Topological energy pumping in a quasi-periodically driven four-level system

We investigate a quasi-periodically driven four-level system that serves as a temporal analog of topological phenomena found in four-band models with intertwined spin and orbital degrees of freedom. Under a two-tone drive in the strong-driving regime, the system realizes a two-dimensional synthetic Floquet lattice, thus facilitating the realization of topological energy pumping. For a temporal quantum spin Hall insulator, we find that the rates of emission and absorption of energy between the two drives are not exactly opposite for a given band. However, when contributions from two chiral symmetric partner bands are added, they become exactly opposite. This quantized rate of energy exchange is a direct consequence of propagating edge modes in the real-space model, which we further characterize by computing the spin-Chern number. Interestingly, our analysis yields zero rate of exchange of energy between the drives for a temporal higher-order topological insulator, suggesting the presence of localized corner modes that we characterize by the mid-gap Wannier spectra. {Our findings uncover the role of chiral, particle-hole and time reversal symmetries on the energy dynamics in temporal quantum spin Hall and higher-order topological insulators.} Finally, we demonstrate that the perfect (imperfect) nature of the fidelity during the time-evolution of the system serves as a characteristic signature of a topological (trivial) phase.