Topological characterization of discontinuous control pulses

We apply a topological analysis to the dynamics of a qubit when driven by discontinuous control pulses that do not end where they start when repeated. This builds on results where multiple drives of incommensurate frequencies create synthetic dimensions which cause the driving dynamics to obtain topological properties [P. J. D. Crowley, et al. Phys. Rev. B 99 064306 (2019)]. We find that truncating a two-frequency quasiperiodic drive and repeating it preserves much of its topological nature, with important changes. The pulse discontinuities shift the quasienergy eigenstate diabatically. The consequence of this depends on the topological class. In the topologically trivial phase, when Chern number C=0, the dynamics are still localized within each control pulse but can become delocalized over multiple time steps. As the system transitions to its non-trivial topological state with C≠0 the dynamics become sensitive to small perturbations of the initial drive even after starting in an eigenstate, suggestive of chaotic dynamics. We then discuss generalizations of these findings to generic control pulses.