Phase space methods for degenerate QUantum gases

Free ebooks ==> www.Ebook777.com www.Ebook777.com Free ebooks ==> www.Ebook777.com INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J BIRMAN S F EDWARDS R FRIEND M REES D SHERRINGTON G VENEZIANO CITY UNIVERSITY OF NEW YORK UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF OXFORD CERN, GENEVA International Series of Monographs on Physics 163 B.J Dalton, J Jeffers, S.M Barnett: Phase space methods for degenerate quantum gases 162 W.D McComb: Homogeneous, isotropic turbulence – phenomenology, renormalization and statistical closures 161 V.Z Kresin, H Morawitz, S.A Wolf: Superconducting state – mechanisms and properties 160 C Barrab` es, P.A Hogan: Advanced general relativity – gravity waves, spinning particles, and black holes 159 W Barford: Electronic and optical properties of conjugated polymers, Second edition 158 F Strocchi: An introduction to non-perturbative foundations of quantum field theory 157 K.H Bennemann, J.B Ketterson: Novel 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141 W.T Grandy: Entropy and the time evolution of macroscopic systems 140 M Alcubierre: Introduction to + numerical relativity 139 A.L Ivanov, S.G Tikhodeev: Problems of condensed matter physics – quantum coherence phenomena in electron-hole and coupled matter-light systems 138 I.M Vardavas, F.W Taylor: Radiation and climate 137 A.F Borghesani: Ions and electrons in liquid helium 135 V Fortov, I Iakubov, A Khrapak: Physics of strongly coupled plasma 134 G Fredrickson: The equilibrium theory of inhomogeneous polymers 133 H Suhl: Relaxation processes in micromagnetics 132 J Terning: Modern supersymmetry 131 M Mari˜ no: Chern-Simons theory, matrix models, and topological strings 130 V Gantmakher: Electrons and disorder in solids 129 W Barford: Electronic and optical properties of conjugated polymers 128 R.E Raab, O.L de Lange: Multipole theory in electromagnetism 127 A Larkin, A Varlamov: Theory of fluctuations in superconductors 126 P Goldbart, N Goldenfeld, D Sherrington: Stealing the 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critical phenomena A Abragam: Principles of nuclear magnetism P.A.M Dirac: Principles of quantum mechanics R.E Peierls: Quantum theory of solids www.Ebook777.com Free ebooks ==> www.Ebook777.com 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 www.Ebook777.com Free ebooks ==> www.Ebook777.com Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Bryan J Dalton, John Jeffers and Stephen M Barnett 2015 The moral rights of 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Free ebooks ==> www.Ebook777.com 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 modes are involved are complemented by phase space distribution functional methods 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 determine the distribution functions or functionals, and Langevin-type equations which 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 field theory, particle physics and statistical physics In those disciplines, path integrals 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, but no attempt is made to review the content of the broad subject of quantum atom optics itself There are other books and reviews that 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 researchers 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; www.Ebook777.com Free ebooks ==> www.Ebook777.com 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 This book would not have been written without helpful discussions with and comments 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 hospitality of the Centre for Cold Matter, Imperial College, London during the writing of this book The authors are grateful to Maureen, Hazel and Claire for their patience during the writing of this book www.Ebook777.com Free ebooks ==> www.Ebook777.com Contents Introduction 1.1 Bosons and Fermions, Commuting and Anticommuting Numbers 1.2 Quantum Correlation and Phase Space Distribution Functions 1.3 Field Operators 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 13 14 17 20 25 27 29 30 32 Complex Numbers and Grassmann Numbers 3.1 Algebra of Grassmann and Complex Numbers 3.2 Complex Conjugation 3.3 Monomials