Semiclassics for chaotic systems with tunnel barriers

TL;DR

This paper develops a comprehensive semiclassical framework for chaotic systems with tunnel barriers, analyzing trajectory structures, probability currents, and scattering matrix correlations, consistent with random matrix theory and applicable to conductance calculations.

Contribution

It provides the first complete semiclassical treatment of open chaotic systems with tunnel barriers, including trajectory analysis, current continuity, and scattering matrix correlations.

Findings

Trajectory structures match random matrix theory predictions.

Continuity equation holds to all orders in semiclassical approximation.

Semiclassical expansion for conductance derived.

Abstract

The addition of tunnel barriers to open chaotic systems, as well as representing more general physical systems, leads to much richer semiclassical dynamics. In particular, we present here a complete semiclassical treatment for these systems, in the regime where Ehrenfest time effects are negligible and for times shorter than the Heisenberg time. To start we explore the trajectory structures which contribute to the survival probability, and find results that are also in agreement with random matrix theory. Then we progress to the treatment of the probability current density and are able to show, using recursion relation arguments, that the continuity equation connecting the current density to the survival probability is satisfied to all orders in the semiclassical approximation. Following on, we also consider a correlation function of the scattering matrix, for which we have to treat a new set of possible trajectory diagrams. By simplifying the contributions of these diagrams, we show that the results obtained here are consistent with known properties of the scattering matrix. The correlation function can be trivially connected to the ac and dc conductances, quantities of particular interest for which finally we present a semiclassical expansion.