Nonlinear Sideband Cooling to a Schr\"odinger Cat State of Motion

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