玻色-爱因斯坦凝聚的基础与前沿

Preface v

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-FluidModel.

1.9 Fragmented Condensate.

1.9.1 Two-statemodel.

1.9.2 Degenerate double-well model.

1.9.3 Spin-1 antiferromagnetic BEC.

1.10 Interference Between Independent Condensates

Preface v

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-FluidModel.

1.9 Fragmented Condensate.

1.9.1 Two-statemodel.

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-PotentialMethod

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-deGennes (BdG) Theory.

2.6 Method of Binary Collision Expansion.

2.6.1 Equation of state

2.6.2 Cluster expansion of partition 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 Scissorsmode

3.7 VariationalMethod

3.7.1 Gaussian variational wave function

3.7.2 Collectivemodes.

3.8 Attractive Bose-Einstein Condensate

3.8.1 Collectivemodes.

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-sumrule

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 CollectiveModes

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 toMoving 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 FractionalVortices

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 functions 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-Schrieffer (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-LinigerModel

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 PolarizedDipolar 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 Manifold

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

《玻色-爱因斯坦凝聚的基础与前沿》首先介绍了玻色-爱因斯坦凝聚（BEC）的基本理论。之后，本书讨论了快速旋转BEC，旋量和偶极BEC，低维BEC等近来发展迅速的方向。本书还介绍了平衡或非平衡费米液体超流，包括BCS-BEC交叉、幺正气体、p波超流等。本书适合本领域的研究者和研究生阅读。

玻色爱因斯坦凝聚是神奇而富有魅力的物理现象。相关研究已经使多位科学家获得了诺贝尔奖。目前。关于冷原子的研究正蓬勃展开，玻色爱因斯坦凝聚正是其理论基础。本书对于相关领域的研究人员来说是不可错过的佳作。