Phase Space Methods for Degenerate Quantum Gases

433 Pages·2015·2.59 MB·English

by  Bryan J. Dalton

INTERNATIONAL SERIES OF MONOGRAPHS
ON PHYSICS
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162. W.D.McComb:Homogeneous,isotropicturbulence–phenomenology,renormalizationandstatistical
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161. V.Z.Kresin,H.Morawitz,S.A.Wolf:Superconducting state–mechanismsandproperties
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159. W.Barford:Electronicandopticalproperties ofconjugatedpolymers,Secondedition
158. F.Strocchi:Anintroduction tonon-perturbativefoundationsofquantumfieldtheory
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156. K.H.Bennemann,J.B.Ketterson:Novelsuperfluids,Volume1
155. C.Kiefer:Quantumgravity,Thirdedition
154. L.Mestel:Stellarmagnetism,Secondedition
153. R.A.Klemm:Layeredsuperconductors, Volume1
152. E.L.Wolf:Principlesofelectrontunnelingspectroscopy, Secondedition
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149. J.Wesson:Tokamaks,Fourthedition
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142. V.M.Agranovich:Excitationsinorganicsolids
141. W.T.Grandy:Entropyandthetimeevolutionofmacroscopic systems
140. M.Alcubierre:Introductionto3+1numericalrelativity
139. A.L.Ivanov,S.G.Tikhodeev:Problemsofcondensedmatterphysics–quantumcoherencephenomena
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Phase Space Methods for Degenerate
Quantum Gases
Bryan J. Dalton
Centre for Quantum and Optical Science, Swinburne University of Technology,
Melbourne, Victoria, Australia
John Jeffers
Department of Physics, University of Strathclyde, Glasgow, UK
Stephen M. Barnett
School of Physics and Astronomy, University of Glasgow, Glasgow, UK
3
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Preface
The aim of this book is to present a comprehensive theoretical description of phase
space methods for both bosonic and fermionic systems in order to provide a useful
textbook for postgraduate students, as well as a reference book for researchers in
the newly emerging field of quantum atom optics. Phase space distribution function
methods involving phase space variables suitable for systems where small numbers of
modesareinvolvedarecomplementedbyphasespacedistributionfunctionalmethods
involving field functions for the study of systems with large mode numbers, such
as when macroscopic numbers of bosons or fermions are present. The approach for
bosonic systems involves c-number quantities, whilst that for fermionic cases involves
Grassmann quantities. The book covers both the Fokker–Planck-type equations that
determinethedistributionfunctionsorfunctionals,andLangevin-typeequationswhich
govern stochastic forms of the variables or fields.
The approach taken to treat bosonic and fermionic systems can be regarded as
complementary to approaches taken in other branches of physics, notably quantum
fieldtheory,particlephysicsandstatisticalphysics.Inthosedisciplines,pathintegrals
and Feynman diagrams rather than Fokker–Planck-type equations are the method of
choice.
Representative applications to physical systems are presented as examples of the
methods,butnoattemptismadetoreviewthecontentofthebroadsubjectofquantum
atom optics itself. There are other books and reviews that do this.
The book provides proofs of important results, with detail presented in the
Appendices. Each chapter contains a number of problems for students to solve. As
Grassmann algebra and calculus will generally be unfamiliar to students and to re-
searchers in quantum atom optics, the main points of this topic are given appropriate
coverage.
Chapters dealing with the following topics are included:
• states and operators in bosonic and fermionic systems;
• complex numbers and Grassmann numbers;
• Grassmann calculus;
• fermion and boson coherent states;
• canonical transformations and their applications;
• phase space distributions for fermions and bosons;
• Fokker–Planck equations;
• Langevin equations;
• application to few-mode systems;
vi Preface
• functional calculus for c-number and Grassmann fields;
• distribution functionals in quantum-atom optics;
• functional Fokker–Planck equations;
• Langevin field equations;
• application to multi-mode systems;
• further developments.
Acknowledgements
Thisbookwouldnothavebeenwrittenwithouthelpfuldiscussionswithandcomments
from colleagues on key theoretical issues over the past several years. In particular,
we wish to acknowledge M. Babiker, R. Ballagh, T. Busch, J. Corney, J. Cresser,
P. Deuar,P.Drummond,A.Filinov,M.Fromhold,B.Garraway,C.Gilson,M.Olsen,
L. Plimak, J. Ruostekowski, K. Rzazewski and R. Walser. This book would not have
been completed without the patience and continued support of OUP. Work on this
book was supported by the Australian Research Council via the Centre of Excellence