and Grassmann Functions Exercises 34 34 37 38 43 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 45 46 49 49 50 53 55 55 59 62 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 64 64 66 69 71 74 www.Ebook777.com 1 Free ebooks ==> www.Ebook777.com viii Contents 5.6 State and Operator Representations via Coherent States 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 75 75 77 79 80 81 82 82 83 85 85 86 87 89 90 90 91 93 93 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 95 96 97 97 101 104 106 106 109 111 114 Phase Space Distributions 7.1 Quantum Correlation Functions 7.1.1 Normally Ordered Expectation Values 7.1.2 Symmetrically Ordered Expectation Values 7.2 Characteristic Functions 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 115 116 116 117 117 117 118 120 121 122 www.Ebook777.com 124 124 127 Free ebooks ==> www.Ebook777.com Contents ix 7.5 7.6 7.7 Combined Systems of Bosons and Fermions Hermiticity of the Density Operator Quantum Correlation Functions 7.7.1 Bosons 7.7.2 Fermions 7.7.3 Combined Case 7.7.4 Uncorrelated Systems 7.8 Unnormalised Distribution Functions 7.8.1 Quantum Correlation Functions 7.8.2 Populations and Coherences Exercises 128 132 134 134 136 138 139 139 140 141 143 Fokker–Planck Equations 8.1 Correspondence Rules 8.2 Bosonic Correspondence Rules 8.2.1 Standard Correspondence Rules for Bosonic Annihilation and Creation Operators 8.2.2 General Bosonic Correspondence Rules 8.2.3 Canonical Bosonic Correspondence Rules 8.3 Fermionic Correspondence Rules 8.3.1 Fermionic Correspondence Rules for Annihilation and Creation Operators 8.4 Derivation of Bosonic and Fermionic Correspondence Rules 8.5 Effect of Several Operators 8.6 Correspondence Rules for Unnormalised Distribution Functions 8.7 Dynamical Processes and Fokker–Planck Equations 8.7.1 General Issues 8.8 Boson Fokker–Planck Equations 8.8.1 Bosonic Positive P Distribution 8.8.2 Bosonic Wigner Distribution 8.8.3 Fokker–Planck Equation in Positive Definite Form 8.9 Fermion Fokker–Planck Equations 8.10 Fokker–Planck Equations for Unnormalised Distribution Functions 8.10.1 Boson Unnormalised Distribution Function 8.10.2 Fermion Unnormalised Distribution Function Exercises 144 144 145 Langevin Equations 9.1 Boson Ito Stochastic Equations 9.1.1 Relationship between Fokker–Planck and Ito Equations 9.1.2 Boson Stochastic Differential Equation in Complex Form 9.1.3 Summary of Boson Stochastic Equations 9.2 Wiener Stochastic Functions 9.3 Fermion Ito Stochastic Equations 9.3.1 Relationship between Fokker–Planck and Ito Equations 9.3.2 Existence of Coupling Matrix for Fermions 9.3.3 Summary of Fermion Stochastic Equations www.Ebook777.com 145 146 148 150 150 151 154 157 158 158 160 160 163 164 167 171 171 172 173 174 175 180 181 181 182 183 187 188 191 Free ebooks ==> www.Ebook777.com 404 Applications to Multi-Mode Systems ds ∂μ δ δψ(s) ∂μ δ δψ + (s) K = {∂μ φ∗k (s)} ds k K = {∂μ φl (s)} ds l = ds ∂μ W [ψ, ψ + ] K ∂ ∂αk {∂μ φ∗k (s)} ∂ w(αk , α+ k) ∂αk l k ∂μ δψ + (s) ∂ w(αk , α+ k) ∂α+ l K ∂ ∂α+ l δ {∂μ φl (s)} δ δψ(s) W [ψ, ψ + ] (I.17) For the third terms, the spatial derivative of the functional derivative can be removed via spatial integration by parts and applied to the spatial function For example, ∂μ ψ + (s) ds ∂μ δ δψ + (s) K = W [ψ, ψ + ] K {∂μ φ∗k (s)}αk+ ds {∂μ φl (s)} k l K =− K {∂μ2 φ∗k (s)}αk+ ds {φl (s)} k =− ds l ∂μ2 ψ + (s) δ δψ + (s) ∂ w(αk , α+ k) ∂α+ l ∂ w(αk , αk+ ) ∂αl+ W [ψ, ψ + ] (I.18) Applying the product rule then enables the functional differentiation to apply to the product of ∂μ2 ψ + (s) with the distribution functional: δ W [ψ, ψ + ] δψ + (s) δ δ ∂ ψ + (s) W [ψ, ψ + ]) − ∂ ψ + (s) W [ψ, ψ + ] δψ + (s) μ δψ + (s) μ δ δ ∂ ψ + (s) W [ψ, ψ + ]) − ∂μ2 + ψ + (s) W [ψ, ψ + ] δψ + (s) μ δψ (s) δ ∂ ψ + (s) W [ψ, ψ + ]) − ∂μ2 δC (s, s) W [ψ, ψ + ], (I.19) δψ + (s) μ ∂μ2 ψ + (s) = = = where the functional derivative of ψ + (s) has been evaluated using (15.12) and involves the restricted delta function (15.4) Thus ds ∂μ ψ + (s) =− ds ∂μ δ δψ + (s) W [ψ, ψ + ] δ ∂ ψ + (s) δψ + (s) μ W [ψ, ψ + ] + ds ∂μ2 δC (s, s) W [ψ, ψ + ] (I.20) www.Ebook777.com Free ebooks ==> www.Ebook777.com Bose Condensate – Derivation of Functional Fokker–Planck