Two-level system with broken inversion symmetry coupled to a quantum harmonic oscillator

We study the generalized Jaynes-Cummings model of quantum optics at the inversion-symmetry-breaking and in the ultrastrong coupling regime. With the help of a generalized multiphoton rotating-wave approximation, we study the stationary solutions of the Schr\"{o}dinger equation. It is shown that the problem is reduced to resonant interaction of two position-displaced harmonic oscillators. Explicit expressions for the eigenstates and eigenvalues of generalized Jaynes-Cummings Hamiltonian are presented. We exemplify our physical model with analytical and numerical considerations regarding collapse and revivals of the initial population of a two-level system and photon distribution function at the direct multiphoton resonant coupling.