase.

E. Numerical simulations FIG. 18. Calculated normalized temperature vs time for three Numerical studies show that, with certain parameters, values of the detuning. ␦=−1 ͑triangles͒,3͑squares͒, and ͑ ͒ ͑ ͒ the kinetically calculated momentum distributions nk t 4.5 meV circles . From Doan, Thien Cao, et al., 2008.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1511

FIG. 19. ͑Color͒ Measured normalized temperature vs time for FIG. 21. ͑Color͒ Measured degeneracy parameter vs time for various values of the detuning for GaAs microcavities with a various values of the detuning measured for GaAs microcavi- Rabi splitting of 14.4 meV and a normalized pump power ties with a Rabi splitting of 14.4 meV and a normalized pump ␦ ͑ ͒ power P/P =3 for a lattice temperature of 4.2 K. ␦=−0.9 P/Pth=3 for a lattice temperature of 4.2 K. =−0.9 squares , th 3.6 ͑triangles͒, 6.6 ͑diamonds͒, and 9.0 meV ͑circles͒. For the ͑squares͒, 3.6 ͑triangles͒, 6.6 ͑diamonds͒, and 9.0 meV ͑circles͒. open symbols a full local equilibrium is not yet reached. From For the open symbols, a full local equilibrium is not yet Deng et al., 2006. reached. The dash-dotted line is the limit of quantum degeneracy with a ground-state population of 1. From Deng et al., 2006. smaller lifetime. This calculated relaxation kinetics of T can be compared directly to the corresponding mea- pol emission from the ground state. The calculated pump surements of Deng et al. ͑2006͒, as shown in Fig. 19. dependences of the temperature and of the degeneracy Only where full symbols are shown, the gas was in good parameter have been shown to agree qualitatively well thermal equilibrium. The measured temperature of the with the corresponding measurements of Deng et al. LP gas also shows an improved temperature relaxation ͑2006͒. These findings show that for favorable param- with increasing detuning. With a detuning of 6.6 meV eters a finite-size thermal-equilibrium BEC of LPs is the lattice temperature is reached after about 40 ps. possible. Even with a detuning of 9 meV the lattice temperature is reached, while the numerical studies yield for this large detuning a less effective relaxation. This difference F. Steady-state momentum distribution is mainly due to the larger Rabi splitting of the real system compared to the model system. In most other Steady-state condensation of LPs was first reported in respects, e.g., the final increase of the temperature, the a CdTe/CdMgTe microcavity with 16 QWs, operating ͑ ͒ measured and calculated transient temperatures show also at positive detunings Kasprzak et al., 2006 . In this ␮ the same trends. experiment, the excitation pulse duration of 1 s is four For the same parameters the calculated degeneracy orders of magnitude longer than the decay and energy ␮͑ ͒ ͑ ͒ relaxation times of the LPs. Therefore the time- parameter − t /kTpol t is shown in Fig. 20 and the measured degeneracy parameters are shown in Fig. 21. integrated measurement reflects the steady-state proper- Although quantum degeneracy is reached for smaller ties of the system. Above the quantum degeneracy detunings at a slightly shorter delay, it stays much longer threshold, the high-energy tail of the time-integrated for larger blue detunings. The temporal range in which momentum distribution decays exponentially, indicating degeneracy exists extends over about 60 ps. The degen- quasiequilibrium among LPs at an effective temperature ϳ ͑ eracy is lost because the LP density decreases due to of 19 K the lattice temperature is 5 K if neglecting heating by the excitation laser͒. The low-energy LP distributions are not described by a simple model, which is attributed to the relatively strong LP-LP interactions. A similar measurement was later performed with a GaAs- based microcavity, where an in-plane harmonic potential was implemented and LPs condense into the potential minimum. An effective LP temperature of ϳ100 K was measured ͑Balili et al., 2007͒.

VI. ANGULAR MOMENTUM AND POLARIZATION KINETICS

Up to now, the spin and orbital angular momenta of FIG. 20. Calculated degeneracy parameter vs time for three excitons have not been considered explicitly. In this sec- values of the detuning. ␦=−1 ͑triangles͒,3͑squares͒, and tion, we formulate a pseudospin formalism and review 4.5 meV ͑circles͒. From Doan, Thien Cao, et al., 2008. experiments with circularly and linearly polarized pump-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1512 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

ings and polarization dependent detection ͑Deng et al., 01 0 − i 10 2003; Renucci et al., 2005; Kasprzak et al., 2006͒. ␴x = ͩ ͪ, ␴y = ͩ ͪ, ␴z = ͩ ͪ. ͑54͒ 10 i 0 0−1 ␳ A. Formulation of the quasispin kinetics The diagonal elements of k,ij corresponds to the distri- ␳ ͑ ͒ ͑ ͒ bution functions k,ii t =nk,i t and we have Shelykh et al. ͑2005͒ reviewed the theory of the exciton angular-momentum Boltzmann kinetics in micro- 2 ͑ ͒ ͑ ͒ ͑ ͒ cavities. The spin kinetics was numerically evaluated Nk t = ͚ nk,i t , 55 taking into account the LP-phonon scattering, but the i=1 even more important LP-LP scattering was omitted. Re- cently Krizhanovskii et al. ͑2006͒ and Solnyshkov et al. 1 ͑ ͒ Sz͑t͒ = ͓n ͑t͒ − n ͑t͔͒. ͑56͒ 2007 studied spin kinetics including the LP-LP scatter- k 2 k,1 k,2 ing for the case of resonant excitations. Here we formulate the kinetic equations for a nonresonantly pumped The off-diagonal elements give the x and y components polariton system. of the spin-polarization amplitude, We consider only optically active exciton states ͑bright ͒ 1 excitons with the total angular momentum m= ±1. The x͑ ͒ ͓␳ ͑ ͒ ␳ ͑ ͔͒ ͑ ͒ ϫ ␳ Sk t = k,12 t + k,21 t , 57 2 2 reduced density matrix k,ij can be composed con- 2 ␴l veniently using the Pauli matrices ij, ␳ ͑ ͒ ͗ † ͑ ͒ ͑ ͒͘ ͑ ͒ i k,ij t = bk,j t bk,i t 52 Sy͑t͒ = ͓␳ ͑t͒ − ␳ ͑t͔͒. ͑58͒ k 2 k,12 k,21 N ͑t͒ k ␦ l ͑ ͒␴l ͑ ͒ According to Cao et al. ͑2008͒, the Hamiltonian of the = ij + ͚ Sk t i,j, 53 2 l system is given by

1 † ͑⍀ † ͒ ␻ † ͑ ͒͑ † ͒ † H = ͚ ekb bk,s + ͚ kb bk,2 + H.c. + ͚ q,q a aq,q + ͚ G k,q,qz a + a−q,q b bk,s k,s 2 k,1 z q,qz z q,qz z k+q,s k,s k q,qz k,q,qz,s 1 ͚ ͑ Ј Ј͒͑ † † ͒ ͚ ͑ Ј Ј͒͑ † † + ͭ V k + q,k − q,k,k bk,sbkЈ,sbkЈ−q,sbk+q,s + U k + q,k − q,k,k bk,sbkЈ,sЈbkЈ−q,sЈbk+q,s 4 k,kЈ,q,s k,kЈ,q,ssЈ † † ͒ ͑ ͒ + bk,sbkЈ,sЈbkЈ−q,sbk+q,sЈ + H.c.ͮ, 59

where a† , a are the creation and annihilation opera- ͑ Ј Ј ͒ ͑ Ј Ј ͒ q,qz q,qz T k,k ,k − q,k + q = V k,k ,k − q,k + q tors of acoustic phonons, respectively. e is the disper- k ϫU͑k,kЈ,kЈ − q,k + q͒, ⍀ sion of the LP branch. k is a complex energy, related to the TE-TM splitting of the microcavity. ␻ is the fre- q,qz where EB and aB are the binding energy and Bohr radius quency of the acoustic phonon, with the in-plane wave of the exciton in two dimensions, respectively. Using the number q and the wave number qz perpendicular to the x and y components, we deﬁne a 2D polarization vector ͑ ͒ quantum well layers. G k,q,qz is the LP-phonon coupling constant. The matrix elements V͑k,kЈ,kЈ−q,k ជ ͑ x y͒ ͑ ͒ Sk = Sk,Sk , 61 +q͒ and U͑k,kЈ,kЈ−q,k+q͒ describe scattering of LPs in the relative triplet and singlet conﬁgurations, respec- ជ ជ ␳ ␳ ␳ ␳ ͑ ͒ tively. They are given by ͑Shelykh et al., 2005͒ 2SkSkЈ = k,21 kЈ,12 + k,12 kЈ,21. 62

Similarly we decompose the TE-TM splitting into x and ͑ ͒ 2 y components which form an effective 2D magnetic ﬁeld V k,kЈ,kЈ − q,k + q =6EBa Xk+qXkЈ−qXkЈXk, B ͑we absorb the g factor and magnetic moment in this effective ﬁeld͒,

͑ Ј Ј ͒ ⍀ ⍀x ⍀y U k,k ,k − q,k + q k = k + i k, ͑ ͒ =−␣V͑k,kЈ,kЈ − q,k + q͒, ␣ Ͼ 0, ͑60͒ 63

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⍀ជ = ͑⍀x,⍀y͒, The in-plane pseudospin lifetime can be estimated as k k k ␶−1 ␶−1 ␶−1 ␶ ks = k + sl , where sl is the characteristic time of the where the components of the splitting field have been spin-lattice relaxation. The scattering terms in Eqs. ͑65͒ given by Kavokin et al. ͑2004, 2005͒ as and ͑68͒ in the cylindrical approximation are put to- 2 2 gether in the Appendix according to Cao et al. ͑2008͒. k − k ⍀x ⍀ x y ⍀ ͑ ␾͒ As a simplifying approximation, we follow Cao et al. k = k 2 = k cos 2 , k ͑2008͒ and assume a cylindrical symmetry ͑Kavokin et ͑64͒ al., 2004͒ so that the polarization has only the radial 2kxky ជ r ⍀y ⍀ ⍀ ͑ ␾͒ component Sk =S eជr and the splitting field is taken to be = k = k sin 2 . k k k2 orthogonal to the polarization vector. This assumption does not take into account certain directional properties Here ␾ is the polar angle of the in-plane momentum of the splitting field, but such effects are relatively small vector, i.e., k =k cos͑␾͒, k =k sin͑␾͒. The momentum x y with respect to the presently investigated aspects of the dependence of the splitting field ͑64͒ is due to the spin kinetics. Using this cylindrical symmetry we obtain TE-TM splitting of the microcavity modes, together with the following set of equations for nk,j and the radial com- the momentum dependence of the optical matrix ele- r ments of the exciton longitudinal-and-transverse split- ponent Sk of the in-plane spin polarization: n LP-LPץ n LP-phץ nץ n nץ -ting. From the equations of the motion, we get the ki k,j =− k,j +ͯ k,j ͯ +ͯ k,j ͯ +ͯ k,j ͯ ץ ץ ץ ␶ ץ -netic equations of the distribution function nk,i and in t k t S-F t scatt t scatt plane pseudospin Sជ . For n , k k,j 1 LP-ph + p P͑k,t͒, j = 1,2, ͑71͒ ץ ץ ץ ץ nk,j nk,j ͯ nk,j ͯ ͯ nk,j ͯ ͯ nk,j ͯ 2 j ץ + ץ + ץ + ␶ −= ץ t k t S-F t mf-s-s t scatt r LP-LP ץ r LP-ph ץ r ץ r r ץ LP-LP 1 Sk Sk Sk Sk Sk ץ ͯ nk,j ͯ ͑ ͒ ͑ ͒ =− +ͯ ͯ +ͯ ͯ +ͯ ͯ ץ ץ ץ ␶ ץ pjP k,t , j = 1,2. 65 + + t ks t S-F t scatt t scatt ץ t scatt 2 1 Here the changes due to the splitting field are given by + ͉eជ ͉P͑k,t͒, ͑72͒ 2 p n 1ץ ͯ k,j ͯ ͑ ͒jជ ͓ជ ϫ⍀ជ ͔ ͑ ͒ ប −1 ez Sk k . 66 where the splitting field terms reduce to−= ץ t S-F n 1ץ The mean-field term due to the interaction of singlet ͯ k,j ͯ ͑ ͒j r ⍀ ͑ ͒ ប −1 Sk k 73−= ץ LPs is given by t S-F n 2 andץ ͯ k,j ͯ ͑ ͒j ͑ ͉͒ជ ϫ ជ ͉ ͑ ͒ = −1 ͚ U k,q,k,q Sk Sq . 67 Sr 1 n − nץ t បץ mf-s-s q ͯ k ͯ ͩ k,1 k,2ͪ⍀ ͑ ͒ ប k. 74 = ץ ជ t S-F 2 For Sk, The mean-field terms due to the LP-LP interactions ͑67͒ ជ LP-phץ ជץ ជץ ជ ជץ Sk Sk ͯ Sk ͯ ͯ Sk ͯ ͯ Sk ͯ and ͑69͒ vanish in the cylindrical approximation due to ץ + ץ + ץ + ␶−= ץ t ks t S-F t mf-LP-LP t scatt the angle integration. For the scattering terms in Eqs. ͑ ͒ ͑ ͒ LP-LP 71 and 72 in the cylindrical approximation we refer to Sជ 1ץ ͯ k ͯ ជ ͑ ͒ ͑ ͒ the appendix of Cao et al. ͑2008͒. epP k,t . 68 + ץ + t scatt 2 Here the mean-field term due to the LP-LP interaction is B. Time-averaged and time-resolved experiments and simulations Sជ 2ץ ͯ k ͯ ͚ ͓ ͑ ͒ ͑ ͔͒ ␶x ␶c ,ប V k,q,k,q − U k,q,k,q To compare with experiments, we use =30 ps = ץ t mf-LP-LP q ␶ =2 ps, and sl=15 ps. The smallness parameter of the ϫ͑f − f ͓͒eជ ϫ Sជ ͔, ͑69͒ interaction between excitons in the relative singlet state q,1 q,2 z k is ␣=5ϫ10−2. The bath temperature is T=4 K. The de- where eជz is the unit vector along the growth axis and pj tuning between the cavity mode and the exciton reso- and eជp denote components of the pump polarization in nance is assumed to be zero. The pump is centered at ͑ ͒ ϫ 5 −1 ⌬ the z direction and x,y plane, respectively. p1 =2, p2 kp =1.7 10 cm with a linewidth E=0.1 meV. The =0 for the circular pump and p1 =p2 =1 for the linear pump pulse width is as in the experiment of Deng et al. ͑ ͒ pump. The changes of the in-plane polarization due to 2003 , tp =3 ps. the splitting field are given by First we examine the time-averaged ground-state LP population for a circularly polarized pump pulse. Here Sជ 1 f − fץ ͯ k ͯ ͩ k,1 k,2ͪជ ϫ⍀ជ ͑ ͒ we use the total number of injected LPs as a measure of ប ez k. 70 = ץ t S-F 2 the pump power. As expected, the LP population at kʈ

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1514 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

co−cir left−cir 100 cross−cir right−cir total total ) ~0

|| 10 k ( n 0 (a) (b)

1 10 1 10 100

Pump Power (mW) Pump Power (mW)

0.8 cirular pump linear pump 0.6

0.4

0.2 FIG. 22. LP ground-state population vs total LP density under (c) ͑ ͒ ͑ ͒ Circular Degree of Polarisation pumping a with circularly polarized and b with linearly po- 0 larized light at T=4 K in a GaAs microcavity. The sum of the 1 10 100 two ground-state populations is given by the dotted line, the Pump Power (mW) spin-up LP population by the solid line, and the spin-down population by the dashed line. ͑c͒ Circular degree of polariza- FIG. 23. The measured time-averaged polarization properties tion of light emitted from the ground state. For linearly polar- of a GaAs microcavity. Emitted light intensity of the LP ized pumping the light intensity is given by the solid line for ground state ͑a͒ for circularly polarized and ͑b͒ for linearly circularly polarized pumping by the dashed line. From Cao et polarized pump light. ͑c͒ The circular degree of polarization is al., 2008. plotted for pumping with linear and circular polarizations. From Deng et al., 2003.

