  We propose and analyze a circuit that implements a nonlinear coupling between
two superconducting microwave resonators. The resonators are coupled through a
superconducting quantum interference device (SQUID) that terminates one of the
resonators. This produces a nonlinear interaction on the standard
optomechanical form, where the quadrature of one resonator couples to the
photon number of the other resonator. The circuit therefore allows for
all-electrical realizations of analogs to optomechanical systems, with coupling
that can be both strong and tunable. We estimate the coupling strengths that
should be attainable with the proposed device, and we find that the device is a
promising candidate for realizing the single-photon strong-coupling regime. As
a potential application, we discuss implementations of networks of
nonlinearly-coupled microwave resonators, which could be used in
microwave-photon based quantum simulation.
