  We investigate a hybrid system consisting of an atomic ensemble trapped
inside a dissipative optomechanical cavity assisted with the perturbative
oscillator-qubit coupling. It is shown that such a hybrid system is very
suitable for generating stationary squeezing of the mirror motion in the
long-time limit under the unresolved sideband regime. Based on the approaches
of master equation and covariance matrix, we discuss the respective squeezing
effects in detail and find that in both approaches, simplifying the system
dynamics with adiabatic elimination of the highly dissipative cavity mode is
very effective. In the approach of master equation, we find that the squeezing
is a resulting effect of cooling process and is robust against the thermal
fluctuations of the mechanical mode. While in the covariance matrix approach,
we can obtain the analytical result of the steady-state mechanical position
variance from the reduced dynamical equation approximately. Finally, we compare
the two different approaches and find that they are completely equivalent for
the stationary dynamics. The scheme may be meaningful for the possible
ultraprecise quantum measurement involved mechanical squeezing.
