Multi-matrix vector coherent states

Abstract

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes-Cummings type Hamiltonians and for a two-level two-mode generalization of the Jaynes-Cummings model.

Landau Levels in a Two-Dimensional Noncommutative Space: Matrix and Quaternionic Vector Coherent States

“…This construction fits the general formulation of VCS by Ali et al [2] instead of the general theory of vector-valued coherent state representations [45] (and references listed therein) and similar developments based on operators of unitary representations of groups, where the VCS are defined as orbits of vectors [5,6,44]. Furthermore, we extend the VCS construction used in [23] to a formal tensor product of quantum Hilbert spaces by using the primary formulation of [55]. This extension is implemented with complex matrices and quaternions as CS variables.…”

“…With this setup and following the construction provided above, based on [2], and taking into account [23,55], the set of vectors formally given by…”

Deepening the vector coherent state analysis: Revisiting the harmonic oscillator

“…In some earlier works, a fairly systematic method has been introduced for constructing VCS over various types of matrix domains [2,3] in analogy with the canonical coherent states (CS), under the additional assumption of the existence of a resolution of the identity. Besides, VCS are also formulated for quantum optical models with spin-orbit interactions among which the Jaynes-Cummings model [4,5,6] and its deformed versions [7,8,9,10]. Furthermore, in [5], the study of the Landau levels has been achieved and different classes of VCS have been rigorously defined by taking into account the degeneracy.…”

“…At the theoretical level, the latter work extends the results given in [12] to a system with several degrees of freedom. An analogous procedure was used in [13] in order to obtain the CS for a free magnetic Schrödinger operator, and in [4] by introducing a class of VCS derived with matrices viewed as simple vectors in an enlarged Hilbert space. The present work deals with an extension of these three contributions by Gazeau and Novaes [11] and Thirulogasanthar et al [4,13].…”

“…In the present study, based on the prime scheme developed in [11,4,13], we perform a systematic analysis of VCS associated with the harmonic oscillator in 2D and then in 3D. The VCS are built using different generalized factorials, are normalizable and have a resolution of unity.…”