Visualization and comparison of classical structures and quantum states of 4D maps

TL;DR

This paper introduces 3D phase-space slices for visualizing and comparing classical and quantum structures in 4D maps, demonstrating their effectiveness over traditional methods and confirming semi-classical eigenfunction hypotheses.

Contribution

It proposes a novel 3D phase-space slice method for visualizing 4D maps and compares classical and quantum structures, validating semi-classical eigenfunction hypotheses.

Findings

3D phase-space slices effectively visualize classical structures.

Husimi functions of eigenstates match classical phase space features.

Method outperforms standard visualization techniques.

Abstract

For generic 4D symplectic maps we propose the use of 3D phase-space slices which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods like 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the first comparison of Husimi functions of eigenstates of 4D maps with classical phase space structures. This confirms the semi-classical eigenfunction hypothesis for 4D maps.