1 Introduction ................................................. 1
PART I
2 Long-range Order, Symmetry Breaking, and Order Parameter ..... 9
2.1 One-body density matrix and long-range order ............ 9
2.2 Order parameter ........................................ 13
3 The Ideal Bose Gas .......................................... 15
3.1 The ideal Bose gas in the grand canonical ensemble ..... 15
3.2 The ideal Bose gas in the box .......................... 19
3.3 Fluctuations and two-body density ...................... 26
4 Weakly Interacting Bose Gas ................................. 29
4.1 Lowest-order approximation: ground state energy and
equation of state ...................................... 29
4.2 Higher-order approximation: excitation spectrum and
quantum fluctuations ................................... 33
4.3 Particles and elementary excitations ................... 37
5 Nonuniform Bose Gases at Zero Temperature ................... 42
5.1 The Gross-Pitaevskii equation .......................... 42
5.2 Thomas-Fermi limit ..................................... 47
5.3 Vortex line in the weakly interacting Bose gas ......... 48
5.4 Vortex rings ........................................... 51
5.5 Solitons ............................................... 55
5.6 Small-amplitude oscillations ........................... 60
6 Superfluidity ............................................... 65
6.1 Landau's criterion of superfluidity .................... 65
6.2 Bose-Einstein condensation and superfluidity ........... 69
6.3 Hydrodynamic theory of superfluids: zero temperature ... 70
6.4 Quantum hydrodynamics .................................. 71
6.5 Beliaev decay of phonons ............................... 74
6.6 Two-fluid hydrodynamics: flrst and second sound ........ 76
6.7 Fluctuations of the phase .............................. 81
6.8 Rotation of superfluids ................................ 85
7 Linear Response Function .................................... 89
7.1 Dynamic structure factor and sum rules ................. 89
7.2 Density response function .............................. 94
7.3 Current response function .............................. 98
7.4 General inequalities .................................. 100
7.5 Response function of the ideal Bose gas ............... 105
7.6 Response function of the weakly interacting Bose gas .. 107
8 Superfluid 4Не ............................................. 110
8.1 Elementary excitations and dynamic structure factor ... 110
8.2 Thermodynamic properties .............................. 118
8.3 Quantized vortices .................................... 121
8.4 Momentum distribution and Bose-Einstein condensation .. 125
9 Atomic Gases: Collisions and Trapping ...................... 130
9.1 Metastability and the role of collisions .............. 130
9.2 Low-energy collisions and scattering length ........... 132
9.3 Low-energy coUisions in two dimensions ................ 141
9.4 Zeeman effect and magnetic trapping ................... 143
9.5 Interaction with the radiation field and optical
traps ................................................. 148
PART II
10 The Ideal Bose Gas in the Harmonic Trap .................... 153
10.1 Condensate fraction and critical temperature .......... 153
10.2 Density of single-particle states and thermodynamics .. 156
10.3 Density and momentum distribution ..................... 158
10.4 Thermodynamic limit ................................... 161
10.5 Release of the trap and expansion of the gas .......... 161
10.6 Bose-Einstein condensation in deformed traps .......... 163
10.7 Adiabatic formation of ВЕС with non-harmonic traps .... 164
11 Ground State of a Trapped Condensate ....................... 168
11.1 An instructive example: the box potential ............. 168
11.2 Interacting condensates in harmonic traps: density
and momentum distribution ............................. 170
11.3 Energy, chemical potential, and virial theorem ........ 173
11.4 Finite-size corrections to the Thomas-Fermi limit ..... 175
11.5 Beyond-mean-field corrections ......................... 180
11.6 Attractive forces ..................................... 182
12 Dynamics of a Trapped Condensate ........................... 184
12.1 Collective oscillations ............................... 184
12.2 Repulsive forces and the Thomas-Fermi limit ........... 187
12.3 Sum rule approach: from repulsive to attractive
forces ................................................ 193
12.4 Finite-size corrections to the Thomas-Fermi limit ..... 196
12.5 Beyond-mean-field corrections ......................... 196
12.6 Large-amplitude oscillations .......................... 198
12.7 Expansion of the condensate ........................... 200
12.8 Dynamic structure factor .............................. 202
12.9 Collective versus single-particle excitations ......... 213
13 Thermodynamics of a Trapped Bose Gas ....................... 217
13.1 Role of interactions, scaling, and thermodynamic
limit ................................................. 217
13.2 The Hartree-Fock approximation ........................ 220
13.3 Shift of the critical temperature ..................... 223
13.4 Critical region near Тc ............................... 226
13.5 Below Тc .............................................. 228
13.6 Equation of state and density profiles ................ 233
13.7 Collective oscillations at a finite temperature ....... 236
14 Superfluidity and Rotation of a Trapped Bose Gas ........... 238
14.1 Critical velocity of a superfluid ..................... 238
14.2 Moment of inertia ..................................... 241
14.3 Scissors mode ......................................... 245
14.4 Expanding a rotating condensate ....................... 247
14.5 Rotation at higher angular velocities ................. 248
14.6 Quantized vortices .................................... 252
14.7 Vortices, angular momentum, and collective
oscillations .......................................... 259
