1．Fundamentals of Bose-Einstein Condensation 1
1.1 Indistinguishability of Identical Particles 1
1.2 Ideal Bose Gas in a Uniform System 3
1.3 Off-Diagonal Long-Range Order：Bose System 6
1.4 Off-Diagonal Long-Range Order：Fermi System 10
1.5 U（1）Gauge Symmetry 11
1.6 Ground-State Wave Function of a Bose System 13
1.7 BEC and Superfluidity 15
1.8 Two-Fluid Model 20
1.9 Fragmented Condensate 23
1.9.1 Two-state model 23
1.9.2 Degenerate double-well model 25
1.9.3 Spin-1 antiferromagnetic BEC 27
1.10 Interference Between Independent Condensates 28
1.11 Feshbach Resonance 31
2．Weakly Interacting Bose Gas 33
2.1 Interactions Between Neutral Atoms 33
2.2 Pseudo-Potential Method 36
2.3 Bogoliubov Theory 40
2.3.1 Bogoliubov transformations 40
2.3.2 Bogoliubov ground state 45
2.3.3 Low-lying excitations and condensate fraction 48
2.3.4 Properties of Bogoliubov ground state 50
2.4 Bogoliubov Theory of Quasi-One-Dimensional Torus 54
2.4.1 Case of BEC at rest：stability of BEC 55
2.4.2 Case of rotating BEC：Landau criterion 56
2.4.3 Ground state of BEC in rotating torus 59
2.5 Bogoliubov-de Gennes（BdG）Theory 60
2.6 Method of Binary Collision Expansion 65
2.6.1 Equation of state 65
2.6.2 Cluster expansion ofpartition function 66
2.6.3 Ideal Bose and Fermi gases 67
2.6.4 Matsubara formula 69
3．Trapped Systems 73
3.1 Ideal Bose Gas in a Harmonic Potential 73
3.1.1 Transition temperature 75
3.1.2 Condensate fraction 76
3.1.3 Chemical potential 77
3.1.4 Specific heat 77
3.2 BEC in One-and Two-Dimensional Parabolic Potentials 79
3.2.1 Density of states 79
3.2.2 Transition temperature 79
3.2.3 Condensate fraction 80
3.3 Semiclassical Distribution Function 81
3.4 Gross-Pitaevskii Equation 83
3.5 Thomas-Fermi Approximation 84
3.6 Collective Modes in the Thomas-Fermi Regime 88
3.6.1 Isotropic harmonic potential 89
3.6.2 Axisymmetric trap 91
3.6.3 Scissors mode 92
3.7 Variational Method 93
3.7.1 Gaussian variational wave function 94
3.7.2 Collective modes 96
3.8 Attractive Bose-Einstein Condensate 98
3.8.1 Collective modes 99
3.8.2 Collapsing dynamics of an attractive condensate 102
4．Linear Response and Sum Rules 105
4.1 Linear Response Theory 105
4.1.1 Linear response of density fluctuations 105
4.1.2 Retarded response function 108
4.2 Sum Rules 109
4.2.1 Longitudinal f-sum rule 110
4.2.2 Compressibility sum rule 112
4.2.3 Zero energy gap theorem 114
4.2.4 Josephson sum rule 115
4.3 Sum-Rule Approach to Collective Modes 120
4.3.1 Excitation operators 121
4.3.2 Virial theorem 122
4.3.3 Kohn theorem 123
4.3.4 Isotropic trap 124
4.3.5 Axisymmetric trap 127
5．Statistical Mechanics of Superfluid Systems in a Moving Frame 129
5.1 Transformation to Moving Frames 129
5.2 Elementary Excitations of a Superfluid 131
5.3 Landau Criterion 133
5.4 Correlation Functions at Thermal Equilibrium 134
5.5 Normal Fluid Density 136
5.6 Low-Lying Excitations of a Superfluid 140
5.7 Examples 141
5.7.1 Ideal Bose gas 141
5.7.2 Weakly interacting Bose gas 143
6．Spinor Bose-Einstein Condensate 145
6.1 Internal Degrees of Freedom 145
6.2 General Hamiltonian of Spinor Condensates 146
6.3 Spin-1 BEC 151
6.3.1 Mean-field theory of a spin-1 BEC 153
6.3.2 Many-body states in single-mode approximation 157
