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Distinctive response of many-body localized systems to strong electric field

We study systems which are close to or within the many-body localized (MBL) regime and are driven by strong electric field. In the ergodic regime, the disorder extends applicability of the equilibrium linear--response theory to stronger drivings, whereas the response of the MBL systems is very distinctive, revealing currents with damped oscillations. The oscillation frequency is independent of driving and the damping is not due to heating but rather due to dephasing. The details of damping depend on the system's history reflecting nonergodicity of the MBL phase, while the frequency of the oscillations remains a robust hallmark of localization. We show that the distinctive characteristic of the driven MBL phase is also a logarithmic increase of the energy and the polarization with time.