for Quantum-Atom Optics (2003–2010). BJD thanks E. Hinds and S. Maricic for the
hospitalityoftheCentreforColdMatter,ImperialCollege,Londonduringthewriting
of this book.
The authors are grateful to Maureen, Hazel and Claire for their patience during
the writing of this book.
Contents
1 Introduction 1
1.1 Bosons and Fermions, Commuting and Anticommuting Numbers 1
1.2 Quantum Correlation and Phase Space Distribution Functions 2
1.3 Field Operators 5
2 States and Operators 8
2.1 Physical States 9
2.2 Annihilation and Creation Operators 13
2.3 Fock States 14
2.4 Two-Mode Systems 17
2.5 Physical Quantities and Field Operators 20
2.6 Dynamical Processes 25
2.7 Normally Ordered Forms 27
2.8 Vacuum Projector 29
2.9 Position Measurements and Quantum Correlation Functions 30
Exercises 32
3 Complex Numbers and Grassmann Numbers 34
3.1 Algebra of Grassmann and Complex Numbers 34
3.2 Complex Conjugation 37
3.3 Monomials and Grassmann Functions 38
Exercises 43
4 Grassmann Calculus 45
4.1 C-number Calculus in Complex Phase Space 46
4.2 Grassmann Differentiation 49
4.2.1 Definition 49
4.2.2 Differentiation Rules for Grassmann Functions 50
4.2.3 Taylor Series 53
4.3 Grassmann Integration 55
4.3.1 Definition 55
4.3.2 Pairs of Grassmann Variables 59
Exercises 62
5 Coherent States 64
5.1 Grassmann States and Grassmann Operators 64
5.2 Unitary Displacement Operators 66
5.3 Boson and Fermion Coherent States 69
5.4 Bargmann States 71
5.5 Examples of Fermion States 74
viii Contents
5.6 State and Operator Representations via Coherent States 75
5.6.1 State Representation 75
5.6.2 Coherent-State Projectors 77
5.6.3 Fock-State Projectors 79
5.6.4 Representation of Operators 80
5.6.5 Equivalence of Operators 81
5.7 Canonical Forms for States and Operators 82
5.7.1 Fermions 82
5.7.2 Bosons 83
5.8 Evaluating the Trace of an Operator 85
5.8.1 Bosons 85
5.8.2 Fermions 86
5.8.3 Cyclic Properties of the Fermion Trace 87
5.8.4 Differentiating and Multiplying a Fermion Trace 89
5.9 Field Operators and Field Functions 90
5.9.1 Boson Fields 90
5.9.2 Fermion Fields 91
5.9.3 Quantum Correlation Functions 93
Exercises 93
6 Canonical Transformations 95
6.1 Linear Canonical Transformations 96
6.2 One- and Two-Mode Transformations 97
6.2.1 Bosonic Modes 97
6.2.2 Fermionic Modes 101
6.3 Two-Mode Interference 104
6.4 Particle-Pair Creation 106
6.4.1 Squeezed States of Light 106
6.4.2 Thermofields 109
6.4.3 Bogoliubov Excitations of a Zero-Temperature Bose Gas 111
Exercises 114
7 Phase Space Distributions 115
7.1 Quantum Correlation Functions 116
7.1.1 Normally Ordered Expectation Values 116
7.1.2 Symmetrically Ordered Expectation Values 117
7.2 Characteristic Functions 117
7.2.1 Bosons 117
7.2.2 Fermions 118
7.3 Distribution Functions 120
7.3.1 Bosons 121
7.3.2 Fermions 122
7.4 Existence of Distribution Functions and Canonical Forms
for Density Operators 124
7.4.1 Fermions 124
7.4.2 Bosons 127
Contents ix
7.5 Combined Systems of Bosons and Fermions 128
7.6 Hermiticity of the Density Operator 132
7.7 Quantum Correlation Functions 134
7.7.1 Bosons 134
7.7.2 Fermions 136
7.7.3 Combined Case 138
7.7.4 Uncorrelated Systems 139
7.8 Unnormalised Distribution Functions 139
7.8.1 Quantum Correlation Functions 140
7.8.2 Populations and Coherences 141
Exercises 143
8 Fokker–Planck Equations 144
8.1 Correspondence Rules 144
8.2 Bosonic Correspondence Rules 145
8.2.1 Standard Correspondence Rules for Bosonic
Annihilation and Creation Operators 145
8.2.2 General Bosonic Correspondence Rules 146
8.2.3 Canonical Bosonic Correspondence Rules 148
8.3 Fermionic Correspondence Rules 150
8.3.1 Fermionic Correspondence Rules for Annihilation
and Creation Operators 150
8.4 Derivation of Bosonic and Fermionic Correspondence Rules 151
8.5 Effect of Several Operators 154
8.6 Correspondence Rules for Unnormalised Distribution Functions 157
8.7 Dynamical Processes and Fokker–Planck Equations 158
8.7.1 General Issues 158
8.8 Boson Fokker–Planck Equations 160
8.8.1 Bosonic Positive P Distribution 160
8.8.2 Bosonic Wigner Distribution 163
8.8.3 Fokker–Planck Equation in Positive Definite Form 164
8.9 Fermion Fokker–Planck Equations 167
8.10 Fokker–Planck Equations for Unnormalised Distribution Functions 171
8.10.1 Boson Unnormalised Distribution Function 171
8.10.2 Fermion Unnormalised Distribution Function 172
Exercises 173
9 Langevin Equations 174
9.1 Boson Ito Stochastic Equations 175
9.1.1 Relationship between Fokker–Planck and Ito Equations 180
9.1.2 Boson Stochastic Differential Equation in Complex Form 181
9.1.3 Summary of Boson Stochastic Equations 181
9.2 Wiener Stochastic Functions 182
9.3 Fermion Ito Stochastic Equations 183
9.3.1 Relationship between Fokker–Planck and Ito Equations 187
9.3.2 Existence of Coupling Matrix for Fermions 188
9.3.3 Summary of Fermion Stochastic Equations 191