  The coherent quantum noise cancellation (CQNC) strategy has been performed in
the single-mode optomechanical systems to promote an ultra-sensitive metrology
protocol to break the standard quantum limit. The key idea of CQNC is that the
backaction noises arising from radiation pressure and driving can be offset by
coupling the optical mode to a near-resonant ancillary mode. In this work, a
continuous weak-force sensing under CQNC is developed in a double-mode
optomechanical system consisted of two optical modes with distinct frequencies
and a mechanical mode. In particular, under the asymmetrical treatment by
driving the higher-frequency optical mode, probing the lower-frequency one, and
coupling the probe mode to the ancillary mode, our configuration can be used to
resemble the conventional CQNC sensing. It is more important to find that the
current CQNC strategy simultaneously stabilizes the double-mode system with
respect to both the constrained driving power (the Routh-Hurwitz criterion) and
the effective positive mechanical damping (the stable optical-spring
condition). Moreover, through exploiting the coupling between the probe mode
and the ancillary mode under this nontrivial extension of the CQNC strategy
(from the single-mode version to the double-mode one), the rotating-wave term
and the counter-rotating term are found to be responsible to the system
stability and the noise cancellation, respectively. In realistic situations,
our scheme can be practiced in a tripartite optomechanical setup with a
membrane in the middle and a twisted-cavity-based weak-torque detector.