=0 with cocircular polarization exhibits a clear condensation threshold ͓solid line in Fig. 22͑a͔͒, while the popu- corresponding threshold pump rates in Figs. 22͑b͒ and lation with cross-circular polarization ͓dashed line in 23͑b͒ agree well. Finally, the degree of polarization for Fig. 22͑a͔͒ stays subcritical up to a total LP density of circular pumping remains high in both theory and ex- 1011 cm−2, where it is about four orders of magnitude periment ͓see Figs. 22͑c͒ and 23͑c͔͒, while the degree of smaller than the cocircular density. polarization for linearly polarized pumping decreases Next we calculate the ground-state LP population for above the threshold. left- and right-circular polarizations when the pump In Fig. 24 the calculated time evolution of the degree pulse is linearly polarized. Figure 22͑b͒ shows that due of circular polarization for a circular pump is shown for to the TE-TM splitting the ground-state population for various total LP densities. The results can be compared the left-circular LPs ͑solid line͒ has a lower threshold, with corresponding measurements of Renucci et al. while the right-circular LPs also condense but at about a factor of 3 higher pump rate. At the highest pump rates, the ground-state populations of both circular polarizations become equal. Finally, we plot in Fig. 22͑c͒ the circular degree of polarization, i.e., the normalized difference between the ground-state populations of the left- and right-circular LPs for circularly and linearly polarized pump light. For circular pump the degree of polarization increases rapidly at the condensation threshold and approaches a value close to 1. For linearly polarized pumping the degree of polarization increases to a value of about 0.95 but decreases rapidly when the right-circular LPs condense and approach a value close to 0 at the highest pump rates. Comparison with the experimental results in Fig. 22 ͑Deng et al., 2003͒ shows good qualitative agreement. In particular, the absence of FIG. 24. Calculated time dependence of the circular degree of a threshold for the cross-circularly polarized LPs for cir- polarization for circularly polarized near-resonant 3 ps pump ͑ ͒ ͑ ͒ ϫ 10 cular pumping is clearly shown in Figs. 22 a and 23 a . pulses which excite a total LP density of Ntot=2.0 10 , 2.5 Similarly the successive condensations of the left- and ϫ1010, 4.6ϫ1010, and 6.0ϫ1010 cm−2. From Doan, Cao, et al., right-circular LPs separated by about a factor of 3 in 2008.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1515

+ excitation

P (W.cm -2 ) 1.0 in 0.3 1 2.25

i 4 7.5

zat i 0.5

rcu

Ci 0.0

0 50 100 150 time (ps)

FIG. 25. Measured time dependence of the circular degree of polarization for circularly polarized pump pulses in GaAs microcavities. From Renucci et al., 2005.

FIG. 26. The polarization measurement setup. Vectors label ͑2005͒ as shown in Fig. 25, where a resonant excitation the fast or polarization axes of the optical components. The ͑␪ ͒ has been used. In both theory and experiment, the de- laser pump is initially horizontally polarized p =90° and is gree of polarization decays rapidly below threshold but incident at an angle of 55° with respect to the growth direction quickly approaches 1 around threshold. With increasing z. Luminescence is collected along the z axis. The first variable pump rate, the degree of polarization remains close to 1 retarder ͑VR1͒ and linear polarizer ͑LP1͒ work as a variable ͑ ͒ for about 30–70 ps, followed by a steep decay. Particu- attenuator. By rotating a half wave plate HWP1 , and using a ͑ ͒ larly striking is the difference in time evolution below removable quarter wave plate QWP1 , we can implement various polarization states for the pump. The second variable and above the threshold, as well as the trend of a longer ͑ ͒ lasting degree of polarization at higher pump rates. retarder VR2 is used as a zero, half, or quarter wave plate. The combination of a quarter wave plate ͑QWP2͒, variable retarder ͑VR3͒, and linear polarizer ͑LP2͒ is used for detection of a particular linear polarization state, depending on the re- C. Stokes vector measurement ␪ ␪ tardance of VR3. Inset: Definition of angles p and d corresponding to the polarization axes of the pump and detection, The polarization state of emission, which has a one- respectively ͑Roumpos, Lai, et al., 2009͒. to-one correspondence to the quasispin state of the internal polariton condensate, is characterized by the ͑ ͒ takes over well above threshold. For circularly polarized Stokes vector S1 ,S2 ,S3 defined as pump ͓Fig. 27͑c͔͒, the emission is circularly polarized up I − I I − I I − I 0° 90° 45° −45° L R ͑ ͒ to S3 =−99.4%. S1 = , S2 = , S3 = . 75 ͑␪ ͒ I0° + I90° I45° + I−45° IL + IR If we rotate the direction p of the linearly polarized pump ͓Figs. 27͑d͒ and 27͑e͔͒, the polarization direction I0°, I90°, I45°, and I−45° are the intensities of the linearly is always rotated by ϳ90° from the pump polarization. ␪ polarized components at detection angles d =0°, 90°, Also, a circularly polarized component has S3 changing ␪ ͓ ͑ ͔͒ 45°, and −45°, respectively, as defined in Fig. 26. IL and sign for varying p Fig. 27 f . The sign change is corre- ␺ ␪ IR are the intensities of the left- and right-circularly po- lated with the deviation of − p from 90° as shown in larized components, respectively. The degree of linear Fig. 27͑e͒. polarization ͑DOLP͒ and the angle of the dominant lin- The above quasispin dynamics can be understood in ear polarization ␺ are calculated as terms of spin-dependent Boltzmann equations ͑Roum- pos, Lai, et al., 2009͒. In particular, the quantum interfer- 1 S ͱ 2 2 ␺ −1ͩ 2 ͪ ͑ ͒ ence effect between the repulsive scattering among isos- DOLP = S1 + S2, = tan . 76 2 S1 pins and the attractive scattering among heterospins is responsible for the observed 90° rotation of the linear The degree of circular polarization is simply given by the polarization shown in Fig. 27͑e͒. This is fully compatible parameter S . 3 with the discussion on the LP-LP interaction in Sec. In Fig. 27͑a͒, the measured intensity for the linearly ␪ III.B. polarized light along d =0° and 90° is shown as a function of pump power in units of the injected polariton ͑ ͒ density per pulse QW Roumpos, Lai, et al., 2009 . The VII. COHERENCE PROPERTIES ͑␪ ͒ pump is horizontally polarized p =90° . The three Stokes parameters defined above are plotted in Fig. In BEC, a macroscopic number of particles occupy a ͑ ͒ ͑ ͒ 27 b . Notice that the circular polarization S3 develops single-quantum state and manifest quantum correlations ͑ ͒ just above threshold but the linear polarization S1 on the macroscopic scale. The wave function of the con-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1516 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

properties of laser light. For polaritons, it was first measured by Deng et al. ͑2002͒ for a GaAs microcavity with 2 ps photon lifetime. Recently it was measured for a GaAs microcavity with 80 ps LP lifetime by Roumpos, Hoefling, et al. ͑2009͒ and also for a CdTe microcavity by Kasprzak et al. ͑2008͒. A first attempt to explain the second-order coherence measurements was given by Laussy et al. ͑2004, 2006͒ in terms of a coupled Boltz- mann equation and master equation kinetics. The model was extended by Doan, Cao, et al. ͑2008͒ to reach better agreement with experiments. The time-domain second-order coherence function g͑2͒͑␶͒ is defined as ͑Glauber, 1963͒

͗Eˆ ͑−͒͑t͒Eˆ ͑−͒͑t + ␶͒Eˆ ͑+͒͑t + ␶͒Eˆ ͑+͒͑t͒͘ g͑2͒͑␶͒ = , ͑77͒ ͗Eˆ ͑−͒͑t͒Eˆ ͑+͒͑t͒͘2

where Eˆ ͑−͒͑t͒ and Eˆ ͑+͒͑t͒ are the negative and positive frequency parts of the electric-ﬁeld operator at time t, respectively. g͑2͒͑␶͒ measures intensity correlation of the ﬁeld between time t and t+␶.IfEˆ ͑t͒ and Eˆ ͑t+␶͒ are uncorrelated but stationary,

͗Eˆ ͑−͒͑t͒Eˆ ͑+͒͑t͒͗͘Eˆ ͑−͒͑t + ␶͒Eˆ ͑+͒͑t + ␶͒͘ g͑2͒͑␶͒ = =1. ͑ ͒ ͑ ͒ FIG. 27. ͑Color͒ Measurement of Stokes parameters ͑markers͒ ͗Eˆ − ͑t͒Eˆ + ͑t͒͘2 compared with the theoretical model ͑solid lines͒. ͑a͒ Horizon- ͑ ͑␪ ͒ ␪ ͑ For a single-mode state, the maximum correlation or tal pumping p =90° . Collected luminescence for d =0° red ͒ ␶ ͑2͒͑␶ ͒ ͒ ␪ ͑ ͒ anticorrelation is obtained at =0, i.e., by g =0 squares and d =90° blue circles vs injected particle density in ␮m−2 per pulse per QW. A clear threshold is observed at which has the following property: ϫ 2 ␮ −2 ͑ ͒ ␪ 5 10 m . Degree of polarization measurement for b p g͑2͒͑0͒ = 2, thermal state, =90° linear pumping and ͑c͒ left circularly polarized pumping. Blue circles: S1, green diamonds: S2, and red squares: S3 de- ͑ ͒ ͑ ͒ 1 fined in Eq. 75 . Calculated polarization parameters from the g 2 ͑0͒ =1− , number state ͉n͘, measurement of the Stokes parameters for linear pumping n ͓Eqs. ͑75͒ and ͑76͔͒. ͑d͒ DOLP. ͑e͒ Angle for major axis of ␺ ␪ ͑ ͒ ͑ ͒ linear polarization relative to p. f Degree of circular po- g 2 ͑0͒ = 1, coherent state. ͑ ͒͑ ͒ larization S3 Roumpos, Lai, et al., 2009 . Thus defined normalized second-order coherence functions are independent of linear losses between the densate serves as the order parameter; its density matrix source and detector. Moreover, while spectrally filtered has finite off-diagonal elements, often referred to as off- ͑ thermal light can be brought into interference after a diagonal long-range order Penrose and Onsager, 1956; ͑2͒͑ ͒ Beliaev, 1958; Yang, 1962͒, which can be measured Michelson interferometer, its g 0 value remains un- ͑2͒͑ ͒ through the first-order coherence functions. Second- and changed. These properties make g 0 especially useful higher-order coherence functions of the state further in identifying the quantum-statistical nature of a state. characterize the nature of a quantum state and distin- Caution is needed, however, given the finite experi- ␶ guish it from a thermal mixture ͑Glauber, 1963; Gar- mental time resolution. When is much longer the co- ␶ ͑2͒͑␶͒ diner and Zoller, 2000͒. We review in this section studies herence time c of the field, g decays to 1. Thus a ␶ of the coherence functions of the LP condensates and detector time resolution much larger than c will make ͑2͒ discuss next special properties of the macroscopic quan- accurate measurement of g ͑0͒ difficult. tum state, such as the first-sound mode and superfluidity due to broken gauge symmetry, appearance of quantized (2) vortices, and phase locking among multiple condensates. 1. g (0) of the LP ground state Deng et al. ͑2002͒ first measured the second-order A. The second-order coherence function temporal coherence function of a condensate. A Han- bury Brown–Twiss-type setup ͑Hanbury Brown and The temporal intensity-intensity correlation function Twiss, 1956͒ is adopted, mainly consisting of a 50/50 is the lowest-order coherence function which character- beam splitter and two single-photon counters ͑Fig. 28͒. izes the statistical nature of a quantum state. It has been In this pulsed experiment, since the polariton dynamics of crucial importance in characterizing the coherence is much faster than the time resolution of the photon

