Quantum Lissajous Scars

TL;DR

This paper demonstrates that certain eigenstates of a perturbed anisotropic harmonic oscillator exhibit quantum scars aligned with classical Lissajous orbits, revealing a novel form of quantum-classical correspondence influenced by system anisotropy and perturbations.

Contribution

It introduces the concept of quantum Lissajous scars, showing their persistence under perturbation and linking their geometry to system anisotropy, with potential experimental observation methods.

Findings

Eigenstates show strong scars along Lissajous orbits.

Scarring depends on anisotropy and persists under small perturbations.

Proposes experimental schemes for observing these scars.

Abstract

A quantum scar - an enhancement of a quantum probability density in the vicinity of a classical periodic orbit - is a fundamental phenomenon connecting quantum and classical mechanics. Here we demonstrate that some of the eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic oscillator are strongly scarred by the Lissajous orbits of the unperturbed classical counterpart. In particular, we show that the occurrence and geometry of these quantum Lissajous scars are connected to the anisotropy of the harmonic confinement, but unlike the classical Lissajous orbits the scars survive under a small perturbation of the potential. This Lissajous scarring is caused by the combined effect of the quantum (near) degeneracies in the unperturbed system and the localized character of the perturbation. Furthermore, we discuss experimental schemes to observe this perturbation-induced scarring.