Quantum Optics

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.- 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 States.- 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.- 4. Representations of the Electromagnetic Field.- 4.1 Expansion in Number States.- 4.2 Expansion in Coherent States.- 4.2.1 P Representation.- a) Correlation Functions.- b) Covariance Matrix.- c) Characteristic Function.- 4.2.2 Wigner’s Phase-Space Density.- a) Coherent State.- b) Squeezed State.- c) Number State.- 4.2.3 Q Function.- 4.2.4 R Representation.- 4.2.5 Generalized P Representations.- a) Number State.- b) Squeezed State.- 4.2.6 Positive P Representation.- Exercises.- 5. Quantum Phenomena in Simple Systems in Nonlinear Optics.- 5.1 Single-Mode Quantum 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.- Problems.- 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 Represention.- a) Complex P Representation.- b) Positive 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.- Exercises.- 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.- 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.- 8.2.3 Dispersive Optical Bistability.- 8.3 Applications of Squeezed Light.- 8.3.1 Interferometric Detection of Gravitational Radiation.- 8.3.2 Sub-Shot-Noise Phase Measurements.- Exercises.- 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 Bistability.- 9.5 Comment on the Use of the Q and Wigner Representations.- Exercises.- 9.A Appendix.- 9.A.1 Evaluation of Moments for the Complex P function for Parametric Oscillation (9.17).- 9.A.2 Evaluation of the Moments for the Complex P Function for Optical Bistability (9.48).- 10. Interaction of Radiation with Atoms.- 10.1 Quantization of the Electron Wave Field.- 10.2 Interaction Between the Radiation Field and the Electron Wave Field.- 10.3 Interaction of a Two-Level Atom with a Single Mode Field.- 10.4 Quantum Collapses and Revivals.- 10.5 Spontaneous Decay of a Two-Level Atom.- 10.6 Decay of a Two-Level Atom in a Squeezed Vacuum.- 10.7 Phase Decay in a Two-Level System.- Exercises.- 11. Resonance Fluorescence.- 11.1 Master Equation.- 11.2 Spectrum of the Fluorescent Light.- 11.3 Photon Correlations.- 11.4 Squeezing Spectrum.- Exercises.- 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.- 13. Intracavity Atomic Systems.- 13.1 Optical Bistability.- 13.2 Nondegenerate Four Wave Mixing.- 13.3 Experimental Results.- Exercises.- 14. Bells Inequalities in Quantum Optics.- 14.1 The Einstein-Podolsky-Rosen (EPR) Argument.- 14.2 Bell Inequalities and the Aspect Experiment.- 14.3 Violations of Bell’s Inequalities Using a Parametric Amplifier Source.- 14.4 One-Photon Interference.- Exercises.- 15. Quantum Nondemolition Measurements.- 15.1 Concept of a QND measurement.- 15.2 Back Action Evasion.- 15.3 Criteria for a QND Measurement.- 15.4 The Beam Splitter.- 15.5 Ideal Quadrature QND Measurements.- 15.6 Experimental Realisation.- 15.7 A Photon Number QND Scheme.- Exercises.- 16. Quantum Coherence and Measurement Theory.- 16.1 Quantum Coherence.- 16.2 The Effect of Fluctuations.- 16.3 Quantum Measurement Theory.- 16.4 Examples of Pointer Observables.- 16.5 Model of a Measurement.- Exercises.- 17. Atomic Optics.- 17.1 Young’s Interference with Path Detectors.- 17.1.1 The Feynman Light Microscope.- 17.2 Atomic Diffraction by a Standing Light Wave.- 17.3 Optical Stern-Gerlach Effect.- 17.4 Quantum Non-Demolition Measurement of the Photon Number by Atomic Beam Deflection.- 17.5 Measurement of Atomic Position.- 17.5.1 Atomic Focussing and Contractive States.- Exercises.- 17.A Appendix.- References.