  Floquet theory combined with the generalized Van Vleck nearly degenerate
perturbation theory, has been widely employed for studying various two-level
systems that are driven by external fields via the time-dependent longitudinal
(i.e., diagonal) couplings. However, three-level systems strongly driven by the
time-dependent transverse (i.e., off-diagonal) couplings have rarely been
investigated, due to the breakdown of the traditional rotating wave
approximation. Meanwhile, the conventional perturbation theory is not directly
applicable, since a small parameter for the perturbed part is no longer
apparent. Here we develop a double-unitary-transformation approach to deal with
the periodically modulated and strongly driven systems, where the
time-dependent Hamiltonian has large off-diagonal elements. The first unitary
transformation converts the strong off-diagonal elements to the diagonal ones,
and the second enables us to harness the generalized Van Vleck perturbation
theory to deal with the transformed Floquet matrix and also allows us to reduce
the infinite-dimensional Floquet Hamiltonian to a finite effective one. For a
strongly modulated three-level system, with the combination of the Floquet
theory and the transformed generalized Van Vleck perturbation theory, we obtain
analytical results of the system, which agree well with exact numerical
solutions. This method offers a useful tool to analytically study the
multi-level systems with strong transverse couplings.