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1517

2.0

1.8

P/Pth=1 1.6 2.0

(0) 1.5

(2) 1.0 g 1.4 P/Pth=15 0.5 2.0 1.5 0 400 800 1.0 1.2 Channel Number 0.5 0 400 800 1.0 10 20 Pump Intensity FIG. 28. ͑Color online͒ A realization Hanbury Brown–Twiss P/Pth measurement with single-photon counters. For autocorrelation ͑ ͒ FIG. 29. The second-order coherence function g 2 ͑0͒ vs pump measurement, use a single-core fiber, beam splitter, and mirror intensity P/P . From Deng et al., 2002. ͑solid path͒. For cross correlation, use a double-core fiber th bundle to carry the two input and output signals ͑dashed path͒. g͑2͒͑0͒ decreases only very slowly with increasing density above threshold, indicating a relatively small con- counters, essentially the integrated area under each densate fraction which builds up only gradually with in- pulse was measured, which gives the numerator in the creasing total density of LPs. This is typical of two- following equation: dimensional systems with relatively strong interactions, ͗ ͑ ͒ ͑ ͒͘ where there is large quantum depletion of the conden- ͑ ͒ n1 i n2 i + j i g 2 ͑j͒ = , ͑78͒ sate. Theoretical and numerical studies have predicted ͗n ͗͘n ͘ 1 2 the same qualitative behavior ͑Sarchi et al., 2008; ͑ ͒ ͑ ͒ Schwendimann and Quattropani, 2008͒. where n1 i and n2 i+j are the photon numbers detected by the two-photon counters in pulses i and i+j, Recently a similar measurement was performed with a ͑ ͒ respectively. In the limit of low average count rates ͑n CdTe-based microcavity Kasprzak et al., 2008 . In this ͑ ͒ experiment, emission from the LP ground state, not fil- Ͻ0.01 per pulse in our experiment͒, g 2 ͑j͒ approximates ͑ ͒ tered in energy, has a coherence time of ϳ2 ps below time-averaged g 2 ͑␶͒ over each pulse. For j0, coinci- ͑2͒͑ ͒ dence counts are recorded from adjacent pulses which and around threshold. g 0 close to 1 was observed for are uncorrelated and thus g͑2͒͑j͒ϵ1 for j0. Assuming excitation densities from below the threshold Pth to ϳ ͑2͒͑ ͒ the same statistical properties of all pulses, the area of 2Pth. At higher excitation densities, g 0 increases Ͼ ϳ these peaks at j0 gives the denominator in Eq. ͑78͒. significantly to 2at 10Pth. ͑ ͒ ͑2͒͑ ͒ The emission pulse of the LP ground state was first Kasprzak et al. 2008 also measured g 0 under ͑ filtered by a monochromator with a resolution of ⌬␭ quasi-continuous-wave excitation with an excitation ϳ ␮ ͒ =0.1 nm, resulting in a pulse with a coherence time of pulse duration of 1 s . In this case, the time reso- ⌬␶ =ͱ8ln2␭2 /c⌬␭ϳ4 ps. Below threshold, the full lution is limited by that of the photon counters, which is c ͑2͒͑ ͒ϳ width at half maximum ͑FWHM͒ temporal width of the 120 ps. A small bunching with gm 0 1.03 was mea- pulse is about 350 ps, much longer than the coherence sured below and around Pth. If correcting for the coher- time of the pulse ⌬␶ ϳ4 ps. Since the measured g͑2͒͑0͒ is ence time and photon counter resolution, it corresponds c ͑2͒͑ ͒Ϸ an integration of g͑2͒͑0͒ over the whole pulse as given in to an actual g 0 2.8±1. No bunching above the ͑ ͒ ͑ ͒ noise level was observed at 1.5P , which indicates a de- Eq. ͑78͒ and g 2 ͑␶͒Ϸ1 during most of the pulse, g 2 ͑0͒ is th ͑2͒͑ ͒ ͑2͒͑ ͒Ͼ also close to 1 even though the system is expected to be crease in g 0 . gm 0 1 is again measured above in a thermal state. Poor time resolution is a common 2.3Pth, consistent with the pulsed measurement. limitation in Hanbury Brown–Twiss-type measurements. Fortunately, at threshold, the pulse width shortens to 2. Kinetic models of the g(2) ϳ ⌬␶ ͑2͒͑ ͒ 8 ps and becomes comparable to c. Hence g 0 becomes a good approximation of g͑2͒͑0͒. Here bunching Using a stochastic Boltzmann kinetic equation and of the emitted photons was observed, with a maximum master equation, we study the probability to find n LPs ͑ ͒ ͑ ͒ g 2 ͑0͒ of 1.77 at P/P ϳ1.1 ͑Fig. 29͒. This bunching ef- in the ground state Doan, Cao, et al., 2008 . The th ͑2͒ fect, which is obscured by the time-integration effect be- quantum-statistical correlation function g is given by low threshold, is the signature of bosonic final-state the correlations of the condensate operators ͑Glauber, stimulation. Far above threshold, the pulse width only 1963͒ ͑ ͒ decreases further, therefore g 2 ͑0͒ remains a good esti- ͗ †͑ ͒ †͑ ␶͒ ͑ ␶͒ ͑ ͒͘ ͑ ͒ b0 t b0 t + b0 t + b0 t mate for g͑2͒͑0͒. However, the g͑2͒͑0͒ measured in this g 2 ͑␶͒ = . ͑79͒ ͗b†͑t + ␶͒b ͑t + ␶͒͗͘b†͑t͒b ͑t͒͘ region decreases, which demonstrates the formation of 0 0 0 0 second-order coherence in the system or, in other words, Note that the field operators are in the normal form with the polariton gas starts to acquire macroscopic coher- all creation operators to the right. In the number repre- ence. sentation for zero delay

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1518 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

Ӷ ͑ ͒ ͗n ͑n −1͒͘ sate density varies from small values of n 1 below g 2 ͑0͒ = 0 0 . ͑80͒ ӷ ͗n ͘2 threshold to large values above n 1 this approxima- 0 tion is not well justified below and at threshold ͗ ͑ ͒͘ ͗ †͑ ͒ ͑ ͒͘ The mean condensate density n0 t = b0 t b0 t is where n=O͑1͒. determined by a Boltzmann equation which governs the • The third possible stochastic extension of the scattering kinetics in and out of the ground state, ground-state kinetics is the master equation of the -n ͘ probability to find n particles in this state. The mas͗ץ 0 = ͗R ͑t͓͒1+n ͑t͔͒͘ − ͗R ͑t͒n ͑t͒͘. ͑81͒ t in 0 out 0 ter equation takes into account the discrete nature ofץ the number of condensed particles ͑Carmichael, The rate in is given by the scattering processes from the 2003͒ but disregards the phase of the condensed excited states to the ground state by LP-LP and LP- state. This semiclassical approach has been studied phonon scatterings, by Laussy et al. ͑2004͒. They calculated g͑2͒͑0͒ for CdTe microcavities from numerical solutions of the LP-LP ͑ ͒ Rin = ͚ w0,k;kЈ,k−kЈ 1+nk nkЈnk−kЈ time-dependent master equation and considered LP- k,kЈ phonon and LP-electron scatterings. Here we use a LP-ph ͑ ͒ well-converging continued fraction method for the + ͚ w0,q␴ nqNq,−␴. 82 q,␴=±1 solution of the stationary master equation and calculate the results for GaAs microcavities, including The rate out is given by both LP-phonon and LP-LP scattering. The results ͑ 1 are compared with experimental observations Deng R = + ͚ wLP-LP n ͑1+n ͒͑1+n ͒ et al., 2002͒. out ␶ 0,k;kЈ,k−kЈ k kЈ k−kЈ 0 Ј k,k • Still another possibility is to treat the kinetics of the LP-ph͑ ͒ ͑ ͒ second-order function ͗b†͑t͒b†͑t͒b ͑t͒b ͑t͒͘ in addi- + ͚ w0,q␴ 1+nq Nq,−␴. 83 0 0 0 0 q,␴=±1 tion to the kinetics ͑Sarchi et al., 2008; Schwendi- mann and Quattropani, 2008͒. Formally one has exchange rates due to the scattering process kជ ,0→0,kជ. These exchange rates do not give rise to changes of the populations and thus should not be included in the kinetics. The mean rate equation can 3. Master equation with gain saturation then be written as We now formulate the master equation for the prob- ͑ ͒ ͘ ability Wn t to find n particles in the condensate at time ͗ץ n0 ͗ ͑ ͓͒͗͘ ͑ ͔͒͘ ͗ ͑ ͒͗͘ ͑ ͒͘ ͑ ͒ = Rin t 1+n0 t − Rout t n0 t . 84 t, dropping the index 0 to ease the notation. With the ͒ t in͑ץ gain saturation rate Gn =Rn n+1 and decay rate Dn out To calculate the second moments of the condensate, we =Rn , we have the master equation need a stochastic extension of the condensate’s kinetic dWn ͑ ͒ equation. There are at least four possible formulations =− Gn + Dn Wn + Dn+1Wn+1 + Gn−1Wn−1 of such an extension: dt m=n+1 • One can supplement the Boltzmann equation for n0 ͚ ͑ ͒ by Langevin fluctuations with shot-noise character = Mn,mWm. 85 m=n−1 ͑Haug and Haken, 1967; Lax, 1967; Haug, 1969͒. The ͑ ͒ second moments of density are then determined by Mn,m is a tridiagonal matrix Risken, 1984 . Its elements the second moments of these shot-noise fluctuations. are As it is known from laser theory, Langevin equations ͑ ͒ can be solved approximately by linearization below Mn,n =− Gn − Dn , Mn,n+1 = Dn+1, Mn,n−1 = Gn−1. and above threshold, but there is no simple way to ͑86͒ find solutions in the whole density regime. The rate equation does not provide the full informa- • This is different for the associate Fokker-Planck tion for constructing the corresponding master equation equation in which the probability for a certain den- because the transition rates are only known for the sity value is calculated. In stationary equilibrium, one mean ͗n͘ but not for n ͓see, e.g., Carmichael ͑2003͔͒. gets an analytic solution in terms of an exponential Systematically, one would have to determine the popu- of a generalized Ginzburg-Landau potential ͓see lations of the excited states as a function of the pump Risken ͑1984͔͒. The change of the potential from be- rate and the number of LPs in the ground state. It can be low threshold with a minimum at n=0 to a potential done analytically for a homogeneously broaden three- with a sharp minimum at n0 above threshold de- level laser but only numerically in our case from the scribes the probabilities for all densities. In both ap- solution of the Boltzmann equation. Simple procedures proaches, the Langevin and Fokker-Planck methods, to include this correction ͑Laussy et al., 2004͒ are to ex- the density is considered to be a continuous variable pand the distributions of the excited states linearly ͒͘ ͗ ͑͒ ץ ץ͑ ӷ ͗ ͘ ͑ ͒ ͗ ͘ which is valid if n 1. Because in a BEC the conden- around n as nk n = nk + nk / n n− n and estimate

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1519

Wn+1

Gn Dn+1

DGn n−1

W n−1

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 ͗ץ ͯ nk n ͯ nk ͑ ͒ ͘ , 87 ͗ = ץ n ͗n͘ n − N Mn+1,n ͑ ͒ where N is the total number of polaritons. This yields Sn =− . 93 Mn+1,n+1 + Mn+1,n+2Sn+1 n − ͗n͘ ͗ ͑͘ ͒ ͗ ͩ͘ ͪ ͑ ͒ ͑ ͒ nk n = nk 1+͗ ͘ . 88 Equation 93 is well suited for numerical iteration. By n − N analytical iteration one obtains continued fractions for These correction terms of the scattering rates in the the ratio Sn and Wn is then given by master equation vanish approximately if one calculates k=n−1 ͗ ͘ with the modiﬁed master equation and with n W = ͟ S W . ͑94͒ ͚ n k 0 = nnWn. We use these corrections of the scattering rates k=0 ͗ ͘Ӷ in the linear approximation. Because nk 1 for all excited states, stimulated scattering into these states is neg- The value of W0 follows again from the normalization ͗ ͘ϳ͗ ͘ condition. One can start the iteration at a value Nӷ͗n͘ ligible and 1+ nk n . Thus the LP-phonon-scattering LP-ph ͑ ͒ ͑ ͗ ͒͘ ͑͗ ͘ when WN+1=WN+2= ¯ =0, so that SN =0. This iterative rate Rin has a correction term F n = n− n / n ͒ LP-LP procedure can also be generalized for the solution of the −N , while the LP-LP scattering rate Rin has a cor- ͑ ͒ ͑ ͒ time-dependent master equation Risken, 1984 where rection factor 2F n . In the rate out, only the LP-LP detailed balance is no longer fulﬁlled. LP-LP ͑ ͒ scattering rate Rout is corrected by a factor F n , while other rates remain unchanged. 5. Numerical solutions of the master equation

4. Stationary solution The stationary probability distributions obtained by iteration are shown in Fig. 31 for three pump powers. Under stationary pumping, if a stationary state is Below threshold, the distribution peaks at n=0 and de- reached, the rates in and out of the ground state no cays monotonically. Slightly above threshold, the distri- longer change with time. The stationary solution is bution peaks at a value ͗n͘ӷ1 but is still rather broad. maintained if the rates between two successive states Well above threshold for P/Pth=4.1, the distribution balance. From Fig. 30 we see that the following detailed peaks sharply around a larger density value of about balance relation holds: ͗n͘Ӎ104. We checked that the solutions obtained itera- ͑ ͒ DnWn = Gn−1Wn−1, 89 tively agree indeed with those obtained from the detailed balance condition. which yields the detailed balance solution Once all Wn are calculated including the normaliza- k=n ͑2͒͑ ͒ ͗ ͑ ͒͘ ͗ ͘2 G G tion W0, one obtains g 0 = n n−1 / n . We see in n−1 k−1 ͑ ͒ ͑2͒ Wn = Wn−1 = ͟ W0. 90 Fig. 32 how the thermal limit with g ͑0͒=2 is smoothly D D ͑ ͒ n k=1 k connected with g 2 ͑0͒=1. Surprisingly, the transition W is obtained from the normalization ͚ W =1. takes place not in the threshold region but around a 0 n n Ӎ In general, the stationary solutions are obtained from pump power of P 2Pth. This delayed transition from m=n+1 the thermal to the coherent limit is due to the semiclas- ͑ ͒ sical approach which assumes an n-diagonal density ma- ͚ Mn,mWm =0. 91 m=n−1 trix. As discussed, the fast decay of the correlations above ͑ ͒ Following Risken 1984 , we introduce the ratio Sn threshold is not observed in the experiments ͑Deng et ͑ ͒ =Wn+1/Wn and rewrite Eq. 91 as al., 2002͒. This can be understood qualitatively, taking ͑ ͒ ͑ ͒ ͑ ͒␶ into account that g 2 ͑␶͒ varies as g 2 ͑␶͒=2e Rin−Rout as Mn,n−1 ͑ ͒ Mn,n + Mn,n+1Sn + =0. 92 one can show, e.g., in a Langevin approach. An average Sn−1 ͑ ͒ϰ ͑2͒͑ ͒ ͑ ͒ over the delay times yields g¯ 2 0 g 0 / Rout−Rin .As → By changing n n+1, we obtain the pump power decreases away from threshold Rout

