  We study an optomechanical system for the purpose of generating a
nonclassical mechanical state when a mechanical oscillator is quadratically
coupled to a single-mode cavity field driven by a squeezed optical field. The
system corresponds to a regime where the optical dissipation dominates both the
mechanical damping and the optomechanical coupling. We identify that
multi-phonon processes emerge in the optomechanical system and show that a
mechanical oscillator prepared in the ground state will evolve into an
amplitude-squared squeezed vacuum state. The Wigner distribution of the steady
state of the mechanical oscillator is non-Gaussian exhibiting quantum
interference and four-fold symmetry. This nonclassical mechanical state,
generated via reservoir engineering, can be used for quantum correlation
measurements of the position and momentum of the mechanics below the standard
quantum limit.
