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ОбложкаDalton B.J. Phase space methods for degenerate quantum gases / B.J.Dalton, J.Jeffers, S.M.Barnett. - Oxford: Oxford university press, 2015. - xiv, 417 p. - (International series of monographs on physics; 163). - Bibliogr.: p.410-412. - Ind.: p.413-417. - ISBN 978-0-19-956274-9
Шифр: (И/В31-D16) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / 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
   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
   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
   9.4  Ito Stochastic Equations for Fermions - Unnormalised
        Distribution Functions ................................ 191
   9.5  Fluctuations and Time Dependence of Quantum
        Correlation Functions ................................. 196
        9.5.1  Boson Fluctuations ............................. 196
        9.5.2  Boson Correlation Functions .................... 197
        9.5.3  Fermion Fluctuations ........................... 200
        9.5.4  Fermion Correlation Functions .................. 201
   Exercises .................................................. 204
10 Application to Few-Mode Systems ............................ 205
   10.1 Boson Case - Two-Mode ВЕС Interferometry .............. 205
        10.1.1 Introduction ................................... 205
        10.1.2 Modes and Hamiltonian .......................... 206
        10.1.3 Fokker-Planck and Ito Equations - P+ ........... 206
        10.1.4 Fokker-Planck and Ito Equations - Wigner ....... 208
        10.1.5 Conclusion ..................................... 209
   10.2 Fermion Case - Cooper Pairing in a Two-Fermion
        System ................................................ 209
        10.2.1 Introduction ................................... 209
        10.2.2 Modes and Hamiltonian .......................... 210
        10.2.3 Initial Conditions ............................. 211
        10.2.4 Fokker-Planck Equations - Unnormalised В ....... 211
        10.2.5 Ito Equations - Unnormalised В ................. 212
        10.2.6 Populations and Coherences ..................... 215
        10.2.7 Conclusion ..................................... 217
   10.3 Combined Case - Jaynes-Cummings Model ................. 217
        10.3.1 Introduction ................................... 217
        10.3.2 Physics of One-Atom Cavity Mode System ......... 218
        10.3.3 Fermionic and Bosonic Modes .................... 219
        10.3.4 Quantum States ................................. 220
        10.3.5 Population and Transition Operators ............ 220
        10.3.6 Hamiltonian and Number Operators ............... 221
        10.3.7 Probabilities and Coherences ................... 222
        10.3.8 Characteristic Function ........................ 223
        10.3.9 Distribution Function .......................... 224
        10.3.10 Probabilities and Coherences as Phase Space
                Integrals ..................................... 226
        10.3.11 Fokker-Planck Equation ........................ 227
        10.3.12 Coupled Distribution Function Coefficients .... 228
        10.3.13 Initial Conditions for Uncorrelated Case ...... 229
        10.3.14 Rotating Phase Variables and Coefficients ..... 229
        10.3.15 Solution to Fokker-Planck Equation ............ 231
        10.3.16 Comparison with Standard Quantum Optics
                Result ........................................ 233
        10.3.17 Application of Results ........................ 236
        10.3.18 Conclusion .................................... 236
   Exercises .................................................. 237
11 Functional Calculus for C-Number and Grassmann Fields ...... 238
   11.1 Features .............................................. 238
   11.2 Functionals of Bosonic C-Number Fields ................ 239
        11.2.1 Basic Idea ..................................... 239
   11.3 Examples of C-Number Functionals ...................... 241
   11.4 Functional Differentiation for C-Number Fields ........ 242
        11.4.1 Definition of Functional Derivative ............ 242
        11.4.2 Examples of Functional Derivatives ............. 243
        11.4.3 Functional Derivative and Mode Functions ....... 244
        11.4.4 Taylor Expansion for Functionals ............... 249
        11.4.5 Basic Rules for Functional Derivatives ......... 249
        11.4.6 Other Rules for Functional Derivatives ......... 251
   11.5 Functional Integration for C-Number Fields ............ 253
        11.5.1 Definition of Functional Integral .............. 253
        11.5.2 Functional Integrals and Phase Space
               Integrals ...................................... 254
        11.5.3 Functional Integration by Parts ................ 257
        11.5.4 Differentiating a Functional Integral .......... 258
        11.5.5 Examples of Functional Integrals ............... 260
   11.6 Functionals of Fermionic Grassmann Fields ............. 261
