Approximating two-mode two-photon Hamiltonian

The Rabi model describes the simplest nontrivial interaction between a few-level system and a bosonic mode, featuring in multiple seemingly unrelated systems of importance to quantum science and technology. While exact expressions for the energies of this model and its few-mode extensions have been obtained, they involve roots of transcendental functions and are thus cumbersome and unintuitive. Utilizing the symmetric generalized rotating wave approximation (S-GRWA), we develop a family of approximations for the energies of the two-mode two-photon Rabi model. The simplest elements of the family are analytically tractable and physically intuitive, providing good approximations in regimes of interest such as ultra- and deep-strong coupling. Higher-order approximate energies can be used if more accuracy is required.