Topological transport of a classical droplet in a lattice of time

Abstract: Thouless charge pumps are quantum mechanical devices whose operation relies on topology. They provide the means for transporting quantum matter in space lattices with a single quantum precision. Contrasting space crystals that spontaneously break a continuous spatial translation symmetry and form crystals in space,time crystals have emerged as novel states of matter that organize into time lattices and spontaneously break a discrete time translation symmetry. The utility of Thouless pumps that enable topologically protected quantised transport of electrons and neutral atoms in spatial superlattices leads to the question if corresponding devices exist for time crystals? Here we show that topological pumps can be realized for time solids by transporting droplets of a liquid forward and backward in time lattices and we measure the topological index that characterises such pumping processes. By exploiting a synthetic time dimension classical time crystals can circumvent the quantum tunneling that underpins Thouless charge pumps. Our results establish topological pumping through time instead of space and pave the way for applications of time crystals.