1 Introduction 2 Quantisation of the Electromagnetic Field 2.1 Field Quantisation 2.2 Fock or Number States 2.3 Coherent States 2.4 Squeezed States 2.5 Two-Photon Coherent States 2.6 Variance in the Electric Field 2.7 Multimode Squeezed States 2.8 Phase Properties of the Field Exercises References Further Reading 3 Coherence Properties of the Electromagnetic Field 3.1 Field-Correlation Functions 3.2 Properties of the Correlation Functions 3.3 Correlation Functions and Optical Coherence 3.4 First-Order Optical Coherence 3.5 Coherent Field 3.6 Photon Correlation Measurements 3.7 Quantum Mechanical Fields 3.7.1 Squeezed State 3.7.2 Squeezed Vacuum 3.8 Phase-Dependent Correlation Functions 3.9 Photon Counting Measurements 3.9.1 Classical Theory 3.9.2 Constant Intensity 3.9.3 Fluctuating Intensity-Short-Time Limit 3.10 Quantum Mechanical Photon Count Distribution 3.10.1 Coherent Light 3.10.2 Chaotic Light 3.10.3 Photo-Electron Current Fluctuations Exercises References Further Reading 4 Representations of the Electromagnetic Field 4.1 Expansion in Number States 4.2 Expansion in Coherent States 4.2.1 PRepresentation 4.2.2 Wigner's Phase-Space Density 4.2.3 Q Function 4.2.4 R Representation 4.2.5 Generalized P Representations 4.2.6 Positive P Representation Exercises References 5 Quantum Phenomena in Simple Systems in Nonlinear Optics 5.1 Single-ModeQuantum Statistics 5.1.1 Degenerate Parametric Amplifier 5.1.2 Photon Statistics
5.1.3 Wigner Function 5.2 Two-Mode Quantum Correlations 5.2.1 Non-degenerate Parametric Amplifier 5.2.2 Squeezing 5.2.3 Quadrature Correlations and the Einstein-Podolsky-Rosen Paradox 5.2.4 Wigner Function 5.2.5 Reduced Density Operator 5.3 Quantum Limits to Amplification 5.4 Amplitude Squeezed State with Poisson Photon Number Statistics Exercises References 6 Stochastic Methods 6.1 Master Equation 6.2 Equivalent c-Number Equations 6.2.1 Photon Number Representation 6.2.2 P Representation 6.2.3 Properties of Fokker-Planck Equations 6.2.4 Steady State Solutions - Potential Conditions 6.2.5 Time Dependent Solution 6.2.6 Q Representation 6.2.7 Wigner Function 6.2.8 Generalized P Representation 6.3 Stochastic Differential Equations 6.3.1 Use of the Positive P Representation 6.4 Linear Processes with Constant Diffusion 6.5 Two Time Correlation Functions in Quantum Markov Processes.. 6.5.1 Quantum Regression Theorem 6.6 Application to Systems with a P Representation 6.7 Stochastic Unravellings 6.7.1 Simulating Quantum Trajectories Exercises References Further Reading 7 Input-Output Formulation of Optical Cavities 7.1 Cavity Modes 7.2 Linear Systems 7.3 Two-Sided Cavity 7.4 Two Time Correlation Functions 7.5 Spectrum of Squeezing 7.6 Parametric Oscillator 7.7 Squeezing in the Total Field 7.8 Fokker-Planck Equation Exercises References. Further Reading 8 Generation and Applications of Squeezed Light 8.1 Parametric Oscillation and Second Harmonic Generation 8.1.1 Semi-Classical Steady States and Stability Analysis 8.1.2 Parametric Oscillation 8.1.3 Second Harmonic Generation
8.1.4 Squeezing Spectrum 8.1.5 Parametric Oscillation 8.1.6 Experiments 8.2 Twin Beam Generation and Intensity Correlations 8.2.1 Second Harmonic Generation 8.2.2 Experiments 8.3 Applications of Squeezed Light 8.3.1 Interferometric Detection of Gravitational Radiation 8.3.2 Sub-Shot-Noise Phase Measurements 8.3.3 Quantum Information Exercises References Further Reading 9 Nonlinear Quantum Dissipative Systems 9.1 Optical Parametric Oscillator: Complex P Function 9.2 Optical Parametric Oscillator: Positive P Function 9.3 Quantum Tunnelling Time 9.4 Dispersive Optical Bistahility 9.5 Comment on the Use of the Q and Wigner Representations Exercises 9.A Appendix 9.A.I Evaluation of Moments for the Complex P function for Parametric Oscillation (9.1 7) 9.A.2 Evaluation of the Moments for the Complex P Function for Optical Bistability (9.4 8) References Further Reading 10 Interaction of Radiation with Atoms 10.1 Quantization of the Many-Electron System 10.2 Interaction of a Single Two-Level Atom with a Single Mode Field. 