Two-Dimensional Topological Edge States in Periodic Space-Time Interfaces

Topological edge states in systems of two (or more) dimensions offer scattering-free transport, exhibiting robustness to inhomogeneities and disorder. In a different domain, time-modulated systems, such as photonic time crystals, offer nonresonant amplification drawing energy from the modulation. Combining these concepts, we explore topological systems that vary periodically in both time and space, manifesting the best of both worlds. We present topological phases and topological edge states in photonic space-time crystals-materials in which the refractive index varies periodically in both space and time, displaying band gaps in both frequency and momentum. The topological nature of this system leads to topological invariants that govern the phase between refracted and reflected waves generated from both the spatial and the temporal interfaces. The 2D (1D space + 1D time) nature of this system leads to propagating edge states, and a unique edge state that grows exponentially in amplitude while following the space-time edge.