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 Schr\"odinger 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 Schr\"odinger 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.Comment: 6 pages, 4 figures with additional supplementary information (12
pages, 2 figures