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1520 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

out,͑2͒ LP-LP ͑ ͒͑ ͒ R = ͚ w0,0;q,−q 1+nq 1+n−q . q

2 2 2 The last ﬁnite decay rates are D1 =D0 =0 and D2 =2Rout,͑2͒ because at least two LPs are needed in the ground state for a quadratic scattering process out of the ground state. The two quadratic terms extend the three-diagonal master equation to a ﬁve-diagonal one,

dW ͑ ͒ ͑ ͒ n =−͑G + D + G 2 + D 2 ͒W + D W dt n n n n n n+1 n+1 FIG. 32. Calculated second-order coherence function g͑2͒͑0͒ vs ͑2͒ ͑2͒ normalized pump power P/Pth. The full line is obtained by + Dn+2Wn+2 + Gn−1Wn−1 + Gn−2Wn−2 iteration and the circles by the detailed balance relation. From m=n+2 Doan, Cao, et al., 2008. ͑ ͒ = ͚ Mn,mWm. 99 m=n−2 ¯ ͑ ͒ −Rin increases. Thus the averaged g2 0 decreases rap- Solving this pentadiagonal master equation iteratively idly below threshold, as seen in the experiment with the ͓for details see Doan, Thien Cao, et al. ͑2008͔͒ one finds strongly fluctuating 2 ps GaAs microcavities. In a recent ͑ for realistic values of the collision broadening a rather experiment with improved GaAs microcavities Roum- small increase of g͑2͒ above threshold. In contrast, ͒ ͑2͒ pos, Hoefling, et al., 2009 , the minimum of g as a func- Schwendimann and Quattropani ͑2008͒ found additional tion of pump power seems to occur at threshold, and the correlations above the coherent limit and a minimum of range of correlations above the Poisson limit extends g͑2͒ around threshold, as has been observed, under the over a much wider pump range. Similar observations assumption of a thermal exciton reservoir. If the kinetics have been made in CdTe microcavities ͑Kasprzak et al., ͑ ͒ of the excited states is included Sarchi and Savona, 2008 . Further many-body effects have to be found in 2008͒, the minimum in the threshold region is nearly order to account for the difference between these ex- eliminated but the extra correlations remain. The differ- perimental findings and the theory. ences between the results of Doan, Thien Cao, et al. ͑2008͒ and Sarchi and Savona ͑2008͒ could be partly due 6. Nonresonant two-polariton scattering (quantum depletion) to the fact that the quadratic corrections have been neglected in the Boltzmann equation. ͑2͒ One way to improve the g calculations is to include In order to give an alternative explanation, Doan, quantum depletion processes. Following the idea of Thien Cao, et al. ͑2008͒ considered a phenomenological Schwendimann and Quattropani ͑2008͒, one can con- kinetic saturation model. It is known that the 2D exci- sider a quadratic correction term in the form of a scat- tons ionize once their density is so high that the exciton tering of two condensate LPs to the excited states with wave functions start to overlap ͑Schmitt-Rink et al., the momenta q and −q. This process is nonresonant and 1985͒. The critical saturation density due to phase-space only takes place when the linewidth is finite. The corre- filling has been evaluated from the reduction of the ex- sponding transition rate is 32␲ 2 citon oscillator strength to be 1/ns = 7 a2D. This will ͚ wLP-LP ͓n͑n −1͒͑1+n ͒͑1+n ͒ also result in a reduction of the exciton density n to 0,0;q,−q q −q ͓ ͑ ͒2͔ q n/ 1+ n/ns . In order to incorporate the saturation effect kinetically in the master equation, we introduce ͑ ͒͑ ͒ ͔ ͑ ͒ − 1+n 2+n nqn−q , 95 phenomenologically a two-polariton ionization rate. with nജ1. The energy-conserving delta function has to Due to the overlap of the wave functions in a collision of be replaced, e.g., by a Gaussian resonance two polaritons, the two polaritons are assumed to by ionized and thus lost for the condensate, 2 ͑ ͒2 ␥2 ␲␦͑ ͒ ⇒ −4 e0 − eq / ͑ ͒ 2 2e0 −2eq ␥e . 96 1 r D2 = n͑n −1͒ s . ͑100͒ n ␶ n This yields two-particle transition rates of the form 0 s ͑ ͒ 2 in, 2 ͑ ͒͑ ͒ We have included a small numerical factor rs in order to Gn = R n +1 n +2 , ͑ ͒ avoid an unphysically large ionization rate far below 97 ␶ in,͑2͒ LP-LP saturation. We used the lifetime 0 to set the scale for R = ͚ w0,0;q,−qnqn−q the ionization rate. Any direct backscattering from the q ionization continuum to the condensate can be ne- and glected. In Fig. 33 the results are plotted for the saturation D2 = Rout,͑2͒n͑n −1͒, n model. We see that with increasing interaction strength ͑98͒ ͑2͒ ജ rs the range where g 1 extends to larger pumping

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1521

-1 2 rs=10 rs=10-2 -6 1.8 rs=10 rs=0 1.6 (0) 2

g 1.4

1.2

0 0.5 1 1.5 2 2.5 3 3.5

P/Pth

FIG. 33. Calculated second-order coherence function g͑2͒͑0͒ vs FIG. 34. ͑Color online͒ Interference fringes between two frag- ϳ ␮ ͑ ͒ normalized pump power P/Pth for the kinetic saturation ments 6 m apart, measured below open triangles and model. above ͑open circles͒ the nonlinear threshold, with contrasts of 5% and 46%, respectively. Horizontal axis is the relative phase. Adapted from Kasprzak et al., 2006. rates as has been observed in the experiment ͑Deng et al., 2002͒. This simple model brings the results closer to those of the experiment. ␳ ͑ ͒ ͗⌿†͑ ͒⌿͑ ͒͘ 1 r,rЈ = rЈ r = ͗⌿†͑rЈ͒⌿ ͑r͒͘ + ͗⌿† ͑rЈ͒⌿ ͑r͒͘ B. First-order temporal coherence 0 0 T T N0 ͑ ͒ 1 ͑ ͒ ip0 r−rЈ ͗ + ͘ ipi r−rЈ ͑ ͒ The first-order temporal coherence is characterized by = e + ͚ aˆ i aˆ i e . 102 V V the energy linewidth of the LP emission. Interestingly, i 0 the linewidth above threshold does not decrease accord- ␳ ͑ Ј͒ ing to the Shallow-Townes formula but increases again 1 r,r describes the first-order coherence between the ͑Porras and Tejedor, 2003͒ with increasing density of the fields at positions r and rЈ. We again can separate it into LPs in the ground state. This observed effect can be terms of the excited states and a term from the ground ͉ ͉→ϱ ͑ ͒ calculated by the Langevin equations for the Bogoliubov state. When r−rЈ , the phase pi r−rЈ of the model of weakly interacting bosons. The LP population excited-state terms becomes random, and the summa- in the ground state gives rise to a Kerr-effect-like behav- tion of these terms vanishes when V→ϱ. The ground- ior which in turn causes the linewidth increase ͑Tassone state term, however, has a finite and constant amplitude Ͻ ഛ and Yamamoto, 2000͒. of 0 n0 n in the BEC phase regardless of the separa- ͉ ͉ ͉␳ ͑ ͉͒ ͉⌿ ͉2 tion r−rЈ . Hence 1 r,rЈ =n0 = 0 is finite and uniform even over macroscopic distances. In other words, C. First-order spatial coherence ␳ 1 has an ODLRO. This is an essential character of BEC. First-order spatial coherence or the so-called off- In two dimensions, long-wavelength thermal fluctua- diagonal long-range order ͑ODLRO͒ is studied by Pen- tions of the phase destroy a genuine long-range order in rose and Onsager ͑1956͒, Beliaev ͑1958͒, and Yang the thermodynamic limit ͑Mermin and Wagner, 1966; ͑1962͒. Hohenberg, 1967͒. But quasi-BEC is established when coherence is established throughout the system size in 1. Long-range spatial coherence in a condensate an interacting Bose gas ͑Lauwers et al., 2003͒, such as The Bose field operator ⌿͑r͒ of an ideal gas is defined manifested in extended spatial coherence. as

ip0r 1 a0e 1 ⌿͑ ͒ ipir ipir r = ͱ ͚ aˆ ie = ͱ + ͱ ͚ aˆ ie 2. Spatial coherence among bottleneck LPs V i V V i0 = ⌿ ͑r͒ + ⌿ ͑r͒, ͑101͒ Spatial coherence of LPs was first studied by Richard 0 T et al. ͑2005͒. In this experiment, a CdTe-based microcav- where r is the coordinate vector and pi is the momentum ity was used. Above a threshold excitation density Pth, ⌿ ͑ ͒ vector of the state i. T r consists of terms of the ex- LP lasing was observed in bottleneck states with in- →ϱ ⌿ ͑ ͒ ͑ ͒ cited states. In the thermodynamic limit of V , T r plane wave number kʈ =kbot and energy E kbot . The 2D Ͼ ⌿ ͑ ͒ Ͻ vanishes for r 0. 0 r is the ground-state term. In a Fourier plane image of the LPs was uniform at P Pth. Ͼ condensed phase, a0 can be approximated by a c-number At P Pth, speckles appeared in the images, which was ⌿ ͑ ͒ amplitude and 0 r has a finite and constant amplitude related to an increase in spatial coherence among the LP given by the square root of the condensate fraction. emitters. At the same time, the emission linewidth at ⌿͑ ͒ ͑ ͒ Thus r also serves as the order parameter of BEC. E kbot was broadened in kʈ, and interference in kʈ was The off-diagonal element of the first reduced density observed by a Billet interferometer setup. It again indi- matrix of ⌿͑r͒ is cated increased spatial coherence ͑Fig. 34͒.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1522 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

0 10 sured for r=1.3 ␮m, which is approximately the intrinsic coherence length of a single LP. At r=8 ␮m, which is the 1/e size of the excitation spot, g͑1͒͑r͒ remains above e−␲. From r=1.3 to 8 ␮m, decay of g͑1͒͑r͒ is well modeled by

(r) −1 ) the Fourier transform of the momentum distribution of

1 10

( P/P th −␲ ͑1͒ g LPs. The e decay length of g ͑r͒ was about eight 6.7 times the thermal de Broglie wavelength above thresh- 2.0 ͑ ͒ 1.7 old Fig. 35 . This demonstrates that coherence extends

−2 1.3 to a much longer range than both the average LP-LP 10 0 2 4 6 8 spacing and LP’s intrinsic coherence length. Finite co- r(µm) herence is maintained throughout the condensate. At the same time, there is a significant thermal excitation ͑ ͒ ͑1͒͑ ͒ FIG. 35. Color online g r vs r for different pump rates from the condensate. ͑1͒͑ ͒ P/Pth as given in the legend. The symbols are measured g r . The solid lines are fits by the modified Bessel function of the first kind, corresponding to the 2D Fourier transform of the 5. Theory and simulations ϰ ͓ ͑ ͒ ␮͔ momentum distribution of kBT/ E k − . The dashed line at P/Pth=7 is fit by the 2D Fourier transform of a Maxwell- An early attempt to calculate the first-order spatial Boltzmann distribution. The dash-dotted line marks where correlation function is due to Sarchi and Savona ͑2007a, g͑1͒͑r͒=1/e ͑Deng et al., 2007͒. 2007b, 2008͒. With the condensation kinetics they obtained relatively large condensate densities which re- 3. Spatial coherence between fragments of a condensate sulted in a strong off-diagonal long-range order. They incorporated many-body effects using the Popov ap- Later Kasprzak et al. ͑2006͒ measured directly the spa- proximation ͑Shi and Griffin, 1998͒ in order to calculate tial coherence of a fragmented condensate. LPs con- the depletion of the condensate. Here we calculate g͑1͒ densed into the kʈ ϳ0 state but at multiple localized ͑ ͒ ͑ ͒ using nk t and n0 t obtained through the small areas of ϳ4 ␮m in diameter within the 20–30 ␮m Boltzmann equations. excitation spot. PL from different localized condensates We write the first-order spatial coherence function in was brought into interference after a Michelson interfer- terms of the LP field operator ␺͑rជ͒ as ometer. The fringe visibility gives the first-order coher- ͑1͒ † ence function g ͑r͒, ͑ ͒ ͗␺ ͑rជ ͒␺͑rជ ͒͘ g 1 ͑rជ ,rជ ͒ = 1 2 . ͑104͒ 1 2 ͗␺†͑rជ ͒͗͘␺͑rជ ͒͘ ͗Eˆ ͑−͒͑r ͒Eˆ ͑+͒͑r + r͒͘ 1 2 ͑1͒͑ ͒ 0 0 g r = Inserting a plane-wave expansion ͱ͗ ˆ ͑−͒͑ ͒ ˆ ͑+͒͑ ͒͗͘ ˆ ͑−͒͑ ͒ ˆ ͑+͒͑ ͒͘ E r0 E r0 E r0 + r E r0 + r ␳ ͑ ͒ 1 ជ ជ 1 r0,r0 + r ␺͑ជ͒ ͚ ik·r ͑ ͒ = . ͑103͒ r = ͱ ͩb0 + bke ͪ, 105 ͱ␳ ͑ ͒␳ ͑ ͒ S kជ 1 r0,r0 1 r0 + r,r0 + r Interestingly, between two fragments of condensate r we find in the free-particle approximation Ϸ ␮ ͑1͒͑ ͒ 6 m apart, g r increased sharply across the thresh- ជ ͑ ͒ ik· rជ1−rជ2 old from 5% to 8% below threshold to a maximum n0 + ͚ nke ϳ ͑ ͒ kជ value of 46% at 2Pth Fig. 34 . This verifies that al- ͑1͒͑ ͒ ͑ ͒ g rជ1,rជ2 = . 106 though local disorder caused spatial fragmentation of n0 the LP gas, different fragments are phase locked with each other. With the calculated populations we can now evaluate ͑1͒͑ ͉ ͉͒ the pump power dependence of g r= rជ1 −rជ2 for various values of the distance r. We use a positive detuning 4. Spatial coherence of a single-spatial mode condensate of 4 meV and a system size of 100 ␮m in cross section. A more systematic study of spatial coherence was car- The pump power P is normalized to the threshold value ͑ ͒ ried out by Deng et al. 2007 . A GaAs-based microcav- Pth. Figure 36 shows how the range of the spatial coher- ity was used, where the LP condensate forms a single- ence increases above threshold. These results are in ex- spatial mode. A double slit was placed in the near-field cellent agreement with the double-slit experiment by image of the LPs, which passes PL from different spatial Deng et al. ͑2007͒ as shown in Fig. 37. regions of the condensate. Far-field interference be- As a next test we plot the distance dependence of tween light through the two slits was then recorded, g͑1͒͑r͒ in Fig. 38 for various values of the pump power. which consists of a cosine oscillation imposed on a sinc We see a more or less exponential decay of the coher- function envelope due to the finite slit width. The ampli- ence function. The coherence length increases with ͑ ͒ tude of the cosine oscillation corresponds to g 1 ͑r͒, r pump power, i.e., with the degeneracy of the condensed given by the slit separation. LPs. The corresponding measurements shown in Fig. 35 g͑1͒͑r͒ increases sharply across the threshold and satu- are again in quantitative agreement with the calculated Ͼ ͑1͒͑ ͒ ϳ rates at P 3Pth. The maximum g r of 0.8 was mea- values of the coherence function.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1523

͑1͒ FIG. 36. Calculated ﬁrst-order coherence function g͑1͒͑r͒ vs FIG. 38. Calculated ﬁrst-order coherence function g ͑r͒ vs distances r for various normalized pump powers P/P . From normalized pump power P/Pth for various distances r. From th Doan, Cao, et al., 2008. Doan, Cao, et al., 2008.

