Semiclassical theory of transport

TL;DR

This paper develops a semiclassical framework for quantum transport in chaotic systems, connecting trajectory sums, diagrammatic methods, and matrix integrals to extend understanding beyond traditional random matrix theory.

Contribution

It introduces a diagrammatic and matrix integral approach to semiclassical quantum transport, incorporating effects like tunnel barriers and superconductors.

Findings

Semiclassical approach aligns with random matrix theory predictions.

Diagrammatic formulations facilitate perturbative calculations.

Matrix integrals provide a versatile algebraic solution method.

Abstract

We discuss the semiclassical approximation to transport problems in quantum chaotic systems. The figures of merit are moments of the transmission matrix and of the time delay matrix. After reviewing a few results obtained by treating these matrices are random matrices, we show how expressions for their elements in terms of sums over trajectories lead to diagrammatic formulations that correspond to perturbative calculations. This semiclassical approach agrees with random matrix theory when it should, and allows further elements to be incorporated, like tunnel barriers, superconductors, absorption effects. We also discuss how this approach can be encoded in matrix integrals, resulting in a powerful and versatile theory that is amenable to algebraic solutions.