Short periodic orbits theory for partially open quantum maps

TL;DR

This paper extends the semiclassical short periodic orbit theory to partially open quantum maps, analyzing how classical trajectories influence quantum resonances and the transition from open to closed system behavior.

Contribution

It introduces a modified theory for partially open quantum maps and demonstrates its effectiveness on the tribaker map, revealing the role of periodic orbits in resonance support.

Findings

Long-lived resonances are supported by periodic orbits of the classical repeller.

Including trajectories outside the repeller extends the theory's validity across different reflectivity R values.

The transition from open to closed behavior is explained through short periodic orbits.

Abstract

We extend the semiclassical theory of short periodic orbits [Phys. Rev. E {\bf 80}, 035202(R) (2009)] to partially open quantum maps. They correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity $R$. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of these kind of systems. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belong to the classical repeller of the open map ($R=0$) are able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of $R$ values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behaviour, providing an explanation in terms of short periodic orbits.