Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices

Photonic simulators using synthetic frequency dimensions have enabled flexible experimental analogues of condensed-matter systems, realizing phenomena that are impractical to observe in real-space systems. However, to date such photonic simulators have been limited in scale, yielding results that suffer from finite-size effects. Here, we present an analog simulator capable of simulating a variety of large two-dimensional (2D) and three-dimensional (3D) lattices, as well as lattices with non-planar connectivity. Our simulator takes advantage of the broad bandwidth achievable in photonics, allowing our experiment to realize programmable lattices with over 100,000 lattice sites. We showcase the scale of our simulator by demonstrating the extension of band-structure spectroscopy from 1D to 2D and 3D lattices, direct observation of time-reversal-symmetry breaking in a 2D triangular lattice in both momentum- and real-space, and site-resolved occupation measurements in a tree-like geometry that serves as a toy model in quantum gravity. We also propose and demonstrate a method to excite arbitrary multi-site states -- in contrast to the current standard approach of single-site excitation -- which we use to study the response of a 2D lattice to both conventional and exotic input states. Our work highlights the scalability and flexibility of optical synthetic frequency dimensions. We anticipate that future experiments building on our approach will be able to explore non-equilibrium phenomenain high-dimensional lattices, and leverage Kerr-frequency-comb technologies to simulate models with non-local higher-order interactions.