SU(2) and SU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

Abstract

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.

LaTeX, 24 pages, 1 figure (compressed PostScript, available at http://www.technion.ac.il/~brif/abstracts/AES.html ). More information on http://www.technion.ac.il/~brif/science.html

Cited by in corpus (32)

• Coherent states à la Klauder-Perelomov for the Pöschl-Teller potentials
• $SU(1,1)$ solution for the Dunkl-Coulomb problem in two dimensions and its coherent states
• Coherent states for exactly solvable potentials
• Quantum Fisher information on two manifolds of two-mode Gaussian states
• Generalized uncertainty relations and coherent and squeezed states
• Generalized Intelligent States for an Arbitrary Quantum System
• Generalized coherent and intelligent states for exact solvable quantum systems
• Characteristic uncertainty relations
• Generalized Intelligent States for Nonlinear Oscillators
• Exact Solutions for the General Nonstationary Oscillator with a Singular Perturbation
• Coherent States for Generalized Laguerre Functions
• Generalized coherent and squeezed states based on the $h(1)\otimes su(2)$ algebra
• Interpolating Coherent States for Heisenberg-Weyl and Single-Photon SU(1,1) Algebras
• The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
• Generation of single-mode SU(1,1) intelligent states and an analytic approach to their quantum statistical properties
• On the squeezed states for n observables
• Quasienergy operators and generalized squeezed states for systems of trapped ions
• New SU(1, 1) Position-Dependent Effective Mass Coherent States for the Generalized Shifted Harmonic Oscillator
• $SU(1,1)$ covariant $s$-parametrized maps
• Barut-Girardello coherent states for u(p,q) and sp(N,R) and their macroscopic superpositions
• Unitary equivalence between ordinary intelligent states and generalized intelligent states
• sh(2/2) Superalgebra eigenstates and generalized supercoherent and supersqueezed states
• Quantum statistical properties of some new classes of intelligent states associated with special quantum systems
• Coupled Supersymmetry and Ladder Structures Beyond the Harmonic Oscillator
• Generalized intelligent states of the su(N) algebra
• Reduction of optimum light power with Heisenberg-limited photon-counting noise in interferometric gravitational-wave detectors
• $SU(1,1)$ and $SU(2)$ Perelomov number coherent states: algebraic approach for general systems
• Analytic representations based on su(3) coherent states and Robertson intelligent states
• Coherent and squeezed states of quantum Heisenberg algebras
• $SU(1,1)$ coherent states for Dirac-Kepler-Coulomb problem in $D+1$ dimensions with scalar and vector potentials
• Governing dynamics by squeezing in a system of cold trapped ions
• Qubit Geometry through Holomorphic Quantization