Hidden periodicities allow the prediction of locked particle motions on quasicrystalline surfaces

Motion of particles across quasicrystalline surfaces exhibits peculiar features due to the presence of long-range order without translational periodicity. Under time-periodic forcing, this motion can become locked in directions thatn deviate strongly from the mean driving direction. We show that for surface potentials with a quasicrystalline pattern of minima generated by a superposition of plane waves, particle trajectories are nonperiodic, yet their mean direction and speed are determined by hidden periodic potentials. The lattice vectors of these underlying potentials define characteristic velocities that dictate both directional and speed locking. The particle motion does not synchronize with the driving, and it is possible for the mean speed to remain nonlocked even in directionally locked states. These findings are demonstrated using a model directly amenable to experimental realization.