Two-photon Rabi model: analytic solutions and spectral collapse
Liwei Duan, You-Fei Xie, Daniel Braak, Qing‐Hu Chen · Journal of Physics A Mathematical and Theoretical · 2016
The two-photon quantum Rabi model is analyzed using the Z 4 -symmetry and s u ( 1 , 1 ) -algebra. We derive the G -function whose zeros give the exact eigenvalues of the Hamiltonian. The singularity structure of the G -function allows to draw conclusions about the distribution of these eigenvalues along the real axis and we derive the spectral collapse phenomenon at critical coupling g c , found numerically before. The spectrum at g c consists of a discrete and a continuous part: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense. For large qubit splitting, also other low lying states split off from the continuum. However, perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. We corroborate this result with a variational calculation for the ground state.