Coherent States in Quantum Physics

This self-contained introduction discusses the evolution of the notion of coherent states, from the early works of Schrodinger to the most recent advances, including signal analysis. An integrated and modern approach to the utility of coherent states in many different branches of physics, it strikes a balance between mathematical and physical descriptions. Split into two parts, the first introduces readers to the most familiar coherent states, their origin, their construction, and their application and relevance to various selected domains of physics. Part II, mostly based on recent original results, is devoted to the question of quantization of various sets through coherent states, and shows the link to procedures in signal analysis. Coherent States in Quantum Physics......Page 5 Contents......Page 7 Preface......Page 15 Part One Coherent States......Page 17 1.1 The Motivations......Page 19 2.2 Four Representations of Quantum States......Page 29 2.2.2 Momentum Representation......Page 30 2.2.3 Number or Fock Representation......Page 31 2.2.5 Analytical or Fock–Bargmann Representation......Page 32 2.2.6 Operators in Fock–Bargmann Representation......Page 33 2.3.1 Bergman Kernel as a Coherent State......Page 34 2.3.3 Schrödinger Coherent States in the Two Other Representations......Page 35 2.5 Why the Adjective Coherent?......Page 36 3.1 Introduction......Page 41 3.2.2 "Coherent'' Resolution of the Unity......Page 42 3.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane (as a Euclidean Space)......Page 43 3.2.4 Analytical Bridge......Page 44 3.2.5 Overcompleteness and Reproducing Properties......Page 45 3.3.2 Lower Symbols......Page 46 3.3.3 Heisenberg Inequalities......Page 47 3.3.4 Time Evolution and Phase Space......Page 48 3.4.1 The Vacuum as a Transported Probe.........Page 51 3.4.2 Under the Action of.........Page 52 3.4.4 Symplectic Phase and the Weyl–Heisenberg Group......Page 53 3.4.5 Coherent States as Tools in Signal Analysis......Page 54 3.5 Quantum Distributions and Coherent States......Page 56 3.5.2 The Density Matrix and the Representation "Q''......Page 57 3.5.3 The Density Matrix and the Representation "P''......Page 58 3.5.4 The Density Matrix and the Wigner(–Weyl–Ville) Distribution......Page 59 3.6 The Feynman Path Integral and Coherent States......Page 60 4.1 Quantum States for Information......Page 65 4.2 Optical Coherent States in Quantum Information......Page 66 4.3.2 Uncertainties on POVMs......Page 67 4.3.3 The Quantum Error Probability or Helstrom Bound......Page 68 4.3.5 Helstrom Bound for Coherent States......Page 69 4.4.1 The Principle......Page 70 4.4.2 Kennedy Receiver Error......Page 71 4.5.2 Sasaki–Hirota Receiver Error......Page 72 4.6.1 The Principle......Page 73 4.6.3 Decision Criterion of the Dolinar Receiver......Page 74 4.6.4 Optimal Control......Page 75 4.6.5 Dolinar Hypothesis Testing Procedure......Page 76 4.7.1 A Theoretical Preliminary......Page 77 4.7.2 Closed-Loop Experiment: the Apparatus......Page 79 4.7.3 Closed-Loop Experiment: the Results......Page 81 4.8 Conclusion......Page 83 5.1 Introduction......Page 85 5.2.1 Poisson and Gamma Distributions......Page 86 5.2.3 The Fock–Bargmann Option......Page 87 5.3 General Setting: "Quantum'' Processing of a Measure Space......Page 88 5.4 Coherent States for the Motion of a Particle on the Circle......Page 92 5.5 More Coherent States for the Motion of a Particle on the Circle......Page 94 6.2 Preliminary Material......Page 95 6.3 The Construction of Spin Coherent States......Page 96 6.5 Spin Coherent States: Group-Theoretical