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.
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