Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.