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| Contents | ||||||
| 1 | Introduction | 1 | ||||
| 2 | Quantisation of the Electromagnetic Field | 7 | ||||
| 2.1 | Field Quantisation | 7 | ||||
| 2.2 | Fock or Number States | 10 | ||||
| 2.3 | Coherent States | 12 | ||||
| 2.4 | Squeezed States | 15 | ||||
| 2.5 | Two-Photon Coherent States | 18 | ||||
| 2.6 | Variance in the Electric Field | 20 | ||||
| 2.7 | Multimode Squeezed States | 22 | ||||
| 2.8 | Phase Properties of the Field | 23 | ||||
| Exercises | 26 | |||||
| References | 26 | |||||
| Further Reading | 27 | |||||
| 3 | Coherence Properties of the Electromagnetic Field | 29 | ||||
| 3.1 | Field-Correlation Functions | 29 | ||||
| 3.2 | Properties of the Correlation Functions | 31 | ||||
| 3.3 | Correlation Functions and Optical Coherence | 32 | ||||
| 3.4 | First-Order Optical Coherence | 34 | ||||
| 3.5 | Coherent Field | 37 | ||||
| 3.6 | Photon Correlation Measurements | 38 | ||||
| 3.7 | Quantum Mechanical Fields | 41 | ||||
| 3.7.1 | Squeezed State | 42 | ||||
| 3.7.2 | Squeezed Vacuum | 44 | ||||
| 3.8 | Phase-Dependent Correlation Functions | 44 | ||||
| 3.9 | Photon Counting Measurements | 46 | ||||
| 3.9.1 | Classical Theory | 46 | ||||
| 3.9.2 | Constant Intensity | 48 | ||||
| 3.9.3 | Fluctuating Intensity-Short-Time Limit | 48 | ||||
| 3.10 | Quantum Mechanical Photon Count Distribution | 50 | ||||
| 3.10.1 | Coherent Light | 51 | ||||
| 3.10.2 | Chaotic Light | 51 | ||||
| 3.10.3 | Photo-Electron Current Fluctuations | 52 | ||||
| Exercises | 54 | |||||
| References | 55 | |||||
| Further Reading | 55 | |||||
| 4 | Representations of the Electromagnetic Field | 57 | ||||
| 4.1 | Expansion in Number States | 57 | ||||
| 4.2 | Expansion in Coherent States | 58 | ||||
| 4.2.1 | P Representation | 58 | ||||
| 4.2.2 | Wigner's Phase-Space Density | 62 | ||||
| 4.2.3 | Q Function | 65 | ||||
| 4.2.4 | R Representation | 67 | ||||
| 4.2.5 | Generalized P Representations | 68 | ||||
| 4.2.6 | Positive P Representation | 71 | ||||
| Exercises | 72 | |||||
| References | 72 | |||||
| 5 | Quantum Phenomena in Simple Systems in Nonlinear Optics | 73 | ||||
| 5.1 | Single-Mode Quantum Statistics | 73 | ||||
| 5.1.1 | Degenerate Parametric Amplifier | 73 | ||||
| 5.1.2 | Photon Statistics | 75 | ||||
| 5.1.3 | Wigner Function | 76 | ||||
| 5.2 | Two-Mode Quantum Correlations | 77 | ||||
| 5.2.1 | Non-degenerate Parametric Amplifier | 77 | ||||
| 5.2.2 | Squeezing | 80 | ||||
| 5.2.3 | Quadrature Correlations and the Einstein-Podolsky-Rosen Paradox | 82 | ||||
| 5.2.4 | Wigner Function | 83 | ||||
| 5.2.5 | Reduced Density Operator | 84 | ||||
| 5.3 | Quantum Limits to Amplification | 86 | ||||
| 5.4 | Amplitude Squeezed State with Poisson Photon Number Statistics | 88 | ||||
| Exercises | 91 | |||||
| References | 91 | |||||
| 6 | Stochastic Methods | 93 | ||||
| 6.1 | Master Equation | 93 | ||||
| 6.2 | Equivalent c-Number Equations | 99 | ||||
| 6.2.1 | Photon Number Representation | 99 | ||||
| 6.2.2 | P Representation | 100 | ||||
| 6.2.3 | Properties of Fokker-Planck Equations | 102 | ||||
| 6.2.4 | Steady State Solutions - Potential Conditions | 103 | ||||
| 6.2.5 | Time Dependent Solution | 104 | ||||
| 6.2.6 | Q Representation | 105 | ||||
| 6.2.7 | Wigner Function | 107 | ||||
