In this work, we investigate the delocalization-localization transition of Floquet eigenstates in a driven chain with an incommensurate Aubry-André (AA) on-site potential and a small non-reciprocal hopping term that is driven periodically in time. The driving protocol is chosen such that the Floquet Hamiltonian corresponds to a localized phase in the high-frequency limit and a delocalized phase in the low-frequency limit. By numerically calculating the inverse participation ratio and the fractal dimension $D_q$, we identify a clear delocalization-localization transition of the Floquet eigenstates at a critical frequency $\omega_{c}\approx0.318\pi$. This transition aligns with the real-to-complex spectrum transition of the Floquet Hamiltonian. For the driven frequency $\omega>\omega_{\mathrm{c}}$, the system resides in a localized phase, and we observe the emergence of CAT states—linear superposition of localized single particle states—in the Floquet spectrum. These states exhibit weight distributions concentrated around a few nearby sites of the chain, forming two peaks of unequal weight due to the non-reciprocal effect, distinguishing them from the Hermitic case. In contrast, for $\omega<\omega_{\mathrm{c}}$, we identify the presence of a mobility edge over a range of driving frequencies, separating localized states (above the edge) from multifractal and extended states (below the edge). Notably, multifractal states are observed in the Floquet eigenspectrum across a broad frequency range. Importantly, we highlight that the non-driven, non-reciprocal AA model does not support multifractal states nor a mobility edge in its spectrum. Thus, our findings reveal unique dynamical signatures that do not exist in the non-driven non-Hermitian scenario, providing a fresh perspective on the localization properties of periodically driven systems. Finally, we provide a possible circuit experiment scheme for the periodically driven non-reciprocal AA model. In the following work, we will extend our research to clean systems, such as Stark models, to explore the influence of periodic driving on their localization properties.
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