  We consider the dynamics of a two-level system (qubit) driven by strong and
short resonant pulses in the framework of Floquet theory. First we derive
analytical expressions for the quasienergies and Floquet states of the driven
system. If the pulse amplitude varies very slowly, the system adiabatically
follows the instantaneous Floquet states, which acquire dynamical phases that
depend on the evolution of the quasienergies over time. The difference between
the phases acquired by the two Floquet states corresponds to a qubit state
rotation, generalizing the notion of Rabi oscillations to the case of large
driving amplitudes. If the pulse amplitude changes very fast, the evolution is
non-adiabatic, with transitions taking place between the Floquet states. We
quantify and analyze the nonadiabatic transitions during the pulse by employing
adiabatic perturbation theory and exact numerical simulations. We find that,
for certain combinations of pulse rise and fall times and maximum driving
amplitude, a destructive interference effect leads to a remarkably strong
suppression of transitions between the Floquet states. This effect provides the
basis of a quantum control protocol, which we name Floquet Interference
Efficient Suppression of Transitions in the Adiabatic basis (FIESTA), that can
be used to design ultra-fast high-fidelity single-qubit quantum gates.
