 | Dalton 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
|
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
|
|
