1 Introduction 1
1.1 Bose-Einstein condensation in atomic clouds 4
1.2 Superfluid 4He 7
1.3 Other condensates 9
1.4 Overview 10
Problems 15
References 15
2 The non-interacting Bose gas 17
2.1 The Bose distribution 17
2.1.1 Density of states 19
2.2 Thansition temperature and condensate fraction 21
2.2.1 Condensate fraction 24
2.3 Density profile and velocity distribution 25
2.3.1 The semi-classical distribution 28
2.4 Thermodynamic quantities 33
2.4.1 Condensed phase 33
2.4.2 Normal phase 35
2.4.3 Specific heat close to Tc 36
2.5 Effect of finite particle number 38
Problems 39
References 40
3 Atomic properties 41
3.1 Atomic structure 41
3.2 The Zeeman effect 45
3.3 Response to an electric field 50
3.4 Energy scales 56
Problems 58
References 59
4 Trapping and cooling of atoms 60
4.1 Magnetic traps 61
4.1.1 The quadrupole trap 62
4.1.2 The TOP trap 64
4.1.3 Magnetic bottles and the Ioffe-Pritchard trap 66
4.1.4 Microtraps 69
4.2 Influence of laser light on an atom 71
4.2.1 Forces on an atom in a laser field 75
4.2.2 Optical traps 77
4.3 Laser cooling:the Doppler process 78
4.4 The magneto-optical trap 82
4.5 Sisyphus cooling 84
4.6 Evaporative cooling 96
4.7 Spin-polarized hydrogen 103
Problems 106
References 107
5 Interactions between atoms 109
5.1 Interatomic potentials and the van der Waals interaction 110
5.2 Basic scattering theory 114
5.2.1 Effective interactions and the scattering length 119
5.3 Scattering length for a model potential 125
5.4 Scattering between different internal states 130
5.4.1 Inelastic processes 135
5.4.2 Elastic scattering and Feshbach resonances 143
5.5 Determination of scattering lengths 151
5.5.1 Scattering lengths for alkali atoms and hydrogen 154
Problems 156
References 156
6 Theory of the condensed state 159
6.1 The Gross-Pitaevskii equation 159
6.2 The ground state for trapped bosons 162
6.2.1 A variational calculation 165
6.2.2 The Thomas-Fermi approximation 168
6.3 Surface structure of clouds 171
6.4 Healing of the condensate wave function 175
6.5 Condensates with dipolar interactions 176
Problems 179
References 180
7 Dynamics of the condensate 182
7.1 General formulation 182
7.1.1 The hydrodynamic equations 184
7.2 Elementary excitations 188
7.3 Collective modes in traps 196
7.3.1 Traps with spherical symmetry 197
7.3.2 Anisotropic traps 200
7.3.3 Collective coordinates and the variational method 204
7.4 Surface modes 21l 213
7.5 Free expansion of the condensate 213
7.6 Solitons 215
7.6.1 Dark solitons 216
7.6.2 Bright solitons 222
Problems 223
References 224
8 Microscopic theory of the Bose gas 225
8.1 The uniform Bose gas 226
8.1.1 The Bogoliubov transformation 229
8.1.2 Elementary excitations 230
8.1.3 Depletion of the condensate 231
8.1.4 Ground-state energy 233
8.1.5 States with definite particle number 234
8.2 Excitations in a trapped gas 236
8.3 Non-zero temperature 241
8.3.1 The Hartree-Fock approximation 242
8.3.2 The Popov approximation 248
8.3.3 Excitations in non-uniform gases 250
8.3.4 The semi-classical approximation 251
Problems 253
References 253
9 Rotating condensates 255
9.1 Potential flow and quantized circulation 255
9.2 Structure of a single vortex 257
9.2.1 A vortex in a uniform medium 257
9.2.2 Vortices with multiple quanta of circulation 261
9.2.3 A vortex in a trapped cloud 262
9.2.4 An off-axis vortex 265
9.3 Equilibrium of rotating condensates 265
9.3.1 Traps with an axis of symmetry 266
9.3.2 Rotating traps 267
9.3.3 Vortex arrays 270
9.4 Experiments on vortices 273
9.5 Rapidly rotating condensates 275
9.6 Collective modes in a vortex lattice 280
Problems 286
References 288
10 Superfluidity 290
10.1 The Landau criterion 291
10.2 The two-component picture 294
