  We analyze the dynamics of a two-level system subject to driving by
large-amplitude external fields, focusing on the resonance properties in the
case of driving around the region of avoided level crossing. In particular, we
consider three main questions that characterize resonance dynamics: (1) the
resonance condition, (2) the frequency of the resulting oscillations on
resonance and (3) the width of the resonance. We identify the regions of
validity of different approximations. In a large region of the parameter space,
we use a geometric picture in order to obtain both a simple understanding of
the dynamics and quantitative results. The geometric approach is obtained by
dividing the evolution into discrete time steps, with each time step described
by either a phase shift on the basis states or a coherent mixing process
corresponding to a Landau-Zener crossing. We compare the results of the
geometric picture with those of a rotating-wave approximation. We also comment
briefly on the prospects of employing strong driving as a useful tool to
manipulate two-level systems.
