Vector coherent states with matrix moment problems

TL;DR

This paper introduces new classes of vector coherent states using matrix-valued variables and moments, extending the canonical coherent states framework with applications to quantum models like Jaynes-Cummings and supersymmetric oscillators.

Contribution

It develops vector coherent states with matrix moments and explores associated oscillator algebras, providing new mathematical tools and physical examples.

Findings

Constructed vector coherent states with matrix moments.

Applied to Jaynes-Cummings model in weak coupling.

Analyzed coherent states for supersymmetric radial oscillator.

Abstract

Canonical coherent states can be written as infinite series in powers of a single complex number $z$ and a positive integer $\rho(m)$. 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 $\{\rho(m)\}_{m=0}^\infty$. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number $z$ and the moments $\rho(m)$ of the canonical coherent states by $n \times n$ 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.