I give a mathematical procedure to obtain the adiabatic approximation for the generalized quantum Rabi Hamiltonian both without and with a quadratic interaction. We consider the Hamiltonian as the energy of a model describing the interaction system of a two-level artificial atom and a one-mode microwave photon in circuit QED.
In the case without the quadratic interaction, I show in the adiabatic approximation that whether each bare eigenstate forms a Schroedinger-cat-like entangled state or not depends on whether the energy bias of the atom is zero or non-zero, and then, the effect of the tunnel splitting of the atom is ignored.
On the other hand, in the case with the quadratic interaction, I show in the adiabatic approximation that all the physical eigenstates obtained by the (meson) pair theory form individual Schroedinger-cat-like entangled states for every energy bias.
I conclude that this fact comes from the effect of the tunnel splitting.