The Quantum World of Ultra-Cold Atoms and Light Book III: Ultra-Cold Atoms

The Table of Contents for the book is as follows:

• I FOUNDATIONS
  • The Quest for Cold Atoms
    • Ultra-Cold Atoms—a New Field of Physics
      • The Legacy of the Past
      • Bose–Einstein Condensation of Atomic Hydrogen
    • The Road to Bose–Einstein Condensation
      • Bose–Einstein Condensation in Rubidium—the First Bose–Einstein Condensate
    • Trapping of Neutral Atoms
      • Optical Molasses and the Magneto-Optical Trap
      • Magnetic Traps
      • Optical Trapping
    • The New Paradigms
      • Atoms, Light and Electromagnetism
      • The Quantum-Stochastic Paradigm
      • Effective Field Theory
      • The Scope of this Book
  • Statistical Mechanics of an Ultra-Cold Gas
    • Kinetic Processes Leading to Bose–Einstein Condensation
    • Bose–Einstein Condensation of an Ideal Gas
      • Statistical Mechanics of the Ideal Bose Gas
      • Bose–Einstein Condensation
      • Other Kinds of Trap
    • Quantum Field Theory of Ultra-Cold Atoms
      • Many-Body Wavefunctions
      • Hamiltonian for an Interacting Quantum Gas
      • Many-Body Schrödinger Equation
      • The Grand Canonical Hamiltonian
    • Kinetic Theory and the Ultra-Cold Gas
      • Quantum Kinetic Theory
    • Degenerate Fermi Gases
  • The Effective Hamiltonian for an Ultra-Cold Gas
    • The Effective Hamiltonian for an Ultra-Cold Gas
      • The Effective Hamiltonian
      • The Interaction Expressed in Terms of the Scattering Length
      • Applicability of Perturbation Theory
      • The Fermi Pseudopotential
    • Heisenberg Equations of Motion for Field Operators
      • Expression as a Projector
    • Using the Effective Hamiltonian
• II BOSE–EINSTEIN CONDENSATION
  • Pure Bose–Einstein Condensates
    • Modelling a Pure Bose–Einstein Condensate
      • Heuristic Justification of the Gross–Pitaevskii Equation
      • The Classical Field Interpretation
    • Stationary Solutions of the Gross–Pitaevskii Equation
      • The Chemical Potential
      • Methods for Stationary Solutions when the Scattering Length is Positive
      • The Hard Wall Potential
      • The Potential Box
      • Solitons
      • The Case of Negative Scattering Length—Bright Solitons
      • The Harmonic Potential
      • Momentum Space Wavefunction for a Harmonic Trap
    • The Excitation Spectrum of the Gross–Pitaevskii Equation
      • The Bogoliubov–de Gennes Equations
      • Properties of the Eigenfunctions
      • Orthogonal Eigenfunctions
      • Excitation Spectrum for a Homogeneous Condensate
      • Excitation Spectrum for Harmonically Trapped Systems
      • Generalizations of the Bogoliubov–de Gennes Equations
  • The Bogoliubov Theory
    • Bogoliubov Theory for a Trapped Condensate
      • Asymptotic Expansion in Powers of the Number of Atoms
      • Separation into Condensate and Above-Condensate Operators
      • The Transfer Operators
      • Expansion of the Hamiltonian
      • Ground State and Excitation Spectrum
    • The Homogeneous Condensate
      • The Bogoliubov Hamiltonian
      • The Bogoliubov Transformation
      • Quasiparticles
      • Ground State Properties
      • Properties at Non-Zero Temperatures
    • Trapped Condensates
      • Expansion in Terms of the Orthogonal Eigenfunctions
      • Thermal Properties
    • Bose Broken Symmetry
      • Condensate Coherent States
      • The Grand Canonical Hamiltonian
      • Application to the Untrapped Bose Gas
      • The Concept of “Bose Broken Symmetry”
    • Beyond the Bogoliubov Theory
  • The Hartree–Fock–Bogoliubov Method
    • The Hartree–Fock–Bogoliubov Equations