10.3 Spontaneous Emission from a Two-Level Atom 10.4 Phase Decay in a Two-Level System 10.5 Resonance Fluorescence Exercises References Further Reading 11 CQED 11.1 Cavity QED 11.1.1 Vacuum Rabi Splitting 11.1.2 Single Photon Sources 11.1.3 Cavity QED with N Atoms 11.2 Circuit QED Exercises References Further Reading 12 Quantum Theory of the Laser 12.1 Master Equation 12.2 Photon Statistics 12.2.1 Spectrum of Intensity Fluctuations 12.3 Laser Linewidth 12.4 Regularly Pumped Laser 12.A Appendix: Derivation of the Single-Atom Increment Exercises
References 13 Bells Inequalities in Quantum Optics 13.1 The Einstein-Podolsky-Rosen (EPR) Argument 13.2 Bell Inequalities and the Aspect Experiment 13.3 Violations of Bell's Inequalities Using a Parametric Amplifier Source 13.4 One-Photon Interference Exercises References 14 Quantum Nondemolition Measurements 14.1 Concept of a QND Measurement 14.2 Back Action Evasion 14.3 Criteria for a QND Measurement 14.4 The Beam Splitter 14.5 Ideal Quadrature QND Measurements 14.6 Experimental Realisation 14.7 A Photon Number QND Scheme Exercises References 15 Quantum Coherence and Measurement Theory 15.1 Quantum Coherence 15.2 The Effect of Dissipation 15.2.1 Experimental Observation of Coherence Decay 15.3 Quantum Measurement Theory 15.3.1 General Measurement Theory 15.3.2 The Pointer Basis 15.4 Examples of Pointer Observables 15.5 Model of a Measurement 15.6 Conditional States and Quantum Trajectories 15.6.1 Homodyne Measurement of a Cavity Field Exercises References 16 Quantum Information 16.1 Introduction 16.1.1 The Qubit 16.1.2 Entanglement 16.2 Quantum Key Distribution 16.3 Quantum Teleportation 16.4 Quantum Computation 16.4.1 Linear Optical Quantum Gates 16.4.2 Single Photon Sources Exercises References Further Reading 17 Ion Traps 17.1 Introduction 17.2 Trapping and Cooling 17.3 Novel Quantum States 17.4 Trapping Multiple Ions 17.5 Ion Trap Quantum Information Processing Exercises
References 18 Light Forces 18.1 Radiative Forces in the Semiclassical Limit 18.2 Mean Force for a Two-Level Atom Initially at Rest 18.3 Friction Force for a Moving Atom 18.3.1 Laser Standing Wave--Doppler Cooling 18.4 Dressed State Description of the Dipole Force 18.5 Atomic Diffraction by a Standing Wave 18.6 Optical Stern--Gerlach Effect 18.7 Quantum Chaos 18.7.1 Dynamical Tunnelling 18.7.2 Dynamical Localisation 18.8 The Effect of Spontaneous Emission References Further Reading 19 Bose-Einstein Condensation 19.1 Hamiltonian: Binary Collision Model 19.2 Mean-Field Theory -- Gross-Pitaevskii Equation 19.3 Single Mode Approximation 19.4 Quantum State of the Condensate 19.5 Quantum Phase Diffusion: Collapses and Revivals of the Condensate Phase 19.6 Interference of Two Bose-Einstein Condensates and Measurement-Induced Phase 19.6.1 Interference of Two Condensates Initially in Number States 19.7 Quantum Tunneling of a Two Component Condensate 19.7.1 Semiclassical Dynamics 19.7.2 Quantum Dynamics 19.8 Coherence Properties of Bose-Einstein Condensates 19.8.1 1st Order Coherence 19.8.2 Higher Order Coherence Exercises References Further Reading Index
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