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

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 read-out oscillator.

Figure 1

Schematic of phonon-number measurement for a coupled mechanical oscillator. Oscillators 0 and 1 are anharmonically coupled with coupling strength ${\lambda }_{01}$. Both oscillators are subjected to thermal-noise injection and dissipation. The oscillator 1 is driven and a read-out apparatus is attached to it.

Figure 2

The ratio of $\Gamma ∕{\Gamma }_0$ as a function of detuning at different values of driving strength and self-anharmonicity (Kerr effect coupling). The parameters are presented in units of the damping rate $\kappa$.