1．Fundamentals of Bose-Einstein Condensation

1.1 Indistinguishability of Identical Particles

1.2 Ideal Bose Gas in a Uniform System

1.3 Off-Diagonal Long-Range Order：Bose System

1.4 Off-Diagonal Long-Range Order：Fermi System

1.5 U（1）Gauge Symmetry

1.6 Ground-State Wave Function of a Bose System

1.7 BEC and Superfluidity

1.8 Two-Fluid Model

1.9 Fragmented Condensate

1.9.1 Two-state model

1.9.2 Degenerate double-well model

1.9.3 Spin-1 antiferromagnetic BEC

1.10 Interference Between Independent Condensates

1.11 Feshbach Resonance

2．Weakly Interacting Bose Gas

2.1 Interactions Between Neutral Atoms

2.2 Pseudo-Potential Method

2.3 Bogoliubov Theory

2.3.1 Bogoliubov transformations

2.3.2 Bogoliubov ground state

2.3.3 Low-lying excitations and condensate fraction

2.3.4 Properties of Bogoliubov ground state

2.4 Bogoliubov Theory of Quasi-One-Dimensional Torus

2.4.1 Case of BEC at rest：stability of BEC

2.4.2 Case of rotating BEC：Landau criterion

2.4.3 Ground state of BEC in rotating torus

2.5 Bogoliubov-de Gennes（BdG）Theory

2.6 Method of Binary Collision Expansion

2.6.1 Equation of state

2.6.2 Cluster expansion ofpartition function

2.6.3 Ideal Bose and Fermi gases

2.6.4 Matsubara formula

3．Trapped Systems

3.1 Ideal Bose Gas in a Harmonic Potential

3.1.1 Transition temperature

3.1.2 Condensate fraction

3.1.3 Chemical potential

3.1.4 Specific heat

3.2 BEC in One-and Two-Dimensional Parabolic Potentials

3.2.1 Density of states

3.2.2 Transition temperature

3.2.3 Condensate fraction

3.3 Semiclassical Distribution Function

3.4 Gross-Pitaevskii Equation

3.5 Thomas-Fermi Approximation

3.6 Collective Modes in the Thomas-Fermi Regime

3.6.1 Isotropic harmonic potential

3.6.2 Axisymmetric trap

3.6.3 Scissors mode

3.7 Variational Method

3.7.1 Gaussian variational wave function

3.7.2 Collective modes

3.8 Attractive Bose-Einstein Condensate

3.8.1 Collective modes

3.8.2 Collapsing dynamics of an attractive condensate

4．Linear Response and Sum Rules

4.1 Linear Response Theory

4.1.1 Linear response of density fluctuations

4.1.2 Retarded response function

4.2 Sum Rules

4.2.1 Longitudinal f-sum rule

4.2.2 Compressibility sum rule

4.2.3 Zero energy gap theorem

4.2.4 Josephson sum rule

4.3 Sum-Rule Approach to Collective Modes

4.3.1 Excitation operators

4.3.2 Virial theorem

4.3.3 Kohn theorem

4.3.4 Isotropic trap

4.3.5 Axisymmetric trap

5．Statistical Mechanics of Superfluid Systems in a Moving Frame

5.1 Transformation to Moving Frames

5.2 Elementary Excitations of a Superfluid

5.3 Landau Criterion

5.4 Correlation Functions at Thermal Equilibrium

5.5 Normal Fluid Density

5.6 Low-Lying Excitations of a Superfluid

5.7 Examples

5.7.1 Ideal Bose gas

5.7.2 Weakly interacting Bose gas

6．Spinor Bose-Einstein Condensate

6.1 Internal Degrees of Freedom

6.2 General Hamiltonian of Spinor Condensates

6.3 Spin-1 BEC

6.3.1 Mean-field theory of a spin-1 BEC

