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A generalized quantum Rabi Hamiltonian with both one- and two-photon termshas emerged in the circuit quantum electrodynamics system for a decade. Theusual parity symmetry is broken naturally in the simultaneous presence of bothcouplings, which complicates analytical treatments, even in the rotating waveapproximations. In this paper, we propose an adiabatic approximation to thisgeneric model by using Bogoliubov operators, and obtain a very conciseanalytical solution for both eigenvalues and eigenstates. Although theadiabatic approximation is only exact in the vanishing limit of the qubitfrequency, the results for some physical observables nevertheless agree wellwith the numerical ones in a wide parameter regime. In the rotating-waveapproximations, we also derive an analytical eigensolution. Two dominant Rabifrequencies are found in the Rabi oscillations of this generalized model. Wealso apply the present analytical theory to the vacuum Rabi splitting. It isfound that some new phenomena emerge just because of the presence of theadditional two-photon coupling term.

Generalized quantum Rabi model with both one- and two-photon terms: A concise analytical study