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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 | |||||
