Analytical eigenstates for the quantum Rabi model

TL;DR

This paper presents a unified method to find analytical eigenstates of the quantum Rabi model using confluent Heun functions, clarifying the spectrum conditions and connecting to known Judd solutions.

Contribution

It introduces a comprehensive analytical approach for the quantum Rabi model's eigenstates, including regular and exceptional solutions, and clarifies the spectrum determination conditions.

Findings

Unified form for regular and exceptional solutions

Analytic conditions for energy spectrum derived

Judd solutions as truncations of Heun functions

Abstract

We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak [Phys. Rev. Lett. \textbf{107}, 100401 (2011)] are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.