Quantum Transport Through Open Systems

Abstract

To study transport properties, one needs to investigate the system of inter- est when coupled to biased external baths. This requires solving a master equation for this open quantum system. Obtaining this solution is very chal- lenging, especially for large systems. This limits applications of the theories of open quantum systems, especially insofar as studies of transport in large quantum systems, of interest in condensed matter, is concerned. In this thesis, I propose three efficient methods to solve the Redfield equation — an example of such a master equation. The first is an open- system Kubo formula, valid in the limit of weak bias. The second is a solu- tion in terms of Green’s functions, based on a BBGKY (Bogoliubov–Born– Green–Kirkwood–Yvon)-like hierarchy. In the third, the Redfield equation is mapped to a generalized Fokker-Planck equation using the coherent-state representation. All three methods, but especially the latter two, have much better efficiency than direct methods such as numerical integration of the Redfield equation via the Runge-Kutta method. For a central system with a d-dimensional Hilbert space, the direct methods have complexity of d3, while that of the latter two methods is on the order of polynomials of log d. The first method, besides converting the task of solving the Redfield equation to solving the much easier Schrodinger’s equation, also provides an even more important conceptual lesson: the standard Kubo formula is not applicable to open systems. Besides these general methodologies, I also investigate transport proper- ties of spin systems using the framework of the Redfield equation and with direct methods. Normal energy and spin transport is found for integrable but interacting systems. This conflicts with the well-known conjecture linking anomalous conductivity to integrability, and it also contradicts the relation- ship, suggested by some, between gapped systems (Jz > Jxy) and normal spin conductivity. I propose a new conjecture, linking anomalous transport to the existence of a mapping of the problem to one for non-interacting particles.