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Bose-Einstein Condensation in Dilute Gases, Second Edition

Book Description

Since an atomic Bose-Einstein condensate, predicted by Einstein in 1925, was first produced in the laboratory in 1995, the study of ultracold Bose and Fermi gases has become one of the most active areas in contemporary physics. This book explains phenomena in ultracold gases from basic principles, without assuming a detailed knowledge of atomic, condensed matter, and nuclear physics. This new edition has been revised and updated, and includes new chapters on optical lattices, low dimensions, and strongly-interacting Fermi systems. This book provides a unified introduction to the physics of ultracold atomic Bose and Fermi gases for advanced undergraduate and graduate students, as well as experimentalists and theorists. Chapters cover the statistical physics of trapped gases, atomic properties, cooling and trapping atoms, interatomic interactions, structure of trapped condensates, collective modes, rotating condensates, superfluidity, interference phenomena, and trapped Fermi gases. Problems are included at the end of each chapter.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Introduction
    1. 1.1 Bose–Einstein condensation in atomic clouds
    2. 1.2 Superfluid 4He
    3. 1.3 Other condensates
    4. 1.4 Overview
    5. Problems
    6. References
  8. 2. The non-interacting Bose gas
    1. 2.1 The Bose distribution
      1. 2.1.1 Density of states
    2. 2.2 Transition temperature and condensate fraction
      1. 2.2.1 Condensate fraction
    3. 2.3 Density profile and velocity distribution
      1. 2.3.1 The semi-classical distribution
    4. 2.4 Thermodynamic quantities
      1. 2.4.1 Condensed phase
      2. 2.4.2 Normal phase
      3. 2.4.3 Specific heat close to Tc
    5. 2.5 Effect of finite particle number
    6. Problems
    7. References
  9. 3. Atomic properties
    1. 3.1 Atomic structure
    2. 3.2 The Zeeman effect
    3. 3.3 Response to an electric field
    4. 3.4 Energy scales
    5. Problems
    6. References
  10. 4. Trapping and cooling of atoms
    1. 4.1 Magnetic traps
      1. 4.1.1 The quadrupole trap
      2. 4.1.2 The TOP trap
      3. 4.1.3 Magnetic bottles and the Ioffe–Pritchard trap
      4. 4.1.4 Microtraps
    2. 4.2 Influence of laser light on an atom
      1. 4.2.1 Forces on an atom in a laser field
      2. 4.2.2 Optical traps
    3. 4.3 Laser cooling: the Doppler process
    4. 4.4 The magneto-optical trap
    5. 4.5 Sisyphus cooling
    6. 4.6 Evaporative cooling
    7. 4.7 Spin-polarized hydrogen
    8. Problems
    9. References
  11. 5. Interactions between atoms
    1. 5.1 Interatomic potentials and the van der Waals interaction
    2. 5.2 Basic scattering theory
      1. 5.2.1 Effective interactions and the scattering length
    3. 5.3 Scattering length for a model potential
    4. 5.4 Scattering between different internal states
      1. 5.4.1 Inelastic processes
      2. 5.4.2 Elastic scattering and Feshbach resonances
    5. 5.5 Determination of scattering lengths
      1. 5.5.1 Scattering lengths for alkali atoms and hydrogen
    6. Problems
    7. References
  12. 6. Theory of the condensed state
    1. 6.1 The Gross–Pitaevskii equation
    2. 6.2 The ground state for trapped bosons
      1. 6.2.1 A variational calculation
      2. 6.2.2 The Thomas–Fermi approximation
    3. 6.3 Surface structure of clouds
    4. 6.4 Healing of the condensate wave function
    5. 6.5 Condensates with dipolar interactions
    6. Problems
    7. References
  13. 7. Dynamics of the condensate
    1. 7.1 General formulation
      1. 7.1.1 The hydrodynamic equations
    2. 7.2 Elementary excitations
    3. 7.3 Collective modes in traps
      1. 7.3.1 Traps with spherical symmetry
      2. 7.3.2 Anisotropic traps
      3. 7.3.3 Collective coordinates and the variational method
    4. 7.4 Surface modes
    5. 7.5 Free expansion of the condensate
    6. 7.6 Solitons
      1. 7.6.1 Dark solitons
      2. 7.6.2 Bright solitons
    7. Problems
    8. References
  14. 8. Microscopic theory of the Bose gas
    1. 8.1 The uniform Bose gas
      1. 8.1.1 The Bogoliubov transformation
      2. 8.1.2 Elementary excitations
      3. 8.1.3 Depletion of the condensate
      4. 8.1.4 Ground-state energy
      5. 8.1.5 States with definite particle number
    2. 8.2 Excitations in a trapped gas
    3. 8.3 Non-zero temperature
      1. 8.3.1 The Hartree–Fock approximation
      2. 8.3.2 The Popov approximation
      3. 8.3.3 Excitations in non-uniform gases
      4. 8.3.4 The semi-classical approximation
    4. Problems
    5. References
  15. 9. Rotating condensates
    1. 9.1 Potential flow and quantized circulation
