Landauer, Kubo, and microcanonical approaches to quantum transport and noise: A comparison and implications for cold-atom dynamics

We compare the Landauer, Kubo, and microcanonical [J. Phys. Cond. Matter {\bf 16}, 8025 (2004)] approaches to quantum transport for the average current, the entanglement entropy and the semiclassical full-counting statistics (FCS). Our focus is on the applicability of these approaches to isolated quantum systems such as ultra-cold atoms in engineered optical potentials. For two lattices connected by a junction, we find that the current and particle number fluctuations from the microcanonical approach compare well with the values predicted by the Landauer formalism and FCS assuming a binomial distribution. However, we demonstrate that well-defined reservoirs (i.e., particles in Fermi-Dirac distributions) are not present for a substantial duration of the quasi-steady state. Thus, the Landauer assumption of reservoirs and/or inelastic effects is not necessary for establishing a quasi-steady state. Maintaining such a state indefinitely requires an infinite system, and in this limit well-defined Fermi-Dirac distributions can occur. A Kubo approach -- in the spirit of the microcanonical picture -- bridges the gap between the two formalisms, giving explicit analytical expressions for the formation of the steady state. The microcanonical formalism is designed for closed, finite-size quantum systems and is thus more suitable for studying particle dynamics in ultra-cold atoms. Our results highlight both the connection and differences with more traditional approaches to calculating transport properties in condensed matter systems, and will help guide the way to their simulations in cold-atom systems.