Non-Hermitian jumps in disordered lattices

In the context of non-Hermitian Anderson localization, we study the physics of wave propagation in one-dimensional waveguide lattices, with randomly distributed gain or loss. Despite the Anderson localization of all eigenstates, the system exhibits counterintuitive propagation by quantized jumps between states located around distant sites. Such a novel effect was recently experimentally demonstrated in optical fiber loop networks. We provide a systematic way of understanding the underlying physical mechanism of such an effect. The role of eigenstates' nonorthogonality and biorthogonal projection is of central importance to our work, and is systematically examined in the symmetric non-Hermitian model, as well as the nonsymmetric Hatano-Nelson Hamiltonian. Our methodology can be applied to any non-Hermitian disordered system that contains complex elements with loss and/or gain, and thus exploits the meaning of wave transport in complex open systems.