1 Overview of Microcavities .................................... 1
1.1 Properties of microcavities ............................. 2
1.1.1 Q-factor and finesse ............................. 2
1.1.2 Intracavity field enhancement and field
distribution ..................................... 3
1.1.3 Tuneability and mode separation .................. 3
1.1.4 Angular mode pattern ............................. 4
1.1.5 Low-threshold lasing ............................. 4
1.1.6 Purcell factor and lifetimes ..................... 5
1.1.7 Strong vs. weak coupling ......................... 5
1.2 Microcavity realizations ................................ 5
1.3 Planar microcavities .................................... 6
1.3.1 Metal microcavities .............................. 8
1.3.2 Dielectric Bragg mirrors ......................... 9
1.4 Spherical mirror microcavities ......................... 10
1.5 Pillar microcavities ................................... 12
1.6 Whispering-gallery modes ............................... 15
1.6.1 Two-dimensional whispering galleries ............ 16
1.6.2 Three-dimensional whispering-galleries .......... 18
1.7 Photonic-crystal cavities .............................. 19
1.7.1 Random lasers ................................... 19
1.8 Plasmonic cavities ..................................... 20
1.9 Microcavity lasers ..................................... 21
1.10 Conclusion ............................................. 21
2 Classical description of light .............................. 23
2.1 Free space ............................................. 24
2.1.1 Light-field dynamics in free space .............. 24
2.2 Propagation in crystals ................................ 27
2.2.1 Plane waves in bulk crystals .................... 27
2.2.2 Absorption of light ............................. 31
2.2.3 Kramers-Kronig relations ........................ 32
2.3 Coherence .............................................. 32
2.3.1 Statistical properties of light ................. 32
2.3.2 Spatial and temporal coherence .................. 33
2.3.3 Wiener-Khinchin theorem ......................... 38
2.3.4 Hanbury Brown-Twiss effect ...................... 41
2.4 Polarisation-dependent optical effects ................. 43
2.4.1 Birefringence ................................... 43
2.4.2 Magneto-optical effects ......................... 44
2.5 Propagation of light in multilayer planar structures ... 45
2.6 Photonic eigenmodes of planar systems .................. 49
2.6.1 Photonic bands of ID periodic structures ........ 52
2.7 Planar microcavities ................................... 59
2.8 Stripes, pillars, and spheres: photonic wires and
dots ................................................... 64
2.8.1 Cylinders and pillar cavities ................... 66
2.8.2 Spheres ......................................... 69
2.9 Further reading ........................................ 73
3 Quantum description of light ................................ 75
3.1 Pictures of quantum mechanics .......................... 76
3.1.1 Historical background ........................... 76
3.1.2 Schrödinger picture ............................. 76
3.1.3 Antisymmetry of the wavefunction ................ 85
3.1.4 Symmetry of the wavefunction .................... 86
3.1.5 Heisenberg picture .............................. 87
3.1.6 Dirac (interaction) picture ..................... 92
3.2 Other formulations ..................................... 94
3.2.1 Density matrix and Liouvillian .................. 94
3.2.2 Second quantization ............................. 97
3.2.3 Quantization of the light field ................ 100
3.3 Quantum states ........................................ 101
3.3.1 Fock states .................................... 101
3.3.2 Coherent states ................................ 101
3.3.3 Glauber-Sudarshan representation ............... 103
3.3.4 Thermal states ................................. 104
3.3.5 Mixture states ................................. 105
3.3.6 Quantum correlations of quantum fields ......... 106
3.3.7 Power spectra .................................. 111
3.3.8 Statistics of the field ........................ 113
3.3.9 Polarisation ................................... 115
3.4 Outlook on quantum mechanics for microcavities ........ 117
3.5 Further reading ....................................... 117
4 Semiclassical description of light-matter coupling ......... 119
4.1 Light-matter interaction .............................. 120
4.1.1 Classical limit ................................ 120
4.1.2 Einstein coefficients .......................... 122
4.2 Optical transitions in semiconductors ................. 125
4.3 Excitons in semiconductors ............................ 129
