Non-adiabatic topological energy pumps with three incommensurate frequencies

Recent developments have shown topological energy pumping in quantum systems driven with either one or two incommensurate frequencies near their adiabatic limit. In this talk, we generalize such multichromatic Floquet techniques to study the topological properties of one-dimensional systems driven by two non-adiabatic drives and two-level systems driven by three incommensurate drives. We find a well-quantized, topologically non-trivial plateau, given by the topology of the micromotion. As the ratios among the frequencies is increased, the photonic modes become delocalized before a topological transition is reached. At fixed ratio of the frequencies, changing the magnitude of the frequencies gives transitions from topologically trivial to non-trivial and back to trivial again in the large frequency limit. We will also discuss plausible experimental possibilities to observe these novel topological effects.