Symmetric Rotating Wave Approximation for the Generalized Two-Mode Two-Photon System Coupled with a Bosonic Field

We study the analytically approximated eigenenergies of the two-mode two-photon quantum Rabi model coupled with a Bosonic field. By equipping the previously established generalized symmetric rotating wave approximation (G-SRWA) method used on single-mode spin-boson systems with a unitary squeezing operator and applying it onto the two-mode two-photon system, we obtain a more generalized and simplified closed-form expression for eigenenergies of the system compared to other expressions derived using the functional Bethe ansatz method. We further investigate the effectiveness of the squeezing operator by examining the eigenenergies of the generalized resonant and nonresonant case for the two-level two-photon system. The accuracy of the updated G-SRWA method in the ultra and deep-strong coupling regimes reveals interesting symmetries within the mathematical structure of the two-level two-photon systems.