Quantum Lissajous Scars

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.

Figure 1

Alpha scar visible in the probability density of the eigenstate $n=3453$ in an elliptical harmonic potential (1, 2) perturbed by Gaussian-like bumps. The state is strongly scarred by the alpha-shape Lissajous orbit of the corresponding unperturbed potential represented as a solid red line. Blue markers denote the locations of the bumps. It is noteworthy that multiple bumps are located on the scar path.

Figure 2

Examples of Lissajous scars in a two-dimensional anisotropic harmonic oscillator with commensurable frequencies perturbed by potential bumps. The geometries of the scars depend on the confinement potential ($p,q$), which also defines the shape of the POs in the unperturbed system. At a fixed ($p,q$), the scars can be divided into two subgroups: strings (upper row) and loops (lower row).

Figure 3

(a) Density of states of the unperturbed two-dimensional harmonic oscillator as a function of anisotropy parameter ${\omega }_x/{\omega }_y$. The dashed vertical lines indicate the commensurable ($p,q$) that correspond to a significant abundance of scarred eigenstates in the perturbed case (see Fig. 2). Two distinct limits are also seen in (a): namely, the unbounded case (${\omega }_x/{\omega }_y\to 0$) and the isotropic oscillator (${\omega }_x/{\omega }_y=1$). (b) Density of states of the corresponding perturbed system as a function frequency ratio in the neighborhood of the commensurable frequency (1, 2) demonstrating that the bumps are sufficiently weak enough not to fully destroy the (near) degeneracy of the unperturbed system. (c) Examples of Lissajous scars in the vicinity of the commensurable frequency (1,2) labeled with the value $\delta$ describing the deviation from the ideal frequency ratio ${\omega }_x/{\omega }_y=0.5$. The scarring level of the quantum state is estimated by the $\alpha$ value. Note that the scars exist, although the corresponding unperturbed classical PO does not. (d) Normalized average of $\alpha$ value as a function of the deviation $\delta$. The scarring weakens as the deviation $\delta$ increases according to the normalized average, as well as the $\alpha$ value of the individual example scars in (c).