Topological classification of quasiperiodically driven quantum systems

Few-level quantum systems driven by n(f) incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in n(f) spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. We organize the fundamental solutions of the frequency lattice model for n(f) = 2 into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.