Context......Page 98 6.7 Spin Coherent States: Spherical Harmonics Aspects......Page 102 6.8.1 Matrix Elements of the SU(2) Unitary Irreducible Representations......Page 103 6.8.3 Spin Spherical Harmonics......Page 105 6.8.5 The Important Case: σ = j......Page 107 6.8.7 Infinitesimal Transformation Laws......Page 108 6.8.8 "Sigma-Spin'' Coherent States......Page 109 6.8.9 Covariance Properties of Sigma-Spin Coherent States......Page 111 7.1 Introduction......Page 113 7.2 Coherent States and the Driven Oscillator......Page 114 7.3.1 The Dicke Model......Page 119 7.3.2 The Partition Function......Page 121 7.3.3 The Critical Temperature......Page 122 7.3.4 Average Number of Photons per Atom......Page 124 7.4 Application of Spin Coherent States to Quantum Magnetism......Page 125 7.5 Application of Spin Coherent States to Classical and Thermodynamical Limits......Page 127 7.5.1 Symbols and Traces......Page 128 7.5.2 Berezin–Lieb Inequalities for the Partition Function......Page 130 7.5.3 Application to the Heisenberg Model......Page 132 8.2 The Unit Disk as an Observation Set......Page 133 8.3 Coherent States......Page 135 8.4 Probabilistic Interpretation......Page 136 8.5 Poincaré Half-Plane for Time-Scale Analysis......Page 137 8.6 Symmetries of the Disk and the Half-Plane......Page 138 8.7.1 Cartan Factorization......Page 139 8.7.2 Discrete Series of SU(1,1)......Page 140 8.7.3 Lie Algebra Aspects......Page 142 8.7.4 Coherent States as a Transported Vacuum......Page 143 8.8 A Few Words on Continuous Wavelet Analysis......Page 145 9.2 Classical Motion in the Infinite-Well and Pöschl–Teller Potentials......Page 151 9.2.1 Motion in the Infinite Well......Page 152 9.2.2 Pöschl–Teller Potentials......Page 154 9.3.1 In the Infinite Well......Page 157 9.3.2 In Pöschl–Teller Potentials......Page 158 9.4 The Dynamical Algebra su(1,1)......Page 159 9.5 Sequences of Numbers and Coherent States on the Complex Plane......Page 162 9.6.1 For the Infinite Well......Page 166 9.6.2 For the Pöschl–Teller Potentials......Page 168 9.7.1 Quantum Revivals......Page 169 9.7.2 Mandel Statistical Characterization......Page 171 9.7.3 Temporal Evolution of Symbols......Page 174 9.7.4 Discussion......Page 178 10.1 Introduction......Page 181 10.2.1 The Construction within a Physical Context......Page 182 10.2.2 Algebraic (su(1,1)) Content of Squeezed States......Page 187 10.2.3 Using Squeezed States in Molecular Dynamics......Page 191 11.2 Coherent States for One Fermionic Mode......Page 195 11.3.1 Fermionic Symmetry SU(r)......Page 196 11.3.2 Fermionic Symmetry SO(2r)......Page 201 11.3.3 Fermionic Symmetry SO(2r+1)......Page 203 11.3.4 Graphic Summary......Page 204 11.4 Application to the Hartree–Fock–Bogoliubov Theory......Page 205 Part Two Coherent State Quantization......Page 207 12.2 The Berezin–Klauder Quantization of the Motion of a Particle on the Line......Page 209 12.3.1 Van Hove Canonical Quantization Rules [161]......Page 212 12.4 More Upper and Lower Symbols: the Angle Operator......Page 213 12.5 Quantization of Distributions: Dirac and Others......Page 215 12.6 Finite-Dimensional Canonical Case......Page 218 13.2 Some Ideas on Quantization......Page 223 13.3 One more Coherent State Construction......Page 225 13.4 Coherent State Quantization......Page 227 13.5.1 Quantization and Symbol Calculus......Page 230 13.5.2 Probabilistic Aspects......Page 232 13.6 Quantization with k-Fermionic Coherent States......Page 234 13.7 Final Comments......Page 236 14.2 Coherent State Quantization of a Finite Set with Complex 22 Matrices......Page 239 14.3.1 Quantization with Finite Subfamilies of Haar