| 6.2.8 | Generalized P Representation | 109 | ||||
| 6.3 | Stochastic Differential Equations | 112 | ||||
| 6.3.1 | Use of the Positive P Representation | 115 | ||||
| 6.4 | Linear Processes with Constant Diffusion | 116 | ||||
| 6.5 | Two Time Correlation Functions in Quantum Markov Processes ... | 117 | ||||
| 6.5.1 | Quantum Regression Theorem | 118 | ||||
| 6.6 | Application to Systems with a P Representation | 118 | ||||
| 6.7 | Stochastic Unravellings | 119 | ||||
| 6.7.1 | Simulating Quantum Trajectories | 123 | ||||
| Exercises | 124 | |||||
| References | 125 | |||||
| Further Reading | 125 | |||||
| 7 | Input-Output Formulation of Optical Cavities | 127 | ||||
| 7.1 | Cavity Modes | 127 | ||||
| 7.2 | Linear Systems | 131 | ||||
| 7.3 | Two-Sided Cavity | 132 | ||||
| 7.4 | Two Time Correlation Functions | 133 | ||||
| 7.5 | Spectrum of Squeezing | 135 | ||||
| 7.6 | Parametric Oscillator | 136 | ||||
| 7.7 | Squeezing in the Total Field | 138 | ||||
| 7.8 | Fokker-Planck Equation | 138 | ||||
| Exercises | 141 | |||||
| References | 141 | |||||
| Further Reading | 141 | |||||
| 8 | Generation and Applications of Squeezed Light | 143 | ||||
| 8.1 | Parametric Oscillation and Second Harmonic Generation | 143 | ||||
| 8.1.1 | Semi-Classical Steady States and Stability Analysis | 145 | ||||
| 8.1.2 | Parametric Oscillation | 146 | ||||
| 8.1.3 | Second Harmonic Generation | 146 | ||||
| 8.1.4 | Squeezing Spectrum | 147 | ||||
| 8.1.5 | Parametric Oscillation | 148 | ||||
| 8.1.6 | Experiments | 149 | ||||
| 8.2 | Twin Beam Generation and Intensity Correlations | 151 | ||||
| 8.2.1 | Second Harmonic Generation | 156 | ||||
| 8.2.2 | Experiments | 157 | ||||
| 8.3 | Applications of Squeezed Light | 158 | ||||
| 8.3.1 | Interferometric Detection of Gravitational Radiation | 158 | ||||
| 8.3.2 | Sub-Shot-Noise Phase Measurements | 171 | ||||
| 8.3.3 | Quantum Information | 173 | ||||
| Exercises | 174 | |||||
| References | 174 | |||||
| Further Reading | 175 | |||||
| 9 | Nonlinear Quantum Dissipative Systems | 177 | ||||
| 9.1 | Optical Parametric Oscillator: Complex P Function | 177 | ||||
| 9.2 | Optical Parametric Oscillator: Positive P Function | 181 | ||||
| 9.3 | Quantum Tunnelling Time | 186 | ||||
| 9.4 | Dispersive Optical Bistability | 190 | ||||
| 9.5 | Comment on the Use of the Q and Wigner Representations | 192 | ||||
| 9.6 | Exercises | 192 | ||||
| 9.A | Appendix | 193 | ||||
| 9.A.1 | Evaluation of Moments for the Complex P function for Parametric Oscillation (9.17) | 193 | ||||
| 9.A.2 | Evaluation of the Moments for the Complex P Function for Optical Bistability (9.48) | 194 | ||||
| References | 195 | |||||
| Further Reading | 195 | |||||
| 10 | Interaction of Radiation with Atoms | 197 | ||||
| 10.1 | Quantization of the Many-Electron System | 197 | ||||
| 10.2 | Interaction of a Single Two-Level Atom with a Single Mode Field.. | 201 | ||||
| 10.3 | Spontaneous Emission from a Two-Level Atom | 203 | ||||
| 10.4 | Phase Decay in a Two-Level System | 204 | ||||
| 10.5 | Resonance Fluorescence | 205 | ||||
| Exercises | 210 | |||||
| References | 210 | |||||
| Further Reading | 211 | |||||
| 11 | CQED | 213 | ||||
| 11.1 | Cavity QED | 213 | ||||
| 11.1.1 | Vacuum Rabi Splitting | 217 | ||||
