Level crossings and new exact solutions of the two-photon Rabi model

TL;DR

This paper uncovers an infinite family of exact solutions for the two-photon Rabi model, revealing new energy level crossings beyond the Juddian class, expressed via special functions and analyzed through multiple mathematical approaches.

Contribution

It introduces a novel set of solutions for the two-photon Rabi model, expanding understanding of its spectral structure beyond known elementary solutions.

Findings

Discovered an infinite family of exact solutions.

Identified new energy level crossings not in Juddian class.

Expressed solutions using parabolic cylinder and Bessel functions.

Abstract

An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.