Quantum Optics, 4th edition

This revised new edition gives a unique and broad coverage of basic laser-related phenomena that allow graduate students, scientists and engineers to carry out research in quantum optics and laser physics. It covers quantization of the electromagnetic field, quantum theory of coherence, atom-field interaction models, resonance fluorescence, quantum theory of damping, laser theory using both the master equation and the Langevin theory, the correlated emission laser, input-output theory with applications to non-linear optics, quantum trajectories, quantum non-demolition measurements and generation of non-classical vibrational states of ions in a Paul trap. This fourth edition provides a new chapter on weak measurement, as well as a new chapter on complementarity. There is also new material included for atom optics and new problems have been added. Each topic is presented in a unified and didactic manner, and is accompanied by specific problems and hints to solutions to deepen the knowledge. Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Contents 1 Einstein's Theory of Atom–Radiation Interaction 1.1 The A and B Coefficients 1.2 Thermal Equilibrium 1.3 Photon Distribution and Fluctuations 1.4 Light Beam Incident on Atoms 1.5 An Elementary Laser Theory 1.5.1 Threshold and Population Inversion 1.5.2 Steady State 1.5.3 Linear Stability Analysis References 2 Atom–Field Interaction: Semiclassical Approach 2.1 Broad-Band Radiation Spectrum 2.2 Rabi Oscillations 2.3 Bloch's Equations 2.4 Decay to an Unobserved Level 2.5 Decay Between Levels 2.6 Optical Nutation References 3 Quantization of the Electromagnetic Field 3.1 Fock States 3.2 Density of Modes 3.3 Commutation Relations Reference 4 States of the Electromagnetic Field I 4.1 Further Properties 4.1.1 Coherent States Are Minimum Uncertainty States 4.1.2 Coherent States Are Not Orthogonal 4.1.3 Coherent States Are Overcomplete 4.1.4 The Displacement Operator 4.1.5 Photon Statistics 4.1.6 Coordinate Representation 4.2 Mixed State: Thermal Radiation References 5 States of the Electromagnetic Field II 5.1 Squeezed States: General Properties and Detection 5.1.1 The Squeeze Operator and the Squeezed State 5.1.2 The Squeezed State Is an Eigenstate of A 5.1.3 Calculation of Moments with Squeezed States 5.1.4 Quadrature Fluctuations 5.1.5 Photon Statistics 5.2 Multimode Squeezed States 5.3 Detection of Squeezed States 5.3.1 Ordinary Homodyne Detection 5.3.2 Balanced Homodyne Detection 5.3.3 Heterodyne Detection References 6 Quantum Theory of Coherence 6.1 One-Atom Detector 6.2 The nn-Atom Detector 6.3 General Properties of the Correlation Functions 6.4 Young's Interference and First-Order Correlation 6.5 Second-Order Correlations: Photon Bunching and Antibunching 6.5.1 Classical Second-Order Coherence 6.5.2 Quantum Theory of Second-Order Coherence 6.5.3 The Handbury Brown-Twiss Effect for Fock States, Thermal and diffused Laser Lightch6scully 6.6 Photon Counting 6.6.1 Some Simple Examples 6.6.2 Quantum Mechanical Photon Count Distribution 6.6.3 Particular Examples References 7 Phase Space Description 7.1 upper QQ-Representation: Antinormal Ordering 7.1.1 Normalization 7.1.2 Average of Antinormally Ordered Products 7.1.3 Some Examples 7.1.4 The Density Operator in Terms of the Function upper QQ 7.2 Characteristic Function 7.3 upper PP Representation: Normal Ordering 7.3.1 Normalization 7.3.2 Averages of Normally Ordered Products 7.3.3 Some Interesting Properties 7.3.4 Some Examples 7.4 The Wigner Distribution: Symmetric Ordering 7.4.1 ps: [/EMC pdfmark [/StPop pdfmark =0ps: [/Subtype /P /StPNE pdfmark [/StBMC pdfmark Marginals 7.4.2 Product Rule 7.4.3 Moments References 8 Atom–Field Interaction 8.1 Atom–Field Hamiltonian and the Dipole Approximation 8.2 A Two-Level Atom Interacting with a Single Field Mode 8.3 The Dressed State Picture: Quantum Rabi Oscillations 8.4 Collapse and Revivals References 9 System–Reservoir Interactions 9.1 Quantum Theory of Damping 9.2 General Properties 9.3 Expectation Values of Relevant Physical Quantities 9.4 Time Evolution of the Density