Exact Solutions of the Full Two-Photon Rabi Model

The full two-photon quantum Rabi model describes the nonlinear coupling between a superconducting qubit and a resonator. It possesses ℤ 2 symmetry, in contrast to the conventional two-photon quantum Rabi model in cavity quantum electrodynamics systems, which exhibits ℤ 4 symmetry. Using the su(1,1) Lie algebra and ℤ 2 symmetry, we derive a transcendental function whose zeros yield the exact eigenvalues. The spectral collapse phenomenon at the critical coupling g c is derived analytically. At g c , the model can be mapped to an inverse-square potential well, where an infinite number of bound states approach the threshold energy exponentially. Our results may provide new insights into spectral phenomena in practical experimental realizations of superconducting circuits with nonlinear coupling.