Floquet engineering of topological localization transitions in non-Hermitian quasicrystals

Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force is found to dress the hopping amplitudes between lattice sites, yielding alternate transitions between localized and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to three representative Aubry-Andr\'e-Harper non-Hermitian quasicrystals, and obtain the conditions of "photon-assisted" localization transitions and the expressions of Lyapunov exponents analytically. We further introduce topological winding numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline phases with different localization nature. Our discovery thus extend the study of quasicrystals to non-Hermitian Floquet systems, and provide an effective way of modulating the topological and transport properties of these unique phases.