      • The Separation into Condensate and Non-Condensate
      • Derivation of Basic Hartree–Fock–Bogoliubov Equations
      • Excitation Spectrum for Homogeneous Condensate
      • The Hartree–Fock–Bogoliubov Method in the Popov Approximation
      • The Hartree–Fock–Bogoliubov Hamiltonian
      • Fundamental Difficulties
    • Using the Hartree–Fock–Bogoliubov Method
      • Consistent Perturbation Theory
      • The Extension to the ZNG Theory
  • Coherence, Correlations and Condensation
    • Measurable Correlation Functions
      • The First Order Correlation Function
      • The Second-Order Correlation Function
      • The One-Point Second-Order Correlation Function
      • The Third-Order Correlation Function
    • The Penrose–Onsager Definition of a Condensate
      • The Penrose–Onsager Criterion for Bose–Einstein Condensation
      • Using the Penrose–Onsager Formulation
      • Non-Ground-State Condensates
      • Off-Diagonal Long-Range Order
      • The Jastrow Wavefunction
    • Time Correlation Functions and Spectroscopy
      • Ramsey Spectroscopy
      • Description of a Ramsey Experiment
• III QUANTUM KINETIC THEORY
  • Quantum Kinetic Theory of Homogeneous Systems
    • Wavelet Expansion of Field Operators
      • The Wavelets
      • Wavelet Expansion of the Field Operators
      • Momentum-Resolved Field Operators
      • Free Hamiltonian in Terms of Momentum-Resolved Field Operators
      • Interaction Hamiltonian in Terms of the Momentum-Resolved Field Operators
    • Quantum Stochastic Equations Resulting from Collisions
      • Projectors and Eigenstates
      • Formal Derivation of the Quantum Kinetic Master Equation
      • Markov Approximation
      • Explicit Form of the Quantum Kinetic Master Equation
      • Stationary Solutions
      • Fermionic Quantum Kinetic Theory
      • Assessment of the Approximations
    • The Effects of Bose–Einstein Condensation on the Quantum Kinetic Master Equation
      • The Effect of Bose–Einstein Condensation on the Value of Δ
      • The Condensate Band and its Hamiltonian
      • Dynamics of the Condensate Band
    • Applications of the Quantum Kinetic Master Equation
      • The Quantum Boltzmann Master Equation
      • The Quantum Boltzmann Equation
    • Summary
  • The Quantum Boltzmann Equation
    • Formulation of the Quantum Boltzmann Equation
      • The Classical Boltzmann Equation
      • The Quantum Boltzmann Equation
      • Stationary Solutions
    • The High-Density Quantum Boltzmann Equation
      • Limiting Forms
      • Relationship to the Gross–Pitaevskii Equation
    • The Ergodic Form of the Quantum Boltzmann Equation
      • The Homogeneous Case
      • Harmonic Trapping
      • Applications
    • Appendix: The Boltzmann Equation
      • Quantum Kinetic Notation
  • Quantum Kinetic Master Equations
    • Formulation of the Coherent and Incoherent Bands
      • Projectors into the Bands
      • Separation of the Hamiltonian
      • Intra-Band Hamiltonians
      • The Interaction Hamiltonians
    • The Case of a Fully Thermalized Incoherent Band
      • Derivation of the Master Equation
      • Hamiltonian Terms
      • Kinetic Terms
      • Evaluating the Kinetic Coefficients
    • Explicit Form of the Master Equation
      • Notation—the L_C Operator
      • Kinetic Terms in the Master Equation
      • The Full Master Equation and its Stationary Solution
      • The Projection into the Coherent Band
      • Application of the Quantum Kinetic Master Equation
    • The Finite-Temperature Master Equation
      • Use of the Backward–Forward Relations
      • Growth Terms
      • Scattering Terms
    • The Simple Growth Equation
