Two-dimensional Thouless pumping of light in photonic moir\'e lattices

Continuous and quantized transports are profoundly different. The latter is determined by the global rather than local properties of a system, it exhibits unique topological features, and its ubiquitous nature causes its occurrence in many areas of science. Here we report the first observation of fully-two-dimensional Thouless pumping of light by bulk modes in a purpose-designed tilted moir\'e lattices imprinted in a photorefractive crystal. Pumping in such unconfined system occurs due to the longitudinal adiabatic and periodic modulation of the refractive index. The topological nature of this phenomenon manifests itself in the magnitude and direction of shift of the beam center-of-mass averaged over one pumping cycle. \rev{Our experimental results are supported by systematic numerical simulations in the frames of the continuous Schr\"{o}dinger equation governing propagation of probe light beam in optically-induced photorefractive moir\'e lattice.} Our system affords a powerful platform for the exploration of topological pumping in tunable commensurate and incommensurate geometries.