Electromagnetic Noise and Quantum Optical Measurements

4.6 Langevin Sources in a Waveguide with Loss.- 4.7 Lossy Linear Multiports at Thermal Equilibrium.- 4.8 The Probability Distribution of Photons at Thermal Equilibrium.- 4.9 Gaussian Amplitude Distribution of Thermal Excitations.- 4.10 Summary.- Problems.- Solutions.- 5. Linear Noisy Multiports.- 5.1 Available and Exchangeable Power from a Source.- 5.2 The Stationary Values of the Power Delivered by a Noisy Multiport and the Characteristic Noise Matrix.- 5.3 The Characteristic Noise Matrix in the Admittance Representation Applied to a Field Effect Transistor.- 5.4 Transformations of the Characteristic Noise Matrix.- 5.5 Simplified Generic Forms of the Characteristic Noise Matrix.- 5.6 Noise Measure of an Amplifier.- 5.6.1 Exchangeable Power.- 5.6.2 Noise Figure.- 5.6.3 Exchangeable Power Gain.- 5.6.4 The Noise Measure and Its Optimum Value.- 5.7 The Noise Measure in Terms of Incident and Reflected Waves.- 5.7.1 The Exchangeable Power Gain.- 5.7.2 Excess Noise Figure.- 5.8 Realization of Optimum Noise Performance.- 5.9 Cascading of Amplifiers.- 5.10 Summary.- Problems.- Solutions.- 6. Quantum Theory of Waveguides and Resonators.- 6.1 Quantum Theory of the Harmonic Oscillator.- 6.2 Annihilation and Creation Operators.- 6.3 Coherent States of the Electric Field.- 6.4 Commutator Brackets, Heisenberg's Uncertainty Principle and Noise.- 6.5 Quantum Theory of an Open Resonator.- 6.6 Quantization of Excitations on a Single-Mode Waveguide.- 6.7 Quantum Theory of Waveguides with Loss.- 6.8 The Quantum Noise of an Amplifier with a Perfectly Inverted Medium.- 6.9 The Quantum Noise of an Imperfectly Inverted Amplifier Medium.- 6.10 Noise in a Fiber with Loss Compensated by Gain.- 6.11 The Lossy Resonator and the Laser Below Threshold.- 6.12 Summary.- Problems.- Solutions.- 7. Classical and Quantum Analysis of Phase-Insensitive Systems.- 7.1 Renormalization of the Creation and Annihilation Operators.- 7.2 Linear Lossless Multiports in the Classical and Quantum Domains.- 7.3 Comparison of the Schrödinger and Heisenberg Formulations of Lossless Linear Multiports.- 7.4 The Schrödinger Formulation and Entangled States.- 7.5 Transformation of Coherent States.- 7.6 Characteristic Functions and Probability Distributions.- 7.6.1 Coherent State.- 7.6.2 Bose-Einstein Distribution.- 7.7 Two-Dimensional Characteristic Functions and the Wigner Distribution.- 7.8 The Schrödinger Cat State and Its Wigner Distribution.- 7.9 Passive and Active Multiports.- 7.10 Optimum Noise Measure of a Quantum Network.- 7.11 Summary.- Problems.- Solutions.- 8. Detection.- 8.1 Classical Description of Shot Noise and Heterodyne Detection.- 8.2 Balanced Detection.- 8.3 Quantum Description of Direct Detection.- 8.4 Quantum Theory of Balanced Heterodyne Detection.- 8.5 Linearized Analysis of Heterodyne Detection.- 8.6 Heterodyne Detection of a Multimodal Signal.- 8.7 Heterodyne Detection with Finite Response Time of Detector.- 8.8 The Noise Penalty of a Simultaneous Measurement of Two Noncommuting Observables.- 8.9 Summary.- Problems.- Solutions.- 9. Photon Probability Distributions and Bit-Error Rate of a Channel with Optical Preamplification.- 9.1 Moment Generating Functions.- 9.1.1 Poisson Distribution.- 9.1.2 Bose-Einstein Distribution.- 9.1.3 Composite Processes.- 9.2 Statistics of Attenuation.- 9.3 Statistics of Optical Preamplification with Perfect Inversion.- 9.4 Statistics of Optical Preamplification with Incomplete Inversion.- 9.5 Bit-Error Rate with Optical Preamplification.- 9.5.1 Narrow-Band Filter, Polarized Signal, and Noise.- 9.5.2 Broadband Filter, Unpolarized Signal.- 9.6 Negentropy and Information.- 9.7 The Noise Figure of Optical Amplifiers.- 9.8 Summary.- Problems.- Solutions.- 10. Solitons and Long-Distance Fiber Communications.- 10.1 The Nonlinear Schrödinger Equation.- 10.2 The First-Order Soliton.