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Chapter 10 Dynamic Condensation of Exciton-Polaritons

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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 0034-6861/2010/82͑2͒/1489͑49͒ 1489 ©2010 The American Physical Society 1490 Deng, Haug, and Yamamoto: Exciton-polariton Bose-Einstein condensation 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 or- ders 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 dy- namical 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 picosec- onds. The dynamical nature of a polariton condensate I. INTRODUCTION places a limitation for studying the standard BEC phys- ics, 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
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