  Cavity optomechanical systems are one of the leading experimental platforms
for controlling mechanical motion in the quantum regime. We exemplify that the
control over cavity optomechanical systems greatly increases by coupling the
cavity also to a two-level system, thereby creating a hybrid optomechanical
system. If the two-level system can be driven largely independently of the
cavity, we show that the non-linearity thus introduced enables us to steer the
extended system to non-classical target states of the mechanical oscillator
with Wigner functions exhibiting significant negative regions. We illustrate
how to use optimal control techniques beyond the linear regime to drive the
hybrid system from the near ground state into a Fock target state of the
mechanical oscillator. We base our numerical optimization on realistic
experimental parameters for exemplifying how optimal control enables the
preparation of decidedly non-classical target states, where naive control
schemes fail. Our results thus pave the way for applying the toolbox of optimal
control in hybrid optomechanical systems for generating non-classical
mechanical states.

Quantum Physics