Two-photon Rabi model: Analytic solutions and spectral collapse
  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. 
Liwei Duan, You-Fei Xie, Daniel Braak, and Qing-Hu Chen
quant-ph