We propose a scheme involving a Cooper pair transistor (CPT) embedded in asuperconducting microwave cavity, where the CPT serves as a charge tunablequantum inductor to facilitate ultra-strong coupling between photons in thecavity and a nano- to meso-scale mechanical resonator. The mechanical resonatoris capacitively coupled to the CPT, such that mechanical displacements of theresonator cause a shift in the CPT inductance and hence the cavity's resonantfrequency. The amplification provided by the CPT is sufficient for the zeropoint motion of the mechanical resonator alone to cause a significant change inthe cavity resonance. Conversely, a single photon in the cavity causes a shiftin the mechanical resonator position on the order of its zero point motion. Asa result, the cavity-Cooper pair transistor (cCPT) coupled to a mechanicalresonator will be able to access a regime in which single photons can affectsingle phonons and vice versa. Realizing this ultra-strong coupling regime willfacilitate the creation of non-classical states of the mechanical resonator, aswell as the means to accurately characterize such states by measuring thecavity photon field.
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