  We investigate 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 cross-Kerr type interactions arise and are modified by
asymmetry. Cross-Kerr type coupling is the leading term at the extremum points
where radiation pressure is zero. In the second case, the main interaction is
single-photon beam splitter, which exists only at finite asymmetry. Another
interaction in this regime is of cross-Kerr type, which exists at all
asymmetries, but generally much weaker than the beam splitter interaction.
Increasing magnetic field can substantially enhance optomechanical couplings
strength with the 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.
