Far-field phonon coupling in valley metamaterial circuits.

On-chip whispering-gallery-mode cavities enable versatile bosonic wave manipulation but typically rely on near-field evanescent coupling. Here we experimentally demonstrate broadband far-field phonon coupling in a valley metamaterial cavity integrated with a Dirac-cone waveguide-termed a "Dirac strip". The far-field coupling is confirmed by transmission spectroscopy and spatiotemporal field mapping over distances up to approximately five wavelengths, enabling multiplexed, distance-robust coupling pathways that overcome near-field limitations. By combining near-field and far-field cavities on the same substrate, we achieve amplification and control of non-Hermitian dynamics through loss and distance modulation of sympathetic resonances, directly resolved via piezo-laser interferometry. This work establishes a scalable phononic platform for far-field coupling and paves the way for parallel topological wave processors.