Equations 405 A similar treatment shows that δ δψ(s) ds (∂μ ψ(s)) ∂μ =− ds W [ψ, ψ + ] δ ∂ ψ(s) δψ(s) μ W [ψ, ψ + ] + ds ∂μ2 δC (s, s) W [ψ, ψ + ], (I.21) so the terms involving ∂μ2 δC (s, s) cancel out For ρ → (−i/ )[T , ρ], the kinetic-energy term in the functional Fokker–Planck equation then is −i W [ψ(r), ψ+ (r)] → − i − + ds ds δ δψ + (s) 2m μ δ δψ(s) μ 2m ∂μ2 ψ + (s) W [ψ, ψ + ] ∂μ2 ψ(s) W [ψ, ψ + ]) (I.22) The first-order functional derivative terms combine to remove the factors 12 Thus only a first-order functional derivative term occurs The result is the same as for the positive P case If ρ → V ρ then Wigner Trap Potential Terms W [ψ(r), ψ + (r)] → ψ + (s) − ds δ δψ(s) V (s) ψ(s) + δ δψ + (s) W [ψ, ψ + ] ds ψ + (s)V (s)ψ(s) W [ψ, ψ + ] = + ds ψ + (s)V (s) + ds + ds δ δψ(s) δ − δψ(s) − δ δψ + (s) W [ψ, ψ + ] V (s)ψ(s) W [ψ, ψ + ] δ δψ + (s) V (s) W [ψ, ψ + ], (I.23) and for ρ → ρV we have W [ψ(r), ψ+ (r)] → ds ψ(s) − δ + δψ (s) V (s) ψ + (s) + δ δψ(s) W [ψ, ψ + ] ds (ψ(s)) V (s) ψ + (s) W [ψ, ψ + ] = + ds (ψ(s)) V (s) + ds − + δ δψ + (s) δ ds − δψ + (s) δ δψ(s) W [ψ, ψ + ] V (s) ψ + (s) W [ψ, ψ + ] V (s) δ δψ(s) W [ψ, ψ + ] www.Ebook777.com (I.24) Free ebooks ==> www.Ebook777.com 406 Applications to Multi-Mode Systems For ρ → (−i/ )[T , ρ], the terms involving {ψ + (s)V (s)ψ(s)} cancel, and it is straightforward to show that the terms involving two functional derivatives are equal and also cancel The treatment of the terms involving a single functional derivative is essentially the same as for the positive P case, and the trap potential-energy term in the functional Fokker–Planck equation then is W [ψ, ψ + ] → −i − i δ {V (s)ψ(s)} W [ψ, ψ + ] δψ(s) δ ds {V (s)ψ + (s)} W [ψ, ψ + ] + δψ (s) − ds + Wigner Boson–Boson Interaction Terms W [ψ, ψ + ] → g ds ψ + (s) − × ψ(s) + δ δψ(s) δ + δψ (s) (I.25) If ρ → U ρ then ψ + (s) − δ δψ(s) ψ(s) + δ + δψ (s) W [ψ, ψ + ], (I.26) and if ρ → ρU then W [ψ(r), ψ + (r)] → g δ δ ψ(s) − + + δψ (s) δψ (s) δ × ψ + (s) + W [ψ, ψ + ] δψ(s) ds ψ(s) − ψ + (s) + δ δψ(s) (I.27) There are a total of 32 terms to consider These can all be developed using the product rules for functional differentiation in (15.13) together with the rules (δ/δψ(s))ψ + (s) = (δ/δψ + (s))ψ(s) = and (δ/δψ(s))ψ(s) = (δ/δψ + (s))ψ + (s) = δC (s, s) from (15.12) As with the previous terms, the overall strategy is to move all the functional derivatives to the left – as required for functional Fokker–Planck equations It turns out that all of the even-order functional derivative terms cancel, as the terms with no functional derivative Unlike the positive P case, not all of the terms involving δC (s, s) cancel out The derivation is left as an exercise For ρ → (−i/ )[U , ρ], the boson–boson interaction energy term in the Fokker– Planck equation is W [ψ(r), ψ + (r)] → −i − i − i − i −g +g +g −g δ (ψ + (s)ψ(s) − δC (s, s))ψ(s) W [ψ, ψ + ] δψ(s) δ ds (ψ + (s)ψ(s) − δC (s, s))ψ + (s) W [ψ, ψ + ] + δψ (s) δ δ δ ds {ψ(s)}W [ψ, ψ + ] + δψ(s) δψ(s) δψ (s) δ δ δ ds {ψ + (s)}W [ψ, ψ + ] , (I.28) + + δψ (s) δψ (s) δψ(s) ds which involves first-order and third-order functional derivatives The appearance of the term δC (s, s) may be surprising at first, but we recall that ψ + (s)ψ(s) represents www.Ebook777.com Free ebooks ==> www.Ebook777.com Fermi Gas – Derivation of Functional Fokker–Planck Equations 407 a symmetrically ordered number density, so ψ + (s)ψ(s) − δC (s, s) corresponds to the true atom number density I.2 I.2.1 Fermi Gas – Derivation of Functional Fokker–Planck Equations Unnormalised B Case In this section, we denote the distribution functional B[ψu (r), ψu+ (r), ψd (r), ψd+ (r)] as B[ψ(r)] for short The Hamiltonian is given in (15.42) The correspondence rules are given in (13.51)–(13.58) We will assume the modes are restricted by a cut-off K B+ Kinetic-Energy Terms If ρ → T ρ then B[ψ(r)] → ds 2m α ∂μ μ − → δ δψα1 (s) (∂μ ψα1 (s)) B[ψ(s)] and if ρ → ρT then B[ψ(r)] → ds B[ψ(s)] (∂μ ψα2 (s)) ∂μ 2m α μ ← − δ δψα2 (s) , so for ρ → (−i/ )[T , ρ] we have B[ψ(r)] → − − i ds 2m i − α μ α − → δ δψα1 (s) (∂μ ψα1 (s)) B[ψ(s)] ← − δ ds B[ψ(s)] (∂μ ψα2 (s)) ∂μ δψα2 (s) 2m ∂μ μ (I.29) Using the mode expansions in (11.149) and (11.156) for the functional