  The theoretical treatment of quasi-periodically driven quantum systems is
complicated by the inapplicability of the Floquet theorem, which requires
strict periodicity. In this work we consider a quantum system driven by a
bi-harmonic driving and examine its asymptotic long-time limit, the limit in
which features distinguishing systems with periodic and quasi-periodic driving
occur. Also, in the classical case this limit is known to exhibit universal
scaling, independent of the system details, with the system's reponse under
quasi-periodic driving being described in terms of nearby periodically driven
system results. We introduce a theoretical framework appropriate for the
treatment of the quasi-periodically driven quantum system in the long-time
limit, and derive an expression, based on Floquet states for a periodically
driven system approximating the different steps of the time evolution, for the
asymptotic scaling of relevant quantities for the system at hand. These
expressions are tested numerically, finding excellent agreement for the
finite-time average velocity in a prototypical quantum ratchet consisting of a
space-symmetric potential and a time-asymmetric oscillating force.
