Location of crossings in the Floquet spectrum of a driven two-level system

The calculation of the Floquet quasi-energies of a system driven by a time-periodic field is an efficient way to understand its dynamics. In particular, the phenomenon of dynamical localization can be related to the presence of close approaches between quasienergies ~either crossings or avoided crossings!. Here we consider a driven two-level system and study how the locations of crossings in the quasienergy spectrum alter as the field parameters are changed. A perturbational scheme provides a direct connection between the form of the driving field and the quasienergies which is exact in the limit of high frequencies. We first obtain relations for the quasienergies for some common types of applied field in the high-frequency limit, and then show how the locations of the crossings drift as the frequency is reduced. We find a simple empirical formula which describes this drift extremely well in general, and which we conjecture is exact for the specific case of square-wave driving.