derivatives, we have for the first term → − δ ds ∂μ (∂μ ψα1 (s)) B[ψ(s)] δψα1 (s) → − ∂ 1 = ds ∂μ ξαi (s) ∂μ ξαj (s) gαj B(g) ∂g αi i j =− ds i j ξαi (s) − → δ δψα1 (s) − → ∂ ∂gαi =− ds =− ds B[ψ(s)] ∂μ2 ψα1 (s) 1 ∂μ2 ξαj (s) gαj ∂μ2 ψα1 (s) B(g) B[ψ(s)] ← − δ δψα1 (s) , www.Ebook777.com (I.30) Free ebooks ==> www.Ebook777.com 408 Applications to Multi-Mode Systems where we have used spatial integration by parts and then (11.169) to reverse the functional derivative, noting that ∂μ2 ψα1 (s) B[ψ(s)] is an odd Grassmann function Similarly, ← − δ ds B[ψ(s)] (∂μ ψα2 (s)) ∂μ δψα2 (s) =− ∂μ2 ψα2 (s) ds B[ψ(s)] ← − δ δψα2 (s) , (I.31) though here no reversal of the functional derivative is needed Combining these results for ρ → (−i/ )[T , ρ] gives the kinetic-energy term in the functional Fokker–Planck equation as ∂ B[ψ(s)] ∂t = −i − K ds α − μ 2m ∂μ2 ψα1 (s)B[ψ(s)] i + ds α μ 2m ← − δ δψα1 (s) ∂μ2 ψα2 (s)B[ψ(s)] ← − δ δψα2 (s) (I.32) B+ Potential-Energy Terms B[ψ(r)] → ds α If ρ → V ρ then − → δ δψα1 (s) Vα (s) (ψα1 (s)) B[ψ(s)] ds {Vα (s) (ψα1 (s))} B[ψ(s)] = α ← − δ , δψα1 (s) (I.33) where the functional derivative in the first term has been reversed, and we have used the fact that {Vα (s) (ψα1 (s))}B[ψ(s)] is an odd Grassmann function If ρ → ρV then B[ψ(r)] → ds B[ψ(s)] (ψα2 (s)) Vα (s) α ← − δ δψα2 (s) , so for ρ → (−i/ )[V , ρ] the potential-energy term in the functional Fokker–Planck equation becomes B[ψ(r)]V → − i + α − i ← − δ δψα1 (s) ← − δ ds (Vα (s) ψα2 (s)B[ψ(s)]) δψα2 (s) ds (Vα (s) ψα1 (s)B[ψ(s)]) − α www.Ebook777.com (I.34) Free ebooks ==> www.Ebook777.com Fermi Gas – Derivation of Functional Fokker–Planck Equations B+ Interaction Energy Terms g B[ψ(s)] → = − = − g α g If ρ → U ρ then − → δ ds + δψα1 (s) → − δ ds + δψα1 (s) α ds 409 + − → δ δψ−α1 (s) ψ−α1 (s)ψα1 (s) B[ψ(s)] ← − δ ψ−α1 (s)ψα1 (s)B[ψ(s)] + ψ−α1 (s)ψα1 (s)B[ψ(s)] + α δψ−α1 (s) ← − δ + δψα1 (s) ← − δ δψ−α1 (s) , (I.35) where the second line is obtained using (11.169), noting that ψ−α1 (s)ψα1 (s)B[ψ(s)] is an even Grassmann function, and the third line is obtained from the same equation, ← − but noting that ψ−α1 (s)ψα1 (s)B[ψ(s)] + δ /δψ−α1 (s) is an odd Grassmann function If ρ → ρU then B[ψ(s)] → g = − ds B[ψ(s)] ψα2 (s)ψ−α2 (s) + δψ−α2 (s) ← − δ ds B[ψ(s)] ψ−α2 (s)ψα2 (s) + δψ−α2 (s) α g ← − δ α ← − δ δψα2 (s) ← − δ + δψα2 (s) + , (I.36) where we have reversed the order of the fields Hence, if ρ → (−i/ )[U , ρ], then the interaction energy term in the functional Fokker–Planck equation becomes ig B[ψ(r)]U → = − ds ψ−α1 (s)ψα1 (s)B[ψ(s)] + α ig α ← − δ δψ−α1 (s) ← − δ ds ψ−α2 (s)ψα2 (s)B[ψ(s)] + δψ−α2 (s) ← − δ + δψα1 (s) ← − δ + δψα2 (s) (I.37) www.Ebook777.com Free ebooks ==> www.Ebook777.com References [1] R J Glauber, Phys Rev 131, 2766 (1963) [2] R J Glauber, Phys Rev 130, 2529 (1963) [3] W H Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973) [4] S M Barnett and P M Radmore, Methods in Theoretical Quantum Optics (Clarendon Press, Oxford, 1997) [5] Z Ficek and S Swain, Quantum Interference and Coherence: Theory and Experiments (Springer, Berlin, 2004) [6] J Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, 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H Masumoto and M Tachiki, Thermofield Dynamics and Condensed States (North-Holland, Amsterdam, 1982) [47] L Pitaevskii and S Stringari, Bose–Einstein Condensation (Oxford University Press, Oxford, 2003) [48] C J Pethick and H Smith, Bose–Einstein Condensation in Dilute Bose Gases (Cambridge University Press, Cambridge, 2002) [49] F A M de Oliveira, Phys Rev A 45, 5104 (1992) [50] B J Dalton, J Phys C: Conf Ser 67, 012059 (2007) [51] S E Hoffmann, J F Corney and P D Drummond, Phys Rev A 78, 013622 (2008) [52] B J Dalton, Ann Phys 326, 668 (2011) [53] B J Dalton, Ann Phys 327, 2432 (2012) [54] P Deuar and P D Drummond, Phys Rev A 66, 033812 (2002) [55] P D Drummond and P Deuar, J Opt B: Semiclass Opt 5, S281 (2003) [56] J F Corney and P D Drummond, Phys Rev A 68, 063822 (2003) [57] J F Corney and P D Drummond, Phys Rev B 73, 125112 (2006) [58] J F Corney and P D Drummond, J Phys A: Math Gen 39, 269 (2006) www.Ebook777.com Free ebooks ==> www.Ebook777.com 412 References [59] I S Gradsteyn and I M Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965), p4 [60] R Schack and A Schenzle, Phys Rev A 44, 682 (1991) [61] A Gilchrist, C W Gardiner and P D Drummond, Phys Rev A 55, 3014 (1997) [62] T Takagi, Japan J Math 1, 83 (1925) [63] R A Horn and C R Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985) [64] The authors are grateful to Dr C Gilson, Department of Applied Mathematics, University of Glasgow, for the first derivation of a solution to this problem [65] M A Kasevich, Science 298, 1363 (2002) [66] A D Cronin, J Schmiedmayer and D E Pritchard, Rev Mod Phys 81, 1051 (2009) [67] C Gross, J Phys B: At Mol Opt Phys 45, 103001 (2012) [68] A Sinatra and Y Castin, Eur Phys J D 8, 319 (2000) [69] M Egorov, R P Anderson, V Ivannikov, B Opanchuk, P Drummond, B V Hall and A I Sidorov, Phys Rev A 84, 021605 (2011) [70] B Opanchuk, M Egorov, S Hoffmann, A I Sidorov and P D Drummond, Europhys Lett 97, 50003 (2012) [71] E T Jaynes and F W Cummings, Proc IEEE 51, 89 (1963) [72] B R Mollow, Phys Rev 188, 1969 (1969) [73] C Cohen-Tannoudji and S Reynaud, J Phys B: At Mol Opt Phys 10, 345 (1976) [74] F Scuda, C R Stroud Jr and M Hercher, J Phys B: At Mol Opt Phys 7, L198 (1974) [75] F Y Wu, R E Grove and S Ezekiel, Phys Rev Lett 35, 1426 (1975) [76] W Hartig, W Rasmussen, R Schieder and H Walther, Z Phys A 278, 205 (1976) [77] C C Gerry and P L Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005) [78] G Rempe, H Walther and N Klein, Phys Rev Lett 58, 353 (1987) [79] M Brune, F Schmidt-Kaler, A Maaili, J Dreyer, E Hagley, J M Raimond and S Haroche, Phys Rev Lett 76, 1800 (1996) [80] S Stenholm, Opt Commun 36, 75 (1981) [81] J Eiselt and H Risken, Phys Rev A 43, 346 (1991) [82] B J Dalton, B M Garraway, J Jeffers and S M Barnett, Ann Phys 334, 100 (2013) [83] S Stenholm, Phys Rep 6, (1973) [84] R P Feynman and A R Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965) [85] J Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford University Press, Oxford, 2005) [86] A M Smith and C W Gardiner, Phys Rev A 39, 3511 (1989) [87] Q.-Y He, M D Reid, B Opanchuk, R Polkinghorne, L E C Rosales-Zarate and P D Drummond, Front Phys 7, 16 (2012) www.Ebook777.com Free ebooks ==> www.Ebook777.com Index A addition, 34–36, 40 algebra Grassmann, 34–43, 122 Lie 98, 101 analytic function, 7, 48, 71–4, 117–18, 122, 145, 147, 151, 161, 171, 174, 279–80, 363, 366, 377–8 annihilation operator, 1–8, 13–15, 19–28, 32, 46–7, 55, 65–74, 79–81, 91–108, 112, 116–9, 131, 134–6, 142–5, 150, 154–9, 171, 191, 197–203, 206, 210, 218–26, 262, 278–9, 285, 293, 298, 316–7, 320, 326–7, 334, 337–40, 345, 349–40, 345, 349, 353, 360, 365, 369, 371 anticommutation relation (anticommutator), 6–7, 14, 19, 24–8, 32–5, 40, 43, 50, 55–7, 65, 92, 95–7, 101, 104, 114, 171, 327, 338–9, 343, 350, 355, 361, 369, 372 anticommutation relation (rules), 1, 6–7, 19, 24, 27–8, 32, 34, 43, 57, 95, 97, 104, 114, 171, 327, 338–9, 343, 350, 355, 361, 372 anticommuting numbers see Grassmann numbers antinormal order, 3, 5, 47, 68, 116–7, 175, 183, 198, 200, 202, 204, 279, 359 antisymmetrising operator, 10 associative law, 2, 35, 38 B Baker–Hausdorff theorem, 42–44, 68, 93, 118, 276 Bargmann projector, 115, 140, 145–7, 151, 154, 157, 336–7, 349, 361, 369, 377–8 state, 4, 64, 71–84, 94, 126, 145–8, 217, 219, 237, 344, 349 B distribution see unnormalised distribution beam splitter, 104, 205 Bloch energy, 335 Bogoliubov excitations, 111–14, 319 transformation, 113 Bose–Einstein condensate (BEC), 1, 10, 12, 16–18, 31, 106, 116, 205, 238, 319, 336, 386, 399–407 interferometry, 205–9 trapped, 319–26 Bose–Einstein distribution, 110 gas, 106, 109 Bose enhancement, 27 Bose gas, 106, 109, 111–14 Bose–Fermi distribution, 132, 138–9, 143, 226–7 box normalisation, 210, 334–5 Brownian motion, 161, 182, 196–7, 383 C canonical density operator, 124–130, 145–7, 360, 363–5 distribution, 122, 133, 148–9, 165, 231, 233, 236–7, 349–50, 361, 376-7 form, 4, 64, 78, 82–4, 124–8 132, 140, 143, 151, 153–4, 349–50, 360–1, 377 P (+) representation, 159–61, 173, 228–9, 237, 377 thermal state, 26 transformation, 95–114 cavity quantum electrodynamics, 217 Cauchy–Riemann equations, 48 characteristic function, 3–5, 88, 115, 117–34, 138–9, 144–7, 150, 152–4, 