Wavelets......Page 243 14.3.2 A Two-Dimensional Noncommutative Quantization of the Unit Interval......Page 244 14.4.1 A Retrospective of Various Approaches......Page 245 14.4.2 Pegg–Barnett Phase Operator and Coherent State Quantization......Page 250 14.4.3 A Phase Operator from Two Finite-Dimensional Vector Spaces......Page 251 14.4.4 A Phase Operator from the Interplay Between Finite and Infinite Dimensions......Page 253 15.2.1 The Cylinder as an Observation Set......Page 257 15.2.2 Quantization of Classical Observables......Page 258 15.2.3 Did You Say Canonical?......Page 259 15.3 From the Motion of the Circle to the Motion on 1 + 1 de Sitter Space-Time......Page 260 15.4.1 Introduction......Page 261 15.4.2 The Standard Quantum Context......Page 262 15.4.3 Two-Component Coherent States......Page 263 15.4.4 Quantization of Classical Observables......Page 265 15.4.5 Quantum Behavior through Lower Symbols......Page 269 15.4.6 Discussion......Page 270 15.5 Motion on a Discrete Set of Points......Page 272 16.2 The Torus as a Phase Space......Page 275 16.3 Quantum States on the Torus......Page 277 16.4 Coherent States for the Torus......Page 281 16.5.2 Weyl Quantization of the Torus......Page 283 16.6.1 Quantization of Irrational and Skew Translations......Page 285 16.6.2 Quantization of the Hyperbolic Automorphisms of the Torus......Page 286 16.6.3 Main Results......Page 287 17.2 Quantizations of the 2-Sphere......Page 289 17.2.2 The Hilbert Space and the Coherent States......Page 290 17.2.4 Quantization of Observables......Page 291 17.2.7 Quantization in the Simplest Case: j = 1......Page 292 17.2.9 The Spin Angular Momentum Operators......Page 293 17.3.1 The Construction of the Fuzzy Sphere à la Madore......Page 294 17.3.2 Operators......Page 296 17.4 Summary......Page 298 17.5 The Fuzzy Hyperboloid......Page 299 18 Conclusion and Outlook......Page 303 A.1.1 Examples......Page 305 A.3 Measurable Function......Page 306 A.5 Probability Axioms......Page 307 A.7 Bayes's Theorem......Page 308 A.9 Probability Distribution......Page 309 A.11 Conditional Probability Densities......Page 310 A.12 Bayesian Statistical Inference......Page 311 A.13.2 Uniform Distribution......Page 312 B.1 Group Transformations and Representations......Page 319 B.2 Lie Algebras......Page 320 B.3 Lie Groups......Page 322 B.3.1 Extensions of Lie algebras and Lie groups......Page 326 C.2 Matrix Elements of SU(2) Unitary Irreducible Representation......Page 329 C.3 Orthogonality Relations and 3j Symbols......Page 330 C.4 Spin Spherical Harmonics......Page 331 C.5 Transformation Laws......Page 333 C.6 Infinitesimal Transformation Laws......Page 334 C.7 Integrals and 3j Symbols......Page 335 C.8 Important Particular Case: j = 1......Page 336 C.9 Another Important Case: σ = j......Page 337 Appendix D Wigner–Eckart Theorem for Coherent State Quantized Spin Harmonics......Page 339 Appendix E Symmetrization of the Commutator......Page 341 References......Page 345 Index......Page 355 Quantum Optics in Phase Space provides a concise introduction to the rapidly moving field of quantum optics from the point of view of phase space. Modern in style and didactically skillful, Quantum Optics in Phase Space prepares students for their own research by presenting detailed derivations, many illustrations and a large set of workable problems at the end of each chapter. Often, the theoretical treatments are accompanied by the corresponding experiments. An exhaustive list of references provides a guide to the literature. Quantum Optics in Phase Space also serves advanced researchers as a comprehensive reference book.

