  We propose and analyze nonlinear optomechanical protocols that can be
implemented by adding a single atom to an optomechanical cavity. In particular,
we show how to engineer the environment in order to dissipatively prepare the
mechanical oscillator in a superposition of Fock states with fidelity close to
one. Furthermore, we discuss how a single atom in a cavity with several
mechanical oscillators can be exploited to realize nonlinear many-body physics
by stroboscopically driving the mechanical oscillators. We show how to prepare
non-classical many-body states by either applying coherent protocols or
engineering dissipation. The analysis of the protocols is carried out using a
perturbation theory for degenerate Liouvillians and numerical tools. Our
results apply to other systems where a qubit is coupled to a mechanical
oscillator via a bosonic mode, e.g., in cavity quantum electromechanics.
