Bose-Einstein condensation in dilute gases

Cover......Page 1
Title......Page 4
Contents......Page 6
Preface......Page 13
1 Introduction......Page 15
1.1 Bose–Einstein condensation in atomic clouds......Page 18
1.2 Superfluid…......Page 21
1.3 Other condensates......Page 23
1.4 Overview......Page 24
References......Page 29
2.1 The Bose distribution......Page 31
2.1.1 Density of states......Page 33
2.2 Transition temperature and condensate fraction......Page 35
2.2.1 Condensate fraction......Page 38
2.3 Density profile and velocity distribution......Page 39
2.3.1 The semi-classical distribution......Page 42
Free expansion......Page 45
2.4.1 Condensed phase......Page 47
2.4.2 Normal phase......Page 49
2.4.3 Specific heat close to T......Page 50
2.5 Effect of finite particle number......Page 52
Problems......Page 53
References......Page 54
3.1 Atomic structure......Page 55
3.2 The Zeeman effect......Page 59
3.3 Response to an electric field......Page 64
Oscillating electric fields......Page 68
3.4 Energy scales......Page 70
Problems......Page 72
References......Page 73
4 Trapping and cooling of atoms......Page 74
4.1 Magnetic traps......Page 75
4.1.1 The quadrupole trap......Page 76
4.1.2 The TOP trap......Page 78
4.1.3 Magnetic bottles and the Ioffe–Pritchard trap......Page 80
4.1.4 Microtraps......Page 83
4.2 Influence of laser light on an atom......Page 85
4.2.1 Forces on an atom in a laser field......Page 89
4.2.2 Optical traps......Page 91
4.3 Laser cooling: the Doppler process......Page 92
4.4 The magneto-optical trap......Page 96
4.5 Sisyphus cooling......Page 98
The periodic potential......Page 100
Transitions between substates......Page 102
The friction coefficient......Page 103
Kinetic theory approach......Page 104
Temperature of atoms......Page 109
4.6 Evaporative cooling......Page 110
A simple model......Page 114
4.7 Spin-polarized hydrogen......Page 117
Problems......Page 120
References......Page 121
5 Interactions between atoms......Page 123
5.1 Interatomic potentials and the van der Waals interaction......Page 124
Magnitude of the van der Waals interaction......Page 127
5.2 Basic scattering theory......Page 128
5.2.1 Effective interactions and the scattering length......Page 133
Energy shift......Page 134
Momentum space analysis......Page 136
5.3 Scattering length for a model potential......Page 139
5.4 Scattering between different internal states......Page 144
The central part of the interaction......Page 146
The magnetic dipole–dipole interaction......Page 147
Low-energy collisions......Page 148
5.4.1 Inelastic processes......Page 149
Spin-exchange processes......Page 151
Dipolar processes......Page 152
Three-body processes......Page 156
5.4.2 Elastic scattering and Feshbach resonances......Page 157
Basic formalism......Page 158
A simple example......Page 160
General solution......Page 161
Tuning effective interactions......Page 162
Measurement of the collision cross section......Page 165
Photoassociative spectroscopy......Page 166
Hydrogen......Page 168
Rubidium......Page 169
References......Page 170
6.1 The Gross–Pitaevskii equation......Page 173
6.2 The ground state for trapped bosons......Page 176
6.2.1 A variational calculation......Page 179
6.2.2 The Thomas–Fermi approximation......Page 182
6.3 Surface structure of clouds......Page 185
6.4 Healing of the condensate wave function......Page 189
6.5 Condensates with dipolar interactions......Page 190
Problems......Page 193
References......Page 194
7.1 General formulation......Page 196
7.1.1 The hydrodynamic equations......Page 198
A uniform gas......Page 202
The Bogoliubov equations......Page 205
Attractive interactions......Page 209
7.3 Collective modes in traps......Page 210
7.3.1 Traps with spherical symmetry......Page 211
7.3.2 Anisotropic traps......Page 214
Low-lying modes......Page 215
The scissors mode......Page 217
Collective coordinates......Page 218
Variational approach......Page 222
7.4 Surface modes......Page 225
7.5 Free expansion of the condensate......Page 227
7.6 Solitons......Page 229
7.6.1 Dark solitons......Page 230
7.6.2 Bright solitons......Page 236
Problems......Page 237
References......Page 238
8 Microscopic theory of the Bose gas......Page 239
8.1 The uniform Bose gas......Page 240
8.1.1 The Bogoliubov transformation......Page 243
8.1.2 Elementary excitations......Page 244
8.1.3 Depletion of the condensate......Page 245
8.1.4 Ground-state energy......Page 247
8.1.5 States with definite particle number......Page 248
8.2 Excitations in a trapped gas......Page 250
8.3 Non-zero temperature......Page 255
8.3.1 The Hartree–Fock approximation......Page 256
Thermal equilibrium......Page 260
8.3.2 The Popov approximation......Page 262
8.3.3 Excitations in non-uniform gases......Page 264
8.3.4 The semi-classical approximation......Page 265
References......Page 267
9.1 Potential flow and quantized circulation......Page 269
9.2.1 A vortex in a uniform medium......Page 271
9.2.2 Vortices with multiple quanta of circulation......Page 275
9.2.3 A vortex in a trapped cloud......Page 276
9.3 Equilibrium of rotating condensates......Page 279
9.3.1 Traps with an axis of symmetry......Page 280
9.3.2 Rotating traps......Page 281
A single vortex......Page 282
9.3.3 Vortex arrays......Page 284
9.4 Experiments on vortices......Page 287
9.5 Rapidly rotating condensates......Page 289
Beyond the Gross-Pitaevskii approximation......Page 293
9.6 Collective modes in a vortex lattice......Page 294
Problems......Page 300