Starting with an extensive review of the experiments that define quantum optics and a brief summary of the foundations of quantum mechanics the author Wolfgang P. Schleich illustrates the properties of quantum states with the help of the Wigner phase space distribution function. His description of waves ala WKB connects semi-classical phase space with the Berry phase. These semi-classical techniques provide deeper insight into the timely topics of wave packet dynamics, fractional revivals and the Talbot effect.

Whereas the first half of the book deals with mechanical oscillators such as ions in a trap or atoms in a standing wave the second half addresses problems where the quantization of the radiation field is of importance. Such topics extensively discussed include optical interferometry, the atom-field interaction, quantum state preparation and measurement, entanglement, decoherence, the one-atom maser and atom optics in quantized light fields.

Quantum Optics in Phase Space presents the subject of quantum optics astransparently as possible. Giving wide-ranging references, it enables students to study and solve problems with modern scientific literature. The result is a remarkably concise yet comprehensive and accessible text- and reference book - an inspiring source of information and insight for students, teachers and researchers alike.
Quantum Optics in Phase Space provides a concise introduction to the rapidly moving field of quantum optics from the point of view of phase space. Modern in style and didactically skillful, Quantum Optics in Phase Space prepares students for their own research by presenting detailed derivations, many illustrations and a large set of workable problems at the end of each chapter. Often, the theoretical treatments are accompanied by the corresponding experiments. An exhaustive list of references provides a guide to the literature. Quantum Optics in Phase Space also serves advanced researchers as a comprehensive reference book. Starting with an extensive review of the experiments that define quantum optics and a brief summary of the foundations of quantum mechanics the author Wolfgang P. Schleich illustrates the properties of quantum states with the help of the Wigner phase space distribution function. His description of waves ala WKB connects semi-classical phase space with the Berry phase. These semi-classical techniques provide deeper insight into the timely topics of wave packet dynamics, fractional revivals and the Talbot effect. Whereas the first half of the book deals with mechanical oscillators such as ions in a trap or atoms in a standing wave the second half addresses problems where the quantization of the radiation field is of importance. Such topics extensively discussed include optical interferometry, the atom-field interaction, quantum state preparation and measurement, entanglement, decoherence, the one-atom maser and atom optics in quantized light fields. Quantum Optics in Phase Space presents the subject of quantum optics as transparently as possible. Giving wide-ranging references, it enables students to study and solve problems with modern scientific literature. The result is a remarkably concise yet comprehensive and accessible text- and reference book - an inspiring source of information and insight for students, teachers and researchers alike. Quantum Theory Of Optical Coherence Is A Compilation Of Roy J. Glauber's Most Renowned And Groundbreaking Articles And Lectures, Among Them His Famous Lectures Held At The Les Houches Summer School In 1964 On 'optical Coherence And Photon Statistics', Which Has Been A Milestone For Future Generations Of Students And Researchers In This Field. The Book Is Intended Not Only As A Reference For Experts, But Also Addresses Graduate Students And Beginning Researchers Who Wish To Gain An Insight Into The Basic Theories Of The Field.--jacket. The Quantum Theory Of Optical Coherence -- Optical Coherence And Photon Statistics -- Correlation Functions For Coherent Fields -- Density Operators For Coherent Fields -- Classical Behavior Of Systems Of Quantum Oscillators -- Quantum Theory Of Parametric Amplification I -- Quantum Theory Of Parametric Amplification Ii -- Photon Statistics -- Ordered Expansions In Boson Amplitude Operators -- Density Operators And Quasiprobability Distributions -- Coherence And Quantum Detection -- Quantum Theory Of Coherence -- The Initiation Of Superfluorescence -- Amplifiers, Attenuators And Schrödingers Cat -- The Quantum Mechanics Of Trapped Wavepackets -- Density Operators For Fermions. Roy J. Glauber. Includes Bibliographical References And Index. This is the third, revised and extended edition of the acknowledged "Lectures on Quantum Optics" by W. Vogel and D.-G. Welsch. It offers theoretical concepts of quantum optics, with special emphasis on current research trends. A unified concept of measurement-based nonclassicality and entanglement criteria and a unified approach to medium-assisted electromagnetic vacuum effects including Van der Waals and Casimir Forces are the main new topics that are included in the revised edition. The rigorous development of quantum optics in the context of quantum field theory and the attention to details makes the book valuable to graduate students as well as to researchers. Voices to the new "There are many good books in this area, but this one really excels in terms of broad coverage, choice of topics, and precision. It is very useful as a textbook for a quantum optics course, and also as a general reference for researchers in quantum optics. ... Also, the new edition includes some subtle and fundamental material about non-classicality, medium-assisted electromagnetic vacuum effects, and leaky cavities, based on research developed by the authors." Prof. Luiz Davidovich, Rio de Janeiro

