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We determine the small signal gain and noise response of an amplifier based on the nonlinear response of a quantum nanomechanical resonator. The resonator is biased in the nonlinear regime by a strong harmonic bias force and we determine the response to a small additional driving signal detuned with respect to the bias force. energy. This system is dynamically equivalent to the Duffing oscillator, which exhibits a fixed point bifurcation: a loss of stability in a fixed point as a control parameter (in this case a driving force) is varied [2]. Near the bifurcation, the system becomes very sensitive to fluctuations, a sensitivity that can be harnessed to make a signal amplifier [3], or sensitive charge detector [4]. Such a bifurcation amplifier has recently been used in the study of superconducting qubits [5].

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