160, 223–6, 349–60, 365–8, 371–8 functional, 7, 278–84 characteristics, method of, 164 chemical potential, 26, 109 coherences, 4–5, 115, 141–3, 174, 196, 209, 215–7, 220, 222, 226–7, 278, 286 coherent state, 2, 4, 64–94, 101, 107, 115, 120, 126, 145, 148, 217, 219, 236, 345–8, 349 operator equivalence, 81 projector, 77–8, 336–7, 345–7 representation, 80–81, 85 collapse and revival, 217, 219, 236 collision, 25–6, 205–6, 208 commutation relation, 1, 6–7, 14, 19, 24, 27–8, 32, 95, 97, 104, 106–7, 112–4, 350, 361 completeness, 64, 71, 74, 82, 85–6, 91–2, 94, 142, 239, 244, 246, 257, 266, 272, 294, 296, 320, 327, 347–9, 387, 395 www.Ebook777.com Free ebooks ==> www.Ebook777.com 414 Index complex conjugate, 3, 13–14, 37–8, 43, 53, 59–61, 63, 64, 67, 71, 74, 77, 91–2, 118–9, 121, 129, 162, 169, 225–6, 240, 253, 261, 271, 280–2, 316–7 differentiation, 72 numbers (c-numbers), 1–2, 34–44 plane, 46–8, 64, 71, 82, 118–23, 225 P, 121–3, 129, 225 variables, 47–8, 64, 84, 118, 146, 148, 150, 158, 162, 167, 181 Cooper pair, 205, 209–17 Corney and Drummond, 116, 191 correspondence rules, 4–5, 90, 115, 140, 144–59, 161, 165, 171–3, 218, 227, 233, 237, 278–9, 287–93, 321–2, 360–78, 399, 407 creation operator see annihilation operator D delta function, 8, 22–3, 111, 241, 243–4, 251, 262, 264–5, 299, 308–9, 314, 324, 327, 334, 386, 390, 393 restricted, 238, 320, 324, 327, 386–9, 404 density operator, 3–4, 6–7, 11–13, 16, 26, 31–2, 64, 78, 80–1, 84, 90, 109–10, 115–17, 120, 124–40, 144–7, 151, 153–8, 171, 191, 206, 220, 222–3, 225, 229, 233–4, 237–8, 278–9, 285–7, 293, 336–7, 340, 349, 360–1, 363, 365, 367–8, 371, 377, 407 diffusion, 162, 169, 182, 196–7, 199, 201, 203, 207–8, 307 matrix, 115, 145, 148, 163–6, 170–3, 181, 188–9, 192–3, 212–14, 237, 293–300, 302–3, 307–16, 324–5, 329–30, 336, 380, 383 non-local, 295, 297, 308, 313–14 positive definite, 161 displacement operator, 65–9, 77, 93, 118, 345 dissipation, 158 distribution function, 2–5, 26, 46–8, 78, 84, 90, 115–84, 191–6, 203, 217–19, 224–9, 231–4, 236–7, 304, 336, 349–64, 366–8, 370–383 functional, 6–7, 238, 278–99, 316, 319, 321, 325–6, 328, 330, 334, 399–400, 403–4, 407 unnormalised, 4–5, 115, 139–44, 157, 171–4, 182, 191–6, 209, 211–12, 278, 285–7, 292–3, 298–9, 316, 326, 328, 330–5, 407 distributive law, 2, 35–6, 38, 41 double-space, 4, 48, 110, 115–7, 121–2, 128, 279, 282, 354, 368, 379, 381 drift, 161–6, 169–70, 182, 188, 191, 194, 196–9, 201–3, 207–8, 211, 295, 297, 302, 307, 315–16, 336, 380, 383 vector, 172–3, 181, 192, 293–4, 296–9, 303, 311–12, 329 Drummond and Gardiner, 5, 84, 127, 164 E electric field, 106–8 electron, 1, 10, 106, 209 eigenvalue (eigenvector), 4, 9, 15–18, 22–3, 64, 69–71, 73–7, 80, 90–3, 126, 142–3, 164, 211, 213, 221–2, 286, 335, 345, 349 entanglement, 12, 18, 99, 102, 105, 109, 211 environment (reservoir), 5, 25–6, 109, 115, 158–9, 196, 206, 340 even and odd Grassmann function, 4, 58, 70, 77, 82–3, 119, 122–126, 130–3, 142–3, 145, 150–1, 155–6, 168–70, 188, 191–2, 203, 225, 268–70, 276–7, 285, 291–2, 343, 349–50, 358, 368–70, 377–8, 409 operator, 67, 81, 86–90, 142–3, 171, 361–3 permutation, 10, 28, 40–4, 50–2, 61 vector, 65–6, 70, 73 F Fermi gas, 109, 286, 326–35, 407–9 –Dirac distribution, 110 field boson, 2, 90–1, 280–1, 293–5, 300–10, 316–17, 386–98 classical, 319, 324 c-number, 238–79, 282 electromagnetic, 2, 106–9, 217 fermion, 66, 91–2, 261–2, 281–2, 296–7, 310–15, 317–8, 386–98 Grassman, 2, 238–79, 282, 284, 327, 333 mode, 1, 107, 218, 224 operator, 5–8, 20–5, 27, 30–2, 64, 66, 80, 90–2, 210, 277, 284–93, 297–8, 319–20, 326–8, 336 stochastic, 18, 299–318, 324–6, 330–5 thermo-, 109–111 Fock state, 4–5, 8, 14–17, 23, 32–33, 64–5, 72–4, 76–7, 80–1, 85–6, 90, 93–4, 99–100, 103, 106, 110, 114, 115, 120, 141–3, 174, 196, 206, 211, 215, 217, 219–20, 278, 334, 345–7, 355–6 projector, 79 Fokker–Planck equation, 4–5, 47, 115, 121–4, 129, 134, 139, 144–76, 181–4, 187, 191–2, 196–8, 201, 206, 208–9, 211–2, 217–18, 227–8, 231–3, 233, 237, 336–7, 359–83, 399–409 functional, 7, 278–9, 282, 287–304, 307–18, 321–5, 328–30, 335 higher-order derivatives, 159–60, 293 www.Ebook777.com Free ebooks ==> www.Ebook777.com Index Fourier Grassmann integral, 125, 276, 277, 342–3, 350 integral, 259, 