Synthetic Non-Abelian Electric Fields and Spin-Orbit Coupling in Photonic Synthetic Dimensions.

We theoretically propose a scheme to synthesize photonic non-Abelian electric field and spin-orbit coupling (SOC) in the synthetic frequency dimension based on a polarization-multiplexed time-modulated ring resonator. Inside the ring resonator, the cascade of polarization-dependent phase modulation, polarization rotation, and phase retardation enables a photonic realization of minimal SOC with Peierls substitution composed of equal Rashba and Dresselhaus coupling. The synthetic bands and associated spin textures can be effectively probed by the continuous-wave polarization- and time-resolved band structure measurements. Meanwhile, the pulse dynamics in the frequency dimension can be manipulated by both synthetic Abelian and non-Abelian electric fields, leading to Bloch oscillation, Zitterbewegung, and their interplay dynamics. Our proposed setup can be realized with free-space optics, fibers, and integrated photonics, and provides pseudospin control knobs for large-scale synthetic lattices based on frequency combs.