Direct Observation of a Localized Flat-Band State in a Mapped Moiré Hubbard Photonic Lattice

Abstract

When two identical periodic structures are stacked at a series of discrete rotation angles, the resulting moir\'e pattern brings an extreme flat band with transport enhancement. It provides insights into the fascinating physics of insulating states and unconventional superconductivity with delocalization-localization transitions and commensurable-incommensurable phases. However, the exploration of symmetry- and geometry-independent flat-band physics with moir\'e patterns is still rare, limited by the stringent requirement in high dimension. Here, we experimentally observe the localized flat-band state by mapping a moir\'e model into a one-dimensional photonic lattice using a femtosecond laser direct writing technique. By accurately controlling the external periodic field, we construct moir\'e photonic lattices with different moir\'e band structures. We successfully observe the photon-walker evolution from splitting with dispersion to nondispersive propagation with larger localization in the input ports as the result of a flat band induced by moir\'e patterns with larger period. Our approach to engineering moir\'e model, together with the integrated photonic implementation, establishes a powerful tool for exploring the effects accompanying the transition from commensurate to incommensurate phases.