Contents -- Foreword -- Preface -- Participants of FINESS 2009 (Durham) -- Common Symbols/Expressions and their Meanings -- Part I. Introductory Material -- Editorial Notes -- I.A. Quantum Gases: The Background -- 1. Quantum Gases: Setting the Scene N.P. Proukakis & K. Burnett -- 1.1. Introduction: Background to Quantum Fluids and Gases -- 1.2. History of Non-Equilibrium and Finite-Temperature Pure BEC Experiments -- 1.2.1. The Search for Idealised Systems: Spin-Polarised Hydrogen -- 1.2.2. The Twist to an Unlikely Candidate: The Scene Opens up for Alkali Atoms -- 1.2.3. Rival Candidates Gaining Ground? -- 1.3. Modelling Quantum Degenerate Gases -- 1.3.1. The Success of Phenomenology -- 1.3.2. Ab Initio Modelling -- 1.3.2.1. The Gross-Pitaevskii Equation -- 1.3.2.2. Generalised Kinetic Theories -- 1.3.3. Classical-Field and Stochastic Approaches -- 1.3.4. Modelling Related Systems -- 1.4. Unified Features of Quantum Gases -- 1.4.1. Non-Equilibrium BECs and the Thermal Phase Transition -- 1.4.2. Thermal and Quantum Fluctuations -- 1.4.3. Quantum Phase Transitions and Disorder -- 1.4.4. The Superfluid Fraction, its Relation to the Condensate and the Issue of Fragmentation -- 1.4.5. Strongly Correlated Physics -- 1.4.6. Ultracold Fermions -- 1.4.7. Potential Applications -- 1.4.8. Other Systems Exhibiting Condensation -- Acknowledgements -- I.B. Quantum Gases: Experimental Considerations -- 2. Ultracold Quantum Gases: Experiments with Many-Body Systems in Controlled Environments P. Kruger -- 2.1. Introduction -- 2.2. Condensate Formation and Growth -- 2.3. Excitations of Bose-Einstein Condensates -- 2.4. Strongly Correlated and Phase-Fluctuating Systems -- 2.4.1. Feshbach Resonances -- 2.4.2. Optical Lattices -- 2.4.3. Low-Dimensional Systems -- Acknowledgements -- 3. Ultracold Quantum Gases: Key Experimental Techniques S.A. Hopkins & S.L. Cornish.

3.1. Introduction -- 3.2. Basic Experimental Techniques -- 3.2.1. Overview -- 3.2.2. Laser Cooling and Trapping of Atoms -- 3.2.3. Magnetic Traps -- 3.2.4. Dipole Traps -- 3.2.5. Evaporative (and Sympathetic) Cooling -- 3.2.6. Feshbach Resonances -- 3.2.7. Manipulation and Visualisation -- 3.2.8. Cold Molecules -- 3.3. High-Level Techniques -- 3.3.1. Interferometry -- 3.3.2. Optical Lattices -- 3.3.3. Rotation, Vortices, and Phase Imprinting -- 3.3.4. Microtraps (or 'Atom Chips') -- 3.3.5. Matter-Wave Lasers (or 'Atom Lasers') -- 3.4. New Tools and Topical Areas -- 3.5. Summary and Outlook -- Acknowledgements -- I.C. Quantum Gases: Background Key Theoretical Notions -- 4. Introduction to Theoretical Modelling M.J. Davis, S.A. Gardiner, T.M. Hanna, N. Nygaard, N.P. Proukakis & M.H. Szymanska -- 4.1. Introduction -- 4.2. Second Quantisation -- 4.3. Effective Interactions -- 4.4. Broken Symmetry Versus Number Conservation -- 4.5. Fluctuations and Degeneracy in Low Dimensions -- 4.6. Periodic Potentials ('Optical Lattices') -- 4.7. Fermionic Issues -- 4.8. Feshbach Resonances -- 4.9. Summary -- Acknowledgements -- Part II. Ultracold Bosonic Gases: Theoretical Modelling -- Editorial Notes -- II.A. Kinetic and Many-Body Approaches -- Editorial Notes -- 5. A Dynamical Self-Consistent Finite-Temperature Kinetic Theory: The ZNG Scheme A.J. Allen, C.F. Barenghi, N.P. Proukakis & E. Zaremba -- 5.1. Introduction -- 5.2. Methodology -- 5.2.1. Mean-Field Coupling: First-Order Effects -- 5.2.2. Particle-Exchanging Collisions: Second-Order Effects -- 5.2.3. Numerical Implementation -- 5.2.4. ZNG Condensate Hydrodynamics -- 5.3. Validity Issues -- 5.3.1. Validity Domain -- 5.3.2. Relevance to Other Theories -- 5.4. Applications -- 5.4.1. Damping of Condensate Scissors Mode -- 5.4.2. Decaying Vortex Dynamics -- 5.5. Relevance to Other Systems -- Acknowledgements.

