Fundamentals and New Frontiers of Bose-Einstein Condensation

The following sections are included:

• Preface

• Contents

The following sections are included:

• Indistinguishability of Identical Particles

• Ideal Bose Gas in a Uniform System

• Off-Diagonal Long-Range Order: Bose System

• Off-Diagonal Long-Range Order: Fermi System

• U(1) Gauge Symmetry

• Ground-State Wave Function of a Bose System

• BEC and Superfluidity

• Two-Fluid Model

• Fragmented Condensate
  • Two-state model
  • Degenerate double-well model
  • Spin-1 antiferromagnetic BEC

• Interference Between Independent Condensates

• Feshbach Resonance

The following sections are included:

• Interactions Between Neutral Atoms

• Pseudo-Potential Method

• Bogoliubov Theory
  • Bogoliubov transformations
  • Bogoliubov ground state
  • Low-lying excitations and condensate fraction
  • Properties of Bogoliubov ground state

• Bogoliubov Theory of Quasi-One-Dimensional Torus
  • Case of BEC at rest: stability of BEC
  • Case of rotating BEC: Landau criterion
  • Ground state of BEC in rotating torus

• Bogoliubov–de Gennes (BdG) Theory

• Method of Binary Collision Expansion
  • Equation of state
  • Cluster expansion of partition function
  • Ideal Bose and Fermi gases
  • Matsubara formula

The following sections are included:

• Ideal Bose Gas in a Harmonic Potential
  • Transition temperature
  • Condensate fraction
  • Chemical potential
  • Specific heat

• BEC in One- and Two-Dimensional Parabolic Potentials
  • Density of states
  • Transition temperature
  • Condensate fraction

• Semiclassical Distribution Function

• Gross–Pitaevskii Equation

• Thomas–Fermi Approximation

• Collective Modes in the Thomas–Fermi Regime
  • Isotropic harmonic potential
  • Axisymmetric trap
  • Scissors mode

• Variational Method
  • Gaussian variational wave function
  • Collective modes

• Attractive Bose–Einstein Condensate
  • Collective modes
  • Collapsing dynamics of an attractive condensate

The following sections are included:

• Linear Response Theory
  • Linear response of density fluctuations
  • Retarded response function

• Sum Rules
  • Longitudinal f-sum rule
  • Compressibility sum rule
  • Zero energy gap theorem
  • Josephson sum rule

• Sum-Rule Approach to Collective Modes
  • Excitation operators
  • Virial theorem
  • Kohn theorem
    • Case of m = 0
    • Case of m = ±1
  • Isotropic trap
    • Monopole mode
    • Quadrupole mode
  • Axisymmetric trap

The following sections are included:

• Transformation to Moving Frames

• Elementary Excitations of a Superfluid

• Landau Criterion

• Correlation Functions at Thermal Equilibrium

• Normal Fluid Density

• Low-Lying Excitations of a Superfluid

• Examples
  • Ideal Bose gas
  • Weakly interacting Bose gas

The following sections are included:

• Internal Degrees of Freedom

• General Hamiltonian of Spinor Condensates

• Spin-1 BEC
  • Mean-field theory of a spin-1 BEC
  • Many-body states in single-mode approximation
  • Superflow, spin texture, and Berry phase

• Spin-2 BEC

The following sections are included:

• Hydrodynamic Theory of Vortices

• Quantized Vortices

• Interaction Between Vortices

• Vortex Lattice
  • Dynamics of vortex nucleation
  • Collective modes of a vortex lattice

• Fractional Vortices

• Spin Current

• Fast Rotating BEC_S
  • Lowest Landau level approximation
  • Mean field quantum Hall regime
  • Many-body wave functions of a fast rotating BEC

The following sections are included:

• Ideal Fermi Gas

• Fermi Liquid Theory

• Cooper Problem
  • Two-body problem
  • Many-body problem

• Bardeen–Cooper–Schrieffer (BCS) Theory

• BCS–BEC Crossover at T = 0

• Superfluid Transition Temperature

• BCS–BEC Crossover at T ≠ 0

• Gor'kov–Melik–Barkhudarov Correction

• Unitary Gas

• Imbalanced Fermi Systems

• P-Wave Superfluid
  • Generalized pairing theory
  • Spin-triplet p-wave states

The following sections are included:

• Non-interacting Systems

• Hohenberg–Mermin–Wagner Theorem

• Two-Dimensional BEC at Absolute Zero

• Berezinskii–Kosterlitz–Thouless Transition
  • Universal jump
  • Quasi long-range order
  • Renormalization-group analysis

• Quasi One-Dimensional BEC

• Tonks–Girardeau Gas

• Lieb–Liniger Model

The following sections are included:

• Dipole–Dipole Interaction
  • Basic properties
  • Order of magnitude and length scale
  • D-wave nature
  • Tuning the dipole–dipole interaction

• Polarized Dipolar BEC
  • Nonlocal Gross–Pitaevskii equation
  • Stability
  • Thomas–Fermi limit
  • Quasi two-dimensional systems
    • Roton–maxon spectrum
    • Soliton
    • Quantum ferrofluidity

• Spinor-Dipolar BEC
  • Einstein–de Haas effect
  • Flux closure and ground-state circulation

The following sections are included:

• Optical Potential
  • Optical trap
  • Optical lattice

• Band Structure
  • Bloch theorem
  • Brillouin zone
  • Bloch oscillations
  • Wannier function

• Bose–Hubbard Model
  • Bose–Hubbard Hamiltonian
  • Superfluid–Mott-insulator transition
  • Phase diagram
  • Mean-field approximation
  • Supersolid

The following sections are included:

• Homotopy Theory
  • Homotopic relation
  • Fundamental group
  • Higher homotopy groups

• Order Parameter Manifold
  • Isotropy group
    • Spin-1 BEC
    • Spin-2 BEC

• Classification of Defects
  • Domains
  • Line defects
    • U(1) vortex
    • SO(3) vortex
    • Half-quantum (Alice) vortex
  • Point defects
    • 'tHooft–Polyakov monopole
    • Dirac monopole
  • Skyrmions
    • Shankar Skyrmion
    • Knots
  • Influence of different types of defects
    • Influence of π₁ on π₂
    • Alice ring
  • Topological charges
    • Line defects: π₁
    • Point defects: π₂
    • Skyrmions: π₃

The following sections are included:

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

• Appendix B Bogoliubov Wave Functions in Coordinate Space

• 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