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