We generalize a proposal for detecting single phonon transitions in a single nanoelectromechanical system (NEMS) to include the intrinsic anharmonicity of each mechanical oscillator. In this scheme two NEMS oscillators are coupled via a term quadratic in the amplitude of oscillation for each oscillator. One NEMS oscillator is driven and strongly damped and becomes a transducer for phonon number in the other measured oscillator. We derive the conditions for this measurement scheme to be quantum limited and find a condition on the size of the anharmonicity. We also derive the relation between the phase diffusion back-action noise due to number measurement and the localization time for the measured system to enter a phonon number eigenstate. We relate both these time scales to the strength of the measured signal, which is an induced current proportional to the position of the readout oscillator.
@article{arxiv.quant-ph/0410039,
title = {Anharmonic effects on a phonon number measurement of a quantum mesoscopic mechanical oscillator},
author = {D. H. Santamore and Hsi-Sheng Goan and G. J. Milburn and M. L. Roukes},
journal= {arXiv preprint arXiv:quant-ph/0410039},
year = {2009}
}