  In this work we study an optomechanical system in which there is a purely
quadratic optomechanical coupling between the optical and mechanical modes. The
optical mode is pumped by three coherent fields and the mechanical mode is
parametrically driven. We show that if the frequencies and amplitudes of both
optical and mechanical drivings are properly chosen, the optomechanical
interaction gives rise to an effective interaction, which, in the presence of
optical damping and in the absence of mechanical damping, has the squeezed
vacuum state and the squeezed one phonon state as dark states of the dynamics.
These states are well known for presenting quadrature squeezing and
sub-Poissonian statistics. However, even in the presence of mechanical damping
it is possible to find steady states with large degrees of quadrature squeezing
or strong sub-Poissonian statistics. Furthermore, we find a counter-intuitive
behavior in which a nonzero temperature of the mechanical environment allows
the observation of mechanical states with more pronounced sub-Poissonian
statistics.
