  We propose a scheme to prepare macroscopic quantum superpositions of motion
in optomachanical nano- or micromechanical oscillators quadratically coupled to
an intracavity field. The nonlinear optomechanical coupling leads to an
effective degenerate three-wave mixing interaction between the mechanical and
cavity modes. The quantum superpositions result from the combined effects of
the interaction and cavity dissipation. We show analytically and confirm
numerically that various deterministic quantum superpositions can be achieved,
depending on initial mechanical state. The effect of mechanical damping is also
studied in detail via the negativity of the Wigner function. The present scheme
can be realized in various optomechanical systems with current technology.
