  Quantum squeezing in mechanical systems is not only a key signature of
macroscopic quantum effects, but can also be utilized to advance the metrology
of weak forces. Here we show that strong mechanical squeezing in the steady
state can be generated in an optomechanical system with mechanical nonlinearity
and red-detuned monochromatic driving on the cavity mode. The squeezing is
achieved as the joint effect of nonlinearity-induced parametric amplification
and cavity cooling, and is robust against thermal fluctuations of the
mechanical mode.
