The observation of genuine quantum features of nano-mechanical motion is a key goal for both fundamental and applied quantum science. To this end, a promising approach is the stabilization of nonclassical features in the presence of dissipation, by means of the tunable coupling with a photonic environment. Here we present a scheme that combines dissipative squeezing with a mechanical nonlinearity to stabilize arbitrary approximations of (displaced) mechanical Fock state of any number. We consider an optomechanical system driven by three control lasers---at the cavity resonance and at the two mechanical sidebands---that couple the amplitude of the cavity field to the resonator's position and position squared. When the amplitude of the resonant drive is tuned to some specific values, the mechanical steady state is found in a (displaced) superposition of a finite number of Fock states, which for large enough squeezing achieves near-unit fidelity with a (displaced) Fock state of any desired number.
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