  In the realm weak force sensing, an important issue is to suppress
fundamental noise (quantum noise and thermal noise), as they limit the accuracy
of force measurement. In this paper, we investigate a weak force sensing scheme
that combines a degenerate optical parametric amplifier (OPA) and an auxiliary
mechanical oscillator into a cavity optomechanical system to reduce quantum
noise. We demonstrate that the noise reduction of two coupled oscillators
depends on their norm mode splitting. and provide a classic analogy and quantum
perspective for further clarification. Besides, the noise reduction mechanism
of OPA is to reduce the fluctuation of photon number and enhance the squeezing
of the cavity field. We propose a specific design aimed at enhancing the joint
effect of both, beyond what can be achieved using OPA alone or two series
coupled oscillators. This scheme provides a new perspective for deeper
understanding of cavity field squeezing and auxiliary oscillator in force
sensing.