Matrix Elements 9.5 The Glauber–Sudarshan Representation, and the Fokker–Planck Equation 9.6 Time-Dependent Solution: The Method of the Eigenfunctions 9.6.1 General Solution 9.7 Langevin's Equations 9.7.1 Calculation of the Correlation Function left angle bracket upper F left parenthesis t prime right parenthesis upper F left parenthesis t double prime right parenthesis Superscript dagger Baseline right angle bracket Subscript upper BlangleF(t)F(t)rangleB 9.7.2 Differential Equation for the Photon Number 9.8 Other Master Equations 9.8.1 Two-Level Atom in a Thermal Bath 9.8.2 Damped Harmonic Oscillator in a Squeezed Bath 9.8.3 Application: Spontaneous Decay in a Squeezed Vaccum References 10 Resonance Fluorescence 10.1 Background 10.2 Heisenberg's Equations 10.3 Spectral Density, and the Wiener–Khinchine Theorem 10.4 Emission Spectra from Strongly Driven Two-Level Atoms 10.5 Intensity Correlations References 11 Quantum Laser Theory: Master Equation Approach 11.1 Heuristic Discussion of Injection Statistics 11.2 Master Equation for Generalized Pump Satistics 11.3 The Quantum Theory of the Laser: Random Injection left parenthesis p equals 0 right parenthesis(p=0) 11.3.1 Photon Statistics 11.3.2 The Fokker–Planck Equation: Laser Linewidth 11.3.3 Alternative Derivation of the Laser Linewidth 11.4 Quantum Theory of the Micromaser: Random Injection left parenthesis p equals 0 right parenthesis(p=0) 11.4.1 Generalities 11.4.2 The Micromaser 11.4.3 Trapping States 11.5 Quantum Theory of the Laser and the Micromaser ... References 12 Quantum Laser Theory: Langevin Approach 12.1 Quantum Langevin Equations 12.1.1 The Generalized Einstein's Relations 12.1.2 The Atomic Noise Moments 12.2 upper CC-Number Langevin Equations 12.2.1 Adiabatic Approximation 12.3 Phase and Intensity Fluctuations 12.4 Discussion References 13 Quantum Noise Reduction 1 13.1 Correlated Emission Laser Systems 13.1.1 The Quantum Beat Laser 13.1.2 Other CEL Systems References 14 Quantum Noise Reduction 2 14.1 Introduction to Non-Linear Optics 14.1.1 Multiple-Photon Transitions 14.2 Parametric Processes Without Losses 14.3 The Input–Output Theory 14.4 The Degenerate Parametric Oscillator 14.5 Experimental Results References 15 Quantum Phase 15.1 The Dirac Phase 15.2 The Louisell Phase 15.3 The Susskind–Glogower Phase 15.4 The Pegg–Barnett Phase 15.4.1 Applications 15.5 Phase Fluctuations in a Laser References 16 Quantum Trajectories 16.1 Montecarlo Wavefunction Method 16.1.1 The Montecarlo Method is Equivalent, on the Average, to the Master Equation 16.2 The Stochastic Schrödinger Equation 16.3 Stochastic Schrödinger Equations and Dissipative Systems 16.4 Simulation of a Monte Carlo SSE 16.5 Simulation of the Homodyne SSDE 16.6 Numerical Results and Localization 16.6.1 Quantum Jumps Evolution 16.6.2 Diffusion-Like Evolution 16.6.3 Analytical Proof of Localization 16.7 Conclusions References 17 Atom Optics 17.1 Optical Elements 17.2 Light Forces 17.2.1 Doppler Cooling 17.3 Atomic Diffraction from an Optical Standing Wave 17.3.1 Theory 17.3.2 Particular Cases 17.4 Atomic Focusing 17.4.1 The Model 17.4.2 Initial Conditions and Solution 17.4.3 Quantum and Classical Foci 17.4.4 Thin Versus Thick Lenses 17.4.5 The Quantum Focal Curve 17.4.6 Aberrations References 18 Measurements, Quantum Limits and All That 18.1 Quantum Standard Limit 18.1.1 Quantum Standard Limit for a Free Particle 18.1.2 Standard Quantum Limit for an Oscillator 18.1.3 Thermal Effects 18.2 Quantum Non-demolition (QND) Measurements 18.2.1 The Free System 18.2.2 Monitoring a Classical Force 18.2.3 Effect of the Measuring Apparatus or Probe 18.3 QND Measurement of the Number of Photons in a Cavity 18.3.1 The Model 18.3.2 The System-Probe Interaction 18.3.3 Measuring the Atomic Phase with Ramsey Fields 18.3.4 QND Measurement of the Photon Number 18.4 Quantum Theory of Continuous Photodetection Process 18.4.1 Introduction 18.4.2 Continuous Measurement in a Two-Mode System: Phase Narrowing 18.5 Generalized Measurements. POVM's 18.5.1 Standard Quantum Measurements 18.5.2 Positive Operator Valued Measures. POVM References 