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arXiv:math-ph/0311046 · Journal of Physics A Mathematical and General

Vector coherent states with matrix moment problems

Kengatharam Thirulogasanthar, Amétépé Lionel Hohouéto

Published Sep 23, 2004• 2 citations
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Abstract

Canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ( m ). The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers {ρ( m )} ∞ m =0 . In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ( m ) of the canonical coherent states by n × n matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes–Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.

DOI
10.1088/0305-4470/37/40/014

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Canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ( m ). The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers {ρ( m )} ∞ m =0 . In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ( m ) of the canonical coherent states by n × n matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes–Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.
Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.
A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z. In the present scheme, the variable z is replaced by matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.
Abstract. The canonical coherent states are expressed as infinite series in powers of a complex number z in their infinite series version. In this article we present classes of coherent states by replacing this complex number z by other choices, namely, iterates of a complex function, higher functions and elementary functions. Further, we show that some of these classes do not furnish generalized oscillator algebras in the natural way. A reproducing kernel Hilbert space is discussed to each class of coherent states. 1.
A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable zzz. In the present scheme, the variable zzz is replaced by a matrix valued function over appropriate domains. As particular examples, we analyze the quaternionic extensions of the the canonical coherent states and the Gilmore-Perelomov and Barut-Girardello coherent states arising from representations of SU(1,1).
Through canonical and f deformation quantizations, classes of nonlinear spin-orbit interaction models are considered. These generalized models are relevant in condensed matter physics and quantum optics in domains of nonlinear spin-Hall effect or of the multiphoton Jaynes-Cummings [Proc. IEEE 51, 89 (1963)] model, for instance. A spectral decomposition follows the exact diagonalization of the Hamiltonian and leads to the definition of ladder operators acting on the associated Hilbert space. Nonlinear vector coherent states are then constructed from a general lowering operator action. Explicit solutions of their related moment problems in ordinary and f-deformed theories are displayed, thus providing new classes of coherent states for such generalized spin-orbit models.