A Guide through the Mysteries of Quantum Physics!
Yakir Aharonov is one of the pioneers in measuring theory, the nature of quantum correlations, superselection rules, and geometric phases and has been awarded numerous scientific honors. The author has contributed monumental concepts to theoretical physics, especially the Aharonov-Bohm effect and the Aharonov-Casher effect.
Together with Daniel Rohrlich of the Weizmann Institute, Israel, he has written a pioneering work on the remaining mysteries of quantum mechanics. From the perspective of a preeminent researcher in the fundamental aspects of quantum mechanics, the text combines mathematical rigor with penetrating and concise language. More than 200 problem sets introduce readers to the concepts and implications of quantum mechanics that have arisen from the experimental results of the recent two decades.
With students as well as researchers in mind, the authors give an insight into that part of the field, which led Feynman to declare that "nobody understands quantum mechanics".

* Free solutions manual available for lecturers at www.wiley-vch.de/supplements/

A Guide through the Mysteries of Quantum Physics! Yakir Aharonov is one of the pioneers in measuring theory, the nature of quantum correlations, superselection rules, and geometric phases and has been awarded numerous scientific honors. The author has contributed monumental concepts to theoretical physics, especially the Aharonov-Bohm effect and the Aharonov-Casher effect. Together with Daniel Rohrlich, Israel, he has written a pioneering work on the remaining mysteries of quantum mechanics. From the perspective of a preeminent researcher in the fundamental aspects of quantum mechanics, the text combines mathematical rigor with penetrating and concise language. More than 200 exercises introduce readers to the concepts and implications of quantum mechanics that have arisen from the experimental results of the recent two decades. With students as well as researchers in mind, the authors give an insight into that part of the field, which led Feynman to declare that "nobody understands quantum mechanics". * Free solutions manual available for lecturers at (http://www.wiley-vch.de/supplements/) www.wiley-vch.de/supplements/ 'Elements of Quantum Information' introduces the reader to the fascinating field of quantum information processing, which lives on the interface between computer science, physics, mathematics, and engineering. This interdisciplinary branch of science thrives on the use of quantum mechanics as a resource for high potential modern applications. With its wide coverage of experiments, applications, and specialized topics - all written by renowned experts - 'Elements of Quantum Information' provides an indispensable up-to-date account of the state of the art of this rapidly advancing field and takes the reader straight up to the frontiers of current research. The articles have first appeared as a special issue of the journal 'Fortschritte der Physik/Progress of Physics'. Since then, they have been carefully updated. The book will be an inspiring source of information and insight for anyone researching and specializing in experiments and theory of quantum information.

This self-contained introduction discusses the evolution of the notion of coherent states, from the early works of Schrödinger to the most recent advances, including signal analysis. An integrated and modern approach to the utility of coherent states in many different branches of physics, it strikes a balance between mathematical and physical descriptions.

Split into two parts, the first introduces readers to the most familiar coherent states, their origin, their construction, and their application and relevance to various selected domains of physics. Part II, mostly based on recent original results, is devoted to the question of quantization of various sets through coherent states, and shows the link to procedures in signal analysis.

"Yakir Aharonov is one of the pioneers in measuring theory, the nature of quantum correlations, superselection rules, and geometric phases and, as such, has made monumental contributions to theoretical physics. Together with Daniel Rohrlich of the Weizmann Institute, Israel, he has written here a groundbreaking work on the remaining mysteries of quantum mechanics. With both students as well as researchers in mind, the authors provide an insight into that part of the field that led Feynman to declare "nobody understands quantum mechanics"."--BOOK JACKET Elements Of Quantum Electrodynamics -- Quantum States Of Bosonic Systems -- Bosonic Systems In Phase Space -- Quantum Theory Of Damping -- Photoelectric Detection Of Light -- Quantum-state Reconstruction -- Nonclassicality And Entanglement Of Bosonic Systems -- Leaky Optical Cavities -- Medium-assisted Electromagnetic Vacuum Effects -- Resonance Fluorescence -- A Single Atom In A High-q Cavity -- Laser-driven Quantized Motion Of A Trapped Atom. Werner Vogel And Dirk-gunnar Welsch. Includes Bibliographical References And Index.