Topology in a synthetic dimension as a tool for non-reciprocal photonic transport

The silicon micro-ring resonator is an important ingredient for integrated photonics [1]. In it, light resonates around a ring leading to a discrete set of resonances that are almost equally spaced in frequency. Recently, we have proposed that these resonances can be coupled by modulation of the dielectric tensor of the cavity to form a synthetic dimension, i.e., light will `hop' between these resonance frequencies in a way similar to a particle hopping between neighboring resonators [2], see Fig. 1(a). Additionally, in a periodic chain of such micro-rings, photons can `hop' between different sites (analogous to electrons tunneling between atoms in a crystal). As the overlap between different frequency states on different resonators is small, we thus obtain an effective two-dimensional lattice for photons composed of a single real space plus a synthetic dimension, see Fig. 1(b). Last, applying a different phase to the dielectric tensor modulations on different sites can encode a synthetic gauge to the hopping on the effective 2D model, leading to quantum Hall effect physics in the combined space-frequency plane. These ideas can be generalized to make the realization of the 4D quantum Hall effect possible using 3D micro-ring arrays and a synthetic dimension.