  The extended quantum Rabi models make a significant contribution to
understand the quantum nature of the atom-light interaction. We transform two
kinds of extended quantum Rabi model, anisotropic Rabi model and asymmetric
Rabi model, into rotating frame, and regard them as periodically driven quantum
systems. The analytical solutions of the quasi-energy spectrum as well as the
Floquet modes for both models are constructed by applying the Floquet theory
and the high-frequency expansion, which is applied to the non-stroboscopic
dynamics of physical observables such as atomic inversion, transverse
magnetization, atom-field correlation, etc. For anisotropic Rabi model, the
quasi energy fits well with the numerical results even when the rotating-wave
coupling is in the deep-strong coupling regime $g\simeq\hbar\omega$ if the
counterrotating terms is small enough compared to the driving frequency.
Avoided level crossing may occur for quasi energy with the same parity when the
positive branch spectrum lines for the total excitation number $N$ cross the
negative branch lines for $N+2$, while the high frequency expansion fails to
predict this due to the conservation of the total excitation number.
Furthermore, we present analytical and numerical study of the long-time
evolution of population and figure out the analytical method is credible for
the population dynamics. For asymmetric Rabi model, we find that the external
bias field which breaks the parity symmetry of total excitation number tends to
cluster the upper and lower branches into two bundles, and the detuning induced
gap in the first temporal Brillouin zone shows a quadratic dependence on the
bias. Both models prove that treating the Hamiltonian in the rotating frame by
Floquet theory gives an alternative tool in the study of interaction between
atom and light.