| 11.1.2 | Single Photon Sources | 218 | ||||
| 11.1.3 | Cavity QED with N Atoms | 221 | ||||
| 11.2 | Circuit QED | 225 | ||||
| Exercises | 227 | |||||
| References | 228 | |||||
| Further Reading | 229 | |||||
| 12 | Quantum Theory of the Laser | 231 | ||||
| 12.1 | Master Equation | 231 | ||||
| 12.2 | Photon Statistics | 233 | ||||
| 12.2.1 | Spectrum of Intensity Fluctuations | 234 | ||||
| 12.3 | Laser Linewidth | 237 | ||||
| 12.4 | Regularly Pumped Laser | 238 | ||||
| 12.A | Appendix: Derivation of the Single-Atom Increment | 242 | ||||
| Exercises | 245 | |||||
| References | 245 | |||||
| 13 | Bells Inequalities in Quantum Optics | 247 | ||||
| 13.1 | The Einstein-Podolsky-Rosen (EPR) Argument | 247 | ||||
| 13.2 | Bell Inequalities and the Aspect Experiment | 248 | ||||
| 13.3 | Violations of Bell's Inequalities Using a Parametric Amplifier Source | 254 | ||||
| 13.4 | One-Photon Interference | 259 | ||||
| Exercises | 264 | |||||
| References | 264 | |||||
| 14 | Quantum Nondemolition Measurements | 267 | ||||
| 14.1 | Concept of a QND Measurement | 268 | ||||
| 14.2 | Back Action Evasion | 270 | ||||
| 14.3 | Criteria for a QND Measurement | 270 | ||||
| 14.4 | The Beam Splitter | 273 | ||||
| 14.5 | Ideal Quadrature QND Measurements | 276 | ||||
| 14.6 | Experimental Realisation | 277 | ||||
| 14.7 | A Photon Number QND Scheme | 279 | ||||
| Exercises | 281 | |||||
| References | 282 | |||||
| 15 | Quantum Coherence and Measurement Theory | 283 | ||||
| 15.1 | Quantum Coherence | 283 | ||||
| 15.2 | The Effect of Dissipation | 288 | ||||
| 15.2.1 | Experimental Observation of Coherence Decay | 291 | ||||
| 15.3 | Quantum Measurement Theory | 293 | ||||
| 15.3.1 | General Measurement Theory | 294 | ||||
| 15.3.2 | The Pointer Basis | 296 | ||||
| 15.4 | Examples of Pointer Observables | 299 | ||||
| 15.5 | Model of a Measurement | 299 | ||||
| 15.6 | Conditional States and Quantum Trajectories | 302 | ||||
| 15.6.1 | Homodyne Measurement of a Cavity Field | 303 | ||||
| Exercises | 305 | |||||
| References | 306 | |||||
| 16 | Quantum Information | 307 | ||||
| 16.1 | Introduction | 307 | ||||
| 16.1.1 | The Qubit | 308 | ||||
| 16.1.2 | Entanglement | 310 | ||||
| 16.2 | Quantum Key Distribution | 312 | ||||
| 16.3 | Quantum Teleportation | 318 | ||||
| 16.4 | Quantum Computation | 324 | ||||
| 16.4.1 | Linear Optical Quantum Gates | 327 | ||||
| 16.4.2 | Single Photon Sources | 336 | ||||
| Exercises | 343 | |||||
| References | 344 | |||||
| Further Reading | 346 | |||||
| 17 | Ion Traps | 347 | ||||
| 17.1 | Introduction | 347 | ||||
| 17.2 | Trapping and Cooling | 347 | ||||
| 17.3 | Novel Quantum States | 353 | ||||
| 17.4 | Trapping Multiple Ions | 356 | ||||
| 17.5 | Ion Trap Quantum Information Processing | 359 | ||||
| Exercises | 362 | |||||
| References | 363 | |||||
| 18 | Light Forces | 365 | ||||
| 18.1 | Radiative Forces in the Semiclassical Limit | 366 | ||||
| 18.2 | Mean Force for a Two-Level Atom Initially at Rest | 368 | ||||
| 18.3 | Friction Force for a Moving Atom | 371 | ||||
| 18.3.1 | Laser Standing Wave- Doppler Cooling | 372 | ||||
| 18.4 | Dressed State Description of the Dipole Force | 374 | ||||
| 18.5 | Atomic Diffraction by a Standing Wave | 377 | ||||
| 18.6 | Optical Stern-Gerlach Effect | 381 | ||||
