Coherent States in Quantum Physics -- Contents -- Preface -- Part One Coherent States -- 1 Introduction -- 1.1 The Motivations -- 2 The Standard Coherent States: the Basics -- 2.1 Schrödinger Definition -- 2.2 Four Representations of Quantum States -- 2.2.1 Position Representation -- 2.2.2 Momentum Representation -- 2.2.3 Number or Fock Representation -- 2.2.4 A Little (Lie) Algebraic Observation -- 2.2.5 Analytical or Fock-Bargmann Representation -- 2.2.6 Operators in Fock-Bargmann Representation -- 2.3 Schrödinger Coherent States -- 2.3.1 Bergman Kernel as a Coherent State -- 2.3.2 A First Fundamental Property -- 2.3.3 Schrödinger Coherent States in the Two Other Representations -- 2.4 Glauber-Klauder-Sudarshan or Standard Coherent States -- 2.5 Why the Adjective Coherent? -- 3 The Standard Coherent States: the (Elementary) Mathematics -- 3.1 Introduction -- 3.2 Properties in the Hilbertian Framework -- 3.2.1 A ``Continuity'' from the Classical Complex Plane to Quantum States -- 3.2.2 ``Coherent'' Resolution of the Unity -- 3.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane (as a Euclidean Space) -- 3.2.4 Analytical Bridge -- 3.2.5 Overcompleteness and Reproducing Properties -- 3.3 Coherent States in the Quantum Mechanical Context -- 3.3.1 Symbols -- 3.3.2 Lower Symbols -- 3.3.3 Heisenberg Inequalities -- 3.3.4 Time Evolution and Phase Space -- 3.4 Properties in the Group-Theoretical Context -- 3.4.1 The Vacuum as a Transported Probe… -- 3.4.2 Under the Action of… -- 3.4.3 … the D-Function -- 3.4.4 Symplectic Phase and the Weyl-Heisenberg Group -- 3.4.5 Coherent States as Tools in Signal Analysis -- 3.5 Quantum Distributions and Coherent States -- 3.5.1 The Density Matrix and the Representation ``R'' -- 3.5.2 The Density Matrix and the Representation ``Q'' -- 3.5.3 The Density Matrix and the Representation ``P''.

3.5.4 The Density Matrix and the Wigner(-Weyl-Ville) Distribution -- 3.6 The Feynman Path Integral and Coherent States -- 4 Coherent States in Quantum Information: an Example of Experimental Manipulation -- 4.1 Quantum States for Information -- 4.2 Optical Coherent States in Quantum Information -- 4.3 Binary Coherent State Communication -- 4.3.1 Binary Logic with Two Coherent States -- 4.3.2 Uncertainties on POVMs -- 4.3.3 The Quantum Error Probability or Helstrom Bound -- 4.3.4 The Helstrom Bound in Binary Communication -- 4.3.5 Helstrom Bound for Coherent States -- 4.3.6 Helstrom Bound with Imperfect Detection -- 4.4 The Kennedy Receiver -- 4.4.1 The Principle -- 4.4.2 Kennedy Receiver Error -- 4.5 The Sasaki-Hirota Receiver -- 4.5.1 The Principle -- 4.5.2 Sasaki-Hirota Receiver Error -- 4.6 The Dolinar Receiver -- 4.6.1 The Principle -- 4.6.2 Photon Counting Distributions -- 4.6.3 Decision Criterion of the Dolinar Receiver -- 4.6.4 Optimal Control -- 4.6.5 Dolinar Hypothesis Testing Procedure -- 4.7 The Cook-Martin-Geremia Closed-Loop Experiment -- 4.7.1 A Theoretical Preliminary -- 4.7.2 Closed-Loop Experiment: the Apparatus -- 4.7.3 Closed-Loop Experiment: the Results -- 4.8 Conclusion -- 5 Coherent States: a General Construction -- 5.1 Introduction -- 5.2 A Bayesian Probabilistic Duality in Standard Coherent States -- 5.2.1 Poisson and Gamma Distributions -- 5.2.2 Bayesian Duality -- 5.2.3 The Fock-Bargmann Option -- 5.2.4 A Scheme of Construction -- 5.3 General Setting: ``Quantum'' Processing of a Measure Space -- 5.4 Coherent States for the Motion of a Particle on the Circle -- 5.5 More Coherent States for the Motion of a Particle on the Circle -- 6 The Spin Coherent States -- 6.1 Introduction -- 6.2 Preliminary Material -- 6.3 The Construction of Spin Coherent States -- 6.4 The Binomial Probabilistic Content of Spin Coherent States.