        11.6.1 Basic Idea ..................................... 261
        11.6.2 Examples of Grassmann-Number Functionals ....... 262
   11.7 Functional Differentiation for Grassmann Fields ....... 263
        11.7.1 Definition of Functional Derivative ............ 263
        11.7.2 Examples of Grassmann Functional Derivatives ... 264
        11.7.3 Grassmann Functional Derivative and Mode
               Functions ...................................... 265
        11.7.4 Basic Rules for Grassmann Functional
               Derivatives .................................... 268
        11.7.5 Other Rules for Grassmann Functional
               Derivatives .................................... 270
   11.8 Functional Integration for Grassmann Fields ........... 271
        11.8.1 Definition of Functional Integral .............. 271
        11.8.2 Functional Integrals and Phase Space
               Integrals ...................................... 271
        11.8.3 Functional Integration by Parts ................ 274
        11.8.4 Differentiating a Functional Integral .......... 275
   Exercises .................................................. 277
12 Distribution Functionals in Quantum Atom Optics ............ 278
   12.1 Quantum Correlation Functions ......................... 279
   12.2 Characteristic Functionals ............................ 279
        12.2.1 Boson Case ..................................... 280
        12.2.2 Fermion Case ................................... 281
   12.3 Distribution Functionals .............................. 282
        12.3.1 Boson Case ..................................... 282
        12.3.2 Fermion Case ................................... 283
        12.3.3 Quantum Correlation Functions .................. 284
   12.4 Unnormalised Distribution Functionals - Fermions ...... 285
        12.4.1 Distribution Functional ........................ 285
        12.4.2 Populations and Coherences ..................... 286
13 Functional Fokker-Planck Equations ......................... 287
   13.1 Correspondence Rules for Boson and Fermion Functional
        Fokker-Planck Equations ............................... 288
        13.1.1 Boson Case ..................................... 288
        13.1.2 Fermion Case ................................... 290
        13.1.3 Fermion Case - Unnormalised Distribution
               Functional ..................................... 292
   13.2 Boson and Fermion Functional Fokker-Planck Equations .. 293
        13.2.1 Boson Case ..................................... 293
        13.2.2 Fermion Case ................................... 296
   13.3 Generalisation to Several Fields ...................... 297
14 Langevin Field Equations ................................... 299
   14.1 Boson Stochastic Field Equations ...................... 300
        14.1.1 Ito Equations for Bosonic Stochastic Phase
               Variables ...................................... 300
        14.1.2 Derivation of Bosonic Ito Stochastic Field
               Equations ...................................... 302
        14.1.3 Alternative Derivation of Bosonic Stochastic
               Field Equations ................................ 304
        14.1.4 Properties of Bosonic Noise Fields ............. 308
   14.2 Fermion Stochastic Field Equations .................... 310
        14.2.1 Ito Equations for Fermionic Stochastic Phase
               Space Variables ................................ 310
        14.2.2 Derivation of Fermionic Ito Stochastic Field
               Equations ...................................... 311
        14.2.3 Properties of Fermionic Noise Fields ........... 313
   14.3 Ito Field Equations - Generalisation to Several
        Fields ................................................ 315
   14.4 Summary of Boson and Fermion Stochastic Field
        Equations ............................................. 316
        14.4.1 Boson Case ..................................... 316
        14.4.2 Fermion Case ................................... 317
   Exercises .................................................. 318
15 Application to Multi-Mode Systems .......................... 319
   15.1 Boson Case - Trapped Bose-Einstein Condensate ......... 319
        15.1.1 Introduction ................................... 319
        15.1.2 Field Operators ................................ 319
        15.1.3 Hamiltonian .................................... 320
        15.1.4 Functional Fokker-Planck Equations and
               Correspondence Rules ........................... 321
        15.1.5 Functional Fokker-Planck Equation - Positive
               P Case ......................................... 322
        15.1.6 Functional Fokker-Planck Equation - Wigner
               Case ........................................... 323
        15.1.7 Ito Equations for Positive P Case .............. 324
        15.1.8 Ito Equations for Wigner Case .................. 325
        15.1.9 Stochastic Averages for Quantum Correlation
               Functions ...................................... 326
