Mathematical Methods of Quantum Optics

ISBN-13: 9783540678021 / Angielski / Twarda / 2001 / 289 str.

Ravinder Puri

Opis

Starting from first principles, this reference treats the theoretical aspects of quantum optics. It develops a unified approach for determining the dynamics of a two-level and three-level atom in combinations of quantized field under certain conditions.

Recenzje

From the reviews of the first edition:

"This book provides an excellent introduction to the mathematical methods of quantum optics. It starts from the postulate of quantum mechanics, their mathematical consequences and paradoxes of quantum mechanics. Then, various SU algebras and representations of some Lie algebras are discussed. Then, stochastic processes, electromagnetic field quantization and atom-field interaction are presented. ... This textbook can be recommended to teachers, postgraduate students and researchers as a book covering all the relevant mathematical and theoretical techniques of quantum optics." (Lubomír Skála, Zentralblatt MATH, Vol. 1041 (16), 2004)

"There are several good textbooks on quantum optics and there are also good textbooks on the mathematical theory of coherent states, which is a related area. This book covers the material in the middle and presents the mathematical methods used in quantum optics. It covers the mathematical formalism of quantum mechanics, coherent states and related group theory, stochastic processes, atom-field interactions, two- and three-level systems, dissipation, etc. The material is clearly presented and it is suitable for postgraduate students and researchers in the field." (Apostolos Vourdas, Mathematical Reviews, Issue 2002 j)

"The initial overview of quantum mechanics is quite useful because of its conciseness and I feel that postgraduate students would benefit particularly from this. ... the question is whether this book adds sufficiently to this store of knowledge to be worth buying. Overall I think it does and I can certainly recommend quantum opticians ordering one for their institutional libraries. Those involved more with the theory on a daily basis may find it handy to have a copy in their office." (D. T. Pegg, The Physicist, Vol. 38 (5), 2001)

