  In this paper, the scheme of a force sensor is proposed which has been
composed of a hybrid optomechanical cavity containing an interacting
cigar-shaped Bose-Einstein condensate (BEC) where the \textit{s}-wave
scattering frequency of the BEC atoms as well as the spring coefficient of the
cavity moving end-mirror (the mechanical oscillator) are parametrically
modulated. It is shown that in the red-detuned regime and under the so-called
impedance-matching condition, the mechanical response of the system to the
input signal is enhanced substantially, which leads to the amplification of the
weak input signal while the added noises of measurement (backaction noises) can
be suppressed and lowered much below the standard quantum limit (SQL). In this
way, such a hybrid system operates as an ultra-sensitive force sensor which can
amplify the input signal and simultaneously suppress the added noises by
controlling the amplitudes of modulation and the system cooperativities. The
advantage of the presented nonlinear hybrid system accompanied with the
mechanical and atomic modulations in comparison to the bare optomechanical
cavities is the enhancement of signal amplification as well as the extension of
amplification bandwidth.
