  We prove that the force sensitivity of the conventional optomechanical
detector associated with the optical quadrature measurement of the output beam
is lower bounded by the so-called ultimate quantum limit (UQL), i.e., the
absolute value of the imaginary part of the inverse mechanical susceptibility.
Through the linear response theory, we find that the force sensitivity of any
linear detector is lower bounded by a generalized UQL, which might beat the
usual UQL by properly tailoring the detector-oscillator interaction. We believe
that our results open a new direction for improving the performance of
high-sensitivity detection schemes.