6.3.3 Superflow,spin texture,and Berry phase 161
6.4 Spin-2 BEC 163
7．Vortices 171
7.1 Hydrodynamic Theory of Vortices 171
7.2 Quantized Vortices 174
7.3 Interaction Between Vortices 180
7.4 Vortex Lattice 181
7.4.1 Dynamics of vortex nucleation 181
7.4.2 Collective modes of a vortex lattice 183
7.5 Fractional Vortices 186
7.6 Spin Current 187
7.7 Fast Rotating BECs 189
7.7.1 Lowest Landau level approximation 189
7.7.2 Mean field quantum Hall regime 192
7.7.3 Many-body wave funetions of a fast rotating BEC 194
8．Fermionic Superfluidity 197
8.1 Ideal Fermi Gas 197
8.2 Fermi Liquid Theory 200
8.3 Cooper Problem 205
8.3.1 Two-body problem 205
8.3.2 Many-body problem 209
8.4 Bardeen-Cooper-Schriefier (BCS) Theory 211
8.5 BCS-BEC Crossover at T=0 215
8.6 Superfluid Transition Temperature 219
8.7 BCS-BEC Crossover at T≠0 221
8.8 Gor'kov-Melik-Barkhudarov Correction 225
8.9 Unitary Gas 228
8.10 Imbalanced Fermi Systems 231
8.11 P-Wave Superfluid 234
8.11.1 Generalized pairing theory 234
8.11.2 Spin-triplet p-wave states 238
9．Low-Dimensional Systems 241
9.1 Non-interacting Systems 241
9.2 Hohenberg-Mermin-Wagner Theorem 243
9.3 Two-Dimensional BEC at Absolute Zero 246
9.4 Berezinskii-Kosterlitz-Thouless Transition 247
9.4.1 Universal jump 247
9.4.2 Quasi long-range order 249
9.4.3 Renormalization-group analysis 250
9.5 Quasi One-Dimensional BEC 252
9.6 Tonks-Girardeau Gas 256
9.7 Lieb-Liniger Model 258
10．Dipolar Gases 261
10.1 Dipole-Dipole Interaction 261
10.1.1 Basic properties 261
10.1.2 Order of magnitude and length scale 263
10.1.3 D-wave nature 264
10.1.4 Tuning the dipole-dipole interaction 265
10.2 Polarized Dipolar BEC 266
10.2.1 Nonlocal Gross-Pitaevskii equation 266
10.2.2 Stability 267
10.2.3 Thomas-Fermi limit 269
10.2.4 Quasi two-dimensional systems 271
10.3 Spinor-Dipolar BEC 273
10.3.1 Einstein-de Haas effect 274
10.3.2 Flux closure and ground-state circulation 274
11．Optical Lattices 277
11.1 Optical Potential 277
11.1.1 Optical trap 277
11.1.2 Optical lattice 280
11.2 Band Structure 283
11.2.1 Bloch theorem 283
11.2.2 Brillouin zone 285
11.2.3 Bloch oscillations 286
11.2.4 Wannier function 287
11.3 Bose-Hubbard Model 288
11.3.1 Bose-Hubbard Hamiltonian 288
11.3.2 Superfluid-Mott-insulator transition 289
11.3.3 Phase diagram 291
11.3.4 Mean-field approximation 292
11.3.5 Supersolid 295
12．Topological Excitations 297
12.1 Homotopy Theory 297
12.1.1 Homotopic relation 297
12.1.2 Fundamental group 299
12.1.3 Higher homotopy groups 302
12.2 Order Parameter ManifoId 303
12.2.1 Isotropy group 303
12.2.2 Spin-1 BEC 304
12.2.3 Spin-2 BEC 305
12.3 Classification of Defects 306
12.3.1 Domains 306
12.3.2 Line defects 306
12.3.3 Point defects 311
12.3.4 Skyrmions 313
12.3.5 Influence of different types of defects 316
12.3.6 Topological charges 318
Appendix A Order of Phase Transition，Clausius-Clapeyron Formula,and Gibbs-Duhem Relation 321
Appendix B Bogoliubov Wave Functions in Coordinate Space 323
B.1 Ground-State Wave Function 323
B.2 One-Phonon State 327
Appendix C Effective Mass，Sound Velocity,and Spin Susceptibility of Fermi Liquid 329
Appendix D Derivation of Eq.（8.155） 333
Appendix E f-Sum Rule 335
Bibliography 337
Index 347