282 theorem, 60 transform, 7, 23, 333 functional average, 304–7 calculus (derivative, integral), 7, 92, 238–79, 288–99, 322, 324–6, 334, 353, 386–98, 400–8 distribution, 278–99, 316, 319, 325, 330 G generator, 39, 96–103, 114 Gardiner and Zoller, 151, 174 Gaussian projector, 116, 336–7 Glauber, 2, 48, 62, 66, 118 Cahill and, 2, 359 Sudarshan, 4–5, 46, 115, 359 Grassmann calculus (derivative, integral), 4, 45–63, 74, 82, 121, 123–5, 129, 133–4, 137–9, 142, 168–9, 183, 192, 225–6, 228, 238–9, 261–77, 281–3, 342–4, 348–50, 355, 358, 373–5 function, 37–44, 50–63, 76–7, 81–3, 86, 115, 122–4, 126, 132–3, 145, 150–1, 153, 168–72, 185, 188, 190–1, 193, 212, 224–5, 237, 291, 311, 314, 327, 358, 368–9, 377–8, 381, 408–9 fields, 91–2, 238–9, 261–79, 281–3, 285, 287, 313, 315, 317–8, 327, 330, 333, 386, 398 numbers (g-numbers), variables, 1–3, 19, 34–44, 64–75, 89, 101, 115, 117–19, 122, 136, 143, 174–5, 184, 189–92, 195–6, 202–4, 209, 215–19, 223–6, 228, 235–6, 326–7, 336, 345, 356–7, 359, 370–2, 382 states and operators, 67, 77, 87–9, 94, 126, 130, 153–8, 347–8, 360–3, 365, 369–70 Gross–Pitaevskii equation, 206, 319, 324, 326, 336 H Hamiltonian, 5, 9, 12, 20–1, 25–7, 32–3, 46–7, 66, 96, 109, 111–15, 154, 158–60, 167, 205–6, 210, 218, 221–3, 222–3, 287, 289, 291, 295, 298, 319–20, 327–8, 340–1, 371, 399, 407 Heisenberg picture, 98, 340 uncertainty principle, 46, 107 Hermiticity, 11, 20–1, 38, 77, 84, 96, 101, 113, 127, 132–4, 164, 188, 225 Hilbert space, 8, 11–12, 14, 16, 20, 64–5, 109 415 Hong Ou and Mandel, 109 Husimi distribution see Q distribution I identical particles, 8–12, 16, 20–1, 24, 81, 134, 169 interaction(s) energy, 12, 319, 402, 409 one-body, 20, 159, 209 two-body, 5, 20–1, 25, 111–12, 115, 154, 158–9, 209, 211, 217, 295, 298, 319, 322–4, 328–9, 335, 401, 406 interference, 12, 31, 104–6, 205, 209 Ito equation, 5, 7, 115, 144–5, 159, 161, 164, 167, 169, 171, 174–96, 198, 200, 206–9, 212–15, 217, 278–9, 287, 299–305, 307, 310–19, 324–6, 330–333, 336–7, 380, 382 J Jacobian, 47, 59, 255–7, 272–3 Jaynes–Cummings model, 205, 217–37 Josephson model, 205–6 K kinetic energy, 154, 320, 322–3, 328, 371, 399–400, 403–5, 407–8 L Lagrangian, 241 Langevin equation, 5, 115, 139, 174–204, 237, 278–9, 287, 299–318, 336, 359, 379–85 Laplace transform, 219 Laplacian, 333, 335 laser, 1, 107–8, 217 Liouville–von Neumann equation, 4, 7, 25, 115, 144, 158, 160, 167, 171, 206, 236, 278, 287, 360 M Markovian coupling (approximation), 115, 159, 340–1 master equation, 4, 7, 25–6, 47, 115, 144, 158–63, 167–8, 173, 278, 287, 340–1, 371 matrix coupling, 188–91 element, 13–14, 21, 26–7, 29, 43, 64, 80–1, 84–5, 87, 119, 126–8, 145, 162, 168, 181, 194–5, 220–1, 223, 299–300, 308–11, 313–15, 324–5, 330–1, 340, 347 Matsubara equation, 26, 144, 158–9, 209, 287 mean-field, 116, 206–9 measurement, 6–7, 12, 30–2, 108, 195, 286 mixed state, 6, 11–12, 16, 25, 109–10, 220–1, 223, 237 www.Ebook777.com Free ebooks ==> www.Ebook777.com 416 Index mode, 1, 3–17, 20–30, 53, 56, 60–2, 64, 67–70, 74–6, 79, 84, 88–90, 97–103, 106–21, 129, 143, 159, 162–3, 174, 205–40, 279–85, 287–93, 307, 336–40, 347–8, 359–60, 391 conjugate, 91–2, 388, 390 function, 90–2, 244–8, 254–8, 260–1, 265–6, 271–4, 294–99, 302, 305, 311, 316–17, 386–7, 389 multi-, 72–3, 319–35, 399–409 two-, 17–20, 97–106 molecule, 12–13, 21, 196 momentum, 7, 23, 46, 107, 112–14, 333–4, 386 angular, 9–10 cut-off, 238, 320, 327 monomial, 38–43, 49–51, 55–8, 61, 67, 119–20, 169, 171, 228, 343 multiplication, 2, 4, 7, 34–43, 65, 89–92, 150, 153, 157, 239, 307, 363, 369, 371, 377 N noise, 5, 7, 107–8, 158, 163, 177, 207–9, 330 field, 299–303, 308–18, 324–5 Gaussian–Markov random, 180–1, 185, 190–3, 300–2, 305, 311, 336–7, 382 non-analytic function, 84, 121, 129, 146, 152, 161, 174, 227, 282, 364, 367 non-physical state, 15–19, 76, 223 normal ordering, 2–3, 5–6, 27–9, 47, 68, 79–80, 93, 115–121, 134, 144, 150, 152, 174–6, 183–4, 192, 196, 199, 203, 209, 220, 224, 278–9, 282, 325–6, 350–2, 354, 359, 367 number operator, 15, 17–18, 26, 29, 100, 103, 109, 111–13, 222 -squeezed state, 18 O odd see even and odd open system, 11, 16 optical lattice, 326–35 orthogonality, 13, 24, 29, 64, 69–70, 72–3, 92, 94, 254, 280–1, 294, 296, 298, 327, 390–1 