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The investigation of the dynamical universality classes of quantum systems isan important, and rather less explored, aspect of non-equilibrium physics. Inthis work, considering the out-of-equilibrium dynamics of spinless fermions ina one-dimensional quasiperiodic model with and without a periodic driving, wereport the existence of the dynamical one-parameter based Family-Vicsek (FV)scaling of the "quantum surface-roughness" associated with the particle-numberfluctuations. In absence of periodic driving, the model is interestingly shownto host a subdiffusive critical phase separated by two subdiffusive criticallines and a triple point from other phases. An analysis of the fate of criticalphase in the presence of (inter-phase) driving indicates that the criticalphase is quite fragile and has a tendency to get absorbed into the delocalizedor localized regime depending on the driving parameters. Furthermore, periodicdriving can conspire to show quantum Kardar-Parisi-Zhang (KPZ)-likesuperdiffusive dynamical behavior, which seems to have no classicalcounterpart. We further construct an effective Floquet Hamiltonian, whichqualitatively captures this feature occurring in the driven model

Family-Vicsek dynamical scaling and Kardar-Parisi-Zhang-like  superdiffusive growth of surface roughness in a driven one-dimensional  quasiperiodic model