  The ability to prepare a macroscopic mechanical resonator into a quantum
superposition state is an outstanding goal of cavity optomechanics. Here, we
propose a technique to generate cat states of motion using the intrinsic
nonlinearity of a dispersive optomechanical interaction. By applying a
bichromatic drive to an optomechanical cavity, our protocol enhances the
inherent second-order processes of the system, inducing the requisite
two-phonon dissipation. We show that this nonlinear sideband cooling technique
can dissipatively engineer a mechanical resonator into a cat state, which we
verify using the full Hamiltonian and an adiabatically reduced model. While the
fidelity of the cat state is maximized in the single-photon, strong-coupling
regime, we demonstrate that Wigner negativity persists even for weak coupling.
Finally, we show that our cat state generation protocol is robust to
significant thermal decoherence of the mechanical mode, indicating that such a
procedure may be feasible for near-term experimental systems.
