Phase Space Methods for Degenerate Quantum Gases

Hardcover | December 13, 2014

byBryan J. Dalton, John Jeffers, Stephen M. Barnett

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Recent experimental progress has enabled cold atomic gases to be studied at nano-kelvin temperatures, creating new states of matter where quantum degeneracy occurs - Bose-Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. This book presents thephase space theory approach for treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phasespace variables which replace mode annihilation, creation operators, the dynamical equation for the density operator determines a Fokker-Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, thephase space variables are replaced by time dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker-Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is treated, the density operator being represented by a distribution functional of field functions which replace field annihilation, creation operators, the distribution functional satisfying a functional FPE, etc. A novel feature of this book is that the phase spacevariables for fermions are Grassmann variables, not c-numbers. However, we show that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions. The book includes the necessary mathematics for Grassmann calculus and functionalcalculus, and detailed derivations of key results are provided.

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Recent experimental progress has enabled cold atomic gases to be studied at nano-kelvin temperatures, creating new states of matter where quantum degeneracy occurs - Bose-Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. This book presents thephase space theory approach for treating the...

Bryan Dalton obtained a PhD degree in 1966 from Monash University. Following postdoctoral positions at University of Chicago and Australian National University, he joined the Department of Physics, University of Queensland in 1970, retiring as a Reader in 2000. His research was in theoretical quantum optics on topics such as non-classi...

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Format:HardcoverDimensions:432 pages, 9.69 × 6.73 × 0.1 inPublished:December 13, 2014Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199562741

ISBN - 13:9780199562749

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Table of Contents

1. Introduction2. States and Operators3. Complex Numbers and Grassmann Numbers4. Grassmann Calculus5. Coherent States6. Canonical Transformations7. Phase Space Distributions8. Fokker-Planck Equations9. Langevin Equations10. Application to Few Mode Systems11. Functional Calculus for C-Number and Grassmann Fields12. Distribution Functionals in Quantum-Atom Optics13. Functional Fokker-Planck Equations14. Langevin Field Equations15. Application to Multi-Mode Systems16. Further DevelopmentsAppendix A: Fermion Anti-Commutation RulesAppendix B: Markovian Master EquationAppendix C: Grassmann CalculusAppendix D: Properties of Coherent StatesAppendix E: Phase Space Distributions for Bosons and FermionsAppendix F: Fokker-Planck EquationsAppendix G: Langevin EquationsAppendix H: Functional Calculus for Restricted Boson and Fermion FieldsAppendix I: Applications to Multi-Mode Systems