Coherent and incoherent superposition of transition matrix elements of the squeezing operator

TL;DR

This paper derives new closed-form expressions for matrix elements of the squeezing operator using Gegenbauer polynomials, enabling analysis of superpositions and multiphoton transitions in quantum harmonic oscillators.

Contribution

It introduces a novel representation of squeezing operator matrix elements with Gegenbauer polynomials, facilitating analytic superposition calculations and applications to quantum transition processes.

Findings

Exact matrix element expressions in terms of Gegenbauer polynomials.

Closed-form superpositions of matrix elements.

Comparison shows semi-classical approximation fails outside Rayleigh-Jeans limit.

Abstract

We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system "charged particle + electromagnetic radiation", induced by a (strong) coherent field or by a black-body radiation component (with a Planck-Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it completely fails.