4.3.1 Frenkel and Wannier-Mott excitons .............. 129
4.3.2 Excitons in confined systems ................... 133
4.3.3 Quantum wells .................................. 133
4.3.4 Quantum wires and dots ......................... 137
4.4 Exciton-photon coupling ............................... 139
4.4.1 Surface polaritons ............................. 142
4.4.2 Exciton-photon coupling in quantum wells ....... 144
4.4.3 Exciton-photon coupling in quantum wires and
dots ........................................... 149
4.4.4 Dispersion of polaritons in planar
microcavities .................................. 152
4.4.5 Motional narrowing of cavity polaritons ........ 162
4.4.6 microcavities with quantum wires or dots ....... 166
5 Quantum description of light-matter coupling ............... 171
5.1 Historical background ................................. 172
5.2 Rabi dynamics ......................................... 172
5.3 Bloch equations ....................................... 175
5.4 Full quantum picture .................................. 177
5.5 Dressed bosons ........................................ 180
5.6 Jaynes-Cummings model ................................. 187
5.7 Dicke model ........................................... 193
5.8 Lindblad dissipation .................................. 194
5.9 Quantum dynamics with decay and pumping ............... 200
5.9.1 Single-time dynamics of coupled Bose fields .... 203
5.9.2 Two-time dynamics of coupled Bose fields ....... 208
5.9.3 The two-level system coupled to a Bose field ... 214
5.10 Excitons in semiconductors ............................ 222
5.10.1 Quantization of the exciton field .............. 222
5.10.2 Excitons as bosons ............................. 224
5.10.3 Excitons in quantum dots ....................... 224
5.11 Exciton-photon coupling ............................... 231
5.11.1 Polariton splitting ............................ 232
5.11.2 The polariton Hamiltonian ...................... 233
6 Weak-coupling microcavities ................................ 235
6.1 Purcell effect ........................................ 236
6.1.1 The physics of weak coupling ................... 236
6.1.2 Spontaneous emission ........................... 237
6.1.3 The case of QDs 2D excitons and 2D electron-
hole pairs ..................................... 239
6.1.4 Fermi's golden rule ............................ 240
6.1.5 Dynamics of the Purcell effect ................. 243
6.1.6 Experimental realizations ...................... 246
6.2 Lasers ................................................ 248
6.2.1 The physics of lasers .......................... 248
6.2.2 Semiconductors in laser physics ................ 252
6.2.3 Vertical-cavity surface-emitting lasers ........ 255
6.2.4 Resonant-cavity LEDs ........................... 259
6.2.5 Quantum theory of the laser .................... 260
6.3 Nonlinear optical properties of weak-coupling
microcavities ......................................... 265
6.3.1 Bistability .................................... 266
6.3.2 Phase matching ................................. 268
6.4 Conclusion ............................................ 268
7 Strong coupling: resonant effects .......................... 269
7.1 Optical properties: background ........................ 270
7.1.1 Quantum well microcavities ..................... 270
7.1.2 Variations on a theme .......................... 272
7.1.3 Motional narrowing ............................. 275
7.1.4 Ultra-strong coupling in THz cavities .......... 275
7.1.5 Polariton emission ............................. 276
7.2 Near-resonant-pumped optical nonlinearities ........... 277
7.2.1 Pulsed stimulated scattering ................... 277
7.2.2 Quasimode theory of parametric amplification ... 282
7.2.3 Microcavity parametric oscillators ............. 284
7.3 Resonant excitation case and parametric
amplification ......................................... 286
7.3.1 Semiclassical description ...................... 287
7.3.2 Stationary solution and threshold .............. 288
7.3.3 Theoretical approach: quantum model ............ 289
7.3.4 Three-level model .............................. 290
7.3.5 Threshold ...................................... 293
7.4 Two-beam experiment ................................... 293
7.4.1 One-beam experiment and spontaneous symmetry
breaking ....................................... 293
7.4.2 Dressing of the dispersion induced by
polariton condensates .......................... 295
7.4.3 Bistable behaviour ............................. 295
8 Strong coupling: polariton Bose condensation ............... 299
8.1 Introduction .......................................... 300
8.2 Basic ideas about Bose-Einstein condensation .......... 300
