Thouless Pumping of Wave Action in Photonic Time Crystals

A topological pump transports a conserved quantity. Yet temporal modulation exchanges energy with the field, so the transported quantity must be identified from Maxwell’s equations. Here we connect band topology to the temporal displacement of conserved Maxwell wave action in a photonic time crystal. The temporal Wannier function of an isolated positive-action band carries unit action with zero net energy flux, and its temporal center is a Berry polarization. Over one adiabatic cycle, that center shifts by the band Chern number: one cell for a rigid translation and two for a two-tone cycle within a single band.