Analytical solutions of basic models in quantum optics

TL;DR

This paper reviews analytical solutions of fundamental quantum optics models using complex differential equations and the Bargmann space, highlighting solutions for the quantum Rabi and asymmetric Dicke models.

Contribution

It introduces a method employing Fuchsian equations and Bargmann space to analytically solve key models in quantum optics, including detailed solutions for the quantum Rabi model.

Findings

Solutions derived from singularity structures of differential equations

Spectral properties linked to the singularity structure

Analytical solutions for the quantum Rabi and asymmetric Dicke models

Abstract

The recent progress in the analytical solution of models invented to describe theoretically the interaction of matter with light on an atomic scale is reviewed. The methods employ the classical theory of linear differential equations in the complex domain (Fuchsian equations). The linking concept is provided by the Bargmann Hilbert space of analytic functions, which is isomorphic to $L^2(\mathbb{R})$, the standard Hilbert space for a single continuous degree of freedom in quantum mechanics. I give the solution of the quantum Rabi model in some detail and sketch the solution of its generalization, the asymmetric Dicke model. Characteristic properties of the respective spectra are derived directly from the singularity structure of the corresponding system of differential equations.