  We investigate a scheme that makes a quantum non-demolition measurement of
the excitation level of a mesoscopic mechanical oscillator by utilizing the
anharmonic coupling between two elastic beam bending modes. The non-linear
coupling between the two modes shifts the resonant frequency of the readout
oscillator proportionate to the excitation of the system oscillator. This
frequency shift may be detected as a phase shift of the readout oscillation
when driven on resonance. We show that in an appropriate regime this
measurement approaches a quantum non-demolition measurement of the phonon
number of the system oscillator. As phonon energies in micromechanical
oscillators become comparable to or greater than the thermal energy, the
individual phonon dynamics within each mode can be resolved. As a result it
should be possible to monitor jumps between Fock states caused by the coupling
of the system to the thermal reservoirs.