19 Weak Measurements 19.1 Weak Value Amplification 19.2 Weak Measurement Model 19.3 Post Selection and the Aharonov-Bergmann-Lebowitz (ABL) Rule ch19ah1 19.4 Jozsa's Theorem 19.5 Examples 19.5.1 Example 1-ch19Tamir 19.5.2 Example 2 19.6 Experiment on the Weak-to-Strong Transition 19.7 The Leggett Garg Inequality 19.8 The Quantum Box Problem ch19sergio 19.9 Discussion References 20 Trapped Ions 20.1 Paul Trap ch20ch19:CohenT 20.1.1 General Properties 20.1.2 Stability Analysis 20.2 Trapped Ions 20.2.1 Introduction 20.2.2 The Model and Effective Hamiltonian 20.2.3 The Lamb–Dicke Expansion and Raman Cooling 20.2.4 The Dynamical Evolution 20.2.5 QND Measurements of Vibrational States 20.2.6 Generation of Non-classical Vibrational States References 21 Decoherence 21.1 Dynamics of the Correlations 21.2 How Long Does It Take to Decohere? 21.3 Decoherence Free Subspaces 21.3.1 Simple Example: Collective Dephasing ch21palma 21.3.2 General Treatment ch21lidar1 21.3.3 Condition for DFS: Hamiltonian Approach 21.3.4 Condition for DFS: Lindblad Approach 21.3.5 Example: upper NN Spins in Boson Bath ch21lidar1 21.4 Quantum Maps 21.4.1 Examples ch21Preskill References 22 Quantum Bits, Entanglement and Applications 22.1 Qubits and Quantum Gates 22.2 Entanglement 22.2.1 Pure States 22.2.2 Mixed States 22.2.3 Bell Inequalities 22.3 Quantum Teleportation 22.4 Quantum Cryptography 22.4.1 BB84 22.4.2 B92 22.4.3 Entanglement Purification References 23 Quantum Correlations 23.1 Introduction 23.2 Entropy 23.3 Entanglement of Formation. Concurrence 23.4 Quantum Discord 23.5 Some Simple Examples 23.6 Three Particle Entangled States. Local Transformations 23.7 Distributed Entanglement 23.8 Correlations and Geometry References 24 Quantum Cloning and Processing 24.1 The No-Cloning Theorem 24.2 The Universal Quantum Copying Machine (UQCM) 24.3 Quantum Copying Machine Implemented by a Circuit 24.3.1 Preparation Stage 24.3.2 Copying Stage and Output 24.3.3 Output States 24.3.4 Summary and Discussion 24.4 Quantum Processors 24.4.1 Introduction 24.4.2 One Qubit Stochastic Processor References 25 Complementarity 25.1 Complementarity 25.2 Quantum Interference 25.3 The Quantum Eraser References Appendix A Operator Relations A.1 Theorem 1 A.2 Theorem 2: The Baker–Campbell–Haussdorf Relation A.3 Theorem 3: Similarity Transformation Appendix B The Method of Characteristics Appendix C Proof Appendix D Stochastic Processes in a Nutshell D.1 Introduction D.2 Probability Concepts D.3 Stochastic Processes D.3.1 The Chapman–Kolmogorov Equation D.4 The Fokker–Planck Equation D.4.1 The Wiener Process D.4.2 General Properties of the Fokker–Planck Equation D.4.3 Steady-State Solution D.5 Stochastic Differential Equations D.5.1 Introduction D.5.2 Ito Versus Stratonovich D.5.3 Ito's Formula D.6 Approximate Methods Appendix E Derivation of the Homodyne Stochastic Schrödinger Differential Equation Appendix F Fluctuations Appendix G Discrimination of Quantum States. Applications of the POVM Formalism [1] G.0.1 Unambiguous Discrimination of Two Pure States G.0.2 Minimum-Error Discrimination of Two Quantum States Appendix H The No-Cloning Theorem [1] Appendix I The Universal Quantum Cloning Machine [1] Appendix J Hints to Solve Some of the Problems Index Quantum Optics gives a very broad coverage of basic laser-related phenomena that allow scientists and engineers to carry out research in quantum optics and laser physics. It covers the quantization of the electromagnetic field, quantum theory of coherence, atom-field interaction models, resonance fluorescence, quantum theory of damping, laser theory using both the master equation and the Langevin approach, the correlated-emission laser, input-output theory with application in nonlinear optics, quantum trajectories, atom optics, quantum non-demolition measurements and generation of non-classsical vibrational states of ions in a Paul trap. These topics are presented in a unified and didactic manner. The presentation of the book is clear and pedagogical; it balances the theoretical aspects of the optical phenomena with recent relevant experiments.