Bose-Einstein Condensation in Dilute Gases

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
1 Introduction......Page 15
1.1 Bose–Einstein condensation in atomic clouds......Page 18
1.2 Superfluid…......Page 20
1.3 Other condensates......Page 22
1.4 Overview......Page 24
Problems......Page 27
References......Page 28
2.1 The Bose distribution......Page 30
2.1.1 Density of states......Page 32
2.2 Transition temperature and condensate fraction......Page 35
2.2.1 Condensate fraction......Page 37
2.3 Density profile and velocity distribution......Page 38
2.3.1 The semi-classical distribution......Page 41
2.4 Thermodynamic quantities......Page 43
2.4.1 Condensed phase......Page 44
2.4.3 Specific heat close to T......Page 46
2.5 Effect of finite particle number......Page 49
2.6 Lower-dimensional systems......Page 50
Problems......Page 51
References......Page 52
3.1 Atomic structure......Page 54
3.2 The Zeeman effect......Page 58
3.3 Response to an electric field......Page 63
Oscillating electric flelds......Page 67
3.4 Energy scales......Page 69
References......Page 71
4 Trapping and cooling of atoms......Page 72
4.1 Magnetic traps......Page 73
4.1.1 The quadrupole trap......Page 74
4.1.2 The TOP trap......Page 76
4.1.3 Magnetic bottles and the Ioffe–Pritchard trap......Page 78
4.2 Influence of laser light on an atom......Page 81
4.2.1 Forces on an atom in a laser field......Page 85
4.2.2 Optical traps......Page 87
4.3 Laser cooling: the Doppler process......Page 88
4.4 The magneto-optical trap......Page 92
4.5 Sisyphus cooling......Page 95
The friction coefficient......Page 99
Kinetic theory approach......Page 100
Temperature of atoms......Page 103
4.6 Evaporative cooling......Page 104
A simple model......Page 107
4.7 Spin-polarized hydrogen......Page 110
Problems......Page 113
References......Page 114
5 Interactions between atoms......Page 116
5.1 Interatomic potentials and the van der Waals interaction......Page 117
Magnitude of the van der Waals interaction......Page 120
5.2 Basic scattering theory......Page 121
5.2.1 Effective interactions and the scattering length......Page 125
5.3 Scattering length for a model potential......Page 128
5.4 Scattering between different internal states......Page 134
The magnetic dipole–dipole interaction......Page 136
Low-energy collisions......Page 137
5.4.1 Inelastic processes......Page 139
Spin-exchange processes......Page 140
Dipolar processes......Page 141
Three-body processes......Page 144
5.4.2 Elastic scattering and Feshbach resonances......Page 145
Basic formalism......Page 146
A simple example......Page 148
General solution......Page 149
Tuning effective interactions......Page 151
Measurement of the collision cross section......Page 153
Photoassociative spectroscopy......Page 154
Other methods......Page 155
Lithium......Page 156
Cesium......Page 157
References......Page 158
6.1 The Gross–Pitaevskii equation......Page 160
6.2 The ground state for trapped bosons......Page 163
6.2.1 A variational calculation......Page 165
6.2.2 The Thomas–Fermi approximation......Page 168
6.3 Surface structure of clouds......Page 172
6.4 Healing of the condensate wave function......Page 175
References......Page 177
7.1 General formulation......Page 179
7.1.1 The hydrodynamic equations......Page 181
A uniform gas......Page 185
The Bogoliubov equations......Page 188
Attractive interactions......Page 191
7.3 Collective modes in traps......Page 192
7.3.1 Traps with spherical symmetry......Page 193
7.3.2 Anisotropic traps......Page 196
Low-lying modes......Page 197
The scissors mode......Page 198
7.3.3 Collective coordinates and the variational method......Page 200
Collective coordinates......Page 201
Variational approach......Page 204
7.4 Surface modes......Page 207
7.5 Free expansion of the condensate......Page 209
7.6 Solitons......Page 210
Problems......Page 215
References......Page 216
8 Microscopic theory of the Bose gas......Page 218
8.1 Excitations in a uniform gas......Page 219
8.1.1 The Bogoliubov transformation......Page 221