6.5 Spin Coherent States: Group-Theoretical Context -- 6.6 Spin Coherent States: Fock-Bargmann Aspects -- 6.7 Spin Coherent States: Spherical Harmonics Aspects -- 6.8 Other Spin Coherent States from Spin Spherical Harmonics -- 6.8.1 Matrix Elements of the SU(2) Unitary Irreducible Representations -- 6.8.2 Orthogonality Relations -- 6.8.3 Spin Spherical Harmonics -- 6.8.4 Spin Spherical Harmonics as an Orthonormal Basis -- 6.8.5 The Important Case: =j -- 6.8.6 Transformation Laws -- 6.8.7 Infinitesimal Transformation Laws -- 6.8.8 ``Sigma-Spin'' Coherent States -- 6.8.9 Covariance Properties of Sigma-Spin Coherent States -- 7 Selected Pieces of Applications of Standard and Spin Coherent States -- 7.1 Introduction -- 7.2 Coherent States and the Driven Oscillator -- 7.3 An Application of Standard or Spin Coherent States in Statistical Physics: Superradiance -- 7.3.1 The Dicke Model -- 7.3.2 The Partition Function -- 7.3.3 The Critical Temperature -- 7.3.4 Average Number of Photons per Atom -- 7.3.5 Comments -- 7.4 Application of Spin Coherent States to Quantum Magnetism -- 7.5 Application of Spin Coherent States to Classical and Thermodynamical Limits -- 7.5.1 Symbols and Traces -- 7.5.2 Berezin-Lieb Inequalities for the Partition Function -- 7.5.3 Application to the Heisenberg Model -- 8 SU(1,1) or SL(2,R) Coherent States -- 8.1 Introduction -- 8.2 The Unit Disk as an Observation Set -- 8.3 Coherent States -- 8.4 Probabilistic Interpretation -- 8.5 Poincaré Half-Plane for Time-Scale Analysis -- 8.6 Symmetries of the Disk and the Half-Plane -- 8.7 Group-Theoretical Content of the Coherent States -- 8.7.1 Cartan Factorization -- 8.7.2 Discrete Series of SU(1,1) -- 8.7.3 Lie Algebra Aspects -- 8.7.4 Coherent States as a Transported Vacuum -- 8.8 A Few Words on Continuous Wavelet Analysis -- 9 Another Family of SU(1,1) Coherent States for Quantum Systems.