      • Interpretation of the Simple Growth Equation
  • The Stochastic Gross–Pitaevskii Equation
    • C-Fields, and the Wigner Function Mapping
      • Quantum Noise
      • Coherent State C-Fields
    • Equations of Motion and Operator Mappings
      • Projected Derivative Notation
      • Operator Correspondences
    • Wigner Function Mapping of Hamiltonian Terms
      • The Truncated Wigner Approximation
      • The Projected Gross–Pitaevskii Equation
      • Validity Criteria for the Truncated Wigner Approximation
    • Wigner Function Mapping of the Kinetic Terms
      • Derivation of Fokker–Planck Equations
      • The Quantum Kinetic Fokker–Planck Equation
      • Stationary Solutions of the Quantum Kinetic Fokker–Planck Equation
    • Stochastic Gross–Pitaevskii Equations
      • Terms in the Stochastic Gross–Pitaevskii Equations
      • The Projected Gross–Pitaevskii Equation
      • The Simple Growth Stochastic Gross–Pitaevskii Equation
      • The Scattering Stochastic Gross–Pitaevskii Equation
      • Comparison of the Scattering and Growth Coefficients
    • Comparison with Other Methods
      • The Method of Stoof
      • The Method of Zaremba, Nikuni and Griffin
      • Conclusion
  • Growth of a Bose–Einstein Condensate
    • Modelling Condensate Growth Experiments
      • Modelling the MIT Condensate Growth Experiment
      • The Munich Experiment of 2002
    • Condensate Growth and Phase Coherence
    • Simplified Equations for Growth and Thermalization
      • The Determination of the Energy Cutoff
      • The Growth and Thermalization Equations
      • The Ergodic Distribution Function
      • The Fokker–Planck Equation for Growth and Thermalization
    • The Growth Fokker–Planck Equation
      • Calculating the Helmholtz Free Energy
      • The Simple Growth Fokker–Planck Equation
    • Simulations Using the Growth Equation
      • Simulation Procedure
    • Appendix: Derivation of the Growth Fokker–Planck Equation
• IV QUANTUM MATTER WAVES
  • The Hydrodynamic Approximation
    • Summary of Classical Hydrodynamics
      • The Continuity Equation
      • Equation of Motion—Euler’s Equation
      • Vorticity
      • Irrotational Flow
      • Velocity Potential
    • Hydrodynamic Equations for a Bose–Einstein Condensate
      • Continuity Equation
      • Derivation of the Hydrodynamic Approximation
      • Euler’s Equation in the Hydrodynamic Approximation
      • Local Energy Density in the Hydrodynamic Approximation
      • Static Solution in the Hydrodynamic Approximation
      • Sound Waves
    • Solutions Using the Hydrodynamic Approximation
      • The Bogoliubov–de Gennes Amplitudes
      • Spherically Symmetric Harmonic Trap
      • Sound Propagation in a Cylindrically Symmetric Harmonic Trap
      • An Elongated Ellipsoidal Harmonic Trap
    • Quantized Hydrodynamics
      • The Hydrodynamic Hamiltonian
      • Landau’s Theory of Superfluidity in Helium
    • Exact Time-Dependent Solutions of the Gross–Pitaevskii Equation
      • The Kohn Mode
      • Linear Potential
      • Constant Potential and Solitons
  • Vortices in Bose Systems
    • Vortices and the Gross–Pitaevskii Equation
      • Vortex Solutions of the Gross–Pitaevskii Equation
      • Solutions
      • Properties of the Velocity Field
      • Energy of a Vortex
    • Vortex Dynamics
      • Vortex Motion in a Flow Field
      • A Pair of Vortices
      • The Motion of a Vortex Pair
      • States with Multiple Vortices
    • Vortices in a Trapped Bose–Einstein Condensate
      • Thomas–Fermi Modelling in an Axially Symmetric Trap
      • Vortex Motion in an Axially Symmetric Trap
      • Quantized Vortex Lattices
    • The Rotating Stochastic Gross–Pitaevskii Equation