D. Spatial distributions microcavity was used without in-plane confinement potential. The excitation laser has a Gaussian profile with a ␻ Ϸ ␮ The presence of a condensate with a long coherence FWHM spot size p 15 m. While the LP emission length changes the spatial distribution of LPs in charac- follows the same profile below threshold, the emission ␻ teristic ways. If there is a harmonic potential, the spatial spot size PL shrank steeply when the center of the spot Ͼ distribution features a Thomas-Fermi profile, and the reached the critical density locally. At P Pth, the in- ␻ width of the distribution scales inversely proportional to crease of PL is much slower than predicted by Eq. the width in the momentum space. In the absence of ͑107͒. in-plane confinement potential, the spatial properties Taking into account the size dependence of the critical are governed by two factors: the spatial distribution due density, Eq. ͑107͒ is modified as ͑Deng et al., 2007͒ to the profile of the excitation laser and the finite exten- 1 n ͓␻͑P/P ͔͒ sion of the macroscopic coherence as a 2D system. ␻͑ ͒ ␻ ͱ ͩ c th ͪ P/Pth = p 1 − log2 1+ ͑␻ ͒ . P/Pth nc c 1. Condensate size and critical density ͑108͒ If the excitation laser has a nonuniform profile, such Here ␻ Џ␻͑P/P =1͒ is the FWHM of the condensate ␻ c th as a Gaussian with a FWHM spot size of p, the center when it first appears and corresponds to the smallest of the spot reaches the degeneracy threshold first and measured size. As a simplified model, consider n for a ␻ c forms a quasicondensate with a spatial extension c 2D boson gas confined in a finite size L=2␻ ͑Ketterle Ӷ␻ ␻ p. With increasing excitation density, c also in- ͒ ͑␻͒ ͑ ⌳2 ͒ ͑ ␻ ⌳ ͒ and van Druten, 1996 : nc = 2/ T ln 2 / T . One creases. If the critical density is constant independent of obtains ␻ the system size, c will expand as P ln͑2␻/⌳ ͒ ␻͑P/P ͒ = ␻ ͱ1 − log ͩ1+ th T ͪ. ͑109͒ ␻ ␻ ͱ ͩ Pthͪ ͑ ͒ th p 2 ͑ ␻ ⌳ ͒ = 1 − log 1+ . 107 P ln 2 c/ T c p 2 P ⌳ Here T is the LP thermal de Broglie wavelength. In two dimensions, however, a higher critical density The measured increase of the LP spot size was well de- ͑␻ ͒ of condensation nc c is required to maintain coherence scribed by Eq. ͑109͒, as shown in Fig. 39 ͑Deng et al., ͑ ͒ over a larger area Ketterle and van Druten, 1996 . This 2007͒. effect was first observed by Deng et al. ͑2003͒. A planar 2. Comparison to a photon laser In the case of a photon laser ͑Deng et al., 2003͒,a reduction in the spot size was also observed at the lasing threshold. In sharp contrast to the LP condensation, the increase of the spot size above threshold is much faster and was explained well by the classical local oscillator model of Eq. ͑107͓͒Fig. 39͑c͔͒. This confirms that in a conventional photon laser gain is determined by local density of the electron-hole pairs with a threshold density independent of the system size. Moreover, there are multiple transverse modes in the FIG. 37. ͑Color online͒ Measured first-order coherence func- spatial profile above threshold ͓Fig. 39͑b͔͒, as is typical ͑1͒͑ ͒ tion g r vs normalized pump power P/Pth for various dis- in large-area vertical cavity surface emitting lasers. Co- tances r. From Deng et al., 2007. herence in this case is present only in the photon field

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1524 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

BEC using two-photon Bragg scattering technique ͑Stamper-Kurn et al., 1999͒. The particle-particle interaction and the Bogoliubov excitation spectrum have been studied theoretically for exciton polaritons ͑Shelykh et al., 2005; Sarchi and Savona, 2008͒. Due to a short polariton lifetime, it is expected that the mode becomes diffusive and the dispersion is flat in a small momentum regime ͑Szymanska et al., 2006; Wouters and Carusotto, 2007͒. In this section, we describe the observation of interaction effects on the condensate mean-field energy and the excitation spectra, which are in quantitative agreement with the Bogoliubov theory. The slope of the linear dispersion at the small momentum regime corresponds to a superfluid critical velocity according to Lan- dau’s criterion ͑Landau and Lifshitz, 1987͒. It is eight orders of magnitude faster than that for atomic BEC ͑Stamper-Kurn et al., 1999͒. More direct evidence of polariton superfluidity came from recent observations of a quantized vortex pinned to a potential defect ͑La- goudakis et al., 2008͒ and a vortex-antivortex bound pair in a finite-size polariton BEC ͑Roumpos, Hoefling, et al., 2009͒. Different from superfluidity arising from LP-LP interactions in a condensate, a superfluid LP was also created by coherent parametric pumping ͑Amo et al., 2009͒.

A. Gross-Pitaevskii equation

The Hamiltonian of the exciton-polariton condensate FIG. 39. ͑Color͒ Spatial distributions and spot sizes of a polar- expressed in terms of the LP field operator ␺ is iton laser vs a photon laser. ͑a͒ and ͑b͒ Spatial profiles of LPs and lasing cavity mode at 1.4 times the threshold pump pow- ͑ ͒ ប2 ers, respectively. c Comparison of the expansion of the spot ˆ ˆ + ˆ H = ͵ͩ ٌ ␺ ٌ ␺ͪdr ͒ ͑ size vs pump rate for the LP condensate circles and the pho- 2m ton laser ͑stars͒. The red dashed line is fit by Eq. ͑108͒, with ␻ ␮ ␻ ␮ 1 p =16.5 m, and c =2.8 m. The green dotted lines are fit by ˆ + ˆ + ˆ ˆ ͑ ͒ ␻ ␮ + ͵ ␺ ␺Ј V͑rЈ − r͒␺␺ЈdrЈdr, ͑110͒ Eq. 107 assuming a pump spot size of p =9 m for the LP 2 ␻ ␮ condensate and p =23 m for the photon laser. From Deng et al., 2003. where V͑r͒ is the two-body interaction potential. The Heisenberg equation of motion of ␺ˆ is but not in the electronic media. In the case of polariton ץ condensation, however, a uniform Gaussian profile is maintained up to very high pump rates without obvious iប ␺ˆ ͑r,t͒ = ͓␺ˆ ͑r,t͒,Hˆ ͔ tץ .multiple transverse modes ប2ٌ2 = ͫ− + ͵ ␺ˆ +͑rЈ,t͒ VIII. POLARITON SUPERFLUIDITY 2m

The occurrence of BEC was originally predicted for ϫV͑rЈ − r͒␺ˆ ͑rЈ,t͒drЈͬ␺ˆ ͑r,t͒. ͑111͒ an ideal gas of noninteracting bosonic particles ͑Ein- stein, 1925͒. However, particle-particle interaction and resulting peculiar excitation spectra are keys for under- If the field operator is written in the form standing BEC and superfluidity physics ͑Tilley and ͒ Tilley, 1990; Pitaevskii and Stringari, 2003 . A quantum 1 ␺ˆ ͑ ͒ ip·r/ប ͑ ͒ field-theoretical formulation for a weakly interacting r = ͱ ͚ aˆ pe , 112 Bose condensed system was first developed by Bogoliu- V p bov, which predicted the phononlike linear excitation spectrum at low-momentum regime ͑Bogoliubov, 1947͒. where V is a quantization volume and aˆ p is the annihi- The experimental verification of the Bogoliubov theory lation operator for the single-particle state with a mo- on a quantitative level was first performed for atomic mentum p, Eq. ͑110͒ can be rewritten as

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1525

2 1 2 ˆ p + + + ˆ g 2 p + H = ͚ aˆ paˆ p + ͚ ͚ ͚ Vqaˆ p +qaˆ p −qaˆ p aˆ p , H = N + ͚ aˆ paˆ p 2m 2V 1 2 1 2 2V 2m p p1 p2 q p 0 ͑113͒ 1 + gn ͚ ͑2aˆ +aˆ + aˆ +aˆ + + aˆ aˆ ͒. ͑121͒ ͐ ͑ ͒ ͓ ប͔ p p p −p p −p where Vq = V r exp −iq·r/ dr is the Fourier transform 2 p0 of V͑r͒. The above Hamiltonian can be diagonalized by the lin- As far as the macroscopic properties of the conden- ear transformation ͑Bogoliubov, 1947͒ sate system are concerned, only small momenta are in- volved so that we are allowed to consider the q=0 value ˆ ˆ * ˆ + ˆ + * ˆ + ˆ ͑ ͒ ap = upbp + v−pb−p, ap = upbp + v−pb−p. 122 of Vq, If the two coefficients up and v−p satisfy ͵ ͑ ͒ ͑ ͒ ͉ ͉2 ͉ ͉2 ͑ ͒ V0 = g = V r dr, 114 up + v−p =1, 123 ˆ ˆ + and to rewrite the Hamiltonian in the form a new set of operators bp and bp satisfies the bosonic commutation relation ͓bˆ ,bˆ + ͔=␦ . In order to make p2 g p pЈ ppЈ ˆ + + + + + H = ͚ aˆ aˆ p + ͚ ͚ ͚ aˆ aˆ aˆ p aˆ p . ˆ ˆ ˆ ˆ 2m p 2V p1+q p2−q 1 2 the non-number conserving terms bpb−p and bpb−p in p p1 p2 q ͑ ͒ Eq. 121 vanish, the coefficients up and v−p must satisfy ͑115͒ U͑n͒ p2 ͉͑ ͉2 ͉ ͉2͒ ͩ ͑ ͒ͪ ͑ ͒ According to the Bogoliubov prescription, we replace up + v−p + + U n upv−p =0. 124 2 2m the operator aˆ 0 for the condensate with a c number, From Eqs. ͑123͒ and ͑124͒, we can solve u and v as ϵ ␣ ͑ ͒ p −p aˆ 0 , 116 follows: ͑ ͒ in Eq. 115 . The corresponding c-number field ampli- p2/2m + U͑n͒ 1 ͑ ͒ ␺ ͑ ͒ ͱ ͑ ͒ tude order parameter 0 r,t varies slowly over dis- up = ␧͑ ͒ + , 125 tances comparable to the range of the interparticle p 2 force, thus we substitute rЈ for r in Eq. ͑111͒ and obtain p2/2m + U͑n͒ 1 the Gross-Pitaevskii equation, v =−ͱ − , ͑126͒ −p ␧͑ ͒ 2 ប2ٌ2 p ץ iប ␺ ͑r,t͒ = ͩ− + g͉␺ ͑r,t͉͒2ͪ␺ ͑r,t͒. ͑117͒ t 0 2m 0 0 whereץ U͑n͒ p2 2 ␧͑p͒ = ͱ p2 + ͩ ͪ . ͑127͒ m 2m B. Bogoliubov excitation spectrum With this linear transformation, the Hamiltonian ͑121͒ is reduced to the diagonal form In a dilute and low-temperature limit, the occupation number for excited states with p0 is finite but small. In ˆ ␧͑ ͒ ˆ + ˆ ͑ ͒ H = E0 + ͚ p bpbp. 128 this first-order approximation, we neglect all terms in p0 Eq. ͑115͒ containing aˆ and aˆ + with p0. Then the p p ␧͑p͒ is the well-known Bogoliubov dispersion law for the ground-state energy is elementary excitations from the condensate system. g 2 ͑ ͒ E0 = N . 118 2V C. Blueshift of the condensate mean-field energy The corresponding chemical potential or mean-field en- The k=0 LP mean-field energy is blueshifted with in- ergy shift is given by creasing number of LPs as shown in Fig. 40. This is a ץ E0 direct manifestation of the repulsive interaction among U͑n͒ϵ = gn, ͑119͒ ͑ ͒ .N LPs. The k=0 LP energy shift U n is calculated by Eqץ ͑39͒. If we assume a constant size of the condensate, the where n=N/V is the particle density. theoretical energy shift is shown by the light blue line in In order to obtain the excitation spectrum and study Fig. 40. We can numerically solve the Gross-Pitaevskii the quantum fluctuation, however, this level of approxi- ͑GP͒ equation ͓Eq. ͑117͔͒ to incorporate the condensate mation is insufficient. We have to work with higher ac- expansion due to repulsive interaction. The results are curacy using the normalization relation shown by red dots in Fig. 40. These two theoretical pre- dictions are compared to the experimental results ͑blue + + ͉␣͉2 + ͑ ͒ aˆ 0aˆ 0 + ͚ aˆ paˆ p = + ͚ aˆ paˆ p = N. 120 diamonds͒. Note that the above mentioned nonlinear p 0 p 0 model based on weakly interacting bosons ͑Schmitt- Substituting Eq. ͑120͒ into ͑115͒, we obtain the new Rink et al., 1985; Ciuti et al., 1998; Rochat et al., 2000͒ Hamiltonian can reproduce the experimental data only in low LP

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1526 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

