Three-period evolution in a photonic Floquet extended Su-Schrieffer-Heeger waveguide array

Periodic driving can induce the emergence of topological $\ensuremath{\pi}$ modes, and their superposition with zero modes leads to two-period dynamics. Introducing long-range couplings enables the realization of larger topological winding numbers, which correspond to multiple pairs of degenerate edge states under open boundary conditions. In this work, we construct a Floquet extended Su-Schrieffer-Heeger (SSH) model by introducing a two-step periodic driving and next-nearest-neighbor coupling into the static SSH chain simultaneously. Remarkably, we identify anomalous edge states with quasienergies $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/3T$ and $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3T$. In order to reveal the dynamical features of these anomalous edge states, we elaborately adjust the optical parameters and ultimately achieve a successful mapping of the model onto a photonic waveguide array. Subsequently, through numerical simulation of the wave equation, we observe the unique behavior of three-period evolution. Our work may serve as a reference for realizing period-multiplied dynamics, and the anomalous edge states discussed here might also find applications in quantum computation.