Quantum Optics, Third, revised and extended edition

Coherent States in Quantum Physics......Page 5
Contents......Page 7
Preface......Page 15
Part One Coherent States......Page 17
1.1 The Motivations......Page 19
2.2 Four Representations of Quantum States......Page 29
2.2.2 Momentum Representation......Page 30
2.2.3 Number or Fock Representation......Page 31
2.2.5 Analytical or Fock–Bargmann Representation......Page 32
2.2.6 Operators in Fock–Bargmann Representation......Page 33
2.3.1 Bergman Kernel as a Coherent State......Page 34
2.3.3 Schrödinger Coherent States in the Two Other Representations......Page 35
2.5 Why the Adjective Coherent?......Page 36
3.1 Introduction......Page 41
3.2.2 \"Coherent\'\' Resolution of the Unity......Page 42
3.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane (as a Euclidean Space)......Page 43
3.2.4 Analytical Bridge......Page 44
3.2.5 Overcompleteness and Reproducing Properties......Page 45
3.3.2 Lower Symbols......Page 46
3.3.3 Heisenberg Inequalities......Page 47
3.3.4 Time Evolution and Phase Space......Page 48
3.4.1 The Vacuum as a Transported Probe…......Page 51
3.4.2 Under the Action of…......Page 52
3.4.4 Symplectic Phase and the Weyl–Heisenberg Group......Page 53
3.4.5 Coherent States as Tools in Signal Analysis......Page 54
3.5 Quantum Distributions and Coherent States......Page 56
3.5.2 The Density Matrix and the Representation \"Q\'\'......Page 57
3.5.3 The Density Matrix and the Representation \"P\'\'......Page 58
3.5.4 The Density Matrix and the Wigner(–Weyl–Ville) Distribution......Page 59
3.6 The Feynman Path Integral and Coherent States......Page 60
4.1 Quantum States for Information......Page 65
4.2 Optical Coherent States in Quantum Information......Page 66
4.3.2 Uncertainties on POVMs......Page 67
4.3.3 The Quantum Error Probability or Helstrom Bound......Page 68
4.3.5 Helstrom Bound for Coherent States......Page 69
4.4.1 The Principle......Page 70
4.4.2 Kennedy Receiver Error......Page 71
4.5.2 Sasaki–Hirota Receiver Error......Page 72
4.6.1 The Principle......Page 73
4.6.3 Decision Criterion of the Dolinar Receiver......Page 74
4.6.4 Optimal Control......Page 75
4.6.5 Dolinar Hypothesis Testing Procedure......Page 76
4.7.1 A Theoretical Preliminary......Page 77
4.7.2 Closed-Loop Experiment: the Apparatus......Page 79
4.7.3 Closed-Loop Experiment: the Results......Page 81
4.8 Conclusion......Page 83
5.1 Introduction......Page 85
5.2.1 Poisson and Gamma Distributions......Page 86
5.2.3 The Fock–Bargmann Option......Page 87
5.3 General Setting: \"Quantum\'\' Processing of a Measure Space......Page 88
5.4 Coherent States for the Motion of a Particle on the Circle......Page 92
5.5 More Coherent States for the Motion of a Particle on the Circle......Page 94
6.2 Preliminary Material......Page 95
6.3 The Construction of Spin Coherent States......Page 96
6.5 Spin Coherent States: Group-Theoretical Context......Page 98
6.7 Spin Coherent States: Spherical Harmonics Aspects......Page 102
6.8.1 Matrix Elements of the SU(2) Unitary Irreducible Representations......Page 103
6.8.3 Spin Spherical Harmonics......Page 105
6.8.5 The Important Case: σ = j......Page 107
6.8.7 Infinitesimal Transformation Laws......Page 108
6.8.8 \"Sigma-Spin\'\' Coherent States......Page 109
6.8.9 Covariance Properties of Sigma-Spin Coherent States......Page 111
7.1 Introduction......Page 113
7.2 Coherent States and the Driven Oscillator......Page 114
7.3.1 The Dicke Model......Page 119
7.3.2 The Partition Function......Page 121
7.3.3 The Critical Temperature......Page 122
7.3.4 Average Number of Photons per Atom......Page 124
7.4 Application of Spin Coherent States to Quantum Magnetism......Page 125
7.5 Application of Spin Coherent States to Classical and Thermodynamical Limits......Page 127
7.5.1 Symbols and Traces......Page 128
7.5.2 Berezin–Lieb Inequalities for the Partition Function......Page 130
7.5.3 Application to the Heisenberg Model......Page 132
8.2 The Unit Disk as an Observation Set......Page 133
8.3 Coherent States......Page 135
8.4 Probabilistic Interpretation......Page 136
8.5 Poincaré Half-Plane for Time-Scale Analysis......Page 137
8.6 Symmetries of the Disk and the Half-Plane......Page 138
8.7.1 Cartan Factorization......Page 139
8.7.2 Discrete Series of SU(1,1)......Page 140
8.7.3 Lie Algebra Aspects......Page 142
8.7.4 Coherent States as a Transported Vacuum......Page 143
8.8 A Few Words on Continuous Wavelet Analysis......Page 145
9.2 Classical Motion in the Infinite-Well and Pöschl–Teller Potentials......Page 151
9.2.1 Motion in the Infinite Well......Page 152
9.2.2 Pöschl–Teller Potentials......Page 154
9.3.1 In the Infinite Well......Page 157
9.3.2 In Pöschl–Teller Potentials......Page 158
9.4 The Dynamical Algebra su(1,1)......Page 159
