Phase Space Methods for Degenerate Quantum Gases 1st Edition by Bryan Dalton, John Jeffers, Stephen Barnett ISBN 978-0191028632 0191028632

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Product details: 

ISBN 10:   0191028632 

ISBN 13: 978-0191028632

Author:  Bryan J. Dalton, John Jeffers, Stephen M. Barnett  

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 the phase 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 phase space 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, the phase 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 space variables 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 functional calculus, and detailed derivations of key results are provided.

Table of contents: 

1 Introduction

1.1 Bosons and Fermions, Commuting and Anticommuting Numbers

1.2 Quantum Correlation and Phase Space Distribution Functions

1.3 Field Operators

2 States and Operators

2.1 Physical States

2.2 Annihilation and Creation Operators

2.3 Fock States

2.4 Two-Mode Systems

2.5 Physical Quantities and Field Operators

2.6 Dynamical Processes

2.7 Normally Ordered Forms

2.8 Vacuum Projector

2.9 Position Measurements and Quantum Correlation Functions

Exercises

3 Complex Numbers and Grassmann Numbers

3.1 Algebra of Grassmann and Complex Numbers

3.2 Complex Conjugation

3.3 Monomials and Grassmann Functions

Exercises

4 Grassmann Calculus

4.1 C-number Calculus in Complex Phase Space

4.2 Grassmann Differentiation

4.2.1 Definition

4.2.2 Differentiation Rules for Grassmann Functions

4.2.3 Taylor Series

4.3 Grassmann Integration

4.3.1 Definition

4.3.2 Pairs of Grassmann Variables

Exercises

5 Coherent States

5.1 Grassmann States and Grassmann Operators

5.2 Unitary Displacement Operators

5.3 Boson and Fermion Coherent States

5.4 Bargmann States

5.5 Examples of Fermion States

State and Operator Representations via Coherent States 5.6

5.6.1 State Representation

5.6.2 Coherent-State Projectors

5.6.3 Fock-State Projectors

5.6.4 Representation of Operators

5.6.5 Equivalence of Operators

5.7 Canonical Forms for States and Operators

5.7.1 Fermions

5.7.2 Bosons

5.8 Evaluating the Trace of an Operator

5.8.1 Bosons

5.8.2 Fermions

5.8.3 Cyclic Properties of the Fermion Trace

5.8.4 Differentiating and Multiplying a Fermion Trace

5.9 Field Operators and Field Functions

5.9.1 Boson Fields

5.9.2 Fermion Fields

5.9.3 Quantum Correlation Functions

Exercises

6 Canonical Transformations

6.1 Linear Canonical Transformations

6.2 One- and Two-Mode Transformations

6.2.1 Bosonic Modes

6.2.2 Fermionic Modes

6.3 Two-Mode Interference

6.4 Particle-Pair Creation

6.4.1 Squeezed States of Light

6.4.2 Thermofields

6.4.3 Bogoliubov Excitations of a Zero-Temperature Bose Gas

Exercises

7 Phase Space Distributions

7.1 Quantum Correlation Functions

7.1.1 Normally Ordered Expectation Values

7.1.2 Symmetrically Ordered Expectation Values

Characteristic Functions 7.2

7.2.1 Bosons

7.2.2 Fermions

7.3 Distribution Functions

7.3.1 Bosons

7.3.2 Fermions

7.4 Existence of Distribution Functions and Canonical Forms for Density Operators

7.4.1 Fermions

7.4.2 Bosons

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Tags: Bryan Dalton, John Jeffers, Stephen Barnett, Phase Space Methods, Quantum Gases