Perturbation Theory and the Quantum Rabi-model

TL;DR

This paper develops a perturbative approach to analyze the eigenvalues of the Quantum Rabi model and its generalizations, providing analytic expansions and spectral asymptotics for complex quantum optical systems.

Contribution

It introduces a perturbative framework for the Rabi model, proving the Braak conjecture for finite eigenvalue families and analyzing spectral asymptotics for multi-level atom systems.

Findings

Proved the Braak conjecture for finite eigenvalue families.

Derived analytic eigenvalue expansions using Rellich's theory.

Analyzed spectral counting function asymptotics for generalized models.

Abstract

In the first part of the paper we study a perturbative model of the Rabi system of Quantum Optics. We are therefore able to describe, through Rellich's theory, an analytic expansion of finite families of eigenvalues, of arbitrary fixed length. In particular, we prove that for finite families of eigenvalues the Braak conjecture holds. In the second part we study the asymptotics of the Weyl spectral counting function of a class of systems that generalize the Quantum Rabi Model to an $N$-level atom ($N\geq3$) with $N-1$ cavity modes of the electromagnetic field.