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We generalize a proposal for detecting single phonon transitions in a singlenanoelectromechanical system (NEMS) to include the intrinsic anharmonicity ofeach mechanical oscillator. In this scheme two NEMS oscillators are coupled viaa term quadratic in the amplitude of oscillation for each oscillator. One NEMSoscillator is driven and strongly damped and becomes a transducer for phononnumber in the other measured oscillator. We derive the conditions for thismeasurement scheme to be quantum limited and find a condition on the size ofthe anharmonicity. We also derive the relation between the phase diffusionback-action noise due to number measurement and the localization time for themeasured system to enter a phonon number eigenstate. We relate both these timescales to the strength of the measured signal, which is an induced currentproportional to the position of the readout oscillator.Comment: 13 pages, 2 figure

Anharmonic effects on a phonon-number measurement of a quantum-mesoscopic-mechanical oscillator