Density Matrix Methods for Quantum Transport

The primary goal of a theory of open systems is to calculate the time evolution of the reduced density operator \(\hat {\varrho } = \mathrm {Tr}_{\mathrm {B}}\left \{\hat {\varrho }_{\mathrm {tot}}\right \}\) , where the trace is taken over the degrees of freedom of the bath (or reservoir), i.e. the large environment which influences the system but whose microscopic dynamics we shall ignore. Open systems enter various realms of physics, chemistry and biology. As such, various methods have been proposed. Here we shall follow the projector operator technique, developed by Nakajima and Zwanzig, and adapt it to fermionic environments. An exact generalized master equation for the reduced operator of an interacting nanojunction is derived. Further, the relevant equations for the current and its cumulants are discussed.