8.2.1 Einstein proposal .............................. 300
8.2.2 Experimental realization ....................... 302
8.2.3 Modern definition of Bose-Einstein
condensation ................................... 303
8.3 Specificities of excitons and polaritons .............. 304
8.3.1 Thermodynamic properties of cavity
polaritons ..................................... 306
8.3.2 Interacting bosons and Bogoliubov model ........ 306
8.3.3 Polariton superfluidity ........................ 309
8.3.4 Quasicondensation and local effects ............ 312
8.4 High-power microcavity emission ....................... 315
8.5 Thresholdless polariton lasing ........................ 317
8.6 Kinetics of formation of polariton condensates:
semiclassical picture ................................. 322
8.6.1 Qualitative features ........................... 322
8.6.2 The semiclassical Boltzmann equation ........... 326
8.6.3 Numerical solution of Boltzmann equations,
practical aspects .............................. 327
8.6.4 Effective scattering rates ..................... 328
8.6.5 Numerical simulations .......................... 329
8.7 Kinetics of formation of polariton condensates:
quantum picture in the Born-Markov approximation ...... 331
8.7.1 Density matrix dynamics of the ground-state .... 332
8.7.2 Discussion ..................................... 336
8.7.3 Coherence dynamics ............................. 337
8.8 Kinetics of formation of polariton condensates:
quantum picture beyond the Born-Markov
approximation ......................................... 339
8.8.1 Two-oscillator toy theory ...................... 339
8.8.2 Coherence of polariton laser emission .......... 351
8.8.3 Numerical simulations .......................... 355
8.8.4 Order parameter and phase diffusion
coefficient .................................... 356
8.9 Semiconductor luminescence equations .................. 359
8.10 Experiments on Bose-Einstein condensation of
exciton-polaritons and polariton lasing ............... 361
8.10.1 Experimental observation ....................... 361
8.10.2 Polariton lasing vs Bose-Einstein
condensation ................................... 362
8.10.3 Polariton diodes ............................... 364
8.11 Experiments on superfluidity of exciton-polaritons .... 364
8.12 Superconductivity mediated by exciton-polaritons ...... 367
8.13 Further reading ....................................... 369
9 Spin and polarisation ...................................... 371
9.1 Spin relaxation of electrons, holes and excitons in
semiconductors ........................................ 372
9.2 microcavities in the presence of a magnetic field ..... 377
9.3 Resonant Faraday rotation ............................. 378
9.4 Spin relaxation of exciton-polaritons in
microcavities: experiment ............................. 381
9.5 Spin relaxation of exciton-polaritons in
microcavities: theory ................................. 386
9.6 Optical spin Hall effect .............................. 390
9.7 Optically induced Faraday rotation .................... 392
9.8 Interplay between spin and energy relaxation of
exciton-polaritons .................................... 394
9.9 Polarisation of Bose condensates and polariton
superfluids ........................................... 398
9.10 Magnetic-field effect and superfluidity ............... 402
9.11 Finite-temperature case ............................... 407
9.12 Quantized vortices and half-vortices .................. 409
9.13 Spin dynamics in parametric oscillators ............... 411
9.13.1 Classical nonlinear optics consideration ....... 412
9.13.2 Polarised OPO: quantum model ................... 413
9.14 Conclusions ........................................... 416
9.15 Further reading ....................................... 417
Glossary ................................................... 419
A Linear algebra ............................................. 427
В Scattering rates of polariton relaxation ................... 431
B.l Polariton-phonon interaction .......................... 431
B.l.l Interaction with longitudinal optical
phonons ........................................ 432
B.l.2 Interaction with acoustic phonons .............. 433
B.2 Polariton-electron interaction ........................ 434
B.3 Polariton-polariton interaction ....................... 436
B.3.1 Polariton decay ................................ 436
B.4 Polariton-structural-disorder interaction ............. 437
С Derivation of the Landau criterion of superfluidity and
Landau formula ............................................. 439
D Landau quantization and renormalisation of Rabi
splitting .................................................. 441
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