  A generalized quantum Rabi Hamiltonian with both one- and two-photon terms
has emerged in the circuit quantum electrodynamics system for a decade. The
usual parity symmetry is broken naturally in the simultaneous presence of both
couplings, which complicates analytical treatments, even in the rotating wave
approximations. In this paper, we propose an adiabatic approximation to this
generic model by using Bogoliubov operators, and obtain a very concise
analytical solution for both eigenvalues and eigenstates. Although the
adiabatic approximation is only exact in the vanishing limit of the qubit
frequency, the results for some physical observables nevertheless agree well
with the numerical ones in a wide parameter regime. In the rotating-wave
approximations, we also derive an analytical eigensolution. Two dominant Rabi
frequencies are found in the Rabi oscillations of this generalized model. We
also apply the present analytical theory to the vacuum Rabi splitting. It is
found that some new phenomena emerge just because of the presence of the
additional two-photon coupling term.
