Energy absorption in time-dependent unitary random matrix ensembles: dynamic vs. Anderson localization

We consider energy absorption in an externally driven complex system of noninteracting fermions with the chaotic underlying dynamics described by the unitary random matrices. In the absence of quantum interference the energy absorption rate $W(t)$ can be calculated with the help of the linear-response Kubo formula. We calculate the leading two-loop interference correction to the semiclassical absorption rate for an arbitrary time dependence of the external perturbation. Based on the results for periodic perturbations, we make a conjecture that the dynamics of the periodically-driven random matrices can be mapped onto the one-dimensional Anderson model. We predict that in the regime of strong dynamic localization $W(t)\propto \ln(t)/t^2$ rather than decays exponentially.