Quantum Optics

By: D.F. Walls, G.J. Milburn

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Quantum Optics gives a comprehensive coverage of developments in quantum optics over the past twenty years. In the early chapters the formalism of quantum optics is elucidated and the main techniques are introduced. These are applied in the later chapters to problems such as squeezed states of light, resonance fluorescence, laser theory, quantum theory of four-wave mixing, quantum non-demolition measurements, Bell's inequalities, and atom optics. Experimental results are used to illustrate the theory throughout. This yields the most comprehensive and up-to-date coverage of experiment and theory in quantum optics in any textbook.

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Introduction	p. 1
Quantisation of the Electromagnetic Field	p. 7
Field Quantisation	p. 7
Fock or Number States	p. 10
Coherent States	p. 12
Squeezed States	p. 15
Two-Photon Coherent States	p. 18
Variance in the Electric Field	p. 20
Multimode Squeezed States	p. 22
Phase Properties of the Field	p. 23
Exercises	p. 26
References	p. 26
Further Reading	p. 27
Coherence Properties of the Electromagnetic Field	p. 29
Field-Correlation Functions	p. 29
Properties of the Correlation Functions	p. 31
Correlation Functions and Optical Coherence	p. 32
First-Order Optical Coherence	p. 34
Coherent Field	p. 37
Photon Correlation Measurements	p. 38
Quantum Mechanical Fields	p. 41
Squeezed State	p. 42
Squeezed Vacuum	p. 44
Phase-Dependent Correlation Functions	p. 44
Photon Counting Measurements	p. 46
Classical Theory	p. 46
Constant Intensity	p. 48
Fluctuating Intensity-Short-Time Limit	p. 48
Quantum Mechanical Photon Count Distribution	p. 50
Coherent Light	p. 51
Chaotic Light	p. 51
Photo-Electron Current Fluctuations	p. 52
Exercises	p. 54
References	p. 55
Further Reading	p. 55
Representations of the Electromagnetic Field	p. 57
Expansion in Number States	p. 57
Expansion in Coherent States	p. 58
P Representation	p. 58
Wigner's Phase-Space Density	p. 62
Q Function	p. 65
R Representation	p. 67
Generalized P Representations	p. 68
Positive P Representation	p. 71
Exercises	p. 72
References	p. 72
Quantum Phenomena in Simple Systems in Nonlinear Optics	p. 73
Single-Mode Quantum Statistics	p. 73
Degenerate Parametric Amplifier	p. 73
Photon Statistics	p. 75
Wigner Function	p. 76
Two-Mode Quantum Correlations	p. 77
Non-degenerate Parametric Amplifier	p. 77
Squeezing	p. 80
Quadrature Correlations and the Einstein-Podolsky-Rosen Paradox	p. 82
Wigner Function	p. 83
Reduced Density Operator	p. 84
Quantum Limits to Amplification	p. 86
Amplitude Squeezed State with Poisson Photon Number Statistics	p. 88
Exercises	p. 91
References	p. 91
Stochastic Methods	p. 93
Master Equation	p. 93
Equivalent c-Number Equations	p. 99
Photon Number Representation	p. 99
P Representation	p. 100
Properties of Fokker-Planck Equations	p. 102
Steady State Solutions - Potential Conditions	p. 103
Time Dependent Solution	p. 104
Q Representation	p. 105
Wigner Function	p. 107
Generalized P Representation	p. 109
Stochastic Differential Equations	p. 112
Use of the Positive P Representation	p. 115
Linear Processes with Constant Diffusion	p. 116
Two Time Correlation Functions in Quantum Markov Processes	p. 117
Quantum Regression Theorem	p. 118
Application to Systems with a P Representation	p. 118
Stochastic Unravellings	p. 119
Simulating Quantum Trajectories	p. 123
Exercises	p. 124
References	p. 125
Further Reading	p. 125
Input-Output Formulation of Optical Cavities	p. 127
Cavity Modes	p. 127
Linear Systems	p. 131
Two-Sided Cavity	p. 132
Two Time Correlation Functions	p. 133
Spectrum of Squeezing	p. 135
Parametric Oscillator	p. 136
Squeezing in the Total Field	p. 138
Fokker-Planck Equation	p. 138
Exercises	p. 141
References	p. 141
Further Reading	p. 141
Generation and Applications of Squeezed Light	p. 143
Parametric Oscillation and Second Harmonic Generation	p. 143
Semi-Classical Steady States and Stability Analysis	p. 145
Parametric Oscillation	p. 146
Second Harmonic Generation	p. 146
Squeezing Spectrum	p. 147
Parametric Oscillation	p. 148
Experiments	p. 149
Twin Beam Generation and Intensity Correlations	p. 151
Second Harmonic Generation	p. 156
Experiments	p. 157
Applications of Squeezed Light	p. 158
Interferometric Detection of Gravitational Radiation	p. 158
