BOSE-EINSTEIN CONDENSATION OF

EXCITONS AND BIEXCITONS AND COHERENT NONLINEAR OPTICS

WITH EXCITONS

S.A. MOSKALENKO D.W. SNOKE Academy of Sciences of University of Pittsburgh Moldova

ш C A M B R I D G E UNIVERSITY PRESS

Contents

Preface page xi

1 Introduction 1 1.1 What is an Exciton? 1 1.2 The Exciton as a Boson 6 1.3 Phase Transitions of Excitons 9 1.4 Experimental Evidence for Boson Behavior 13

1.4.1 Phonon-Assisted Luminescence from Excitons 14 1.4.2 Bose Narrowing 20 1.4.3 The Ideal-Gas Model for Excitons 23

1.5 Experiments on Bose-Einstein Condensation of Excitons 26

2 Basic Theory of Bose—Einstein Condensation of Excitons 29 2.1 The Bogoliubov Model of the Weakly Nonideal Bose Gas 29

2.1.1 The Quasi-Averages 30 2.1.2 Coherent States 33 2.1.3 The Excitons as Weakly Interacting, Structureless Bosons 36 2.1.4 The Beliaev Diagram Technique 40 2.1.5 The Excitation Spectrum and Momentum Distribution as Seen

in the Phonon-Assisted Luminescence 46 2.2 Bose-Einstein Condensation of Coupled Electron-Hole Pairs 54

2.2.1 The Keldysh-Kozlov-Kopaev Formulation of Bose Condensation 54

2.2.2 The Hartree-Fock-Bogoliubov Approximation 62 2.2.3 The Screening Effects and the Correlation Energy 66 2.2.4 Collective Elementary and Macroscopic Excitations 73

3 The Interaction of Condensed Excitons with Lattice Phonons 82 3.1 Introduction 82 3.2 The Diagram Technique for the Case T = 0. The Energy Spectrum of

the System 85 3.2.1 The Definition of the Green's Functions 85 3.2.2 The Symmetry Properties of the Green's Functions and their

Self-Energy Parts for the Bose System in the Presence of the Condensate 86

3.2.3 The Structure of the Equations 88

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3.2.4 The Transition from the Variable N to the variable /x. The Feynman Rules for the Exciton-Phonon Green's Functions 92

3.2.5 The Energy Spectrum and the Change of the Sound Velocity in Crystals due to Bose Condensation of Excitons 94

3.3 In-Depth Study: The Green's Function Method for T > 0 95 3.3.1 The Zeroth-Order Approximation 100 3.3.2 Phonons and Hydrons in the Dipole-Allowed Single-Photon

Transition 103 3.4 Experiments on Superfluid Exciton Transport and the Phonon Wind 107

3.4.1 Superfluidity of Excitons in Semiconductors 109 3.4.2 Anomalous Exciton Transport in СигО 112 3.4.3 The Exciton-Condensate Soliton 114 3.4.4 The Phonon Wind 116

4 Bose-Einstein Condensation of Biexcitons 123 4.1 Biexcitons and Exciton-Exciton Interactions in Semiconductors 123

4.1.1 Electron-Hole Structure of the Excitonic Molecule 123 4.1.2 Exciton-Exciton Interactions 129

4.2 Coherent Binding of Electron-Hole Quaternions 136 4.2.1 Formulation of the Method 136 4.2.2 Coherent Pairing of Electrons and Holes with Opposite-Sign

Trions 140 4.3 Coherent Pairing of Excitons 143

4.3.1 Coherent Pairing in the Multicomponent Exciton System 144 4.3.2 In-Depth Study: Coherent Pairing of Excitons Induced by

Laser Radiation 149 4.4 Experimental Studies on Nonequilibrium Bose-Einstein Condensation

of Biexcitons in CuCl Crystals 156 4.4.1 Bose Narrowing of Biexciton Luminescence 156 4.4.2 The Nonequilibrium Bose-Einstein Condensate 157 4.4.3 Optical Phase Conjugation Based on Stimulated Scattering 160

5 Phase Transitions and Thermodynamics of High-Density of Excitons 166 5.1 Introduction 166 5.2 Excitons and Electron-Hole Plasma 168

5.2.1 Electron-Hole Liquid and Electron-Hole Plasma 168 5.2.2 First-Order and Second-Order Phase Transitions in the

Degenerate Electron-Hole System 171 5.2.3 Hysteresis due to Screening Effects 177

5.3 The Approach of TD. Lee and C.N. Yang 182 5.3.1 In-Depth Study: Excitons with Different Dispersion Laws 185 5.3.2 In-Depth Study: Bose Condensation of Excitons in an

External Field - Ferromagnetism and Ferroelectricity 191

6 The Optical Stark Effect and the Virtual Bose Condensate 201 6.1 Nonequilibrium Theory of the Optical Stark Effect in the Exciton

