Optomechanics allows the transduction of weak forces to optical fields, withmany efforts approaching the standard quantum limit. We consider force-sensingusing a mirror-in-the-middle setup and use two coupled cavity modes originatedfrom normal mode splitting for separating pump and probe fields. We find thatthis two-mode model can be reduced to an effective single-mode model, if wedrive the pump mode strongly and detect the signal from the weak probe mode.The optimal force detection sensitivity at zero frequency (DC) is calculatedand we show that one can beat the standard quantum limit by driving the cavityclose to instability. The best sensitivity achievable is limited by mechanicalthermal noise and by optical losses. We also find that the bandwidth whereoptimal sensitivity is maintained is proportional to the cavity damping in theresolved sideband regime. Finally, the squeezing spectrum of the output signalis calculated, and it shows almost perfect squeezing at DC is possible by usinga high quality factor and low thermal phonon-number mechanical oscillator.
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