Quantum Optics

by D. F. Walls and G. J. Milburn

Table of Contents

Figures (98 in all)
Preface
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 Corelation 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
Exercises

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

(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 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 Squeezed 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 Focusing and Contractive States

Exercises
17.A Appendix

References
Subject Index

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