Strong and tunable couplings in flux-mediated optomechanics

We investigate a superconducting interference device (SQUID) with two asymmetric Josephson junctions coupled to a mechanical resonator embedded in the loop of the SQUID. We quantize this system in the case when the frequency of the mechanical resonator is much lower than the cavity frequency of the SQUID and in the case when they are comparable. In the first case, the radiation pressure and the cross-Kerr type interactions arise and are modified by the asymmetry. The cross-Kerr type coupling is the leading term at the extremum points where the radiation pressure is zero. In the second case, the main interaction is the single-photon beam splitter, which exists only at a finite asymmetry. Another interaction in this regime is of cross-Kerr type, which exists at all asymmetries, but is generally much weaker than the beam splitter interaction. Increasing magnetic field can substantially enhance the optomechanical couplings strength with a potential for the radiation pressure coupling to reach the single-photon strong coupling regime, even the ultrastrong coupling regime, in which the single-photon coupling rate exceeds the mechanical frequency.