Sub-Shot-Noise Phase Measurements	p. 171
Quantum Information	p. 173
Exercises	p. 174
References	p. 174
Further Reading	p. 175
Nonlinear Quantum Dissipative Systems	p. 177
Optical Parametric Oscillator: Complex P Function	p. 177
Optical Parametric Oscillator: Positive P Function	p. 181
Quantum Tunnelling Time	p. 186
Dispersive Optical Bistability	p. 190
Comment on the Use of the Q and Wigner Representations	p. 192
Exercises	p. 192
Appendix	p. 193
Evaluation of Moments for the Complex P function for Parametric Oscillation (9.17)	p. 193
Evaluation of the Moments for the Complex P Function for Optical Bistability (9.48)	p. 194
References	p. 195
Further Reading	p. 195
Interaction of Radiation with Atoms	p. 197
Quantization of the Many-Electron System	p. 197
Interaction of a Single Two-Level Atom with a Single Mode Field	p. 201
Spontaneous Emission from a Two-Level Atom	p. 203
Phase Decay in a Two-Level System	p. 204
Resonance Fluorescence	p. 205
Exercises	p. 210
References	p. 210
Further Reading	p. 211
CQED	p. 213
Cavity QED	p. 213
Vacuum Rabi Splitting	p. 217
Single Photon Sources	p. 218
Cavity QED with N Atoms	p. 221
Circuit QED	p. 225
Exercises	p. 227
References	p. 228
Further Reading	p. 229
Quantum Theory of the Laser	p. 231
Master Equation	p. 231
Photon Statistics	p. 233
Spectrum of Intensity Fluctuations	p. 234
Laser Linewidth	p. 237
Regularly Pumped Laser	p. 238
Appendix: Derivation of the Single-Atom Increment	p. 242
Exercises	p. 245
References	p. 245
Bells Inequalities in Quantum Optics	p. 247
The Einstein-Podolsky-Rosen (EPR) Argument	p. 247
Bell Inequalities and the Aspect Experiment	p. 248
Violations of Bell's Inequalities Using a Parametric Amplifier Source	p. 254
One-Photon Interference	p. 259
Exercises	p. 264
References	p. 264
Quantum Nondemolition Measurements	p. 267
Concept of a QND Measurement	p. 268
Back Action Evasion	p. 270
Criteria for a QND Measurement	p. 270
The Beam Splitter	p. 273
Ideal Quadrature QND Measurements	p. 276
Experimental Realisation	p. 277
A Photon Number QND Scheme	p. 279
Exercises	p. 281
References	p. 282
Quantum Coherence and Measurement Theory	p. 283
Quantum Coherence	p. 283
The Effect of Dissipation	p. 288
Experimental Observation of Coherence Decay	p. 291
Quantum Measurement Theory	p. 293
General Measurement Theory	p. 294
The Pointer Basis	p. 296
Examples of Pointer Observables	p. 299
Model of a Measurement	p. 299
Conditional States and Quantum Trajectories	p. 302
Homodyne Measurement of a Cavity Field	p. 303
Exercises	p. 305
References	p. 306
Quantum Information	p. 307
Introduction	p. 307
The Qubit	p. 308
Entanglement	p. 310
Quantum Key Distribution	p. 312
Quantum Teleportation	p. 318
Quantum Computation	p. 324
Linear Optical Quantum Gates	p. 327
Single Photon Sources	p. 336
Exercises	p. 343
References	p. 344
Further Reading	p. 346
Ion Traps	p. 347
Introduction	p. 347
Trapping and Cooling	p. 347
Novel Quantum States	p. 353
Trapping Multiple Ions	p. 356
Ion Trap Quantum Information Processing	p. 359
Exercises	p. 362
References	p. 363
Light Forces	p. 365
Radiative Forces in the Semiclassical Limit	p. 366
Mean Force for a Two-Level Atom Initially at Rest	p. 368
Friction Force for a Moving Atom	p. 371
Laser Standing Wave-Doppler Cooling	p. 372
Dressed State Description of the Dipole Force	p. 374
Atomic Diffraction by a Standing Wave	p. 377
Optical Stern-Gerlach Effect	p. 381
Quantum Chaos	p. 385
Dynamical Tunnelling	p. 387
Dynamical Localisation	p. 389
The Effect of Spontaneous Emission	p. 390
References	p. 394
Further Reading	p. 395
Bose-Einstein Condensation	p. 397
Hamiltonian: Binary Collision Model	p. 398
Mean-Field Theory - Gross-Pitaevskii Equation	p. 399
Single Mode Approximation	p. 400
Quantum State of the Condensate	p. 401
Quantum Phase Diffusion: Collapses and Revivals of the Condensate Phase	p. 401
Interference of Two Bose-Einstein Condensates and Measurement-Induced Phase	p. 405
Interference of Two Condensates Initially in Number States	p. 405
Quantum Tunneling of a Two Component Condensate	p. 409
Semiclassical Dynamics	p. 411
Quantum Dynamics	p. 414
Coherence Properties of Bose-Einstein Condensates	p. 416
1st Order Coherence	p. 416
Higher Order Coherence	p. 417
Exercises	p. 419
References	p. 419
Further Reading	p. 420
Index	p. 421
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Quantum Optics

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