  We consider an optomechanical cavity with a movable end-mirror as a quantum
mechanical oscillator (MO) containing an interacting cigar-shaped
Bose-Eisenstein condensate (BEC). It is assumed that both the MO and the BEC
interact with the radiation pressure of the cavity field in the red-detuned and
weak coupling regimes while the two-body atomic collisions frequency of the BEC
and the mechanical spring coefficient of the MO are coherently modulated. By
analyzing the scattering matrix we find that a frequency-dependent squeezing is
induced to the three subsystems only due to the coherent modulations. In the
largely different cooperativities regime together with strong modulations, the
mechanical mode of the MO and the Bogoliubov mode of the BEC exhibit quadrature
squeezing which can surpass the so-called 3dB limit (up to 75 dB) with high
robustness to the thermal noises. Surprisingly, in this regime by controlling
the system and modulation parameters, a very high degree of squeezing (up to 16
dB) together with high purity of quantum state for the output cavity field is
achievable. Furthermore, one can attain simultaneous strong quantum
amplification, added-noise suppression, and controllable gain-bandwidth for the
complementary quadratures of squeezed ones in the subsystems.
