Two-photon Rabi model: Analytic solutions and spectral collapse

TL;DR

This paper provides analytic solutions and spectral analysis of the two-photon quantum Rabi model, revealing a finite energy gap at the spectral collapse point and highlighting the non-perturbative nature of the excitation gap.

Contribution

It introduces extended squeezed state methods and G-functions for the two-photon Rabi model, offering new insights into spectral collapse and eigenstate structure.

Findings

Finite energy gap persists at spectral collapse for low-lying states.

Eigenstates can be explicitly approximated with perturbative formulas.

Spectral collapse involves a non-perturbative vanishing of the excitation gap.

Abstract

The two-photon quantum Rabi model with quadratic coupling is studied using extended squeezed states and we derive $G$-functions for Bargmann index $q=1/4$ and $3/4$. The simple singularity structure of the $G$-function allows to draw conclusions about the distribution of eigenvalues along the real axis. The previously found picture of the spectral collapse at critical coupling $g_{\mathrm{c}}$ has to be modified regarding the low lying states, especially the ground state: We obtain a finite gap between ground state and the continuum of excited states at the collapse point. For large qubit splitting, also other low lying states may be separated from the continuum at $g_{\mathrm{c}}$. We have carried out a perturbative analysis allowing for explicit and simple formulae of the eigenstates. Interestingly, a vanishing of the gap between ground state and excited continuum at $g_{\mathrm{c}}$ is obtained in each finite order of approximation. This demonstrates cleary the non-pertubative nature of the excitation gap. We corroborate these findings with a variational calculation for the ground state.