14.8 Stability and precession of the vortex line ........... 266
14.9 Quantized vortices and critical velocity in
a toroidal trap ....................................... 269
15 Coherence, Interference, and the Josephson Effect .......... 272
15.1 Coherence and the one-body density matrix ............. 273
15.2 Interference between two condensates .................. 276
15.3 Double-well potential and the Josephson effect ........ 284
15.4 Quantization of the Josephson equations ............... 290
15.5 Decoherence and phase spreading ....................... 296
15.6 Boson Hubbard Hamiltonian ............................. 297
PART III
16 Interacting Fermi Gases and the BCS-BEC Crossover .......... 305
16.1 The ideal Fermi gas ................................... 305
16.2 Dilute interacting Fermi gases ........................ 307
16.3 The weakly repulsive Fermi gas ........................ 307
16.4 Gas of composite bosons ............................... 309
16.5 The BCS limit of a weakly attractive gas .............. 311
16.6 Gas at unitarity ...................................... 313
16.7 The BCS-BEC crossover ................................. 319
16.8 The Bogoliubov-de Gennes approach to the BCS-BEC
crossover ............................................. 323
16.9 Equation of state, momentum distribution, and
condensate fraction of pairs .......................... 328
17 Fermi Gas in the Harmonic Trap ............................. 333
17.1 The harmonically trapped ideal Fermi gas .............. 333
17.2 Equation of state and density profiles ................ 336
17.3 Momentum distribution ................................. 339
18 Tan Relations and the Contact Parameter .................... 341
18.1 Wave function of a dilute Fermi gas near a Feshbach
resonance ............................................. 341
18.2 Tails of the momentum distribution .................... 342
18.3 Dependence of energy on scattering length ............. 344
18.4 Relation between the energy and the momentum
distribution .......................................... 347
18.5 Static structure factor ............................... 348
18.6 The contact of a harmonically trapped gas ............. 350
19 Dynamics and Superfluidity of Fermi Gases .................. 354
19.1 Hydrodynamics at zero temperature: sound and
collective oscillations ............................... 354
19.2 Expansion of a superfluid Fermi gas ................... 360
19.3 Phonon versus pair-breaking excitations and Landau's
critical velocity ..................................... 362
19.4 Dynamic structure factor .............................. 365
19.5 Radiofrequency transitions ............................ 366
19.6 Two-fluid hydrodynamics: first and second sound ....... 370
19.7 Rotations and vortices ................................ 376
20 Spin-polarized Fermi Gases ................................. 383
20.1 Magnetic properties of the weakly repulsive Fermi
gas ................................................... 383
20.2 Superfluidity and magnetization ....................... 385
20.3 Phase separation at unitarity ......................... 387
20.4 Phase separation in harmonic traps at unitarity ....... 390
20.5 The Fermi polaron ..................................... 394
PART IV
21 Quantum Mixtures and Spinor Gases .......................... 401
21.1 Mixtures of Bose-Einstein condensates ................. 401
21.2 Spinor Bose-Einstein condensates ...................... 407
21.3 Coherently coupled Bose-Einstein condensates .......... 409
21.4 Synthetic gauge fields and spin-orbit coupling ........ 414
21.5 Fermi-Bose mixtures ................................... 424
22 Quantum Gases in Optical Lattices .......................... 428
22.1 Single-particle properties in an optical lattice ...... 428
22.2 Equilibrium properties of a Bose-Einstein condensate .. 432
22.3 Localization in one-dimensional quasiperiodic
potentials ............................................ 437
22.4 Equilibrium properties of a Fermi gas in a lattice .... 440
22.5 Bloch oscillations .................................... 442
22.6 Elementary excitations of ВЕС gases in an optical
lattice ............................................... 445
22.7 Centre-of-mass oscillation of a Fermi gas in an
optical lattice ....................................... 451
22.8 Dimer formation in periodic potentials ................ 452
22.9 Quantum fluctuations in optical lattices and the
Bose-Hubbard model .................................... 454
23 Quantum Gases in Pancake and Two-dimensional Regimes ....... 459
23.1 From three-dimensional pancakes to the
two-dimensional regime ................................ 460
23.2 Two-dimensional Bose gas at finite temperatures ....... 465
23.3 Fast-rotating Bose gases and the lowest Landau level
regime ................................................ 471
23.4 Two-dimensional Fermi gas: the BEC-BCS crossover ...... 475
24 Quantum Gases in Cigar and One-dimensional Regimes ......... 482
24.1 Bose gas: from three-dimensional radial cigars to
the one-dimensional mean-field regime ................. 482
24.2 Solitons and vortical configurations in cigar traps ... 487
24.3 Phase fluctuations and long-range behaviour of the
off-diagonal one-body density ......................... 492
24.4 Lieb-Liniger theory: from the one-dimensional mean
field to the Tonks-Girardeau limit .................... 495
24.5 Dynamic structure factor and superfluidity ............ 503
24.6 One-dimensional Fermi gas ............................. 506
25 Dipolar Gases .............................................. 512
25.1 The dipole-dipole force ............................... 512
25.2 Harmonic trapping and stability of dipolar ВЕС gases .. 516
25.3 Dynamic behaviour of dipolar gases .................... 521
25.4 Dipolar Fermi gases ................................... 525
References ................................................. 527
Index ......................................................... 551
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