  Recent work [M. H. Kolodrubetz et al, PRL 120, 150601] has demonstrated that
periodically driven one-dimensional fermionic systems can support quantized
energy pumping resulting from an adiabatic modulation of a second parameter. In
this work, we explore this topological Floquet-Thouless energy pump in the
quasiperiodic driving regime where the parametric driving occurs at finite
frequency. We show that quantization of energy pumping persists for finite
ramping frequencies, as long as they are incommensurate with the driving
frequency, and the system remains localized by spatial disorder. Thus, the
topological Floquet-Thouless energy pump is stable beyond the adiabatic regime,
occupying a finite region of parameter space. Phase transitions away from these
topological phases are accompanied by delocalization in position space, photon
number (energy) space, or both. Using a dimensional reduction scheme, we
demonstrate that a related phase can be realized with a cavity-qubit system
driven by two incommensurate modes.