Range of the Spectrum 201 6.1.1 The Keldysh Diagram Technique 202

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6.1.2 Nonequilibrium Theory of Virtual Excitonic Bose-Einstein Condensation 205

6.1.3 The Coulomb Green's Function Method 211 6.1.4 Linear Response to a Weak Perturbation 214

6.2 The Energy Spectrum and the Exciton Absorption and Gain Bands in the Presence of Laser Radiation 216 6.2.1 Energy Spectrum of the Elementary Excitations in a

Coherently Polarized Crystal 218 6.2.2 Exciton Absorption and Gain Bands in a Coherently Polarized

Crystal 224 6.3 In-Depth Study: Biexcitons in a Coherent Cloud of Virtual Excitons 229 6.4 In-Depth Study: Optical Stark Effect in the Exciton-Biexciton System 234

6.4.1 Biexcitons as Structureless Bosons 234 6.4.2 Nonlinear Optical Properties of Biexcitons 237 6.4.3 The Role of Biexcitonic States in the Excitonic Optical Stark

Effect 243

7 Bose-Einstein Condensation of Mixed States of Excitons and Photons 249 7.1 Introduction. Polaritons and Semiconductor Microcavities 249 7.2 Condensate-Photon Modes and Giant Polaritons 252

7.2.1 The Keldysh Equations Describing the Coherent Excitons and Photons 252

7.2.2 The Excitons as Simple Bosons 259 7.3 The Energy Spectrum of the Noncondensed Excitons and Photons in

the Presence of a Condensate 263 7.3.1 The Absolute and Convective Instabilities. Sturrock's Rules 263 7.3.2 Energy Spectrum of the Elementary Excitations in the

Presence of the Condensate Photon Mode 266 7.3.3 The Instabilities in a System with a Strong Polariton Effect 273 7.3.4 In-Depth Study: The Elementary Excitations in the

Three-Component System of Excitons, Photons, and Phonons 277 7.4 In-Depth Study: Phonoritons, and Nonlinear Absorption of Polaritons 283

7.4.1 Acoustic and Optical Phonoritons 284 7.4.2 Stokes- and Anti-Stokes-Scattering Processes in a

Two-Component Model 285 7.5 In-Depth Study: High-Intensity Polariton Wave near the

Stimulated-Scattering Threshold 289

8 Nonequilibrium Kinetics of High-Density Excitons 295 8.1 Condensate Formation in the Bose Gas 295

8.1.1 Ginzburg-Landau Theory of Superfluidity in the Coherent State Representation 297

8.1.2 The Nucleation and Buildup of the Bose Condensate 300 8.2 Population Dynamics of a Bose Gas 305

8.2.1 Evolution of the Particle Distribution Function 307 8.2.2 Two Kinetic Stages, Before and After Condensation 310 8.2.3 Bose Condensation by means of Particle-Phonon Interaction 311

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8.3 In-Depth Study: Quantum Fluctuations and Induced Bose-Einstein Condensation of Polaritons - Squeezed States of Polaritons in Quantum Dots 313 8.3.1 The Model Hamiltonian, Master, and Kinetic Equations 316 8.3.2 The Fokker-Planck Equation and Squeezed States 323 8.3.3 Stationary Self-Consistent Solution 328

9 Coherent Nonlinear Optics with Excitons 336 9.1 Optical Bistability Effects and Self-Pulsations 337

9.1.1 Self-Pulsations in the Excitonic Range of the Spectrum 337 9.1.2 Self-Pulsating Laser Radiation due to Exciton-Biexciton

Conversion 344 9.2 Self-Induced Transparency, Nutation, and Quantum Beats 346

9.2.1 The Polariton Soliton 349 9.2.2 Exciton-Photon Nutation 354 9.2.3 Quantum Beats of Excitons 358

9.3 Photon-Echo and Transient Effects 362 9.3.1 Photon-Echo Theory Beyond the Fixed-Field Approximation 364 9.3.2 The Transient Stage and the Area Theorems 367

10 New Directions 374 10.1 Trapping Excitons with Stress 374 10.2 Two-Dimensional Systems 377

10.2.1 Dipole Excitons in Coupled Quantum Wells 377 10.2.2 Charged Bosons and Bose Condensate Superconductors 382 10.2.3 Bose Statistics of Excitons and Biexcitons in Single Quantum

Wells 383 10.2.4 Trapping Excitons in Two Dimensions 386

10.3 The Excitonic Insulator, or "Excitonium" 388 10.3.1 Long-Range Order in the Excitonic Insulator 389 10.3.2 Experimental Evidence for the Excitonic-Insulator Phase 395

10.4 Optical Coherence Without basing 396

Appendix A: Properties of Excitons in СигО 402