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| Introduction | |
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| Ultracold Atomic Quantum Gases | |
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| Outline | |
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| Part 1 | |
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| Part 2 | |
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| Part 3 | |
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| Gaussian Integrals | |
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| The Gaussian Integral over Real Variables | |
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| Generating Function | |
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| Multi-Dimensional Gaussian Integral | |
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| Complex Analysis | |
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| Differentiation and Contour Integrals | |
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| Laurent Series and the Residue Theorem | |
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| Gaussian Integrals over Complex Variables | |
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| Grassmann Variables | |
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| Problems | |
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| Quantum Mechanics | |
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| Hilbert Spaces | |
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| Observables | |
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| Schr�dinger vs. Heisenberg Picture | |
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| Bosonic Harmonic Oscillator | |
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| Creation and Annihilation Operators | |
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| Three-Dimensional Harmonic Oscillator | |
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| Coherent States | |
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| Fermionic Harmonic Oscillator | |
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| Spin | |
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| Perturbation Theory | |
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| Scattering Theory | |
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| Many-particle Quantum Mechanics | |
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| Problems | |
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| Statistical physics | |
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| Legendre Transformations | |
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| Statistical Physics | |
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| Spin Chain | |
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| Canonical Ensemble | |
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| Grand-Canonical Ensemble | |
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| Ideal Gases | |
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| Ideal Maxwell-Boltzmann Gas | |
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| Ideal Bose Gas: Bose-Einstein Condensation | |
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| Ideal Fermi Gas | |
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| Density Matrix | |
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| Problems | |
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| Path Integrals | |
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| Functionals and Functional Derivatives | |
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| Principle of Least Action | |
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| Phase-Space Representation | |
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| The Feynman Path Integral | |
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| Continuum Limit and Fluctuation Expansion | |
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| Gel'fand-Yaglom Method | |
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| Matrix Elements and Time Ordering | |
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| Quantum-Mechanical Partition Function | |
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| Expectation Values | |
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| Hubbard-Stratonovich Transformation | |
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| Problems | |
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| Second Quantization | |
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| Many-Body Hamiltonian | |
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| Fock Space | |
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| Creation and Annihilation Operators | |
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| Second-Quantized Hamiltonian | |
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| Field Operators | |
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| Equivalence of First and Second Quantization | |
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| Coherent States | |
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| Problems | |
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| Functional Integrals | |
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| Grand-Canonical Partition Function | |
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| Ideal Quantum Gases | |
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| Semiclassical Method | |
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| Matsubara Expansion | |
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| Green's Function Method | |
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| Wick's Theorem | |
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| Problems | |
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| Interactions and Feynman Diagrams | |
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| Quasiparticles | |
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| The Lehmann Representation | |
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| The Spectral-Weight Function | |
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| Collective Excitations | |
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| Perturbation Theory | |
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| Dyson's Equation | |
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| Hartree-Fock Approximation | |
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| Variational Approach | |
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| Hubbard-Stratonovich Transformation | |
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| Hartree Theory | |
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| Fock Theory | |
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| Hartree-Fock Theory for an Atomic Fermi Gas | |
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| The Jellium Model | |
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| Field-Theory Approach | |
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| Effective Action | |
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| Dispersion and Screened Coulomb Interaction | |
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| Problems | |
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| Landau Theory of Phase Transitions | |
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| Ising Model in d Dimensions | |
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| Landau Approach | |
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| Hubbard-Stratonovich Transformation | |
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| Gaussian Fluctuations | |
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| Spontaneous Symmetry Breaking | |
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| Problems | |
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| Atomic Physics | |
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| Atomic Structure | |
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| Fine Structure | |
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| Hyperfine Structure | |
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| Zeeman Effect | |
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| Two-body Scattering in Vacuum | |
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| Two-Body Transition Matrix | |
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| Partial-Wave Expansion | |
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| Scattering from a Square-Well Potential | |