6. Extended Mean-Field Theory: Reversible and Irreversible Quantum Evolution of Trapped Gases R. Walser -- 6.1. Introduction -- 6.2. Methodology -- 6.2.1. Coarse-Grained Markovian Density Operator -- 6.2.2. Derivation of an Integral Equation -- 6.2.3. Quantum Kinetic Equations -- 6.2.4. Fields and Energies -- 6.2.5. The Set of Relevant Operators -- 6.2.6. Relevant Kinetic Equations for Bose Gases -- 6.3. Validity Issues -- 6.4. Applications -- 6.4.1. Reversible Isentropic Evolution -- 6.4.2. Irreversible Collisional Thermalisation -- 6.5. Relevance to Other Systems -- Acknowledgements -- 7. Cumulant Dynamics of Strongly Interacting Ultracold Gases T.M. Hanna & J. Mur-Petit -- 7.1. Introduction -- 7.2. Methodology -- 7.2.1. Mean-Field Evolution -- 7.2.2. One-Body Density Matrix Evolution -- 7.2.3. Form of Two-Body Potential -- 7.3. Validity Issues -- 7.4. Applications -- 7.4.1. Dynamics of Feshbach Molecule Production -- 7.4.2. Relevance to Other Systems -- Acknowledgements -- 8. Number-Conserving Approaches for Atomic Bose-Einstein Condensates: An Overview S.A. Gardiner & T.P. Billam -- 8.1. Introduction -- 8.2. Methodology: A Number-Conserving Perturbative Approach -- 8.2.1. Number-Conserving versus Symmetry-Breaking Approaches -- 8.2.2. Key Concept: A Number-Conserving Fluctuation Operator -- 8.2.3. Number-Conserving Bogoliubov Treatment -- 8.2.4. Second-Order Self-Consistent Treatment -- 8.2.5. Numerical Implementation -- 8.3. Validity Issues -- 8.4. Applications -- Acknowledgements -- 9. Multiconfigurational Time-Dependent Hartree Methods for Bosonic Systems: Theory and Applications O.E. Alon, A.I. Streltsov, K. Sakmann & L.S. Cederbaum -- 9.1. Introduction -- 9.2. Methodology: Multiconfigurational Time-Dependent Hartree Methods for Bosonic Systems -- 9.2.1. Single-Component Bosons -- 9.2.2. Extension to Bose-Bose Mixtures.

