  We propose a superconducting circuit comprising a dc-SQUID with mechanically
compliant arm embedded in a coplanar microwave cavity that realizes an
optomechanical system with a degenerate or non-degenerate parametric
interaction generated via the dynamical Casimir effect. For experimentally
feasible parameters, this setup is capable of reaching the single-photon,
ultra-strong coupling regime, while simultaneously possessing a parametric
coupling strength approaching the renormalized cavity frequency. This opens up
the possibility of observing the interplay between these two fundamental
nonlinearities at the single-photon level.
