Generalized Aubry-Andr\'e self-duality and Mobility edges in non-Hermitian quasi-periodic lattices

We demonstrate the existence of generalized Aubry-Andr\'e self-duality in a class of non-Hermitian quasi-periodic lattices with complex potentials. From the self-duality relations, the analytical expression of mobility edges is derived. Compared to Hermitian systems, mobility edges in non-Hermitian ones not only separate localized from extended states, but also indicate the coexistence of complex and real eigenenergies, making it possible a topological characterization of mobility edges. An experimental scheme, based on optical pulse propagation in synthetic photonic mesh lattices, is suggested to implement a non-Hermitian quasi-crystal displaying mobility edges.