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| Two-Body Scattering in a Medium | |
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| Physical Regimes | |
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| Problems | |
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| Bose-Einstein Condensation | |
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| Definitions for a Bose-Einstein Condensate | |
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| Superfluidity | |
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| Landau Criterion | |
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| Superfluid Density | |
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| Field-Theory Approach | |
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| Bogoliubov Theory and the Gross-Pitaevskii Equation | |
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| Dyson Equation | |
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| Quasiparticle Dispersion | |
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| Thermodynamic Potential for Bosons | |
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| Bogoliubov-de Gennes Equation | |
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| Popov Theory | |
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| Hydrodynamic-Like Approach | |
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| Time-Dependent Gross-Pitaevskii Equation | |
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| Collective Modes | |
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| Rotating Bose-Einstein Condensates | |
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| Attractive Interactions | |
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| Effective Action | |
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| Breathing Mode | |
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| Metastability of the Condensate | |
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| Problems | |
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| Condensation of Fermionic Pairs | |
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| Introduction | |
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| Thouless Criterion | |
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| Hubbard-Stratonovich Transformation | |
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| Bardeen-Cooper-Schrieffer Theory | |
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| Critical Temperature | |
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| Gap Equation | |
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| Thermodynamic Potential for Fermions | |
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| The BEC-BCS Crossover | |
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| Theoretical Results | |
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| Comparison with Experiment | |
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| Problems | |
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| Symmetries and Symmetry Breaking | |
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| Effective Actions | |
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| Noether's Theorem | |
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| Ward Identities | |
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| Hugenholtz-Pines Theorem | |
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| Bragg Scattering | |
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| RF Spectroscopy | |
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| Second-Order Perturbation Theory | |
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| Ladder Summations | |
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| Absence of Clock Shift | |
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| Absence of Vertex Corrections | |
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| Phase Diffusion | |
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| Problems | |
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| Renormalization Group Theory | |
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| The Renormalization-Group Transformation | |
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| Scaling | |
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| Interactions | |
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| Quantum Effects | |
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| Interactions | |
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| Nonuniversal Quantities | |
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| Renormalization Group for Fermions | |
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| Renormalization-Group Equations | |
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| Extremely-Imbalanced Case | |
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| Homogeneous Phase Diagram | |
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| Problems | |
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| Low-Dimensional Systems | |
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| Modified Popov Theory | |
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| Phase Fluctuations | |
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| Many-Body T Matrix | |
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| Long-Wavelength Physics | |
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| Comparison with Popo Theory | |
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| One Dimension | |
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| Two Dimensions | |
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| Three Dimensions | |
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| Vortices in Two Dimensions | |
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| Kosterlitz-Thouless Phase Transition | |
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| Trapped Bose Gases | |
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| Density Profiles | |
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| Phase Fluctuations | |
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| Comparision with Exact Results | |
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| Problems | |
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| Optical Lattices | |
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| Introduction | |
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| Coupling between Atoms and Light | |
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| Two-Level Approximation | |
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| Fine Structure | |
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| Band Structure | |
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| Hubbard Models | |
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| Superfluid-Mott Insulator Transition | |
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| Bogoliubov Approximation | |
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| Decoupling Approximation | |
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| Hubbard-Stratonovich Transformation | |
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| Fluctuations | |
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| Mott Insulator | |
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| Superfluid Phase | |
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| Bragg Spectroscopy | |
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| Problems | |
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| Feshbach Resonances | |
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| Example of a Feshbach Resonance | |
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| Bare Atom-Molecule Theory | |
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| Ladder Summations | |
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| Effective Atom-Molecule theory | |
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| Scattering Properties | |
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| Bound-State Energy | |
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| Molecular Density of States | |
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| Bogoliubov Theory for the Bose-Einstein Condensed Phase | |
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| Experiments | |
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| Josephson Frequency | |
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| Problems | |
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| References | |
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| Index | |