Synthetic frequency dimension in a spatiotemporally driven phononic ring resonator

The concept of synthetic dimensions offers a unique approach to exploring higher-dimensional physics within lower-dimensional systems. Since its initial demonstration in atomic systems, synthetic dimensions have been implemented in various optical platforms, often by forming a lattice through the coupling of photonic states. In this work, we propose a similar method for realizing synthetic frequency dimensions on a phononic platform. Specifically, we design a parametrically driven phononic ring resonator composed of cantilever beams and conduct numerical investigations into the modal dynamics along this synthetic frequency dimension. Our approach employs detuned spatiotemporal stiffness modulation, which generates synthetic electric fields for the phononic modes. This modulation induces both harmonic and anharmonic Bloch oscillations along the synthetic frequency dimension. To elucidate these modal dynamics, we numerically construct and visualize a dynamic band structure within the Brillouin zone in the presence of synthetic electric fields.