  We show two effects as a result of considering the second-order correction to
the spectrum of a nanomechanical resonator electrostatically coupled to a
Cooper-pair box. The spectrum of the Cooper-pair box is modified in a way which
depends on the Fock state of the resonator. Similarly, the frequency of the
resonator becomes dependent on the state of the Cooper-pair box. We consider
whether these frequency shifts could be utilized to prepare the nanomechanical
resonator in a Fock state, to perform a quantum non-demolition measurement of
the resonator Fock state, and to distinguish the phase states of the
Cooper-pair box.
