  We develop a theory for the non-equilibrium screening of a charged impurity
in a two-dimensional electron system under a strong time-periodic drive. Our
analysis of the time-averaged polarization function and dielectric function
reveals that Floquet driving modifies the screened impurity potential in two
main regimes. In the weak drive regime, the time-averaged screened potential
exhibits unconventional Friedel oscillations with multiple spatial periods
contributed by a principal period modulated by higher-order periods, which are
due to the emergence of additional Kohn anomalies in the polarization function.
In the strong drive regime, the time-averaged impurity potential becomes almost
unscreened and does not exhibit Friedel oscillations. This tunable Friedel
oscillations is a result of the dynamic gating effect of the time-dependent
driving field on the two-dimensional electron system.