   15.2 Fermion Case - Fermions in an Optical Lattice ......... 326
        15.2.1 Introduction ................................... 326
        15.2.2 Field Operators ................................ 326
        15.2.3 Hamiltonian .................................... 328
        15.2.4 Functional Fokker-Planck Equation -
               Unnormalised В ................................. 328
        15.2.5 Ito Equations for Unnormalised Distribution
               Functional ..................................... 330
        15.2.6 Case of Free Fermi Gas ......................... 333
        15.2.7 Case of Optical Lattice ........................ 335
   Exercise ................................................... 335
16 Further Developments ....................................... 336
Appendix A  Fermion Anticommutation Rules ..................... 338
Appendix В  Markovian Master Equation ......................... 340
Appendix С  Grassmann Calculus ................................ 342
   C.l  Double-Integral Result ................................ 342
   C.2  Grassmann Fourier Integral ............................ 342
   C.3  Differentiating Multiple Grassmann Integrals of
        Functions of Two Sets of Grassmann Variables .......... 343
Appendix D  Properties of Coherent States ..................... 345
   D.l Fermion Coherent-State Eigenvalue Equation ............. 345
   D.2  Trace of Coherent-State Projectors .................... 345
   D.3  Completeness Relation for Fermion Coherent States ..... 347
Appendix E  Phase Space Distributions for Bosons and
   Fermions ................................................... 349
   E.l Canonical Forms of Fermion Distribution Function ....... 349
   E.2  Quantum Correlation Functions ......................... 350
        E.2.1  Boson Case - Normal Ordering ................... 350
        E.2.2  Boson Case - Symmetric Ordering ................ 352
        E.2.3  Fermion Case ................................... 355
   E.3  Normal, Symmetric and Antinormal Distribution
        Functions ............................................. 359
Appendix F  Fokker-Planck Equations ........................... 360
   F.l  Correspondence Rules .................................. 360
        F.l.l  Grassmann and Operator Formulae .................360
        F.l.2  Boson Case - Canonical-Density-Operator
               Approach ....................................... 363
        F.1.3  Boson Case - Characteristic-Function Approach .. 365
        F.l.4  Fermion Case - Density Operator Approach ....... 368
        F.l.5  Fermion Case - Characteristic-Function Method .. 371
        F.l.6  Boson Case - Canonical-Distribution Rules ...... 376
   F.2  Successive Correspondence Rules ....................... 376
Appendix G  Langevin Equations ................................ 379
   G.l  Stochastic Averages ................................... 379
        G.l.l Basic Concepts .................................. 379
        G.1.2  Gaussian-Markov Stochastic Process ............. 382
   G.2  Fluctuations .......................................... 383
        G.2.1  Boson Correlation Functions .................... 383
        G.2.2  Fermion Correlation Functions .................. 384
Appendix H  Functional Calculus for Restricted Boson and
   Fermion Fields ............................................. 386
   H.l  General Features ...................................... 386
   H.2  Functional for Restricted C-Number Fields ............. 386
        H.2.1  Restricted Functions ........................... 386
        H.2.2  Functional ..................................... 387
        H.2.3  Related Restricted Sets ........................ 388
   H.3  Functional Differentiation for Restricted C-Number
        Fields ................................................ 390
        H.3.1  Definition of Functional Derivative ............ 390
        H.3.2  Examples of Restricted Functional Derivatives .. 392
        H.3.3  Restricted Functional Derivatives and Mode
               Functions ...................................... 395
   H.4  Functional Integration for Restricted C-Number
        Functions ............................................. 397
        H.4.1  Definition of Functional Integral .............. 397
   H.5  Functional for Restricted Grassmann Fields ............ 398
Appendix I  Applications to Multi-Mode Systems ................ 399
   I.1  Bose Condensate - Derivation of Functional Fokker-
        Planck Equations ...................................... 399
        I.1.1  Positive P Case ................................ 399
        I.1.2  Wigner Case .................................... 403
   I.2  Fermi Gas - Derivation of Functional Fokker-Planck
        Equations ............................................. 407
        I.2.1  Unnormalised В Case ............................ 407
   References ................................................. 410

Index ......................................................... 413

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