9.2.3. Extension to Resonantly Coupled Bosonic Atoms and Molecules -- 9.3. Discussion of Validity -- 9.4. Applications -- 9.4.1. Exact Quantum Dynamics of a One-Dimensional Bosonic Josephson Junction -- 9.4.2. Swift Loss of Coherence of One-Dimensional Bright Soliton Trains -- 9.5. Relevance to Other Systems -- Acknowledgments -- II.B. Classical-Field, Stochastic and Field-Theoretic Approaches -- Editorial Notes -- 10. C-Field Methods for Non-Equilibrium Bose Gases M.J. Davis, T.M. Wright, P.B. Blakie, A.S. Bradley, R.J. Ballagh & C.W. Gardiner -- 10.1. Introduction -- 10.2. Methodology -- 10.2.1. Outline of Derivation -- 10.2.2. Numerical Implementation of the SPGPE -- 10.2.3. Projected Gross-Pitaevskii Equation (PGPE) -- 10.2.4. Truncated Wigner Projected Gross-Pitaevskii Equation (TWPGPE) -- 10.3. Validity Issues -- 10.3.1. Relevance to Other Theories -- 10.4. Application -- 10.5. Relevance to Other Systems -- Acknowledgements -- 11. The Stochastic Gross-Pitaevskii Methodology S.P. Cockburn & N.P. Proukakis -- 11.1. Introduction -- 11.2. Methodology -- 11.2.1. Fokker-Planck Equation for an Ultracold Bose Gas -- 11.2.2. Formulation as a Langevin Equation -- 11.2.3. Stochastic Hydrodynamics -- 11.2.4. Simple Numerical Implementation -- 11.2.5. Interpretation: Single Runs and Extracting Coherence Properties -- 11.2.5.1. Single Versus Averaged Runs -- 11.2.5.2. A Posteriori Condensate/Quasi-Condensate Extraction -- 11.2.5.3. Comparison with the Modified Popov Method -- 11.3. Validity Issues -- 11.3.1. Validity Domain -- 11.3.2. Relevance to Other Theories -- 11.4. Applications -- 11.4.1. Comparison with Quasi-One-Dimensional Bose Gas Experiments -- 11.4.2. Dark-Soliton Dynamics in a Quasi-Condensate -- 11.5. Relevance to Other Systems -- Acknowledgements -- 12. A Classical-Field Approach for Bose Gases M. Brewczyk, M. Gajda & K. Rzazewski.

12.1. Introduction -- 12.2. Methodology: The Classical-Field Approach -- 12.2.1. Choice of Cutoff -- 12.2.2. Generating Equilibrium States on Demand -- 12.2.2.1. Relation to Actual Parameters -- 12.2.2.2. Numerical Details -- 12.3. Applications -- 12.3.1. Quadrupole Oscillations of Condensate and Thermal Cloud -- 12.3.2. Decay of Vortices -- Acknowledgements -- 13. The Truncated Wigner Method for Bose Gases J. Ruostekoski & A.D. Martin -- 13.1. Introduction -- 13.2. Methodology -- 13.2.1. Initial-State Generation in the Truncated Wigner Approximation -- 13.2.1.1. Uniform System -- 13.2.1.2. Trapped System -- 13.2.1.3. Quasi-Condensate Description -- 13.2.1.4. Relaxation -- 13.2.2. Wigner Representation and Symmetric Ordering -- 13.2.3. Numerical Implementation -- 13.3. Validity Issues -- 13.4. Applications -- 13.4.1. Dark Solitons -- 13.4.2. Atom-Number Squeezing -- Acknowledgements -- 14. Number-Conserving Stochastic Approaches for Equilibrium and Time-Dependent Bose Gases A. Sinatra, Y. Castin, I. Carusotto, C. Lobo & E. Witkowska -- 14.1. Introduction -- 14.2. Methodology -- 14.2.1. Number-Conserving Wigner Method -- 14.2.1.1. Sampling of W at Thermal Equilibrium -- 14.2.1.2. Time Evolution -- 14.2.2. Giving up Quantum Fluctuations: A Classical Field Model -- 14.2.2.1. Generation of the Fields in the Canonical Ensemble -- 14.2.2.2. Time Evolution -- 14.2.3. Exact and Semiclassical Methods -- 14.3. Validity Issues -- 14.3.1. Relevance to Other Non-U(1)-Symmetry Preserving Theories -- 14.4. Applications -- 14.4.1. Number-Conserving Wigner Method -- 14.4.2. Classical-Field Method -- 14.4.3. Exact and Semiclassical Methods -- Acknowledgements -- 15. Quantum Dynamics on Extended Phase Space: The Positive-P Representation P.D. Drummond -- 15.1. Introduction -- 15.2. Methodology -- 15.2.1. Operator Identities.

15.2.2. The Fokker-Planck and Stochastic Equations.