Popov theory Numerical model Bogolubov theory Experiment

FIG. 40. ͑Color͒ The measured mean-field energy shift vs total number of polaritons ͑dark blue diamond͒. The analytical ͑blue solid line͒ and numerical ͑red dot͒ results based on Bo- goliubov approximation ͑Utsunomiya et al., 2008͒. The analytical result based on Popov approximation ͑pink solid line͒ is provided by Savona et al. ͑private communication͒. FIG. 41. ͑Color͒ Polarization dependence of the excitation density regimes. Recently the Popov approximation is spectrum for an untrapped condensate system. ͑a͒ A linear applied to the LP system and quantitative agreement plot of the intensity, while ͑b͒–͑d͒ employ three-dimensional with the experimental data on U͑n͒ is obtained as shown logarithmic plots of the intensity to magnify the excitation by the pink solid line ͑Sarchi and Savona, 2008͒. We use spectra. A circularly polarized pump beam was incident with ͑ ͒ ͑ ͒ the experimental values ͑not theoretical values͒ for U͑n͒ an angle of 60°. Three detection schemes: a and b detection of the leakage photons with the cocircular polarization as the as the interaction energy in subsequent discussions for ͑ ͒ the universal feature of the Bogoliubov excitations. pump beam, c detection of the cross-circular polarization, and ͑d͒ detection of a small amount of mixture of the cocircu- D. Energy vs momentum dispersion relation lar polarization with the cross-circular polarization. The theoretical curves represent the Bogoliubov excitation energy EB ͑ ͒ Ј ͑ ͒ Using a microcavity with a very long cavity-photon pink line , the quadratic dispersion relations ELP black line lifetime of 80 ps, Roumpos, Hoefling, et al. ͑2009͒ mea- which starts from the condensate energy, and the noninteracting free LP dispersion relation E ͑white line͒ which is experi- sured again the excited-state energy E dispersion versus LP mentally determined by the data taken far below the threshold in-plane wave number k, as shown in Fig. 41 for a pump ϳ ␮ P=0.001Pth. From Utsunomiya et al., 2008. rate P/Pth=3. The pump spot size is 30 m in diameter. The LP condensate is formed in an area determined by the pump spot size and the pump rate due to with the Bogoliubov excitation spectrum without any fit- the limited lateral diffusion and varying spatial density ting parameter. ͑ ͒ of the LPs Deng et al., 2007; Lai et al., 2007 . Figure A circularly polarized pump-laser beam was used to ͑ ͒ 41 a represents a linear plot of the luminescence inten- inject spin-polarized LPs in this experiment. In this case, ͑ ͒ ͑ ͒ sity, while Figs. 41 b –41 d employ logarithmic plots of the LP condensate preserves the original spin polariza- the intensity to magnify the excitation spectra. Above tion of optically injected LPs as described in Sec. VIII.C threshold, two drastic changes are noticed compared to ͑Deng et al., 2003; Roumpos, Hoefling, et al., 2009͒. The the standard quadratic dispersion observed below result shown in Fig. 41͑b͒ was taken for detecting the threshold. One is the blueshift of the k=0 LP energy as leakage photons with the same circular polarization as discussed in Sec. VIII.C and the other is the phononlike that of the pump ͑cocircular detection͒. A small amount linear dispersion relation at low-momentum regime of cross-circularly polarized photons was also detected ͉ ␰͉Ͻ ␰ ប ͱ ͑ ͒ k 1, where = / 2mU n is the healing length. because of spin relaxation during the cooling process ͑ ͒ White and black lines in Fig. 41 b represent the two ͑Deng et al., 2003; Roumpos, Hoefling, et al., 2009͒. ͑ ͒ ͑ប ͒2 quadratic dispersion relations, ELP=−U n + k /2m However, the standard quadratic dispersion was ob- Ј ͑ប ͒2 and ELP= k /2m, where m is the effective mass of the served but with slightly lower energy, compared to the ͑ k=0 LP. Here we choose the zero energy as the blue- condensate energy, as shown in Fig. 41͑c͒. With increas- ͒ shifted condensate mean-field energy for convenience. ing pump rate, the number of the condensed LPs is in- Neither of the two theoretical curves can explain the creased so that the weak attractive interaction between ͑ ͒ measurement result. A solid pink line in Fig. 41 b is the condensate and excited states with the opposite spin Bogoliubov excitation energy ͑127͒, which can be rewrit- Ј ͑ ͒ ͑ ͒ results in a slight redshift compared to ELP black line ten as Pitaevskii and Stringari, 2003; Ozeri et al., 2005 in Fig. 41͑c͒. In Fig. 41͑d͒ we simultaneously detected a ͱ Ј ͓ Ј ͑ ͔͒ ͑ ͒ͱ͑ ␰͒2͓͑ ␰͒2 ͔ small amount of the cocircular polarized photons with EB = ELP ELP +2U n = U n k k +2 . ͑ ͒ the cross-circular polarized photons. The difference be- 129 tween the Bogoliubov excitations with cocircular polar- The measured dispersion relation for the excitation en- ization and the standard but redshifted quadratic excita- ergy versus in-plane wave number is in good agreement tion with cross-circular polarization is clearly seen.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1527

FIG. 42. ͑Color online͒ Excitations of a condensate. ͑a͒ False color contour plot of the density of states for elementary excitations of a ﬁnite-size polariton condensate ͑Utsunomiya et al., 2008͒. ͑b͒ The dispersion of elementary excitations of a dissipative polariton condensate ͑Wouters and Carusotto, 2007͒.

As shown in Fig. 41͑b͒, the condensate at an energy Ј ELP stretches to a finite wave number k 0, which is not expected from the Bogoliubov dispersion law ͓Eqs. ͑127͒ and ͑128͔͒. This experimental result has two possible ex- FIG. 43. ͑Color͒ Universal features of Bogolinbor excitations. planations. One explanation is the Heisenberg uncer- ͑a͒ Excitation energy normalized by the interaction energy tainty relationship between the position and the momen- E/U͑n͒ as a function of normalized wave number k␰ for four tum of condensate particles. The spatial extent of the different untrapped condensate systems. A ͑blue dot͒: ⌬ ͑ ͒ ͑ ͒ ⌬ polariton condensate is determined by the pump spot =1.41 meV, P=4Pth Pth=6.3 mW ;B light blue dot : ͑ ͒ ͑ ͒ ⌬ size and the size-dependent critical density ͑Deng et al., =0.82 meV, P=8Pth Pth=8.2 mW ;C red dot : =4.2 meV, ͑ ͒ ͑ ͒ ⌬ 2003͒, as discussed in Sec. VII. It is expected that the P=4Pth Pth=6.4 mW ; and D pink dot : =−0.23 meV, P ͑ ͒ condensate has a finite momentum uncertainly ⌬p when =24Pth Pth=8.2 mW . The experimental data far below threshold are also plotted by blue crosses for system A. Three it is confined in a finite space region. Indeed, the mea- ⌬ theoretical dispersion curves normalized by the interaction en- sured product of the spot size x and the spread in mo- ͑ ͒ ergy are plotted; the Bogoliubov excitation energy EB /U n mentum ⌬p is close to the minimum allowable value, ͑ ͒ Ј ͑ ͒ green solid line , the quadratic dispersion curves ELP/U n ⌬x⌬p=ប/2, within a factor of 2–4 above the threshold ͑ ͒ ͑ ͒͑ gray solid line , and the free LP dispersion ELP/U n red ͑ ͒ ͒ ͑ ͒ ͑ ͒ Utsunomiya et al., 2008 . It is also shown by the spa- solid line . b and c The energy shift EB −ELP in the free- tially inhomogeneous Gross-Pitaevskii equation that the particle regime ͉͑k␰͉=1͒ plotted as a function of the interaction density of states for the excitation spectrum disappears energy U͑n͒ for the same four different untrapped systems as in a small momentum regime ͉⌬p͉ഛប/2⌬x as shown in in ͑a͒ and ͑b͒ and four different trapped systems ͑c͒, where ͑ ͒ Ј trapped condensate systems are labeled as below: E ͑blue Fig. 42 a , for which the condensate at ELP fills in the vacant momentum regime ͑Utsunomiya et al., 2008͒. circle͒: d͑diameter͒=7 ␮m, ⌬=3.3 meV; F ͑light blue circle͒: The other possible explanation is the finite lifetime of d=7 ␮m, ⌬=2.9 meV; G ͑red circle͒: d=8 ␮m, ⌬=1.6 meV; ͑ ͒ ␮ ⌬ condensate particles. Using a generalized Gross- and H pink circle : d=8 m, =2.5 meV. The dashed line ͑ ͒ ͑ ͒ represents the theoretical prediction EB −ELP=2U n . d LP Pitaevskii equation with polariton loss and gain terms, it ␰ can be found that the dispersion of elementary excita- population distribution normalized by the value at k =0.3 for the trapped system G ͓in ͑c͔͒ at the pump rate; P=2P ͑P tions around the condensate becomes diffusive, i.e., is th th =4 mW͒. Theoretical 1/k2 dependence for the thermal deple- flat in a small momentum regime and then rises up to tion is shown by the blue line and 1/k dependence for the approach the Bogoliubov spectrum ͑superfluidity- ͒ ͑ ͒ ͑ quantum depletion is shown by the blue dotted line. From diffusion crossover , as shown in Fig. 42 b Szymanska Utsunomiya et al., 2008. et al., 2006; Wouters and Carusotto, 2007͒. It is not well understood at present which of the above two factors, a finite condensate size or a finite condensate lifetime, is versal curve except for the region of small momenta mainly responsible for the observed flat dispersion near ͉k␰͉Ӷ1 as mentioned above. On the other hand, at a k=0. pump rate far below threshold, the measured dispersion relation is completely described by the single LP disper- ͑ ͒ E. Universal features of Bogoliubov excitations sion ELP red solid line . The sound velocity deduced from the phononlike lin- As indicated by Eq. ͑129͒, the Bogoliubov excitation ear dispersion spectrum is of the order of ϳ108 cm/s energy normalized by the mean-field energy shift and is proportional to the square root of the condensate ͑ ͒ ͑ϰͱ ͒ EB /U n is a universal function of the wave number nor- population n0 at reasonable pump rates. This value malized by the healing length k␰. In Fig. 43͑a͒, this uni- is eight orders of magnitude larger than that of atomic versal relation ͑green solid line͒ is compared to the ex- BEC and four orders of magnitude larger than that of perimental results for four different condensate systems superfluid 4He. This enormous difference comes from with varying detuning parameters. In both the phonon- the fact that the LP mass is nine to ten orders of magni- like regime at ͉k␰͉Ͻ1 and the free-particle regime at tude smaller than the atomic mass and the LP interac- ͉k␰͉Ͼ1, the experimental results agree well with the uni- tion energy is six to seven orders of magnitude larger

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1528 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

than the atomic interaction energy. According to Land- this case, the LP population is given by nk ͑ ͒ Ϸ ͑ប ͒2 ͉ ␰͉Ӷ Ϸ͑ ͱ ͒ ␰ au’s criterion Landau and Lifshitz, 1987 , the observa- mkBT/ k at k 1 regime, while nk 1/2 2 /k tion of this linear dispersion at low-momentum regime is if the quantum depletion is dominant ͑Pitaevskii and an indication of superfluidity in the exciton-polariton Stringari, 2003͒. The theoretical prediction of the 1/k2 system. It has recently been reported that the scattering dependence of nk for thermal depletion is compared to rate for the moving exciton-polariton condensate is dra- the experimental data in Fig. 43͑d͒ and reasonable matically suppressed when the initial velocity given to agreement was obtained. the polariton system is lower than this critical velocity ͑Amo et al., 2009͒. It was also observed that the Bogo- liubov theory does not fit to the measured dispersion of G. Quantized vortices elementary excitations at very high pump rates. This is an expected result because the Bogoliubov theory is A hallmark of superfluidity is quantized vortices. In a based on the weakly interacting Bose gas and this ap- BEC phase, vortices appear when finite angular momen- proximation is violated at a pump rate far above thresh- tum is transferred to the condensate. A uniform 2D sys- old. tem of bosons does not undergo BEC at finite tempera- In the free-particle regime ͉͑k␰͉Ͼ1͒, the excitation en- tures. However, such a system turns superfluid in the ͑ ͒ ergy associated with the condensate is larger by 2U͑n͒ BKT phase see discussion in Sec. II.D.1 . Elementary than that of a single LP energy for the same wave num- excitations of this superfluid phase are bound pairs of ber. A factor of 2 difference between the mean-field en- vortices with opposite winding numbers. The BKT to ergy shifts of the condensate and excited states origi- normal-fluid-phase transition is identified by the disso- ͑ nates from a doubled exchange interaction energy ciation of the vortex pair into free vortices Hadzibabic ͒ among different modes. The fact was already manifested et al., 2006; Posazhennikova, 2006 . in the kink observed in Fig. 40. In Figs. 43͑b͒ and 43͑c͒, A quantized vortex of LPs was first observed in a this important prediction of the Bogoliubov theory is CdTe microcavity. The vortex seemed to be created dur- compared to the experimental results for four different ing the thermalization process and was pinned by a crys- ͑ untrapped and trapped condensate systems, respectively. tal defect induced potential fluctuation Lagoudakis et ͒ The experimental data were determined as the differ- al., 2008 . A fixed winding direction of the observed vor- ence between the measured excitation energy with the tex was attributed to a specific potential landscape. presence of the condensate and the standard quadratic Recently vortex-antivortex pairs were observed in a ͑ dispersion for a single LP state, which is determined by finite-size GaAs microcavity Roumpos, Hoefling, et al., ͒ the experimental data obtained for a pump rate far be- 2009 . According to the numerical analysis of the Gross- ͑ Ӷ ͒ Pitaevskii equation, a vortex-antivortex pair in the BKT low threshold P/Pth 1 . The experimental data are in good agreement with the theoretical curve ͑gray dashed topological order moves in parallel with a velocity close line͒ for both untrapped and trapped cases. to the sound velocity of the Bogoliubov excitation spectrum if the 2D system is uniform. If the 2D system is trapped to a finite size, the vortex-antivortex pair is also F. Thermal and quantum depletion trapped with a residual micromotion inside a trap ͑Fraser et al., 2009͒. The experiment of Roumpos, ͑ ͒ Figure 43 d shows the normalized excitation popula- Hoefling, et al. ͑2009͒ observed this behavior. In the ex- 0 ͉ ␰͉ tion nk /nk versus the normalized wave number k for a periment, the ensemble-averaged 2D distributions of the 0 ͉ ␰͉ trapped condensate, where nk is evaluated at k =0.3 population n͑r͒, phase ␾͑r͒, and first-order spatial coher- ͑1͒ for convenience. The LP occupation number nk in the ence function g ͑r͒ are measured simultaneously by a excitation spectrum can be calculated by applying the Michelson interferometer. The existence of a vortex- Bose-Einstein distribution for the Bogoliubov quasipar- antivortex pair is identified by the population dip, ␲ ticles and subsequently taking the inverse Bogoliubov phase shift, and reduced spatial coherence at the bound- transformation ͑Pitaevskii and Stringari, 2003͒, ary of the pair. These experimental results are success- ͉u ͉2 + ͉v ͉2 fully reproduced by the numerical analysis of the Gross- ͉v ͉2 k −k ͑ ͒ Pitaevskii equation with a gain-loss term. nk = −k + ͑␤ ͒ , 130 exp EB −1 where IX. POLARITON CONDENSATION IN SINGLE TRAPS ͑បk͒2/2m + U͑n͒ 2 1/2 AND PERIODIC LATTICE POTENTIALS u ,v =±ͫ ± ͬ k −k 2E 1 B Most of the experiments we discussed in this review ␤ and =1/kBT. The first and second terms on the right- used planar microcavities without an intentional in- hand side of Eq. ͑130͒ represent the real particles ͑LPs͒ plane trapping potential. The size of the system is created by the quantum depletion and the thermal largely determined by the spot size of the pump laser. In depletion, respectively. In the present LP condensate comparison, a trapped polariton system may offer sev- system, the thermal depletion is much stronger than the eral advantages. With a trapping potential, genuine BEC quantum depletion so that the second term on the right- becomes possible in two dimensions, and spatial conden- hand side of Eq. ͑130͒ dominates over the first term. In sation accompanies the configuration-space condensa-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1529

