Topological Pumping in Doubly Modulated Mechanical Systems

Topological pumping allows higher-dimensional topological phenomena to be explored in low-dimensional systems using synthetic dimension with time modulation. Extensive studies have been devoted to the one-dimensional pumping lattice with singly modulated parameters, which was shown to exhibit the mechanical-wave analog of the quantum Hall effect. Here, we study the topological pumping in the lattice chain with doubly modulated parameters, and report the theoretical finding of two additional topological phases with the large-gap Chern number, which has not been well disclosed previously in singly modulated mechanical systems. The edge state protected by the large-gap Chern number exhibits the faster edge-To-edge energy transportation under the slow temporal variation of modulating parameters. The adiabatic approximation of doubly modulated pumping is also analyzed, and it would not be violated in the process of the fast-pumping evolution of edge states. The finding is expected to be used for fast and robust energy transportation of mechanical waves.