10.2.1 Momentum carried by excitations 294
10.2.2 Normal fluid density 295
10.3 Dynamical processes 296
10.4 First and second sound 300
10.5 Interactions between excitations 307
10.5.1 Landau damping 308
Problems 314
References 315
11 Trapped clouds at non-zero temperature 316
11.1 Equilibrium properties 317
11.1.1 Energy scales 317
11.1.2 Transition temperature 319
11.1.3 Thermodynamic properties 321
11.2 Collective modes 325
11.2.1 Hydrodynamic modes above Tc 328
11.3 Collisional relaxation above Tc 334
11.3.1 Relaxation of temperature anisotropies 339
11.3.2 Damping of oscillations 342
Problems 345
References 346
12 Mixtures and spinor condensates 348
12.1 Mixtures 349
12.1.1 Equilibrium properties 350
12.1.2 Collective modes 354
12.2 Spinor condensates 356
12.2.1 Mean-field description 358
12.2.2 Beyond the mean-field approximation 360
Problems 363
References 364
13 Interference and correlations 365
13.1 Tunnelling between two wells 365
13.1.1 Quantum fluctuations 371
13.1.2 Squeezed states 373
13.2 Interference of two condensates 374
13.2.1 Phase-locked sources 375
13.2.2 Clouds with definite particle number 381
13.3 Density correlations in Bose gases 384
13.3.1 Collisional shifts of spectral lines 386
13.4 Coherent matter wave optics 390
13.5 Criteria for Bose-Einstein condensation 394
13.5.1 The density matrix 394
13.5.2 Fragmented condensates 397
Problems 399
References 399
14 Optical lattices 401
14.1 Generation of optical lattices 402
14.1.1 One-dimensional lattices 403
14.1.2 Higher-dimensional lattices 406
14.1.3 Energy scales 407
14.2 Energy bands 409
14.2.1 Band structure for a single particle 409
14.2.2 Band structure for interacting particles 411
14.2.3 Tight-binding model 416
14.3 Stability 418
14.3.1 Hydrodynamic analysis 421
14.4 Intrinsic non-linear effects 423
14.4.1 Loops 423
14.4.2 Spatial period doubling 427
14.5 From superfluid to insulator 431
14.5.1 Mean-field approximation 433
14.5.2 Effect of trapping potential 439
14.5.3 Experimental detection of coherence 439
Problems 441
References 442
15 Lower dimensions 444
15.1 Non-interacting gases 445
15.2 Phase fluctuations 447
15.2.1 Vortices and the Berezinskii-Kosterlitz-Thouless transition 451
15.3 Microscopic theory of phase fluctuations 453
15.3.1 Uniform systems 455
15.3.2 Anisotropic traps 456
15.4 The one-dimensional Bose gas 460
15.4.1 The strong-coupling limit 461
15.4.2 Arbitrary coupling 466
15.4.3 Correlation functions 474
Problems 479
References 480
16 Fermions 481
16.1 Equilibrium properties 483
16.2 Effects of interactions 486
16.3 Superfluidity 489
16.3.1 Transition temperature 491
16.3.2 Induced interactions 496
16.3.3 The condensed phase 498
16.4 Pairing with unequal populations 506
16.5 Boson-fermion mixtures 508
16.5.1 Induced interactions in mixtures 509
Problems 511
References 513
17 From atoms to molecules 514
17.1 Bose-Einstein condensation of molecules 516
17.2 Diatomic molecules 518
17.2.1 Binding energy and the atom-atom scattering length 518
17.2.2 A simple two-channel model 520
17.2.3 Atom-atom scattering 526
17.3 Crossover:From BCS to BEC 527
17.3.1 Wide and narrow Feshbach resonances 528
17.3.2 The BCS wave function 530
17.3.3 Crossover at zero temperature 531
17.3.4 Condensate fraction and pair wave function 535
17.4 Crossover at non-zero temperature 540
17.4.1 Thermal molecules 540
17.4.2 Pair fluctuations and thermal molecules 543
17.4.3 Density of atoms 548
17.4.4 Transition temperature 549
17.5 A universal limit 550
17.6 Experiments in the crossover region 553
17.6.1 Collective modes 553
17.6.2 Vortices 556
Problems 559
References 560
Appendix.Fundamental constants and conversion factors 562
Index 564