Synthetic dimension provides ways to investigate different physical phenomena in a controlable manner. It allows the simulation of complex structure that are hard to achieve in real space or even systems that beyond the three-dimensional Euclidian space. Optics is an ideal platform for synthetic dimension. Building up synthetic dimension in frequency domain has been proposed and widely investigated theoretically over the past decade. In recent years, experimental investigations on frequency synthetic dimension has been explored in fiber system. The thin-film lithium niobate is a natrual candidate to realize frequency synthetic dimension due to the strong electro-optic effect, low loss, and the good scalability. As a result, thin-film lithium niobate can create complicated and large synthetic system in frequency domain.

High-dimensional frequency crystals [Yaowen Hu, et al. Optica 7, 1189 (2020)]: We theoretically propose and experimentally demonstrate high-dimensional frequency crystals using the electro-optic effect on thin-film lithium niobate. We create the crystals and measured the density of states of the crystals in one-, two-, three-, and four- dimensions. We further show coherent random walks in frequency crystals in one- and two- dimensions.