the LP position ⌬x and wave number ⌬k are plotted as a ͑ ͒ function of P/Pth in Fig. 44 b . The sudden decreases in ⌬ ⌬ Х x and k were clearly observed at threshold P Pth. Just above threshold, the measured uncertainty product ⌬x⌬k is ϳ0.98, which is compared to the Heisenberg limit ͑⌬x⌬kϳ0.5͒ for a minimum uncertainty wave packet. Here ϳ103 LPs condense into a nearly minimum uncertainty wave packet at threshold. The monotonic ⌬ ⌬ ⌬ FIG. 44. ͑Color online͒ Polariton condensation in an optical increase in x and x k at higher pump rates stems trap by metal masks. ͑a͒ A schematic of a polariton trap. ͑b͒ from the repulsive interaction among LPs in a conden- The measured standard deviations of LP distribution in coor- sate and is well reproduced by the numerical analysis dinate ⌬x ͑circles͒ and in wave number ⌬k ͑crosses͒ are plot- using the spatially inhomogeneous GP equation as ⌬ ⌬ ͑ ͒ ted as a function of P/Pth. Theoretical values for x and k shown by solid lines in Fig. 44 b . obtained by the GP equation are shown by the solid lines ͑Ut- ͒ sunomiya et al., 2008 . 2. Optical traps by fabrication Strong optical confinement, hence discrete polariton tion. A strong trapping potential may lead to discrete modes, has been realized by modifying the microcavity polariton modes, hence single-mode BEC. Signatures of structure. Earlier efforts focused on etching small pillar BEC should be more pronounced in all these cases. The structures out of a planar microcavity to achieve strong trap can also serve as a tuning knob for the system. With transverse confinement of the photon modes. In pillars a lattice potential, coherence and interactions among of a few microns across, discrete polariton modes ͑Bloch multiple condensates can be studied. In this section, we et al., 1998; Gutbrod et al., 1998; Obert et al., 2004͒ and review the efforts to implement single and lattice poten- polariton parametric scattering ͑Dasbach et al., 2001͒ tials for polaritons. were observed. In very small pillars, leakage of the cavity-photon fields and loss of excitons at the side walls A. Polariton traps prevent strong coupling. Alternatively, in a similar spirit as the metal masks, ͑ ͒ A single isolated trap for polaritons is implemented Daif et al. 2006 devised a way to grow small regions ͑ by modifying either the cavity photon or the QW exci- with thicker cavity layers and lower photon and thus ͒ ton energies. polariton energies. Compared to using metal masks, the energy difference between inside and outside the trap region is large enough to discretize the polariton modes. 1. Optical traps by metal masks Compared to the etching method, the QW excitons are A rather versatile type of polariton trap is based on untouched and will not suffer surface recombination; modulating the cavity layer thickness and transmission transport and interactions are possible between un- with a thin metal mask deposited on the surface of the trapped and trapped polaritons. Discrete polariton spec- microcavity ͑Lai et al., 2007; Kim et al., 2008; Ut- tra were observed in such a structure, in agreement with sunomiya et al., 2008͒. calculations ͑Kaitouni et al., 2006͒. As shown in Fig. 44͑a͒, a trap potential of ϳ200 ␮eV is provided by a hole surrounded by a thin metal 3. Exciton traps by mechanical stress ͑Ti/Au͒ film ͑Lai et al., 2007; Utsunomiya et al., 2008͒. A polariton trap with a relatively larger area has been The cavity resonant field has normally an antinode at formed by mechanically pressing a planar microcavity the AlGaAs-air interface. However, the antinode posi- wafer ͑Balili et al., 2006͒. The mechanical stress intro- tion is shifted inside the AlGaAs layer with the metal duces a strain field which acts as a harmonic potential on film as shown in Fig. 44͑a͒, which results in the blueshift the QW excitons ͑Negoita et al., 1999͒ and thus on the of the cavity photon and subsequently LP resonances. polaritons as well. A potential up to about 5 meV deep The LPs were confined in circular holes of varying diam- ␮ eters from 5 to 100 ␮m. In a trap with 5–10 ␮m diam- was created by a stressing tip of about 50 m in radius. LPs excited away from the trap center drifted over more eter, a single fundamental transverse mode dominates ␮ ͑ ͒ the condensation dynamics over higher-order transverse than 50 m into the trap Balili et al., 2006 . Condensa- modes due to the relatively weak confining potential. A tion of LPs in the trap was also observed, with a thresh- weak confining potential helps to cool polaritons effi- old density lower than that required when without the trap and with a contracted spatial profile ͑Balili et al., ciently because high-energy polaritons created by two- ͒ body scattering are quickly expelled from the trap re- 2007 . gion. The top panels in Fig. 44͑b͒ show the near-field- B. Polariton array in lattice potential emission patterns from a trap with 8 ␮m diameter at pump rates below, just above, and well above condensa- Extending a single trap potential to a lattice potential, tion threshold. The measured standard deviations for the long-range spatial coherence leads to the existence

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1530 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

͑ metallic layer, the cavity resonance energy Ecav for kʈ =0͒ is expected to increase by ϳ400 ␮eV according to the transfer-matrix analytical method ͑Yeh, 2005͒. When ͑ ͒ the cavity photon Ecav is near resonance with the QW ͑ ͒ exciton Eexc at kʈ =0, the shift of the LP energy induced by the metallic layer is ϳ200 ␮eV, approximately one- half of the cavity-photon resonance shift ͓Fig. 45͑b͔͒. Therefore, the LPs are expected to be trapped in the gap region where the LP energy is lower. The measured spatial modulation of LP energy is ϳ100 ␮eV, which is less than the theoretical prediction due to the diffraction- limited spatial resolution ͑ϳ2 ␮m͒ of our optical detection system ͓Fig. 45͑c͔͒. The microcavity is excited by a mode-locked Ti:sap- phire laser with an ϳ2.5 ps pulse near the QW exciton resonance at an incident angle of 60°, corresponding to 4 −1 an in-plane wave number kʈ Ϸ7ϫ10 cm in air. The

large kʈ of the pump assures that the coherence of the pump laser is lost by multiple phonon emissions before

the LPs scatter into kʈ Ϸ0 states. Considering a periodic and coherent array of condensates aligned along the x axis, we can approximate the ͑ ͒ ͑ ͒ FIG. 45. Color online Formation of a polariton array. a A order parameter in momentum space as ͑Pedri et al., schematic of a polariton array formed by depositing periodic 2001͒ thin metallic strips ͑Au/Ti͒ on top of a microcavity structure. ͑ ͒ ͑ ͒ b Dispersion curves of the cavity-photon mode and LP. c ͑ ␾͒ ␺͑ ͒ ␺ ͑ ͒ in kxa+ Spatial LP energy modulation is detected from the position- kx = 0 kx ͚ e dependent central energy of LP emissions. From Lai et al., n=0,±1,...,±nM 2007. sin͓Nk a/2͔ ␺ ͑ ͒ x ͑ ␾ ͒ ͑ ͒ = 0 kx ͓ ͔ if =0 , 131 sin kxa/2 of phase-locked multiple condensates, such as in an array of superfluid helium ͑Davis and Packard, 2002͒, su- where n labels the condensate array element, N=2nM perconducting Josephson junctions ͑Hansen and Linde- +1 is the total number of periodic array elements, a is lof, 1984; Hadley et al., 1988; Cataliotti et al., 2001͒, and the pitch distance between two neighboring elements, ␺ ͑ ͒ atomic BECs ͑Anderson and Kasevich, 1998; Cataliotti and 0 kx is the momentum-space wave function of an et al., 2001; Orzel et al., 2001; Greiner et al., 2002͒. Under individual element. The overall momentum distribution ␳͑ ͒ ͉␺͑ ͉͒2 certain circumstances, a quantum phase difference of ␲ kx = kx reflects the coherence properties and na- is predicted to develop among weakly coupled Joseph- ture of the condensate arrays through distinctive inter- son junctions ͑Bulaevskii et al., 1977͒. Such a metastable ference patterns. In the presence of the periodic lattice, ␲ state was discovered in a weak link of superfluid 3He, the momentum distribution displays narrow peaks at ϫ ␲ ͑ ෈ ͒ which is characterized by a p-wave order parameter kx =m 2 /a m integer with an envelope function ͉␺ ͑ ͉͒2 ͑Backhaus et al., 1998͒. Possible existence of such a ␲ given by 0 kx . If all the elements are locked in phase state in weakly coupled atomic BECs has also been pro- ͑␾=0͒, at a LP wavelength ␭=780 nm and a grating posed ͑Smerzi et al., 1997͒ but remains undiscovered in pitch distance a=2.8 ␮m, a Fraunhofer diffraction pat- atomic systems. Recently Lai et al. ͑2007͒ observed tern has a central peak at ␪=0° and two side lobes at ␪ −1 spontaneous buildup of in-phase ͑zero-state͒ and an- =sin ␭/aϷ ±16° ͑kʈ =±2␲/a, m= ±1 first-order diffrac- tiphase ͑␲-state͒ coherent states in a polariton conden- tion͒. The LP emissions from a polariton condensate ar- sate array connected by weak periodic potential barri- ray are expected to display such an interference pattern ers. These states reflect the band structure of the one- in momentum space, just like in an atomic BEC experi- dimensional periodic potential for polaritons and ment ͑Anderson and Kasevich, 1998; Cataliotti et al., feature the dynamical competition between metastable 2001; Orzel et al., 2001; Greiner et al., 2002͒. and stable polariton condensates. The observed LP distributions in both coordinate and momentum space of the condensate array are shown in Figs. 46͑a͒ and 46͑b͒. Below the condensation threshold, 1. One-dimensional periodic array of polaritons the near-field image reveals the cigar-shaped LP emis- Employing the same metal deposition technique de- sions through the 1.4-␮m-wide gaps between the metal- scribed in Sec. IX.B, an array of one-dimensional cigar- lic strips. The corresponding LP momentum distribution shaped polaritons is created by the deposition of peri- is broad and isotropic, independent of the periodic odic strips of a metallic thin film on the top surface of a modulation of LP population and the elliptically shaped microcavity structure, as shown in Fig. 45͑a͒. Under the pumping spot. The result indicates that there is negli-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1531

FIG. 46. ͑Color͒ Near-field and far-field images of polaritons in a one-dimensional array. ͑a͒ Near-field images showing the LP distribution across a polariton array in coordinate space under pumping powers of 20, 70, and 200 mW ͑left to right͒. The FIG. 47. ͑Color online͒ Energy-momentum dispersions and threshold pumping power is about 45 mW. ͑b͒ Corresponding real space wave functions of polaritons in a one-dimensional far-field images showing the LP distribution in momentum array. ͑a͒ Time-integrated energy vs in-plane momentum of a space. From Lai et al., 2007. Ϸ polariton array at near resonance Ecav Eexc. The pumping power is P=10 mW ͑below threshold͒. ͑b͒ Extended-zone gible phase coherence among different cigar-shaped scheme of the band structure for the polariton array under a LPs. With an increasing pumping rate, two strong and weak periodic potential with a lattice constant a. ͑c͒ Energy vs two weak side lobes emerge at ␪Ϸ ±8° and ±24°, which in-plane momentum of a polariton condensate array at blue Ϸ ͑ ͒ ͑ ͒ detuning Ecav−Eexc 6 meV above threshold P=40 mW . d correspond to kʈ =±␲/a and ±3␲/2a, respectively. The particular diffraction pattern suggests that the phase dif- Schematic of the Bloch wave functions for states labeled as A–C in ͑b͒. From Lai et al., 2007. ference of adjacent condensates in the array is locked exactly to ␲ ͓i.e., ␾=␲ in Eq. ͑131͔͒. Moreover, the cor- ␲ ͑ responding near-field image ͓Fig. 46͑a͔͒ reveals that kʈ =G0 /2= /a one-half of the primitive reciprocal- ͒ ប2͑ ͒2 Ϸ ␮ strong LP emission is generated from under the metallic lattice vector , Ek = G0 /2 /2m* 500 eV, where the strips rather than from the gaps. The dramatic contrast ϫ −5 LP effective mass m* =9 10 me. We deduce the band is better observed near the boundary of the central con- structure of the polariton array assuming a “nearly free densate and the outside thermal LP emissions, which polariton” in the presence of the periodic square-well come through the gaps. The dominant LP emission potential. Given a one-dimensional periodic potential through the metallic layers despite the lower transmis- U͑X͒ with a lattice constant a, the band structure can be sion indicates that LP condensates are strongly localized obtained using the standard Bloch wave-function for- under the metallic strips. With a further increasing malism. Only those LPs with in-plane momentum close pumping rate, the intensity of the central peak near ␪ ͑ ෈ ͒ to the Bragg condition kʈ =mG0 /2, m integer are sub- =0°, with weak side lobes at ␪Ϸ ±16°, gradually sur- ject to backscattering due to the periodic potential, and passes the peaks at ␪Ϸ ±8° and ±24° ͓Fig. 46͑b͔͒. Simul- gaps appear at these Bragg planes where standing waves taneously, the dominant LP emission is observed are formed. Similar to the standard extended-zone through the gaps. scheme ͑Madelung, 1996͒, the band structure can be These two distinct interference patterns suggest the constructed by starting with original and displaced pa- transition from an antiphase state localized under the ͑ ͒ ͑ ͒ rabolas ELP kʈ ±mG0 as shown in Fig. 47 b . Here the metallic strips to an in-phase state localized in the gaps. size of from thermal population of LPs. This is well re- Both the zero and ␲ state can be observed for a polar- produced in the observed LP energy versus in-plane mo- iton condensate array consisting of more than 30 strips mentum with a below threshold pumping rate ͓Fig. of condensates ͑total array dimension ϳ100 ␮m͒. These 47͑a͔͒. The first band gap is on the order of the barrier ͉ ͉Ӎ ␮ collective states are manifestations of the long-range potential, U0 200 eV, between the first and second spatial coherence as well as phase locking across the ar- bands near the zone boundaries ͑Bragg planes͒. This ray. small gap is masked in the measured spectra by the large inhomogeneous broadening of the LP emission lines ͑linewidth ϳ500 ␮eV͒. ␲ 2. Band structure, metastable state, and stable zero-state In Fig. 47͑c͒, the energy versus in-plane momentum of Figure 47͑a͒ shows the energy versus in-plane momen- a polariton condensate array above threshold and Ecav Ϸ tum dispersion relation for a one-dimensional LP array −Eexc ±6 meV is shown. Above threshold, LP emis- Ϸ near resonance Ecav Eexc and below threshold. From sions occur in two states with an energy difference of the dispersion curve, we deduce the LP kinetic energy at about 1 meV. These two states with emission peaks at