9.1 Introduction -- 9.2 Classical Motion in the Infinite-Well and Pöschl-Teller Potentials -- 9.2.1 Motion in the Infinite Well -- 9.2.2 Pöschl-Teller Potentials -- 9.3 Quantum Motion in the Infinite-Well and Pöschl-Teller Potentials -- 9.3.1 In the Infinite Well -- 9.3.2 In Pöschl-Teller Potentials -- 9.4 The Dynamical Algebra su(1,1) -- 9.5 Sequences of Numbers and Coherent States on the Complex Plane -- 9.6 Coherent States for Infinite-Well and Pöschl-Teller Potentials -- 9.6.1 For the Infinite Well -- 9.6.2 For the Pöschl-Teller Potentials -- 9.7 Physical Aspects of the Coherent States -- 9.7.1 Quantum Revivals -- 9.7.2 Mandel Statistical Characterization -- 9.7.3 Temporal Evolution of Symbols -- 9.7.4 Discussion -- 10 Squeezed States and Their SU(1,1) Content -- 10.1 Introduction -- 10.2 Squeezed States in Quantum Optics -- 10.2.1 The Construction within a Physical Context -- 10.2.2 Algebraic (su(1,1)) Content of Squeezed States -- 10.2.3 Using Squeezed States in Molecular Dynamics -- 11 Fermionic Coherent States -- 11.1 Introduction -- 11.2 Coherent States for One Fermionic Mode -- 11.3 Coherent States for Systems of Identical Fermions -- 11.3.1 Fermionic Symmetry SU(r) -- 11.3.2 Fermionic Symmetry SO(2r) -- 11.3.3 Fermionic Symmetry SO(2r+1) -- 11.3.4 Graphic Summary -- 11.4 Application to the Hartree-Fock-Bogoliubov Theory -- Part Two Coherent State Quantization -- 12 Standard Coherent State Quantization: the Klauder-Berezin Approach -- 12.1 Introduction -- 12.2 The Berezin-Klauder Quantization of the Motion of a Particle on the Line -- 12.3 Canonical Quantization Rules -- 12.3.1 Van Hove Canonical Quantization Rules [161] -- 12.4 More Upper and Lower Symbols: the Angle Operator -- 12.5 Quantization of Distributions: Dirac and Others -- 12.6 Finite-Dimensional Canonical Case -- 13 Coherent State or Frame Quantization -- 13.1 Introduction.

13.2 Some Ideas on Quantization -- 13.3 One more Coherent State Construction -- 13.4 Coherent State Quantization -- 13.5 A Quantization of the Circle by 22 Real Matrices -- 13.5.1 Quantization and Symbol Calculus -- 13.5.2 Probabilistic Aspects -- 13.6 Quantization with k-Fermionic Coherent States -- 13.7 Final Comments -- 14 Coherent State Quantization of Finite Set, Unit Interval, and Circle -- 14.1 Introduction -- 14.2 Coherent State Quantization of a Finite Set with Complex 22 Matrices -- 14.3 Coherent State Quantization of the Unit Interval -- 14.3.1 Quantization with Finite Subfamilies of Haar Wavelets -- 14.3.2 A Two-Dimensional Noncommutative Quantization of the Unit Interval -- 14.4 Coherent State Quantization of the Unit Circle and the Quantum Phase Operator -- 14.4.1 A Retrospective of Various Approaches -- 14.4.2 Pegg-Barnett Phase Operator and Coherent State Quantization -- 14.4.3 A Phase Operator from Two Finite-Dimensional Vector Spaces -- 14.4.4 A Phase Operator from the Interplay Between Finite and Infinite Dimensions -- 15 Coherent State Quantization of Motions on the Circle, in an Interval, and Others -- 15.1 Introduction -- 15.2 Motion on the Circle -- 15.2.1 The Cylinder as an Observation Set -- 15.2.2 Quantization of Classical Observables -- 15.2.3 Did You Say Canonical? -- 15.3 From the Motion of the Circle to the Motion on 1+1 de Sitter Space-Time -- 15.4 Coherent State Quantization of the Motion in an Infinite-Well Potential -- 15.4.1 Introduction -- 15.4.2 The Standard Quantum Context -- 15.4.3 Two-Component Coherent States -- 15.4.4 Quantization of Classical Observables -- 15.4.5 Quantum Behavior through Lower Symbols -- 15.4.6 Discussion -- 15.5 Motion on a Discrete Set of Points -- 16 Quantizations of the Motion on the Torus -- 16.1 Introduction -- 16.2 The Torus as a Phase Space -- 16.3 Quantum States on the Torus.

16.4 Coherent States for the Torus.