    2. 9.2 Structure of a single vortex
      1. 9.2.1 A vortex in a uniform medium
      2. 9.2.2 Vortices with multiple quanta of circulation
      3. 9.2.3 A vortex in a trapped cloud
      4. 9.2.4 An off-axis vortex
    3. 9.3 Equilibrium of rotating condensates
      1. 9.3.1 Traps with an axis of symmetry
      2. 9.3.2 Rotating traps
      3. 9.3.3 Vortex arrays
    4. 9.4 Experiments on vortices
    5. 9.5 Rapidly rotating condensates
    6. 9.6 Collective modes in a vortex lattice
    7. Problems
    8. References
  16. 10. Superfluidity
    1. 10.1 The Landau criterion
    2. 10.2 The two-component picture
      1. 10.2.1 Momentum carried by excitations
      2. 10.2.2 Normal fluid density
    3. 10.3 Dynamical processes
    4. 10.4 First and second sound
    5. 10.5 Interactions between excitations
      1. 10.5.1 Landau damping
    6. Problems
    7. References
  17. 11. Trapped clouds at non-zero temperature
    1. 11.1 Equilibrium properties
      1. 11.1.1 Energy scales
      2. 11.1.2 Transition temperature
      3. 11.1.3 Thermodynamic properties
    2. 11.2 Collective modes
      1. 11.2.1 Hydrodynamic modes above Tc
    3. 11.3 Collisional relaxation above Tc
      1. 11.3.1 Relaxation of temperature anisotropies
      2. 11.3.2 Damping of oscillations
    4. Problems
    5. References
  18. 12. Mixtures and spinor condensates
    1. 12.1 Mixtures
      1. 12.1.1 Equilibrium properties
      2. 12.1.2 Collective modes
    2. 12.2 Spinor condensates
      1. 12.2.1 Mean-field description
      2. 12.2.2 Beyond the mean-field approximation
    3. Problems
    4. References
  19. 13. Interference and correlations
    1. 13.1 Tunnelling between two wells
      1. 13.1.1 Quantum fluctuations
      2. 13.1.2 Squeezed states
    2. 13.2 Interference of two condensates
      1. 13.2.1 Phase-locked sources
      2. 13.2.2 Clouds with definite particle number
    3. 13.3 Density correlations in Bose gases
      1. 13.3.1 Collisional shifts of spectral lines
    4. 13.4 Coherent matter wave optics
    5. 13.5 Criteria for Bose–Einstein condensation
      1. 13.5.1 The density matrix
      2. 13.5.2 Fragmented condensates
    6. Problems
    7. References
  20. 14. Optical lattices
    1. 14.1 Generation of optical lattices
      1. 14.1.1 One-dimensional lattices
      2. 14.1.2 Higher-dimensional lattices
      3. 14.1.3 Energy scales
    2. 14.2 Energy bands
      1. 14.2.1 Band structure for a single particle
      2. 14.2.2 Band structure for interacting particles
      3. 14.2.3 Tight-binding model
    3. 14.3 Stability
      1. 14.3.1 Hydrodynamic analysis
    4. 14.4 Intrinsic non-linear effects
      1. 14.4.1 Loops
      2. 14.4.2 Spatial period doubling
    5. 14.5 From superfluid to insulator
      1. 14.5.1 Mean-field approximation
      2. 14.5.2 Effect of trapping potential
      3. 14.5.3 Experimental detection of coherence
    6. Problems
    7. References
  21. 15. Lower dimensions
    1. 15.1 Non-interacting gases
    2. 15.2 Phase fluctuations
      1. 15.2.1 Vortices and the Berezinskii–Kosterlitz–Thouless transition
    3. 15.3 Microscopic theory of phase fluctuations
      1. 15.3.1 Uniform systems
      2. 15.3.2 Anisotropic traps
    4. 15.4 The one-dimensional Bose gas
      1. 15.4.1 The strong-coupling limit
      2. 15.4.2 Arbitrary coupling
      3. 15.4.3 Correlation functions
    5. Problems
    6. References
  22. 16. Fermions
    1. 16.1 Equilibrium properties
    2. 16.2 Effects of interactions
    3. 16.3 Superfluidity
      1. 16.3.1 Transition temperature
      2. 16.3.2 Induced interactions
      3. 16.3.3 The condensed phase
    4. 16.4 Pairing with unequal populations
    5. 16.5 Boson–fermion mixtures
      1. 16.5.1 Induced interactions in mixtures
    6. Problems
    7. References
  23. 17. From atoms to molecules
    1. 17.1 Bose–Einstein condensation of molecules
    2. 17.2 Diatomic molecules
      1. 17.2.1 Binding energy and the atom–atom scattering length
      2. 17.2.2 A simple two-channel model
      3. 17.2.3 Atom–atom scattering
    3. 17.3 Crossover: From BCS to BEC
      1. 17.3.1 Wide and narrow Feshbach resonances
      2. 17.3.2 The BCS wave function
      3. 17.3.3 Crossover at zero temperature
      4. 17.3.4 Condensate fraction and pair wave function
    4. 17.4 Crossover at non-zero temperature
      1. 17.4.1 Thermal molecules
      2. 17.4.2 Pair fluctuations and thermal molecules
      3. 17.4.3 Density of atoms
      4. 17.4.4 Transition temperature
    5. 17.5 A universal limit
    6. 17.6 Experiments in the crossover region
      1. 17.6.1 Collective modes
      2. 17.6.2 Vortices
    7. Problems
    8. References
  24. Appendix. Fundamental constants and conversion factors
  25. Index