References......Page 302
10 Superfluidity......Page 304
10.1 The Landau criterion......Page 305
10.2.1 Momentum carried by excitations......Page 308
10.2.2 Normal fluid density......Page 309
10.3 Dynamical processes......Page 310
Basic results......Page 314
The ideal Bose gas......Page 317
The interacting Bose gas......Page 318
10.5 Interactions between excitations......Page 321
10.5.1 Landau damping......Page 322
Problems......Page 328
References......Page 329
11 Trapped clouds at non-zero temperature......Page 330
11.1.1 Energy scales......Page 331
11.1.2 Transition temperature......Page 333
Low temperatures......Page 335
Higher temperatures......Page 337
11.2 Collective modes......Page 339
11.2.1 Hydrodynamic modes above T......Page 342
General formalism......Page 343
Low-frequency modes......Page 346
11.3 Collisional relaxation above T......Page 348
The Boltzmann equation......Page 349
11.3.1 Relaxation of temperature anisotropies......Page 353
11.3.2 Damping of oscillations......Page 356
The hydrodynamic and intermediate regimes......Page 358
Problems......Page 359
References......Page 360
12 Mixtures and spinor condensates......Page 362
12.1 Mixtures......Page 363
Stability......Page 364
Density profiles......Page 366
12.1.2 Collective modes......Page 368
12.2 Spinor condensates......Page 370
12.2.1 Mean-field description......Page 372
12.2.2 Beyond the mean-field approximation......Page 374
Problems......Page 377
References......Page 378
13.1 Tunneling between two wells......Page 379
13.1.1 Quantum fluctuations......Page 385
13.1.2 Squeezed states......Page 387
13.2 Interference of two condensates......Page 388
13.2.1 Phase-locked sources......Page 389
A thought experiment......Page 390
Phase states......Page 393
13.2.2 Clouds with definite particle number......Page 395
13.3 Density correlations in Bose gases......Page 398
13.3.1 Collisional shifts of spectral lines......Page 400
Phase imprinting......Page 404
Bragg diffraction of matter waves......Page 406
Four-wave mixing......Page 407
13.5.1 The density matrix......Page 408
13.5.2 Fragmented condensates......Page 411
References......Page 413
14 Optical lattices......Page 415
14.1 Generation of optical lattices......Page 416
14.1.1 One-dimensional lattices......Page 417
14.1.2 Higher-dimensional lattices......Page 420
14.1.3 Energy scales......Page 421
14.2.1 Band structure for a single particle......Page 423
14.2.2 Band structure for interacting particles......Page 425
Bloch Oscillations......Page 429
14.2.3 Tight-binding model......Page 430
14.3 Stability......Page 432
Energetic stability......Page 433
Elementary excitations and dynamical stability......Page 434
14.3.1 Hydrodynamic analysis......Page 435
14.4.1 Loops......Page 437
14.4.2 Spatial period doubling......Page 441
Stability of states at the zone boundary......Page 444
14.5 From superfluid to insulator......Page 445
14.5.1 Mean-field approximation......Page 447
14.5.3 Experimental detection of coherence......Page 453
Problems......Page 455
References......Page 456
15 Lower dimensions......Page 458
15.1 Non-interacting gases......Page 459
15.2 Phase fluctuations......Page 461
Phase correlations......Page 463
15.2.1 Vortices and the Berezinskii–Kosterlitz–Thouless transition......Page 465
15.3 Microscopic theory of phase fluctuations......Page 467
Zero temperature......Page 469
15.3.2 Anisotropic traps......Page 470
15.4 The one-dimensional Bose gas......Page 474
15.4.1 The strong-coupling limit......Page 475
15.4.2 Arbitrary coupling......Page 480
Two-particle correlation functions......Page 488
Momentum distribution......Page 491
Problems......Page 493
References......Page 494
16 Fermions......Page 495
16.1 Equilibrium properties......Page 497
16.2 Effects of interactions......Page 500
Equilibrium size and collective modes......Page 501
16.3 Superfluidity......Page 503
16.3.1 Transition temperature......Page 505
Eliminating the bare interaction......Page 507
Analytical results......Page 509
16.3.2 Induced interactions......Page 510
16.3.3 The condensed phase......Page 512
Elementary excitations......Page 513
The gap equation......Page 517
The gap at zero temperature......Page 518
16.4 Pairing with unequal populations......Page 520
16.5 Boson–fermion mixtures......Page 522
16.5.1 Induced interactions in mixtures......Page 523
Problems......Page 525
References......Page 527
17 From atoms to molecules......Page 528
17.1 Bose–Einstein condensation of molecules......Page 530
17.2.1 Binding energy and the atom–atom scattering length......Page 532
Coupling between channels......Page 533
17.2.2 A simple two-channel model......Page 534
Magnetic moment of a molecule......Page 538
17.2.3 Atom–atom scattering......Page 540
17.3 Crossover: From BCS to BEC......Page 541
17.3.1 Wide and narrow Feshbach resonances......Page 542
17.3.2 The BCS wave function......Page 544
17.3.3 Crossover at zero temperature......Page 545
17.3.4 Condensate fraction and pair wave function......Page 549
17.4.1 Thermal molecules......Page 554
17.4.2 Pair fluctuations and thermal molecules......Page 557
17.4.3 Density of atoms......Page 562
17.4.4 Transition temperature......Page 563
17.5 A universal limit......Page 564
Zero temperature......Page 567
17.6.2 Vortices......Page 570
Problems......Page 573
References......Page 574
Appendix. Fundamental constants and conversion factors......Page 576
A-C......Page 578
C-F......Page 579
F-M......Page 580
M-S......Page 581
S-W......Page 582
Y-Z......Page 583