Spectral determinant of the two-photon quantum Rabi model

TL;DR

This paper analyzes spectral determinants of the two-photon quantum Rabi model, revealing that only Chen et al.'s G-function explicitly encodes the spectrum's structure and collapse behavior.

Contribution

It rigorously derives the G-function based on the model's symmetry, clarifying its role in determining the regular spectrum and collapse point.

Findings

Chen et al.'s G-function has a known pole structure related to collapse.

Zeros of the G-function correspond to the complete regular spectrum.

The analysis emphasizes the qualitative spectral features of the model.

Abstract

The various generalized spectral determinants (G-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the G-function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. We derive this function rigorously employing the $\mathbb{Z}_4$-symmetry of the model and show that its zeros correspond to the complete regular spectrum.