orthonormality, 9–11, 15, 19, 22, 24, 90–1, 213, 238–9, 261, 320, 386, 389 P parity, 100, 268–9 particle-pair creation, 106–14 Pauli exclusion principle, 6, 10, 15, 17, 25, 73, 75, 106, 220 blockade, 27 permutation, 10, 20–1, 28, 39, 58, 85, 354 P distribution (function, functional), 5, 46, 84, 115–22, 127–39, 144–63, 165, 167, 174–6, 181–4, 217–18, 224–29, 231–7, 278–89, 293–4, 296, 298, 299, 304, 316, 319, 321, 325, 336, 349–59, 361, 367, 399–400 positive, 4–5, 64, 84, 115–16, 118, 121–2, 128–30, 145, 148, 150, 158–64, 169, 173–4, 206, 209, 217–18, 224–5, 237, 282, 319, 321–6, 336–7, 379, 381, 399–406 photon, 2, 11–12, 106–9, 159, 217–19, 222, 226 physical state, 9–20, 24, 76, 91–2, 116, 119, 124, 222–4 Plimak et al., 2, 140, 151, 191–2, 209 population, 4–5, 115, 141–3, 174, 196, 215–221, 232, 278, 286, 334 position, 5–7, 22–5, 30–3, 46, 90, 107, 238–9, 241, 261, 271, 279, 286 potential well (energy), 25, 85, 111–12, 205–6, 209–10, 241, 319–20, 322–3, 326, 328–9, 333, 335–6, 400–1, 405–8 probability, 2, 4, 6, 26, 31–2, 46, 105–6, 221–3, 226, 286 pure state, 6, 11–12, 15–16, 25, 32, 75–6, 82, 109–10, 220 purification, 110 Q Q distribution (function), 5, 46, 115–22, 359 quadrature, 107–8 quantum correlation function, 5–7, 27, 30–2, 47, 64, 93, 115–17, 120, 127, 134–44, 171, 174–6, 183–4, 192, 195–204, 224–7, 278–9, 284–7, 299, 325–6, 350–9, 379–81 optics, 3, 46–7, 217–19, 233–7 quasi-distribution (probability), 4, 46–8, 53, 182, 224, 326, 379–82 R Rabi frequency (oscillation), 217–9, 236 relaxation, 26, 161–3, 167, 169, 173, 206, 209, 340 reservoir see environment resonance fluorescence, 217 restricted functions, 239–40, 386–98 R representation, 84 S Schră odinger equation (picture), 12, 25, 98, 238 second quantisation, 8, 20–2 shifts, 206–8, 341 single-particle state, 1, 8–10, 13, 19–22, 104–5, 340 www.Ebook777.com Free ebooks ==> www.Ebook777.com Index spin, 1, 9, 17, 25, 209–11, 219–20, 279, 293, 297–9, 315, 326, 328, 333 squeezing (squeezed state), 2, 6, 28, 106–9 Stenholm, 217, 219, 231, 237 stochastic, 2, 5, 7, 75, 115–16, 141, 144–5, 159, 163–4, 167, 169, 171, 174–204, 207–9, 212–17, 229, 237, 278–9, 287, 299–319, 324–7, 330–7, 359, 379–82 Stratonovich form, 380 superconductor, 209 super-selection rule, 11–13, 21, 220–1 s-wave interactions (scattering), 25, 320, 328 symmetric ordering, 5, 68, 116–18, 122, 135, 150, 152, 208–9, 279, 325–6, 352–9, 367–8, 407 symmetrising operator, 10 T Takagi factorisation, 164, 190, 193, 207, 213, 300, 315, 330–1 Taylor series (expansion), 53–4, 177, 184, 249, 305 temperature, 10, 23, 26, 109–14, 158, 209, 319, 336 thermal state, 26, 109–11 thermofields, 109–11 third-order derivative, 159, 163, 208, 324, 406 trace, 3, 6, 11, 26, 64, 77–8, 85–90, 116, 119, 131, 142, 145, 154, 345–7, 349–50, 355, 362, 363, 371–3, 377 cyclic property of, 3, 118, 281–2, 366 trajectory, 197 417 transition, 2, 8, 26, 141, 217–18, 220–1, 230, 238, 286, 340 tunnelling, 205–6, 208–9 two-level atom, 162, 217–9 U ultracold atoms, 1, 23 unitary transformation, 64–8, 95–103, 106, 111, 345 unit operator, 89 V vacuum, 3, 8, 11, 17, 19, 28–32, 64–5, 69–70, 77, 79, 99, 103, 107–8, 110–11, 120, 127, 142, 219–221, 339, 345 W W distribution (function, functional, Wigner distribution), 5, 46, 115–22, 163–4, 289, 319, 325, 359, 368, 403–7 weak coupling, 26, 109, 115, 159, 206, 319, 340 Wiener increment, 174–5, 192, 195, 215–17, 335 stochastic function, 182, 382–3 variables, 301, 311 Wick’s theorem, 28 Z zero-particle state see vacuum zero-range approximation, 206, 295, 320, 327–9 www.Ebook777.com Free ebooks ==> www.Ebook777.com www.Ebook777.com ... of solids www.Ebook777.com Free ebooks ==> www.Ebook777.com Phase Space Methods for Degenerate Quantum Gases Bryan J Dalton Centre for Quantum and Optical Science, Swinburne University of Technology,... field of quantum atom optics Phase space distribution function methods involving phase space variables suitable for systems where small numbers of modes are involved are complemented by phase space. .. CAMBRIDGE UNIVERSITY OF OXFORD CERN, GENEVA International Series of Monographs on Physics 163 B.J Dalton, J Jeffers, S.M Barnett: Phase space methods for degenerate quantum gases 162 W.D McComb: Homogeneous,