  In this review, we study how a hybrid optomechanical system (OMS), in which a
quantum micro- or nano-mechanical oscillator (MO) is coupled to the
electromagnetic (EM) radiation pressure, consisting of an ensemble of ultracold
atoms or an atomic Bose-Einstein condensate (BEC), can be used as an ultra
precision quantum sensor for measuring very weak signals. As is well-known in
any precise quantum measurement the competition between the shot noise (SN) and
the backaction noise of measurement executes a limitation on the measurement
precision which is the so-called standard quantum limit (SQL). In the case
where the intensity of the signal is even lower than the SQL, one needs to
perform an ultra precision quantum sensing to beat the SQL. For this purpose,
we review three important methods for surpassing the SQL in a hybrid OMS: (i)
the backaction evading measurement of a quantum nondemolition (QND) variable of
the system, (ii) the coherent quantum backaction noise cancellation (CQNC), and
(iii) the so-called parametric sensing, the simultaneous signal amplification
and added noise suppression below the SQL. Furthermore, we have shown in this
article for the first time how the classical fluctuation of the driving laser
phase, the so-called laser phase noise (LPN), affects the power spectrum of the
output optical field in a standard OMS and induces an additional impression
noise which makes the total system noise increase above the SQL. Also, for the
first time in this review it has been shown that in the standard OMSs, it is
impossible to amplify signal while suppressing the noise below the SQL
simultaneously.
