Analytic expressions for the steady-state current with finite extended reservoirs
Michael P. Zwolak
Open system simulations of quantum transport enable the computational study of true steady states, Floquet states, and the role of temperature, time-dynamics, and fluctuations, among other physical processes. They are rapidly gaining traction, especially techniques that revolve around ''extended reservoirs'' -- a collection of a finite number of degrees of freedom with relaxation that maintain a bias or temperature gradient -- and have appeared under various guises (e.g., the extended reservoir, auxiliary master equation, and driven Liouville--von Neumann approaches). Yet, there are still a number of open questions regarding the behavior and convergence of these techniques. Here, we derive general analytical solutions, and associated asymptotic analyses, for the steady-state current driven by finite reservoirs with proportional coupling to the system/junction. In doing so, we present a simplified and unified derivation of the non- interacting and many-body steady-state currents through arbitrary junctions, including outside of proportional coupling. We conjecture that the analytic solution for proportional coupling is the most general of its form (i.e., relaxing proportional coupling will remove the ability to find compact, general analytical expressions for finite reservoirs). These results should be of broad utility in diagnosing the behavior and implementation of extended reservoir approaches, including the convergence to the Landauer limit (for non-interacting systems) and the Meir-Wingreen formula (for many-body systems).