Single-step arbitrary control of mechanical quantum states in ultrastrong optomechanics

We describe how ultrastrong interactions in optomechanical systems can be used to force the system ground state to evolve into an arbitrary quantum state of mechanical motion in a completely controlled and deterministic manner. If the target quantum state is a superposition of $N$ Fock states, it can be obtained by applying in single-step $N$ classical optical signals of different frequencies for a common time interval. This protocol can be applied to various strongly interacting quantum systems as trapped ions beyond the Lamb-Dicke regime and cavity QED into the ultrastrong coupling regime.