Scalable and Programmable Emulation of Photonic Hyperbolic Lattices

Non-Euclidean geometry, which violates the traditional parallel postulate, encompasses a wide class of physical systems, including fullerenes, space-time structures, and complex networks. A notable example is the hyperbolic lattice, which exhibits unique band properties and topological phenomena compared with those of Euclidean lattices. Although hyperbolic lattices have been effectively realized on a Euclidean plane using projections onto the Poincaré disk, the poor scalability of this approach hinders large-size implementations essential for examining lattice physics. Here, we demonstrate a scalable and reconfigurable emulation of wave dynamics in a hyperbolic lattice by employing programmable photonics. We develop a one-dimensional coupled-resonator lattice with reconfigurable gauge fields and resonances, which emulates unitary wave evolutions inside a hyperbolic lattice. The programmability of our system allows for modeling time-varying hyperbolic lattices including dynamic defects. The superior scalability of our system compared to the realizations using the Poincaré disk and spatial-domain circuits paves the way to extending non-Euclidean photonics into large-scale and dynamical systems.