  We experimentally study the transient dynamics of a dissipative
superconducting qubit under periodic drive towards its nonequilibrium steady
states. The corresponding stroboscopic evolution, given by the qubit states at
times equal to integer multiples of the drive period, is determined by a
(generically non-Hermitian) Floquet Liouvillian. The drive period controls both
the transients across its non-Hermitian degeneracies and the resulting
nonequilibrium steady states. These steady states can exhibit higher purity
compared to those achieved with a constant drive. We further study the
dependence of the steady states on the direction of parameter variation and
relate these findings to the recent studies of dynamically encircling
exceptional points. Our work provides a new approach to control non-Hermiticity
in dissipative quantum systems and presents a new paradigm in quantum state
preparation and stabilization.
