  Landau-Zener-St\"uckelberg (LZS) interference of continuously driven
superconducting qubits is studied. Going beyond the second order perturbation
expansion, we find a time dependent stationary population evolution as well as
unsymmetrical microwave driven Landau-Zener transitions, resulting from the
nonresonant terms which are neglected in rotating-wave approximation. For the
low-frequency driving, the qubit population at equilibrium is a periodical
function of time, owing to the contribution of the nonresonant terms. In order
to obtain the average population, it is found that the average approximation
based on the perturbation approach can be applied to the low-frequency region.
For the extremely low frequency which is much smaller than the decoherence
rate, we develop noncoherence approximation by dividing the evolution into
discrete time steps during which the coherence is lost totally. These
approximations present comprehensive analytical descriptions of LZS
interference in most of parameter space of frequency and decoherence rate,
agreeing well with those of the numerical simulations and providing a simple
but integrated understanding to system dynamics. The application of our models
to microwave cooling can obtain the minimal frequency to realize effective
microwave cooling.
