  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.

Quantum Physics;Mesoscale and Nanoscale Physics