  Synthetic optical structures have enabled studies of various physical
processes and discoveries of new phenomena in a controllable manner. Past
realizations, using for example coupled optical waveguides, were however
limited in size and dimensionality. Here we theoretically propose and
experimentally demonstrate optical frequency crystals of arbitrary dimensions,
formed by hundreds of coupled spectral modes within an on-chip electro-optic
frequency comb. We show a direct link between the measured optical transmission
spectrum and the density-of-state of frequency crystals, validated by
measurements of one-, two-, three- and four-dimensional crystals with no
restrictions to further expand the dimensionality. We measure signatures of
quantum random walks and Bloch oscillations in the frequency domain of light in
one- and two-dimensional synthetic lattices. Our work employs a quantum
description of phase modulation11 to describe resonant electro-optic frequency
combs, and shows that they enable the study of exotic crystal properties,
topologies and quantum dynamics, while providing a versatile platform to
manipulate quantum information in the optical frequency domain.
