Phonon number measurements using single photon opto-mechanics

We describe a system composed of two coupled optical cavity modes with a coupling modulated by a bulk mechanical resonator. In addition, one of the cavity modes is irreversibly coupled to a single photon source. Our scheme is an opto-mechanical realization of the Jaynes–Cummings model where the qubit is a dual rail optical qubit while the bosonic degree of freedom is a matter degree of freedom realized as the bulk mechanical excitation. We show the possibility of engineering phonon number states of the mechanical oscillator in such a system by computing the conditional state of the mechanics after successive photon counting measurements