On the solvability of the quantum Rabi model and its 2-photon and two-mode generalizations

TL;DR

This paper investigates the solvability of the quantum Rabi model and its generalizations, deriving exact solutions and characterizing eigenfunctions in terms of polynomials with roots from algebraic equations.

Contribution

It demonstrates that the quantum Rabi model and its extensions are quasi-exactly solvable and provides explicit formulas for energies and parameters.

Findings

Exact energy expressions derived for all models

Eigenfunctions are polynomial-based with roots from algebraic equations

Models are shown to be quasi-exactly solvable

Abstract

We study the solvability of the time-independent matrix Schr\"odinger differential equations of the quantum Rabi model and its 2-photon and two-mode generalizations in Bargmann Hilbert spaces of entire functions. We show that the Rabi model and its 2-photon and two-mode analogs are quasi-exactly solvable. We derive the exact, closed-form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces. Up to a normalization factor, the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations.