Optomechanical generation of a Schr\"odinger cat-like mechanical state

We propose a novel scheme to prepare a macroscopic mechanical oscillator in a cat-like state, close to a coherent state superposition. The mechanical oscillator, coupled by radiation-pressure interaction to a field in an optical cavity, is first prepared close to a squeezed vacuum state using a reservoir engineering technique. The system is then probed using a short optical pulse tuned to the lower motional sideband of the cavity resonance, realizing a photon-phonon swap interaction. A photon number measurement of the photons emerging from the cavity then conditions a phonon-subtracted cat-like state with a negative Wigner distribution. We show that this scheme is feasible using state-of-the-art photonic crystal optomechanical system.