  In recent years there has been significant interest in the concepts of
synthetic dimensions, where one couples the internal degrees of freedom of a
particle to form higher-dimensional lattices in lower-dimensional physical
structures. For these systems the concept of band structure along the synthetic
dimension plays a central role in their theoretical description. Here we
provide the first direct experimental measurement of the band structure along
the synthetic dimension. By dynamically modulating a ring resonator at
frequencies commensurate with its mode spacing, we realize a periodically
driven system with a synthetic frequency dimension lattice. The strength and
range of the couplings along the lattice can be dynamically reconfigured by
changing the amplitude and frequency of modulation. We show theoretically and
demonstrate experimentally that time-resolved transmission measurements of this
system result in a direct "read out" of its band structure. We also show how
long-range coupling, photonic gauge potentials and nonreciprocal bands can be
realized in the system by simply incorporating additional frequency drives,
enabling great flexibility in engineering the band structure.