- 10.3 Properties of Solitons.- 10.4 Perturbation Theory of Solitons.- 10.5 Amplifier Noise and the Gordon-Haus Effect.- 10.6 Control Filters.- 10.7 Erbium-Doped Fiber Amplifiers and the Effect of Lumped Gain.- 10.8 Polarization.- 10.9 Continuum Generation by Soliton Perturbation.- 10.10 Summary.- Problems.- Solutions.- 11. Phase-Sensitive Amplification and Squeezing.- 11.1 Classical Analysis of Parametric Amplification.- 11.2 Quantum Analysis of Parametric Amplification.- 11.3 The Nondegenerate Parametric Amplifier as a Model of a Linear Phase-Insensitive Amplifier.- 11.4 Classical Analysis of Degenerate Parametric Amplifier.- 11.5 Quantum Analysis of Degenerate Parametric Amplifier.- 11.6 Squeezed Vacuum and Its Homodyne Detection.- 11.7 Phase Measurement with Squeezed Vacuum.- 11.8 The Laser Resonator Above Threshold.- 11.9 The Fluctuations of the Photon Number.- 11.10 The Schawlow-Townes Linewidth.- 11.11 Squeezed Radiation from an Ideal Laser.- 11.12 Summary.- Problems.- Solutions.- 12. Squeezing in Fibers.- 12.1 Quantization of Nonlinear Waveguide.- 12.2 The x Representation of Operators.- 12.3 The Quantized Equation of Motion of the Kerr Effect in the x Representation.- 12.4 Squeezing.- 12.5 Generation of Squeezed Vacuum with a Nonlinear Interferometer.- 12.6 Squeezing Experiment.- 12.7 Guided-Acoustic-Wave Brillouin Scattering.- 12.8 Phase Measurement Below the Shot Noise Level.- 12.9 Generation of Schrödinger Cat State via Kerr Effect.- 12.10 Summary.- Problems.- Solutions.- 13. Quantum Theory of Solitons and Squeezing.- 13.1 The Hamiltonian and Equations of Motion of a Dispersive Waveguide.- 13.2 The Quantized Nonlinear Schrödinger Equation and Its Linearization.- 13.3 Soliton Perturbations Projected by the Adjoint.- 13.4 Renormalization of the Soliton Operators.- 13.5 Measurement of Operators.- 13.6 Phase Measurement with Soliton-like Pulses.- 13.7 Soliton Squeezing in a Fiber.- 13.8 Summary.- Problems.- Solutions.- 14. Quantum Nondemolition Measurements and the "Collapse" of the Wave Function.- 14.1 General Properties of a QND Measurement.- 14.2 A QND Measurement of Photon Number.- 14.3 "Which Path" Experiment.- 14.4 The "Collapse" of the Density Matrix.- 14.5 Two Quantum Nondemolition Measurements in Cascade.- 14.6 The Schrödinger Cat Thought Experiment.- 14.7 Summary.- Problems.- Solutions.- Epilogue.- Appendices.- A.1 Phase Velocity and Group Velocity of a Gaussian Beam.- A.2 The Hermite Gaussians and Their Defining Equation.- A.2.1 The Defining Equation of Hermite Gaussians.- A.2.2 Orthogonality Property of Hermite Gaussian Modes.- A.2.3 The Generating Function and Convolutions of Hermite Gaussians.- A.3 Recursion Relations of Bessel Functions.- A.4 Brief Review of Statistical Function Theory.- A.5 The Different Normalizations of Field Amplitudes and of Annihilation Operators.- A.5.1 Normalization of Classical Field Amplitudes.- A.5.2 Normalization of Quantum Operators.- A.6 Two Alternative Expressions for the Nyquist Source.- A.7 Wave Functions and Operators in the n Representation.- A.8 Heisenberg's Uncertainty Principle.- A.9 The Quantized Open-Resonator Equations.- A.10 Density Matrix and Characteristic Functions.- A.10.1 Example 1. Density Matrix of Bose-Einstein State.- A.10.2 Example 2. Density Matrix of Coherent State.- A.11 Photon States and Beam Splitters.- A.12 The Baker-Hausdorff Theorem.- A.12.1 Theorem 1.- A.12.2 Theorem 2.- A.12.3 Matrix Form of Theorem 1.- A.12.4 Matrix Form of Theorem 2.- A.13 The Wigner Function of Position and Momentum.- A.14 The Spectrum of Non-Return-to-Zero Messages.- A.15 Various Transforms of Hyperbolic Secants.- A.16 The Noise Sources Derived from a Lossless Multiport with Suppressed Terminals.- A.17 The Noise Sources of an Active System Derived from Suppression of Ports.- A.19 The Heisenberg Equation in the Presence of Dispersion.- References.