Unconditional preparation of nonclassical states via linear-and-quadratic optomechanics

Reservoir engineering enables the robust and unconditional preparation of pure quantum states in noisy environments. We show how a family of nonclassical states of a mechanical oscillator can be stabilized in a cavity that is parametrically coupled to both the mechanical displacement and the displacement squared. The cavity is driven with three tones, on the red sideband, on the cavity resonance, and on the second blue sideband. The states so stabilized are (squeezed and displaced) superpositions of a finite number of phonons. They show the unique feature of encompassing two prototypes of nonclassicality for bosonic systems: by adjusting the strength of the drives, one can in fact move from a single-phonon- to a Schrödinger-cat-like state. The scheme is deterministic, supersedes the need for measurement-and-feedback loops, and does not require initialization of the oscillator to the ground state.