  We propose a superconducting electrical circuit that simulates a quadratic
optomechanical system. A capacitor placed between two transmission-line (TL)
resonators acts like a semi-transparent membrane, and a superconducting quantum
interference device (SQUID) that terminates a TL resonator behaves like a
movable mirror. Combining these circuit elements, it is possible to simulate a
quadratic optomechanical coupling whose coupling strength is determined by the
coupling capacitance and the tunable bias flux through the SQUIDs. Estimates
using realistic parameters suggest that an improvement in the coupling strength
could be realized, to five orders of magnitude from what has been observed in
membrane-in-the-middle cavity optomechanical systems. This leads to the
possibility of achieving the strong-coupling regime of quadratic optomechanics.