6.3.2 Many-body states in single-mode approximation

6.3.3 Superflow,spin texture,and Berry phase

6.4 Spin-2 BEC

7．Vortices

7.1 Hydrodynamic Theory of Vortices

7.2 Quantized Vortices

7.3 Interaction Between Vortices

7.4 Vortex Lattice

7.4.1 Dynamics of vortex nucleation

7.4.2 Collective modes of a vortex lattice

7.5 Fractional Vortices

7.6 Spin Current

7.7 Fast Rotating BECs

7.7.1 Lowest Landau level approximation

7.7.2 Mean field quantum Hall regime

7.7.3 Many-body wave funetions of a fast rotating BEC

8．Fermionic Superfluidity

8.1 Ideal Fermi Gas

8.2 Fermi Liquid Theory

8.3 Cooper Problem

8.3.1 Two-body problem

8.3.2 Many-body problem

8.4 Bardeen-Cooper-Schriefier (BCS) Theory

8.5 BCS-BEC Crossover at T=0

8.6 Superfluid Transition Temperature

8.7 BCS-BEC Crossover at T≠0

8.8 Gor'kov-Melik-Barkhudarov Correction

8.9 Unitary Gas

8.10 Imbalanced Fermi Systems

8.11 P-Wave Superfluid

8.11.1 Generalized pairing theory

8.11.2 Spin-triplet p-wave states

9．Low-Dimensional Systems

9.1 Non-interacting Systems

9.2 Hohenberg-Mermin-Wagner Theorem

9.3 Two-Dimensional BEC at Absolute Zero

9.4 Berezinskii-Kosterlitz-Thouless Transition

9.4.1 Universal jump

9.4.2 Quasi long-range order

9.4.3 Renormalization-group analysis

9.5 Quasi One-Dimensional BEC

9.6 Tonks-Girardeau Gas

9.7 Lieb-Liniger Model

10．Dipolar Gases

10.1 Dipole-Dipole Interaction

10.1.1 Basic properties

10.1.2 Order of magnitude and length scale

10.1.3 D-wave nature

10.1.4 Tuning the dipole-dipole interaction

10.2 Polarized Dipolar BEC

10.2.1 Nonlocal Gross-Pitaevskii equation

10.2.2 Stability

10.2.3 Thomas-Fermi limit

10.2.4 Quasi two-dimensional systems

10.3 Spinor-Dipolar BEC

10.3.1 Einstein-de Haas effect

10.3.2 Flux closure and ground-state circulation

11．Optical Lattices

11.1 Optical Potential

11.1.1 Optical trap

11.1.2 Optical lattice

11.2 Band Structure

11.2.1 Bloch theorem

11.2.2 Brillouin zone

11.2.3 Bloch oscillations

11.2.4 Wannier function

11.3 Bose-Hubbard Model

11.3.1 Bose-Hubbard Hamiltonian

11.3.2 Superfluid-Mott-insulator transition

11.3.3 Phase diagram

11.3.4 Mean-field approximation

11.3.5 Supersolid

12．Topological Excitations

12.1 Homotopy Theory

12.1.1 Homotopic relation

12.1.2 Fundamental group

12.1.3 Higher homotopy groups

12.2 Order Parameter ManifoId

12.2.1 Isotropy group

12.2.2 Spin-1 BEC

12.2.3 Spin-2 BEC

12.3 Classification of Defects

12.3.1 Domains

12.3.2 Line defects

12.3.3 Point defects

12.3.4 Skyrmions

12.3.5 Influence of different types of defects

12.3.6 Topological charges

Appendix A Order of Phase Transition，Clausius-Clapeyron Formula,and Gibbs-Duhem Relation

Appendix B Bogoliubov Wave Functions in Coordinate Space

B.1 Ground-State Wave Function

B.2 One-Phonon State

Appendix C Effective Mass，Sound Velocity,and Spin Susceptibility of Fermi Liquid

Appendix D Derivation of Eq.（8.155）

Appendix E f-Sum Rule

Bibliography

Index