We consider transport through a non-Hermitian conductor connected to a pair of Hermitian leads and analyze the underlying non-Hermitian scattering problem. In a typical non-Hermitian system, such as a Hatano--Nelson-type asymmetric hopping model, the continuity of probability and probability current is broken at a local level. As a result, the notion of transmission and reflection probabilities becomes ill-defined. Instead of these probabilities, we introduce the injection rate $R_{ \rm I}=1-|{ \cal R}|^2$ and the transmission rate $R_{ \rm T}=|{ \cal T}|^2$ as relevant physical quantities, where ${ \cal T}$ and ${ \cal R}$ are the transmission and reflection amplitudes, respectively. In a generic non-Hermitian case, $R_{ \rm I}$ and $R_{ \rm T}$ have independent information. We provide a modified continuity equation in terms of incoming and outgoing currents, from which we derive a global probability conservation law that relates $R_{ \rm I}$ and $R_{ \rm T}$. We have tested the usefulness of our probability conservation law in the interpretation of numerical results for non-Hermitian localization and delocalization phenomena.
Categories: cond-mat.mes-hall quant-ph