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We investigate superconducting interference device (SQUID) with twoasymmetric Josephson junctions coupled to a mechanical resonator embedded inthe loop of the SQUID. We quantize this system in the case when the frequencyof the mechanical resonator is much lower than the cavity frequency of theSQUID and in the case when they are comparable. In the first case, theradiation pressure and cross-Kerr type interactions arise and are modified byasymmetry. Cross-Kerr type coupling is the leading term at the extremum pointswhere radiation pressure is zero. In the second case, the main interaction issingle-photon beam splitter, which exists only at finite asymmetry. Anotherinteraction in this regime is of cross-Kerr type, which exists at allasymmetries, but generally much weaker than the beam splitter interaction.Increasing magnetic field can substantially enhance optomechanical couplingsstrength with the potential for the radiation pressure coupling to reach thesingle-photon strong coupling regime, even the ultrastrong coupling regime, inwhich the single-photon coupling rate exceeds the mechanical frequency.

Strong and tunable couplings in flux-mediated optomechanics