      • Transformation to a Rotating Frame
      • Formulation of the Rotating Stochastic Gross–Pitaevskii Equation
      • Components of the Rotating Stochastic Gross–Pitaevskii Equation
    • The Experimental Production of Vortices
      • Early Experiments
      • The Effect of Anharmonicity
    • Modelling Vortex Generation
      • Stirring a Condensate
      • Dissipation without Noise: Vortex Lattice Formation from a Rotating Thermal Cloud
      • Stirring and Dynamical Instabilities
      • Rotating Bose–Einstein Condensation
    • Appendix: Calculation of Dissipation Rates
      • Growth
      • Scattering
  • Quantum Matter Wave Dynamics
    • C-Fields for Pure and Thermal Condensates
      • Approximating the Wigner Function
      • Stochastic Sampling of the Wigner Function
      • Alternative Sampling Methods
    • The Projected Gross–Pitaevskii Equation
      • The “Classical Region”
      • The Projector and Numerical Grid Methods
      • Harmonically Trapped Systems
    • The Projected Gross–Pitaevskii Equation Formalism for Systems Close to Equilibrium
      • A Realistic Example
      • Thermodynamic Quantities: Temperature and Chemical Potential
      • Including the Atoms in the Incoherent Region
      • Validity Conditions for Projected Gross–Pitaevskii Equation Simulations
    • Applications of the Projected Gross–Pitaevskii Equation
      • Critical Temperature of a Uniform Bose Gas
      • Critical Temperature of a Trapped Bose Gas
      • Condensate Collisions
    • Applications of the Stochastic Gross–Pitaevskii Equation
      • Spontaneous Vortex Formation During Bose–Einstein Condensation
      • Including Three-Body Loss in the Stochastic Gross–Pitaevskii Equation
    • Alternatives to Truncated Wigner Function Methods
      • Accuracy and Interpretation of the Truncated Wigner Method
• V ULTRA-COLD MOLECULES AND SCATTERING
  • Scattering Theory at Very Low Energies
    • The Scattering Amplitude
      • The Differential Cross Section
      • Total Cross Section
      • Meaning of the Phase Shift
      • Scattering Length and Effective Range
      • The Spherical Step
      • The Universal Relationship between Binding Energy and Scattering Length
      • The “Unitarity Limit”
    • The Integral Equations of Scattering Theory
      • Green’s Functions for the Free Particle
      • The Lippmann–Schwinger Equation
      • Asymptotic Analysis
      • Scattering Amplitude
      • The Born Approximation
    • The T-Matrix
      • Lippmann Schwinger Equation for the T-Matrix
      • The Off-Energy-Shell T-Matrix
      • Operator Form of the Lippmann–Schwinger Equation
      • Unitarity and the Optical Theorem
    • Separable Potentials
      • Definition of a Separable Potential
      • Bound States
      • Scattering with a Separable Potential
    • The Contact Potential
      • The T-Matrix
      • The Bound State
      • The Contact Potential Limit
    • Application to Ultra-Cold Gases
  • Modelling the Interatomic Interaction
    • Interatomic Potentials
      • The Born–Oppenheimer Approximation
      • Electronic and Nuclear Spin
      • Wavefunctions and Interatomic Potentials
      • The van der Waals Length and the Dimensionless Schrödinger Equation
      • The van der Waals Parameters
    • Calculating the Scattering Length
      • The Scattering Length
      • Inner and Outer Regions of the Interatomic Potential
      • Matching the Solutions at r_*
    • Bound States
      • Length Scales
      • Bound State Wavefunctions
      • Determination of a Bound State
      • WKB Approximation for the Bound States
      • The Connection between Binding Energy and Scattering Length