Spis treści

1. Basic Quantum Mechanics.- 1.1 Postulates of Quantum Mechanics.- 1.1.1 Postulate 1.- 1.1.2 Postulate 2.- 1.1.3 Postulate 3.- 1.1.4 Postulate 4.- 1.1.5 Postulate 5.- 1.2 Geometric Phase.- 1.2.1 Geometric Phase of a Harmonic Oscillator.- 1.2.2 Geometric Phase of a Two-Level System.- 1.2.3 Geometric Phase in Adiabatic Evolution.- 1.3 Time-Dependent Approximation Method.- 1.4 Quantum Mechanics of a Composite System.- 1.5 Quantum Mechanics of a Subsystem and Density Operator.- 1.6 Systems of One and Two Spin-1/2s.- 1.7 Wave—Particle Duality.- 1.8 Measurement Postulate and Paradoxes of Quantum Theory.- 1.8.1 The Measurement Problem.- 1.8.2 Schrödinger’s Cat Paradox.- 1.8.3 EPR Paradox.- 1.9 Local Hidden Variables Theory.- 2. Algebra of the Exponential Operator.- 2.1 Parametric Differentiation of the Exponential.- 2.2 Exponential of a Finite-Dimensional Operator.- 2.3 Lie Algebraic Similarity Transformations.- 2.3.1 Harmonic Oscillator Algebra.- 2.3.2 The SU(2) Algebra.- 2.3.3 The SU(1,1) Algebra.- 2.3.4 The SU(m) Algebra.- 2.3.5 The SU(m, n) Algebra.- 2.4 Disentangling an Exponential.- 2.4.1 The Harmonic Oscillator Algebra.- 2.4.2 The SU(2) Algebra.- 2.4.3 SU(1,1) Algebra.- 2.5 Time-Ordered Exponential Integral.- 2.5.1 Harmonic Oscillator Algebra.- 2.5.2 SU (2) Algebra.- 2.5.3 The SU(1,1) Algebra.- 3. Representations of Some Lie Algebras.- 3.1 Representation by Eigenvectors and Group Parameters.- 3.1.1 Bases Constituted by Eigenvectors.- 3.1.2 Bases Labeled by Group Parameters.- 3.2 Representations of Harmonic Oscillator Algebra.- 3.2.1 Orthonormal Bases.- 3.2.2 Minimum Uncertainty Coherent States.- 3.3 Representations of SU(2).- 3.3.1 Orthonormal Representation.- 3.3.2 Minimum Uncertainty Coherent States.- 3.4 Representations of SU(1, 1).- 3.4.1 Orthonormal Bases.- 3.4.2 Minimum Uncertainty Coherent States.- 4. Quasiprobabilities and Non-classical States.- 4.1 Phase Space Distribution Functions.- 4.2 Phase Space Representation of Spins.- 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components.- 4.4 Classical and Non-classical States.- 4.4.1 Non-classical States of Electromagnetic Field.- 4.4.2 Non-classical States of Spin-1/2s.- 5. Theory of Stochastic Processes.- 5.1 Probability Distributions.- 5.2 Markov Processes.- 5.3 Detailed Balance.- 5.4 Liouville and Fokker—Planck Equations.- 5.4.1 Liouville Equation.- 5.4.2 The Fokker—Planck Equation.- 5.5 Stochastic Differential Equations.- 5.6 Linear Equations with Additive Noise.- 5.7 Linear Equations with Multiplicative Noise.- 5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations.- 5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations.- 5.8 The Poisson Process.- 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise.- 6. The Electromagnetic Field.- 6.1 Free Classical Field.- 6.2 Field Quantization.- 6.3 Statistical Properties of Classical Field.- 6.3.1 First-Order Correlation Function.- 6.3.2 Second-Order Correlation Function.- 6.3.3 Higher-Order Correlations.- 6.3.4 Stable and Chaotic Fields.- 6.4 Statistical Properties of Quantized Field.- 6.4.1 First-Order Correlation.- 6.4.2 Second-Order Correlation.- 6.4.3 Quantized Coherent and Thermal Fields.- 6.5 Homodvned Detection.- 6.6 Spectrum.- 7. Atom-Field Interaction Hamiltonians.- 7.1 Dipole Interaction.- 7.2 Rotating Wave and Resonance Approximations.- 7.3 Two-Level Atom.- 7.4 Three-Level Atom.- 7.5 Effective Two-Level Atom.- 7.6 Multi-channel Models.- 7.7 Parametric Processes.- 7.8 Cavity QED.- 7.9 Moving Atom.- 8. Quantum Theory of Damping.- 8.1 The Master Equation.- 8.2 Solving a Master Equation.- 8.3 Multi-Time Average of System Operators.- 8.4 Bath of Harmonic Oscillators.- 8.4.1 Thermal Reservoir.- 8.4.2 Squeezed Reservoir.- 8.4.3 Reservoir of the Electromagnetic Field.- 8.5 Master Equation for a Harmonic Oscillator.- 8.6 Master Equation for Two-Level Atoms.- 8.6.1 Two-Level Atom in a Monochromatic Field.- 8.6.2 Collisional Damping.- 8.7 aster Equation for a Three-Level Atom.- 8.8 Master Equation for Field Interacting with a Reservoir of Atoms.- 9. Linear and Nonlinear Response of a System in an External Field.- 9.1 Steady State of a System in an External Field.- 9.2 Optical Susceptibility.- 9.3 Rate of Absorption of Energy.- 9.4 Response in a Fluctuating Field.- 10. Solution of Linear Equations: Method of Eigenvector Expansion.- 10.1 Eigenvalues and Eigenvectors.- 10.2 Generalized Eigenvalues and Eigenvectors.- 10.3 Solution of Two-Term Difference-Differential Equation.- 10.4 Exactly Solvable Two- and Three-Term Recursion Relations.- 10.4.1 Two-Term Recursion Relations.- 10.4.2 Three-Term Recursion Relations.- 11. Two-Level and Three-Level Hamiltonian Systems.- 11.1 Exactly Solvable Two-Level Systems.- 11.1.1 Time-Independent Detuning and Coupling.- 11.1.2 On-Resonant Real Time-Dependent Coupling.- 11.1.3 Fluctuating Coupling.- 11.2 N Two-Level Atoms in a Quantized Field.- 11.3 Exactly Solvable Three-Level Systems.- 11.4 Effective Two-Level Approximation.- 12. Dissipative Atomic Systems.- 12.1 Two-Level Atom in a Quasimonochromatic Field.- 12.1.1 Time-Dependent Evolution Operator Reducible to SU(2).- 12.1.2 Time-Independent Evolution Operator.- 12.1.3 Nonlinear Response in a Bichromatic Field.- 12.2 N Two-Level Atoms in a Monochromatic Field.- 12.3 Two-Level Atoms in a Fluctuating Field.- 12.4 Driven Three-Level Atom.- 13. Dissipative Field Dynamics.- 13.1 Down-Conversion in a Damped Cavity.- 13.1.1 Averages and Variances of the Cavity Field Operators.- 13.1.2 Density Matrix.- 13.2 Field Interacting with a Two-Photon Reservoir.- 13.2.1 Two-Photon Absorption.- 13.2.2 Two-Photon Generation and Absorption.- 13.3 Reservoir in the Lambda Configuration.- 14. Dissipative Cavity QED.- 14.1 Two-Level Atoms in a Single-Mode Cavity.- 14.2 Strong Atom—Field Coupling.- 14.2.1 Single Two-Level Atom.- 14.3 Response to an External Field.- 14.3.1 Linear Response to a Monochromatic Field.- 14.3.2 Nonlinear Response to a Bichromatic Field.- 14.4 The Micromaser.- 14.4.1 Density Operator of the Field.- 14.4.2 Two-Level Atomic Micromaser.- 14.4.3 Atomic Statistics.- Appendices.- A. Some Mathematical Formulae.- B. Hypergeometric Equation.- C. Solution of Twoand Three-Dimensional Linear Equations.- D. Roots of a Polynomial.- References.

Blurb

This book provides an accessible introduction to the mathematical methods of quantum optics. Starting from first principles, it reveals how a given system of atoms and a field is mathematically modelled. The method of eigenfunction expansion and the Lie algebraic method for solving equations are outlined. Analytically exactly solvable classes of equations are identified. The text also discusses consequences of Lie algebraic properties of Hamiltonians, such as the classification of their states as coherent, classical or non-classical based on the generalized uncertainty relation and the concept of quasiprobability distributions. A unified approach is developed for determining the dynamics of a two-level and a three-level atom interacting with combinations of quantized fields under certain conditions. Simple methods for solving a variety of linear and nonlinear dissipative master equations are given. The book will be valuable to newcomers to the field and to experimentalists in quantum optics.