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1532 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation

kʈ =0,±G0 and at kʈ =±G0 /2,±3G0 /2 correspond to the aforementioned zero and ␲ state, respectively. On referring to the multivalley band structure ͓Fig. 47͑b͔͒, the dynamic condensation process of the polariton array becomes transparent. Without the spatial modulation, the quasistationary state of the LP conden-

sate is the lowest-energy state at kʈ =0. Here the bottom of LP dispersion parabola serves as a trap of LPs in momentum space. When the spatial potential modulation is introduced, a metastable dynamic condensate can FIG. 48. ͑Color online͒ Momentum distribution profiles of the occur near the bottom of the second band ͓point A in ␲ phase connected p state and zero phase connected s state of Fig. 47͑b͔͒ when the pumping rate is just above thresh- polaritons in a one-dimensional array. ͑a͒ Profiles of the “␲ old. In these points, LPs experience a relaxation bottle- state” as a function of in-plane momentum taken at a cross neck, resulting in metastable condensates. Eventually, section in the dispersion curve ͑inset͒ for a polariton conden- the LP system will relax to the lowest-energy state ͓point sate array for various pumping rates. ͑b͒ Profiles of the “zero C in Fig. 47͑b͔͒ as the pumping rate is increased. state.” The intensity of the zero state surpasses that of the ␲ To understand the spatial distributions of LPs in the state at around P=45 mW. From Lai et al., 2007. array ͓near-field images shown in Fig. 46͑a͔͒, the Bloch wave functions must be considered for the above two LPs with large in-plane wave number kʈ are optically states. The Bloch wave functions for these eigenstates injected into the system, they cool down to the meta- exhibit not only opposite relative phase between adja- stable ␲ state first by continuously emitting phonons or cent elements in the array but also different orbital wave via LP-LP scattering. If the decay rate of the ␲ state is functions. Schematic Bloch wave functions for selected slower than the loading rate into the ␲ state, the popu- eigenstates labeled as A–C in Fig. 47͑b͒ are shown in lation of the ␲ state exceeds 1 ͑quantum degeneracy Fig. 47͑d͒. The zero state at point C carries to the s-like condition͒ and induces the bosonic final-state stimulated wave with maximal amplitude in potential wells and scattering into this mode. This is similar to a so-called identical phase between adjacent elements across the ar- bottleneck condensation effect ͑Tartakovskii et al., 2000; ray. The wave functions of A and B at kʈ =G0 /2 both ͒ ␲ Huang et al., 2002 . Here the LPs with finite k dynami- exhibit a relative phase difference between adjacent cally condense due to the slow relaxation rate into the elements. The lower energy unstable state at point B kϷ0 region, where the density of states is reduced. corresponds to antibonding of s waves, while the higher When the pump rate is further increased, the zero energy metastable state at point A corresponds to anti- state eventually acquires the population greater than 1. bonding of p waves. As the atomic p wave, the LP den- The onset of stimulated scattering into the zero state ␲ sity vanishes at the center of the potential well for this from the metastable ␲ state favors the condensation at state. This explains the near-field imaging shown in Fig. ͑ ͒ the zero state. The reverse stimulated process from the 46 a . Under the weak potential modulation, the Bloch zero to the ␲ state requires the absorption of phonons wave functions are superpositions of forward-and- and thus is energetically unfavorable. This mode compe- backward traveling plane waves at the zone boundary tition behavior is well described by the coupled Boltz- and have a relatively broad distribution in space com- mann ͑rate͒ equation analysis ͑Lai et al., 2007͒. pared to the standard tight-binding model of a normal Assuming reasonable numerical parameters, the condensed-matter system. qualitative agreement with the experimental results is The metastability of the ␲ state ͑antibonding of p ␲ ͒ obtained. With increasing pumping, the state appears waves is a unique property for dynamic polariton con- first and then is surpassed by the dominant zero state, densates. Under strong pumping rates, the zero state ͑ ͒ consistent with experimental observation. The time- bonding of s waves dominates eventually due to en- resolved LP emissions from the zero and ␲ state are hanced cooling by stimulated LP-LP scattering at high ␲ shown in Fig. 49. The simple rate equations with param- densities. A typical evolution from the metastable eters fixed for all pumping rates can describe the ob- state to the stable zero state is illustrated by the LP served mode competition. momentum distribution profiles for various pumping rates in Fig. 48. At a high pumping rate, the LP emission from the zero state surpasses that from the ␲ state by X. OUTLOOK more than one order of magnitude. With regard to fundamental physics of polariton BEC, it is important to understand the ground state as well as 3. Dynamics and mode competition the excitations of the quantum phases to establish the The ␲ state consisting of local p-type wave functions experimental phase diagram of the polariton system and is a metastable state with a higher energy than the zero to compare experiments with theoretical analysis taking state consisting of local s-type wave functions. Both into account both thermal and quantum depletions of a states decay to the crystal ground state by emitting pho- condensate. The dynamical nature of the polariton con- tons with a finite lifetime. After high-energy excitonlike densate due to the short polariton lifetime imposes chal-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1533

Experiment Simulation 1.2 5 a P=80mW c -state Initial Pump n3=2 1.0 4

0.8 zero-state 3 0.6 2 0.4

1 0.2 Intensity (arb. units) PL Intensity (arb. units)

0.0 0 0 20406080100 020406080100 3.0 10 b d P=160mW Initial Pump n =20 2.5 -state 3 8

2.0 zero-state 6 1.5 4 1.0

0.5 2 Intensity (arb. units) PL Intensity (arb. units) 0.0 0 0 20406080100 020406080100 Time (ps) Time (ps)

FIG. 49. ͑Color online͒ Dynamics of the zero and ␲ state. ͑a͒ LP dynamics measured by a streak camera system. The black dashed curve in ͑a͒ is the system response measured with scattered laser pulse from the sample. ͑b͒ Simulation. From Lai et al., 2007. lenges on this kind of study. Fortunately, the Q value of polariton BEC to electron-hole BCS phase is another a GaAs microcavity has been dramatically improved re- intriguing topic. With increasing exciton density to cently ͑Reitzenstein et al., 2007͒, so one may expect a above the Mott density, excitons overlap with each other quantitative comparison between theory and experiment and both the electron and hole liquids are normal except in the near future. at the Fermi surface. The remaining binding energy is The quantum-statistical properties of the BEC ground expected to create a gap at the Fermi energy. A mea- state are expected to be influenced by phase locking be- surement of the eigenenergies will help elucidate the tween the condensed particles at kʈ =0 and excitations at crossover behavior. ±kʈ 0 ͑Nozieres, 1995͒. The current polariton BEC is The complex phase diagram and rich physics of the still limited by thermal depletion, but with high-Q mi- polariton system make it a unique candidate for the crocavities and more efficient cooling schemes the quan- quantum simulation of many-body Hamiltonians in one tum nature of the BEC ground state should be eluci- and two dimensions. The superfluid to Mott insulator ͑or dated soon. Bose glass͒ phase transition in two dimensions and Superfluidity associated with polariton BEC is an- Tanks gas in one-dimensional polariton systems have other urgent issue. A first sign has been reported on the been addressed theoretically so far, while intense experi- sound velocity ͑Utsunomiya et al., 2008͒, and prelimi- mental research is underway. nary results on quantized vortices have been reported With regard to applications of polariton BEC, most ͑ ͒ recently Lagoudakis et al., 2008 . A more quantitative promising is a polariton BEC as a low-threshold source study on quantized vortices, in particular, dynamical for- of coherent light. A fundamental limit on the threshold mation process of vortices in a rotating polariton con- of a standard semiconductor laser is required to create densate, is needed. an electronic population inversion in the gain medium: The crossover from polariton BEC in a relatively nϳa*−2, where a* is the exciton Bohr radius. On the small 2D polariton system to a BKT phase in a relatively B B large 2D system is an interesting topic in studying the other hand, the polariton BEC threshold is given by n ϳ␭2 ␭ polariton phase diagram. The decay of the first-order T, where T is the thermal de Broglie wavelength. * ␭ −2 spatial coherence, the recombination of thermally ex- Since aB / T is on the order of 10 , the fundamental cited vortices, as well as the temperature dependence of limit of the polariton BEC threshold density is 10−4 of a fractional condensate will be decisive experimental sig- that of a counterpart semiconductor laser. A proof of natures for the BEC-BKT crossover. The crossover from principle comparison between the two types of lasers

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1534 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation was made by Deng et al. ͑2003͒ with a GaAs microcavity ford at present is supported by the JST/SORST program at liquid helium temperature. Recently Tsintzos et al. and Special Coordination Funds for Promoting Science ͑2008͒ demonstrated electronically pumped polariton and Technology from University of Tokyo, Japan. light-emitting diode operating at 235 K. An electrically pumped BEC with a p-n junction diode at high tempera- APPENDIX: CALCULATION OF SCATTERING tures may soon be within experimental reach. COEFFICIENTS

In this appendix we list the LP-phonon and LP-LP ACKNOWLEDGMENTS scattering rates for the quasispin kinetics ͓Eqs. ͑65͒ and ͑68͔͒ of Sec. VI. In the Markov approximation, the LP- We wish to acknowledge and thank present and phonon-scattering rate of the polariton population nk,i is former members of the Stanford group for their contri- given by butions to the polariton research reported herein. These p-ph 2␲ ץ researchers include H. Cao, R. Huang, N. Y. Kim, C. W. ͯ nk,i ͯ ͚ ͕ ͓ ͑ ͒ ជ ជ ͔ ប WkЈ,k nk,i 1+nkЈ,i + SkSkЈ −= ץ -Lai, S. Pau, G. Roumpos, F. Tassone, L. Tian, S. Ut t scatt Ј sunomiya, and G. Weihs. We also wish to acknowledge k and thank T. D. Doan, H. Thien Cao, and D. B. Tran ͓ ͑ ͒ ជ ជ ͔͖ ͑ ͒ − Wk,kЈ nkЈ,i 1+nk,i + SkSkЈ A1 Thoai at the Vietnam Centre for Natural Science and ជ Technology for their contributions to the theoretical and and the LP-phonon-scattering rate of the pseudospin Sk numerical works reported herein. The research at Stan- is given by

p-ph Sជ 2␲ 1 1ץ ͯ k ͯ =− ͚ ͭW ͩSជ + ͚ ͑n Sជ + n Sជ ͒ͪ − W ͩSជ + ͚ ͑n Sជ + n Sជ ͒ͪͮ. ͑A2͒ Ј Ј Ј Ј Ј Ј Ј t ប k ,k k 2 k ,i k k,i k k,k k 2 k,i k k ,i kץ scatt kЈ i=1,2 i=1,2

͑ ͒ Here the LP-phonon transition rate Wk,kЈ is given in Eq. 37 . Similarly, the LP-LP scattering rates are given by n p-p 2␲ץ ͯ k,i ͯ ␦͑ ͒ ͉ ͑ Ј Ј ͉͒2͓ ͑ ͒ =− ͚ ͭ ekЈ−q + ek+q − ek − ekЈ ͫ V k,k ,k − q,k + q nk,inkЈ,i 1+nkЈ−q,i + nk+q,i t បץ scatt kЈ,q ͑ ͔͒ ͉ ͑ Ј Ј ͉͒2 ͑ ជ ជ ͒ ͑ ͒ − nkЈ−q,ink+q,i 1+nk,i + nkЈ,i + U k,k ,k − q,k + q ͫ nk,inkЈ,¯i + SkSkЈ ͩ2+ ͚ nkЈ−q,j + nk+q,j ͪ j=1,2 ͑ ͒ ជ ជ ͑ Ј Ј ͒ ͓͑ ͒ជ ជ − 1+nk,i + nkЈ,¯i ͩ2SkЈ−qSk+q + ͚ nkЈ−q,¯jnk+q,jͪͬ + T k,k ,k − q,k + q ͩ ͚ nkЈ,j − nkЈ−q,j Sk+qSk j=1,2 j=1,2 ͑ ͒ជ ជ ͔ ͓ ͑ជ ជ ជ ជ ͒ ជ ជ ជ ជ ͔ ͑ ͒ + nkЈ,j − nk+q,j SkЈ−qSk +2 nk,i SkЈSkЈ−q + SkЈSk+q − nkЈ−q,iSkЈSk+q − nk+q,iSkЈSkЈ−q ͪͬͮ, A3

p-p Sជ ␲ץ ͯ k ͯ ␦͑ ͒ ͉ ͑ Ј Ј ͉͒2 ជ ͓ ͑ ͒ ͔ =− ͚ ͭ ek+q + ekЈ−q − ek − ekЈ ͫ V k,k ,k − q,k + q ͩSk ͚ nkЈ,i 1+nkЈ−q,i + nk+q,i − nkЈ−q,ink+q,i t បץ scatt kЈ,q i=1,2 ͓ជ ͑ជ ជ ͒ ជ ͑ជ ជ ͒ ជ ͑ជ ជ ͔͒ −2 Sk+q SkЈSkЈ−q + SkЈ−q SkЈSk+q − SkЈ SkЈ−qSk+q ͪ

͉ ͑ Ј Ј ͉͒2 ͑ជ ជ ͒ ͑ ͒ + U k,k ,k − q,k + q ͫ ͚ SknkЈ,i + SkЈnk,i ͩ2+ ͚ nkЈ−q,i + nk+q,i ͪ i=1,2 i=1,2 ͑ជ ជ ͒ ͑ ͒ ជ ជ ͑ Ј Ј ͒ ជ ͑ជ ជ ជ ជ ͒ −2 Sk + SkЈ ͩ ͚ nkЈ−q,ink+q,¯i +2SkЈ−qSk+qͪͬ + T k,k ,k − q,k + q ͭ4Sk SkЈSkЈ−q + SkЈSk+q i=1,2 ជ ͑ ͒ + SkЈ−qͫ ͚ nk,iͩ ͚ nkЈ,i − nk+q,i ͪ i=1,2 i=1,2 ͑ ͒ ជ ͑ ͒ ͑ ͒ ͑ ͒ −2͚ nk+q,i 1+nkЈ,i ͬSk+qͫ ͚ nk,iͩ ͚ nkЈ,i − nkЈ−q,i ͪ −2͚ nkЈ−q,i 1+nkЈ,i ͬͮͬͮ. A4 i=1,2 i=1,2 i=1,2 i=1,2

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

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 1535

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