| 18.7 | Quantum Chaos | 385 | ||||
| 18.7.1 | Dynamical Tunnelling | 387 | ||||
| 18.7.2 | Dynamical Localisation | 389 | ||||
| 18.8 | The Effect of Spontaneous Emission | 390 | ||||
| References | 394 | |||||
| Further Reading | 395 | |||||
| 19 | Bose-Einstein Condensation | 397 | ||||
| 19.1 | Hamiltonian: Binary Collision Model | 398 | ||||
| 19.2 | Mean-Field Theory - Gross-Pitaevskii Equation | 399 | ||||
| 19.3 | Single Mode Approximation | 400 | ||||
| 19.4 | Quantum State of the Condensate | 401 | ||||
| 19.5 | Quantum Phase Diffusion: Collapses and Revivals of the Condensate Phase | 401 | ||||
| 19.6 | Interference of Two Bose-Einstein Condensates and Measurement-Induced Phase | 405 | ||||
| 19.6.1 | Interference of Two Condensates Initially in Number States | 405 | ||||
| 19.7 | Quantum Tunneling of a Two Component Condensate | 409 | ||||
| 19.7.1 | Semiclassical Dynamics | 411 | ||||
| 19.7.2 | Quantum Dynamics | 414 | ||||
| 19.8 | Coherence Properties of Bose-Einstein Condensates | 416 | ||||
| 19.8.1 | 1st Order Coherence | 416 | ||||
| 19.8.2 | Higher Order Coherence | 417 | ||||
| Exercises | 419 | |||||
| References | 419 | |||||
| Further Reading | 420 | |||||
| Index | 421 | |||||
Index
AAdiabatic elimination, 179, 224
Antibunching, photon, 2
- laser, 3
- resonance fluorescence, 205
- second-order correlation function, 41
- sub-Poissonian statistics, 42
Atomic decay rate, transverse, 217
Atomic operators, collective, 222
Atomic optics, 5
BBack action evasion, 270
Bargmann state, 100
Beam splitter, 273
Bell inequality, 4, 249
- CHSH form, 251
- parametric down conversion, 254
Bistability, dispersive (linear), 143
- squeezing spectrum, 147
Bistability, dispersive (nonlinear), 190
- Fokker-Planck equation, 190
Bistability, optical, 224
- atomic model, 232
- noise correlation, 224
Brownian motion, 112
Bunching, photon, 2
- second-order correlation function, 39, 40
- sub-Poissonian statistics, 42
CCauchy-Schwarz inequality, 32, 79
Cavity, boundary condition, 121
Cavity, laser, 229
Cavity, two-sided, 127
Chaotic field, see Thermal field Characteristic function, 62
- optical bistability, 224
- P representation, 62
- Q representation, 65
- quantum coherence, 283
- Wigner representation, 63, 107
CHSH inequality, 251
Clauser-Horne inequality, 255, 259
Coherence, first order optical, 34
- Young's interference experiment, 32
Coherence function, 284
Coherence, optical, 29
Coherence, quantum, 283
- visibility, 283
Coherent field, 37
Coherent state, 12, 109
completeness, 14
- displacement operator, 12
- number state expansion, 12
- Poissonian statistics, 13, 58
- second-order correlation function, 40, 58
Coherent state two photon, 18
Coincidence probability, 261
Collisional broadening limit, 204
Complementarity, 247
Contractive state, 268
Correlated state, 247
Correlated state, polarization, 248
Correlation function, 29
- general inequalities, 29
- phase information, 44
Correlation function, first order, 30
Correlation function, higher order, 30
Correlation function, phase dependent, 44
Correlation function, photon number, 202
Correlation function, QND, 267
Correlation function, second-order, 39
- antibunching, 41
- bunching, 40
coherent state, 41
- number state, 41
- resonance fluorescence, 205
- thermal field, 49
