Photonic gauge potential in a system with a synthetic frequency dimension

We generalize the concept of photonic gauge potential in real space, by introducing an additional "synthetic" frequency dimension in addition to the real space dimensions. As an illustration we consider a one-dimensional array of ring resonators, each supporting a set of resonant modes having a frequency comb with spacing $\Omega$, undergoing a refractive index modulation at the modulation frequency equal to $\Omega$. We show that the modulation phase provides a gauge potential in the synthetic two-dimensional space with the dimensions being the frequency and the spatial axes. Such gauge potential can create a topologically protected one-way edge state in the synthetic space that is useful for high-efficiency generation of higher-order side bands.