Following the recent observation of localised phonon polaritons in
user-defined silicon carbide nano-resonators, here we demonstrate coherent
coupling between those localised modes and propagating phonon polaritons bound
to the surface of the nano-resonator's substrate. In order to obtain
phase-matching, the nano-resonators have been fabricated to serve the double
function of hosting the localised modes, while also acting as grating for the
propagating ones. The coherent coupling between long lived, optically
accessible localised modes, and low-loss propagative ones, opens the way to the
design and realisation of phonon-polariton based quantum circuits