Correlation function, two-time, 53
- Brownian motion, 112
- input-output formalism, 127
- master equation, 93
- Ornstein-Uhlenbeck process, 116
- parametric oscillator, 136
- phase diffusion, 237
- photon counting, 49
- regularly pumped laser, 238
- resonance fluorescence, 205
- two-level atom, 208
Correlation, intensity, 39
- Hanbury-Brown Twiss experiment, 1, 30, 38
- parametric amplifier, 73
Covariance matrix, 61
Covariance matrix, stationary, 116
DDensity operator, reduced, 93
Detailed balance, 100
- harmonic oscillator, 100
- laser, 232
Duality, wave-particle, 327
EEfficiency, quantum, 52
Einstein-Podolsky-Rosen paradox, see
Entangled state, EPR paradox Electric dipole approximation, 203, 225
Electromagnetic field, 7
- commutation relations, 10
- Hamiltonian, 10
- quantisation, 7
- vacuum state, 10
Entangled state, 261, 310
EPR paradox, 80, 247
- parametric amplifier, 74
Error ellipse, 18
- coherent state, 18
- Wigner function, 63
FFock state, see Number state Fokker-Planck equation, 93
- degenerate parametric oscillator, 177
- dispersive bistability, 177, 190
- Green's function, 103
- potential condition, 101
- stochastic differential equation, 111
Four wave mixing, 3, 279
- degenerate operation, 277
- noise correlation, 224
- phase matching, 254
- QND, 276
- side-band modes, 351
squeezing spectrum, 157
GGain, QND, 271, 276-277
Gravitational radiation, 267-268
HHanbury-Brown Twiss experiment, 2, 30, 38
Harmonic oscillator, damped, 102
Harmonic oscillator, master equation, 93
- Fokker-Planck equation, 102, 104
- transition probabilities, 109-110
Harmonic oscillator, 7
- electromagnetic field, 7
- Heat bath, see Reservoir Hidden variable theory, 250
Homodyne detection, 44-46
- four wave mixing, 279-280
- parametric oscillator, 136-138
IInput-output formulation, 127
- boundary condition, 131
- laser fluctuations, 138
- parametric oscillator, 136-138
- photon counting distribution, 47
- squeezing spectrum, 140
- two-time correlation function, 139
Intensity fluctuations, laser, 234-237
Interaction picture, 205
Interferometer, gravity wave, 159
- semiclassical behaviour, 162
- signal variance, 165
- total noise, 167
Interferometer, Mach-Zehnder, 309
Interferometer, polarization, 173
Interferometry, matter, 1, 172, 316
Interferometry, optical, 1
- squeezed light, 3
JJaynes-Cummings model, 202
KKapitza-Dirac regime, 379
Kerr effect, 88, 190
KTP crystal, 277
LLangevin equation, 112
- Brownian motion, 112
- constant diffusion process, 116
- damped harmonic oscillator, 113
- four wave mixing, 279
- input-output formalism, 127
Laser, 2, 231
- diode, 231
- gain, 234
- phase diffusion, 237
- Scully-Lamb form, 233
- sub-Poissonian statistics, 3
- threshold, 234
Laser cooling, 5, 365
Laser, fluctuations, 236
- intensity, 234-236
- photon number, 234-235, 240-241
- pump, 238-241
- sub-shot noise, 238
Laser, quantum theory, 231
Laser, regularly pumped, 239
Laser, semiconductor, 238, 242
Lens, atomic, 342
Linewidth, laser, 237-238
Liouvillian operator, 118
MMarkov approximation, first, 96, 117
Master equation, 93
- gravitational wave interferometer, 158, 171
- optical bistability, 178, 224
- parametric oscillator, 136
- resonance fluorescence, 205
- Scully-Lamb laser, 233
- two-level atom, 205
two-time correlation function, 122
Maxwell's equations, 8
