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Physical review. E

Structure of superscars in a pseudointegrable right-triangular billiard

Zhen-Qi Chen, Rui-Hua Ni, Yalei Song, Hong-Ya Xu, Liang Huang

Published Apr 28, 2025• 1 citation

Abstract

Superscars are quantum eigenstates that are localized in an extended region of a family of parallel periodic orbits. They are prevalent in pseudointegrable systems. Here we consider a pseudo-integrable right-triangular quantum billiard with an acute angle π/5 as an example. In the configuration space, we identify and explain a fringe structure of superscar eigenstates originating from a correlation between the two quantum numbers imposed by the sign change rule of the wave function during unfolding. In the wave-vector space, we unveil unique characteristics of the peaks for superscar states, and in return, exploit these features to determine the family of periodic orbits and the corresponding quantum numbers without ambiguity. We also find that the eigenstates can be superpositions not only of superscar modes corresponding to different families of periodic orbits, but also of superscar modes corresponding to the same family but with different sets of quantum numbers, which poses a challenge for conventional approaches in identifying the superscar modes but can be addressed properly by our analysis. These results are also applicable to superscar states in other pseudointegrable systems.

DOI: 10.1103/physreve.111.044137

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Superscars are quantum eigenstates that are localized in an extended region of a family of parallel periodic orbits. They are prevalent in pseudointegrable systems. Here we consider a pseudo-integrable right-triangular quantum billiard with an acute angle π/5 as an example. In the configuration space, we identify and explain a fringe structure of superscar eigenstates originating from a correlation between the two quantum numbers imposed by the sign change rule of the wave function during unfolding. In the wave-vector space, we unveil unique characteristics of the peaks for superscar states, and in return, exploit these features to determine the family of periodic orbits and the corresponding quantum numbers without ambiguity. We also find that the eigenstates can be superpositions not only of superscar modes corresponding to different families of periodic orbits, but also of superscar modes corresponding to the same family but with different sets of quantum numbers, which poses a challenge for conventional approaches in identifying the superscar modes but can be addressed properly by our analysis. These results are also applicable to superscar states in other pseudointegrable systems.

The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

The recently measured superscars in a pseudointegrable barrier billiard [Phys. Rev. Lett. 97, 254102 (2006)] serve as a paradigm for strength function phenomena known in a variety of open mesoscopic quantum systems ranging from atoms to nuclei. The superscarring state is thereby of a simple structure whose strength is spread over the many other nonscarring chaotic states in the billiard, i.e. it acts as a so called doorway state. In this talk the observed spreading of four different superscars is briefly discussed in terms of a model based on Random Matrix Theory and also by investigating the spatial correlations of the measured wave functions. It is shown that the results derived are fully consistent with each other. Furthermore, some properties of the nodal domains of the experimental wave functions are analyzed. It is found that the mean value of the rescaled nodal domains shows a deviation from what is expected from a percolation model. This deviation can, however, be qualitatively understood in terms of the doorway state picture.

The quantum spectra are derived from the wave-functions and the energy-functions of the isosceles-right triangular model. Although the eigenstates of the billiard system are not separable, the problem of functions with two variables is exactly solvable. The numerical results of the Fourier transform of quantum spectral functions are compared with the results from the classical orbits whose lengths match well with the positions of the spectra peaks. This result gives a new evident for the correspondence of quantum and classical mechanics.

The problem of the quantizations of the

L

-shaped billiards and the like ones, i.e. each angle of which is equal to

π/2

or

3π/2

, is considered using as a tool the Fourier series expansion method. The respective wave functions and the quantization conditions are written and discussed looking for and discussing about the superscars effects in such multi-rectangular billiards (MRB). It is found that a special set of POC modes effect the superscars phenomena in MRB in which the billiards are excited as a whole to the modes closest to the semiclassical ones existing in their approximated copies being MRB in which their parallel sides remain in rational relations between themselves.

Right triangular billiards are very simple systems that are completely integrable in classical mechanics for acute angle pairs (45 degrees,45 degrees ) and (30 degrees,60 degrees ). In quantum mechanics, the energy level spacing distribution of these billiards are neither Poisson-like nor Wigner-like. We use Bohm's formalism to calculate the trajectories, by numerical methods, for a particle inside these billiards. We use a linear combination of the first three energy states as the initial wave function. We show that a particle can have quasiperiodic or chaotic behavior, depending on its initial position in the billiards.

We study the correspondence between quantum spectra and classical orbits in the equilateral triangular billiards. The eigenstates of such systems are not separable functions of two variables even though the problem is exactly solvable. We calculate the Fourier transform of a quantum spectral function and find that the positions of the peaks match well with the lengths of the classical orbits. This is another example showing that the quantum spectral function provides a bridge between quantum and classical mechanics.