  We present experiments on the driven dynamics of a two-level superconducting
artificial atom. The driving strength reaches 4.78 GHz, significantly exceeding
the transition frequency of 2.288 GHz. The observed dynamics is described in
terms of quasienergies and quasienergy states, in agreement with Floquet
theory. In addition, we observe the role of pulse shaping in the dynamics, as
determined by non-adiabatic transitions between Floquet states, and we
implement subnanosecond single-qubit operations. These results pave the way to
quantum control using strong driving with applications in quantum technologies.