- Measurement limit, strong, 301
Measurement theory, quantum, 283
Micro-cavity, 213
Microscope, Feynman light, 307
Minimum uncertainty state, 15
- parametric oscillator, 149
- squeezed state, 15
Mixture, classical, 290
NNoise spectrum, 117
Normal ordering, 39, 50
- P representation, 118
Number state, 10, 57
OObservable, QND, 268
Optical Bloch equations, 368-371
Optical tap, 279, 301
Ornstein-Uhlenbeck process, 112
- Brownian motion, 112
- constant diffusion process, 116
- damped harmonic oscillator, 113
Oscillation threshold, 187
PParametric amplifier, degenerate, 73
Parametric amplifier, nondegenerate, 77
- selected state, 85
- thermal reduced state, 85
- two-mode squeezing, 80
Parametric amplifier, QND, 277
Parametric down conversion, 3, 73
- Bell inequality, 254
Parametric oscillator, 136, 180, 185
- squeezed state, 185
Parametric oscillator (linear), 3, 131, 136
- critical fluctuations, 137
- Fokker-Planck equation, 136, 138
- linearisation, 131, 136, 139
- minimum uncertainty state, 75
- output correlation matrix, 139
- squeezing spectrum, 80, 136
- threshold, 136, 138
Parametric oscillator (nonlinear), 177
- quadrature variance, 183
- semiclassical solution, 215
- threshold distribution, 184
Pauli spin operators, 199
Phase decay, collisionally induced, 204
Phase diffusion, laser, 237
- linewidth, 233
- two-time correlation function, 237
Phase instability, 152
Phase matching, 254
Phase operator, Pegg-Barnett, 25
Phase operator, Susskind-Glogower, 23
Photo-detection, 29, 38, 337
Photon counting, 301
Photon counting, classical, 46
- depletion effects, 50
- ergodic hypothesis, 48
- generating function, 47
Photon counting, conditional, 85
parametric amplifier, 86
Photon counting distribution, 49
Photon counting, nonclassical, 69
- number state, 70
- squeezed state, 70
Photon counting, quantum mechanical, 49
Photon number, QND, 279
Planck distribution, 58
Pointer basis, 296
Poissonian pumping, 231
Polarization rotator, 254
P representation, 58, 61
- Bell inequality, 252
- chaotic state, 59
- coherent state, 59
- harmonic oscillator, 99
- normal ordering, 135
- quadrature variance, 44
P representation, complex, 68
- coherent state, 69
- dispersive bistability, 190, 195
- harmonic oscillator, 107
- number state, 65
P representation, generalized, 71, 109
- bistability, 143
- harmonic oscillator, 107
parametric oscillator, 136, 181
P representation, positive, 71, 115
- Fokker-Planck equation, 72
- harmonic oscillator, 107
- optical bistability, 194
parametric oscillator, 177
- twin beams, 151
Projection postulate, von Neumann, 85
Pump depletion, 181
QQ function, 65
- coherent state, 66
- harmonic oscillator, 106
- number state, 66
- squeezed state, 66-67
QND measurement, 4, 268-270
- pointer basis, 296
- sub-Poissonian statistics, 238
- two photon transition, 279
QND measurement, ideal, 268
Q parameter, 239
Quadrature phase measurement, 44
Quadrature phase operator, 81
Quantisation, electromagnetic field, 7
- Quantisation, electron field, 213
Quantum nondemolition measurement, see
QND measurement Quantum recurrance phenomena, 90
- Kerr effect, 88
Quantum regression theorem, 118
- resonance fluorescence, 205
RRabi frequency, 390
Radiation pressure, 160
Raman-Nath regime, 377
Recoil energy, 365-366, 372
Refractive index, nonlinear, 88
Relative states, 298, 301
