Squeezed States and Hermite polynomials in a Complex Variable

TL;DR

This paper constructs and analyzes three types of coherent states related to Hermite polynomials in a complex variable, exploring their relations to squeezed states and providing new realizations in the Bargmann space.

Contribution

It introduces novel coherent states associated with Hermite polynomials in a complex variable and links them to squeezed states and Bargmann space representations.

Findings

Established relations between the new coherent states and squeezed states.

Provided a second realization of canonical coherent states in Bargmann space.

Connected Hermite polynomial-based states with quantum optics concepts.

Abstract

Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].