9.5 Sequences of Numbers and Coherent States on the Complex Plane......Page 162
9.6.1 For the Infinite Well......Page 166
9.6.2 For the Pöschl–Teller Potentials......Page 168
9.7.1 Quantum Revivals......Page 169
9.7.2 Mandel Statistical Characterization......Page 171
9.7.3 Temporal Evolution of Symbols......Page 174
9.7.4 Discussion......Page 178
10.1 Introduction......Page 181
10.2.1 The Construction within a Physical Context......Page 182
10.2.2 Algebraic (su(1,1)) Content of Squeezed States......Page 187
10.2.3 Using Squeezed States in Molecular Dynamics......Page 191
11.2 Coherent States for One Fermionic Mode......Page 195
11.3.1 Fermionic Symmetry SU(r)......Page 196
11.3.2 Fermionic Symmetry SO(2r)......Page 201
11.3.3 Fermionic Symmetry SO(2r+1)......Page 203
11.3.4 Graphic Summary......Page 204
11.4 Application to the Hartree–Fock–Bogoliubov Theory......Page 205
Part Two Coherent State Quantization......Page 207
12.2 The Berezin–Klauder Quantization of the Motion of a Particle on the Line......Page 209
12.3.1 Van Hove Canonical Quantization Rules [161]......Page 212
12.4 More Upper and Lower Symbols: the Angle Operator......Page 213
12.5 Quantization of Distributions: Dirac and Others......Page 215
12.6 Finite-Dimensional Canonical Case......Page 218
13.2 Some Ideas on Quantization......Page 223
13.3 One more Coherent State Construction......Page 225
13.4 Coherent State Quantization......Page 227
13.5.1 Quantization and Symbol Calculus......Page 230
13.5.2 Probabilistic Aspects......Page 232
13.6 Quantization with k-Fermionic Coherent States......Page 234
13.7 Final Comments......Page 236
14.2 Coherent State Quantization of a Finite Set with Complex 22 Matrices......Page 239
14.3.1 Quantization with Finite Subfamilies of Haar Wavelets......Page 243
14.3.2 A Two-Dimensional Noncommutative Quantization of the Unit Interval......Page 244
14.4.1 A Retrospective of Various Approaches......Page 245
14.4.2 Pegg–Barnett Phase Operator and Coherent State Quantization......Page 250
14.4.3 A Phase Operator from Two Finite-Dimensional Vector Spaces......Page 251
14.4.4 A Phase Operator from the Interplay Between Finite and Infinite Dimensions......Page 253
15.2.1 The Cylinder as an Observation Set......Page 257
15.2.2 Quantization of Classical Observables......Page 258
15.2.3 Did You Say Canonical?......Page 259
15.3 From the Motion of the Circle to the Motion on 1 + 1 de Sitter Space-Time......Page 260
15.4.1 Introduction......Page 261
15.4.2 The Standard Quantum Context......Page 262
15.4.3 Two-Component Coherent States......Page 263
15.4.4 Quantization of Classical Observables......Page 265
15.4.5 Quantum Behavior through Lower Symbols......Page 269
15.4.6 Discussion......Page 270
15.5 Motion on a Discrete Set of Points......Page 272
16.2 The Torus as a Phase Space......Page 275
16.3 Quantum States on the Torus......Page 277
16.4 Coherent States for the Torus......Page 281
16.5.2 Weyl Quantization of the Torus......Page 283
16.6.1 Quantization of Irrational and Skew Translations......Page 285
16.6.2 Quantization of the Hyperbolic Automorphisms of the Torus......Page 286
16.6.3 Main Results......Page 287
17.2 Quantizations of the 2-Sphere......Page 289
17.2.2 The Hilbert Space and the Coherent States......Page 290
17.2.4 Quantization of Observables......Page 291
17.2.7 Quantization in the Simplest Case: j = 1......Page 292
17.2.9 The Spin Angular Momentum Operators......Page 293
17.3.1 The Construction of the Fuzzy Sphere à la Madore......Page 294
17.3.2 Operators......Page 296
17.4 Summary......Page 298
17.5 The Fuzzy Hyperboloid......Page 299
18 Conclusion and Outlook......Page 303
A.1.1 Examples......Page 305
A.3 Measurable Function......Page 306
A.5 Probability Axioms......Page 307
A.7 Bayes\'s Theorem......Page 308
A.9 Probability Distribution......Page 309
A.11 Conditional Probability Densities......Page 310
A.12 Bayesian Statistical Inference......Page 311
A.13.2 Uniform Distribution......Page 312
B.1 Group Transformations and Representations......Page 319
B.2 Lie Algebras......Page 320
B.3 Lie Groups......Page 322
B.3.1 Extensions of Lie algebras and Lie groups......Page 326
C.2 Matrix Elements of SU(2) Unitary Irreducible Representation......Page 329
C.3 Orthogonality Relations and 3j Symbols......Page 330
C.4 Spin Spherical Harmonics......Page 331
C.5 Transformation Laws......Page 333
C.6 Infinitesimal Transformation Laws......Page 334
C.7 Integrals and 3j Symbols......Page 335
C.8 Important Particular Case: j = 1......Page 336
C.9 Another Important Case: σ = j......Page 337
Appendix D Wigner–Eckart Theorem for Coherent State Quantized Spin Harmonics......Page 339
Appendix E Symmetrization of the Commutator......Page 341
References......Page 345
Index......Page 355