Reservoir, 93
- collisional process, 398
- spectrum, 97-98
- spontaneous emission, 203
- two-level atom, 98
Reservoir, squeezed, 96, 144
- correlation function, 96
- two-level atom, 98
Reservoir, thermal, 96
Resonance fluorescence, 2, 205
- elastic scattering, 207
- in elastic scattering, 207
- quantum features, 207-208
- spectrum, 206
Rotating wave approximation, 128, 190, 199, 200, 203, 227, 366
R representation, 67
Rydberg atom, 217, 291, 382
SSchrodinger cat, 283
Second harmonic generation, 146
- critical behaviour, 146-147, 152
Second harmonic generation, 156-157
- twin beams, 155
Self pulsing, 147
Shot-noise limit, 173
- laser, 237, 241
Signal-to-noise ratio, 172-173, 302
Spectroscopy, photon correlation, 40
Spontaneous emission, 203-204
- master equation, 205-206
- Wigner theory, 207
Squeezed state, 15-18, 42^3
- photon number distribution, 21
Squeezed state, multimode, 22-23
Squeezed state, two-mode, 23
- parametric amplifier, 74
Squeezing, 3, 81-82
- four-wave mixing, 279
- input-output formalism, 127
- nonlinear susceptibility, 136
- parametric oscillator, 148
Squeezing, generator, 74
- parametric amplifier, 74, 77
Squeezing spectrum, 147-148
- dispersive bistability, 143
- parametric oscillator, 148-149
- resonance fluorescence, 205-209
Stability analysis, 145-146
Standard quantum limit, 168-171, 267
- gravity wave, 267-268
State preparation, 270, 271, 276, 333, 335, 384
Statistics, photon, 43
- laser, 238
- micromaser cavity, 382
- regularly pumped laser, 238-241
Statistics, photon counting, 46-^17
Statistics, Poissonian, 2, 42
- coherent state, 13, 58
- Kerr effect, 88
- laser, 238
Statistics, sub-Poissonian, 2
- antibunching, 42
- bunching, 41
- laser, 2, 49, 238
- QND, 268
- resonance fluorescence, 205
Statistics, super-Poissonian, 42
Steady-state solution, 145
Stern-Gerlach effect, optical, 381-387
Stochastic differential equation, 112-115
- Fokker-Planck equation, 112-113
- gravity wave interferometer, 171
- Ito equation, 112
- Langevin equation, 112
- optical bistability, 190
- parametric oscillator, 181
Super-Poissonian statistics, 42-44
Superposition, coherent, 401
Superposition, macroscopic, 4, 401
Superposition principle, 283, 325
Superposition state, 177, 185, 283, 288, 305, 325, 326, 401
Susceptibility, nonlinear, 190
Switching time, 188, 191
TThermal field, 49
- laser, 49
- second-order correlation function, 52
Transition probability, 29
Tunnelling time, 177, 186-190
Twin beams, 3, 4, 155
- critical behaviour, 156
- pulsed, 155
- semiclassical steady-state, 208
- squeezing spectrum, 156, 157
Two-level atom, 201
- master equation, 203-204
UUncertainity principle, 7, 12, 16, 83, 267, 270, 301, 303, 312
- gravity waves, 267
VVector potential, 7, 8, 9, 197
Visibility, 32, 34, 289-291
Bell inequality, 251, 256
one-photon interference, 259-260
quantum coherence, 283
WWavelength, de Broglie, 397
Wigner function, 62-65
- coherent state, 64
- covariance matrix, 65
- harmonic oscillator, 107-109
- number state, 65
- parametric amplifier, 83-84
- squeezed state, 64
Wigner theory, 192
- resonance fluorescence, 205, 341
YYoung's interference experiment, 1, 32
- coherent superposition, 281, 297
- correlation function, 32-34, 38
- experiments, 32, 35, 37, 38
- quantum explanation of, 1, 37
- visibility, 32, 34
- with atoms, 261
