Many Body Localized Systems Subjected to Two-tone Drives

Closed systems subjected to a periodic drive, also known as Floquet systems, offer an exciting way to manipulate ordinary materials and yield exotic physical phenomena. Furthermore, these systems provide a promising direction in the search of novel phases, which are absent in equilibrium systems. Unfortunately, the tendency of driven, isolated many-body quantum systems to heat up towards a high entropy state, poses a fundamental obstacle to Floquet engineering. Interestingly, disorder and localization offer a possibility overcome this obstacle. In particular, when a system which is deep enough in the many-body localized (MBL) phase is subjected to a large frequency drive, the system may retain memory of its initial conditions. Can the localized phase survive when subjected to two fast drives which synchronize only after many cycles, thus generating a small effective frequency? We study systems driven by two-tone drives, and investigate the critical driving amplitudes as a function of the commensurability of the two frequencies. We find that the critical driving amplitude saturates at small effective frequencies. Surprisingly, this result implies that the MBL phase might be able to survive even in systems subjected to a quasi-periodic drive, extending the notion of stable non-equilibrium phases even further.