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We consider an optomechanical cavity with a movable end-mirror as a quantummechanical oscillator (MO) containing an interacting cigar-shapedBose-Eisenstein condensate (BEC). It is assumed that both the MO and the BECinteract with the radiation pressure of the cavity field in the red-detuned andweak coupling regimes while the two-body atomic collisions frequency of the BECand the mechanical spring coefficient of the MO are coherently modulated. Byanalyzing the scattering matrix we find that a frequency-dependent squeezing isinduced to the three subsystems only due to the coherent modulations. In thelargely different cooperativities regime together with strong modulations, themechanical mode of the MO and the Bogoliubov mode of the BEC exhibit quadraturesqueezing which can surpass the so-called 3dB limit (up to 75 dB) with highrobustness to the thermal noises. Surprisingly, in this regime by controllingthe system and modulation parameters, a very high degree of squeezing (up to 16dB) together with high purity of quantum state for the output cavity field isachievable. Furthermore, one can attain simultaneous strong quantumamplification, added-noise suppression, and controllable gain-bandwidth for thecomplementary quadratures of squeezed ones in the subsystems.

Strong quadrature squeezing and quantum amplification in a coupled Bose-Einstein condensate- optomechanical cavity via coherent modulation