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 systems 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.
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