Hofstadter butterfly and topological edge states in a quasiperiodic photonic crystal cavity array

Quasiperiodic structures with additional synthetic degrees of freedom have recently been recognized as a promising way for investigating high-dimensional topological phases with lower physical dimensions. Here, we investigated the well-known Harper-Aubry-André model on an integrated photonic platform by proposing a new design of a quasiperiodic photonic crystal (PhC) cavity array. This array is composed of closely coupled H1 PhC cavities with their cavity lengths being periodically modulated in the real space. The frequency spectrum of the structure shows the main features of the Hofstadter butterfly, which is one of the most important phenomena in the Harper-Aubry-André model. By varying the modulation phase, this structure exhibits nontrivial topology, which supports strongly localized topological edge states. These results have shown that quasiperiodic PhC cavity arrays can serve as the testbed for studying topological phases and new topological phenomena on an integrated platform.