    • Scattering Amplitude and Phase Shifts
      • Application of the WKB Approximation to Scattering
    • Feshbach Resonances
      • Separated States and Scattering Channels
      • Energy Levels of the ⁸⁵Rb Dimer
      • The Feshbach Resonance in ⁸⁵Rb at B = 155.041Gauss
  • Effective Field Theory
    • Derivation of the Effective Field Theory
      • Hamiltonian in Momentum Space
      • Separation into High and Low Momenta
      • Derivation of a Projected Schrödinger Equation
      • Explicit Form of the Projected Schrödinger Equation
    • Analysis of the Effective Field Theory
      • Evaluating the Effective Potential—the Influence of a Bound State
      • Consistency
      • Dependence on Centre of Mass Motion
    • Phenomenological Effective Field Theory
      • Renormalization
    • Pseudopotentials
      • The Original Fermi Pseudopotential
      • Huang–Yang Pseudopotential Method
      • Critique of Pseudopotential Methods
    • Appendix: The T-Matrix for a Truncated Potential
      • Evaluation of the Cutoff-Dependent Term
      • Elimination of Centre of Mass Dependence
  • Atom-Molecule Systems
    • The Effective Field Theory for Atoms and Molecules
      • The Molecular Field Hamiltonian
    • Scattering and Bound States
      • Schrödinger Equation in the 2-Atom: 1-Molecule Sector
      • The Yamaguchi Equation
      • The Bound State
      • The Scattering Amplitude
      • Region of Validity of the Expression for the Scattering Amplitude
      • Scattering Length and Effective Range
      • Feshbach Resonance and Molecules of ⁸⁵Rb
      • Elementary and Compound Molecules
    • The Quantum Gas of Interacting Atoms and Molecules
      • C-Field Equations
      • The Gross–Pitaevskii Limit
      • Stationary Solutions and the Thomas–Fermi Limit
      • Bogoliubov Theory of Atom-Molecule Systems
      • Application to a Uniform Condensate
    • Appendix: Orthogonality and Normalization Conditions
  • Field Theory of Feshbach Resonances
    • Essential Features of a Feshbach Resonance
      • Relevant Quantum States
    • The Feshbach Resonance Effective Hamiltonian
      • Atomic and Molecular Fields
      • Effect of the Exchange Interaction
      • Interaction Parameters
    • Scattering Amplitude and Bound States
      • Schrödinger Equation in the 2-Atom: 1-Molecule Sector
      • Elimination of Closed-Channel Atoms
      • Scattering Length
      • Binding Energy
    • Determination of Parameters
      • The Scattering Length
      • The Binding Energy
      • Effective Field Theory and Choice of Cutoff Value
• VI ULTRA-COLD FERMIONS
  • The Ideal Fermi Gas
    • Statistical Mechanics of the Ideal Fermi Gas
      • The Fermi–Dirac Distribution
      • Criterion for a Degenerate Fermi Gas
    • The Ideal Fermi Gas in a Potential Trap
      • Semiclassical Distribution
      • Semiclassical Distribution Functions at Zero Temperature
      • Comparison with Bose–Einstein Condensate Distributions
      • Behaviour at Non-Zero Temperatures
    • Quantum Field Theory of Fermions
      • Spin Components
      • Hamiltonian for an Interacting Fermi Gas
      • Effective Field Theory
    • Quantum Kinetic Theory for Fermions
      • Issues Specific to Fermi Systems
      • Fermionic Quantum Boltzmann Equation
  • The Interacting Fermi Gas
    • Interaction Regimes
      • The Repulsive Gas
      • The Attractive Fermi Gas
    • Cooper Pairing
      • Binding when the Centre of Mass is at Rest
      • The Nature of Cooper Pair Binding
    • The BCS Theory
      • The BCS Hamiltonian
      • The Bogoliubov–de Gennes Hamiltonian
      • Diagonalization Using a Bogoliubov Transformation