8.1.2 Elementary excitations......Page 223
Depletion of the condensate......Page 224
Ground-state energy......Page 226
States with definite particle number......Page 227
8.2 Excitations in a trapped gas......Page 228
8.2.1 Weak coupling......Page 230
8.3 Non-zero temperature......Page 232
8.3.1 The Hartree–Fock approximation......Page 233
Thermal equilibrium......Page 237
8.3.2 The Popov approximation......Page 239
8.3.3 Excitations in non-uniform gases......Page 240
8.3.4 The semi-classical approximation......Page 242
Hartree–Fock theory......Page 244
Collective effects......Page 247
Problems......Page 250
References......Page 251
9.1 Potential flow and quantized circulation......Page 252
9.2.1 A vortex in a uniform medium......Page 254
Multiply-quantized vortices......Page 258
9.2.2 A vortex in a trapped cloud......Page 259
9.2.3 Off-axis vortices......Page 261
9.3.1 Traps with an axis of symmetry......Page 263
9.3.2 Rotating traps......Page 265
9.4 Vortex motion......Page 268
9.4.1 Force on a vortex line......Page 269
9.5 The weakly-interacting Bose gas under rotation......Page 271
Repulsive interactions......Page 273
Attractive interactions......Page 274
Problems......Page 275
References......Page 276
10 Superfluidity......Page 278
10.1 The Landau criterion......Page 279
10.2.1 Momentum carried by excitations......Page 281
10.2.2 Normal fluid density......Page 282
10.3 Dynamical processes......Page 284
Basic results......Page 287
The ideal Bose gas......Page 290
The interacting Bose gas......Page 291
10.5 Interactions between excitations......Page 294
10.5.1 Landau damping......Page 295
Problems......Page 301
References......Page 302
11 Trapped clouds at non-zero temperature......Page 303
11.1.1 Energy scales......Page 304
11.1.2 Transition temperature......Page 306
Low temperatures......Page 308
Higher temperatures......Page 309
11.2 Collective modes......Page 312
11.2.1 Hydrodynamic modes above T......Page 315
General formalism......Page 316
Low-frequency modes......Page 318
11.3 Collisional relaxation above T......Page 320
The Boltzmann equation......Page 321
11.3.1 Relaxation of temperature anisotropies......Page 324
The classical limit......Page 327
11.3.2 Damping of oscillations......Page 329
The hydrodynamic and intermediate regimes......Page 331
Problems......Page 332
References......Page 333
12 Mixtures and spinor condensates......Page 334
12.1 Mixtures......Page 335
Stability......Page 336
Density profiles......Page 338
12.1.2 Collective modes......Page 340
12.2 Spinor condensates......Page 342
12.2.1 Mean-field description......Page 344
12.2.2 Beyond the mean-field approximation......Page 347
Problems......Page 349
References......Page 350
13.1 Interference of two condensates......Page 352
13.1.1 Phase-locked sources......Page 353
Phase states......Page 355
13.1.2 Clouds with definite particle number......Page 357
13.2 Density correlations in Bose gases......Page 362
13.3 Coherent matter wave optics......Page 364
Phase imprinting......Page 365
Bragg diffraction of matter waves......Page 366
13.4 The atom laser......Page 368
13.5 The criterion for Bose–Einstein condensation......Page 369
13.5.1 Fragmented condensates......Page 371
References......Page 373
14 Fermions......Page 375
14.1 Equilibrium properties......Page 376
14.2 Effects of interactions......Page 380
Equilibrium size and collective modes......Page 381
Stability of uniform matter......Page 383
14.3 Superfluidity......Page 384
14.3.1 Transition temperature......Page 385
Eliminating the bare interaction......Page 387
Analytical results......Page 389
14.3.2 Induced interactions......Page 390
14.3.3 The condensed phase......Page 392
Elementary excitations......Page 394
The gap equation......Page 397
The gap at zero temperature......Page 398
14.4 Boson–fermion mixtures......Page 399
14.4.1 Induced interactions in mixtures......Page 400
14.5 Collective modes of Fermi superfluids......Page 402
Problems......Page 405
References......Page 406
Appendix. Fundamental constants and conversion factors......Page 408
Index......Page 411