The semiclassical continuity equation for open chaotic systems

TL;DR

This paper derives a semiclassical expansion for the probability current density in open chaotic quantum systems, demonstrating that the continuity equation relating current density, survival probability, and scattering matrix correlations holds to all orders.

Contribution

It develops a recursive semiclassical framework connecting trajectory structures for survival probability and current density, ensuring the continuity equation's validity to all orders.

Findings

Semiclassical expansion for probability current density derived

Continuity equation verified to all orders in semiclassical approximation

Trajectory-based relations connect survival probability and scattering matrix correlations

Abstract

We consider the continuity equation for open chaotic quantum systems in the semiclassical limit. First we explicitly calculate a semiclassical expansion for the probability current density using an expression based on classical trajectories. The current density is related to the survival probability via the continuity equation, and we show that this relation is satisfied within the semiclassical approximation to all orders. For this we develop recursion relation arguments which connect the trajectory structures involved for the survival probability, which travel from one point in the bulk to another, to those structures involved for the current density, which travel from the bulk to the lead. The current density can also be linked, via another continuity equation, to a correlation function of the scattering matrix whose semiclassical approximation is expressed in terms of trajectories that start and end in the lead. We also show that this continuity equation holds to all orders.