      • The Fermionic Bogoliubov–de Gennes Equations
    • The BCS Theory for a Homogeneous Fermi Gas
      • Evaluation of the BCS Wavefunctions
      • The BCS Ground State Wavefunction
    • Calculating the Mean Field Potentials
      • Renormalization
    • Qualitative Behaviour of Solutions of the Gap Equation
      • Criterion for a Non-Zero Gap at Absolute Zero
      • The Gap for Non-Zero Temperatures
    • Ground State Energy and Renormalization
      • Renormalization of the Gap-Dependent Term
      • Renormalization of the Hartree-Dependent Term
      • Renormalized Ground State Energy
    • Limitations and Extensions of the BCS Theory
      • The Variational Basis for the BCS Theory
      • Unbalanced Fermi Gases
    • Appendix: Cooper Pairs with Non-Zero Centre of Mass Momentum
      • Minimum Q for Binding to Occur
  • Fermionic Superfluidity
    • Hydrodynamics and the Phase of the Gap Function
      • Hydrodynamic Variables
      • Construction of a Matrix Wigner Function
      • Solutions of the Equations
    • The Superfluid Hydrodynamic Equations
      • Euler’s Equation
      • The Continuity Equation
      • Irrotational Flow
      • Validity of the Approximations
      • The Phase in Fermionic Superfluidity
      • The “Fermionic Gross–Pitaevskii Equation”
      • Sound Waves and Collective Oscillations
  • Resonance Superfluidity and the BEC–BCS Crossover
    • Modelling the BEC–BCS Crossover
      • Typical Fermionic Feshbach Resonances
      • Modelling Philosophies
      • The Unitary Fermi Gas
    • Formulating the Molecular Field Model
      • Fermions and the Molecular Field
      • The BCS Hamiltonian as a Limiting Case
      • A Simplified Model at T = 0
      • The Full Model at Finite Temperature
    • Summary
    • A Appendix: Scattering Parameters for Atoms and Molecules
• VII ATOMS IN OPTICAL LATTICES
  • Bose Atoms in an Optical Lattice
    • Physics of Trapping in an Optical Lattice
      • Bloch Bands
      • Wannier Functions
    • The Bose–Hubbard Model
      • Locality Approximations
      • The Bose–Hubbard Hamiltonian
      • The Tight-Binding Approximation
    • The Ground State of the Bose–Hubbard Hamiltonian
      • The Nature of the Ground States
      • The Gutzwiller Approximation
      • The Self-Consistent Mean Field Method
      • Validity and Limitations
    • Quantum Phases of the Bose–Hubbard Hamiltonian
      • A Homogeneous Lattice
      • Solutions in the Presence of a Harmonic Trap
      • Measurements by Ballistic Expansion
      • Measurements of Atom Numbers
    • The Physics of Lattice Models
      • Numerical Techniques
    • Experimental Realization of Lattice Models
  • The Cold Atom Lattice Toolbox
    • Features of Optical Potentials
      • Lattice Geometry
    • The Influence of Atomic Structure
      • State-Dependent Lattice Features
      • The Effect of Spontaneous Emission
      • The Microscopic Picture
    • Lattice Quantum Computing
      • Entanglement Creation Using Coherent Ground State Collisions
      • Use as a Quantum Gate
      • Lattice Quantum Simulators
  • Engineering Lattice Electromagnetism
    • Periodic Potentials and Magnetic Fields
      • The Uniform Magnetic Field
      • Non-Uniform Magnetic Fields
      • Validity of the Peierls Substitution
      • The Plaquette Formulation
      • The Uniform magnetic Field—the Harper Hamiltonian
    • Emulation by Cold Atoms in an Optical Lattice
      • Description of Atoms in the Optical Lattice
      • Additional Features Needed to Produce Phase-Dependent Hopping
    • A Appendix: Raman Laser Configurations
  • References
  • Bibliography
  • Author Index
  • Subject Index