  Cavity electro-(opto-)mechanics allows us to access not only single isolated
but also multiple mechanical modes in a massive object. Here we develop a
multi-mode electromechanical system in which a several membrane vibrational
modes are coupled to a three-dimensional loop-gap superconducting microwave
cavity. The tight confinement of the electric field across a
mechanically-compliant narrow-gap capacitor brings the system into the quantum
strong coupling regime under a red-sideband pump field. We demonstrate strong
coupling between two mechanical modes, which is induced by two-tone parametric
drives and mediated by a virtual photon in the cavity. The tunable
inter-mechanical-mode coupling can be used to generate entanglement between the
mechanical modes.
