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Entropy

Scarring in Rough Rectangular Billiards

Felix M. Izrailev, Germán A. Luna-Acosta, Jose Antonio Mendez-Bermudez

Published Jan 18, 2023• 0 citations
Publisher

Abstract

We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential.

DOI
10.3390/e25020189

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We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential.
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even- and odd-parity pair whose linear combination gives the localized state.
Quantum scars are tracks of enhanced probability densities along unstable classical periodic orbits in quantum chaotic systems. They are a visual representation of classical-quantum correspondence as well as an example of quantum suppression of chaos. A new type of quantum scarring was recently discovered in radially symmetric two-dimensional potential wells perturbed by localized impurities. These perturbation-induced scars show potential to be utilized in controlled applications in classically chaotic quantum systems due to the ability to efficiently propagate wave packets along the scars. In this thesis, the focus is on special linearly shaped perturbation-induced quantum scars which correspond to the classical motion of a ball bouncing between two walls. These bouncing ball quantum scars have potential to be used as efficient channels in quantum transport, but their behavior is still mostly unexplored. This thesis studies how bouncing ball scars could be controlled by changing the perturbation parameters and how stable these scars are against external noise. Both of these features are relevant when utilization in experimental applications is considered. Bouncing ball scars are studied in a radially symmetric two-dimensional (2D) potential well perturbed by a singular Gaussian-like potential bump or a dip, which are placed into the same location. For the stability calculations, randomly placed smaller Gaussian bumps are also placed into the potential with a constant uniform mean density in addition to the main perturbation. Imaginary time propagation algorithm is then applied to solve the 2D Schrödinger equation in our model systems. Bouncing ball scars are detected from the solved eigenstates by integrating the probability densities of the states inside narrow detection boxes. The results indicate that the singular potential bump and the dip affect the same states of the clean system in different ways. The bump seems to pin the bouncing ball scars onto itself, when the dip causes the scars to avoid it. This means that the different perturbation types have the opposite scar orientation preferences. The abundance of the bouncing ball scars seems to also be connected to the perturbation parameters, and the results indicate that the parameters could be optimized in order to maximize the abundance. The stability results show impressive robustness against external noise. The number of bouncing ball scars remain nearly constant with weaker noise and even when the noise bumps were 10% from the main perturbation size, a notable amount of bouncing ball scars persist. It also seems that the scars retain their orientations even in the noisy systems.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ≃0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)≃1150 compared to the rectangular billiard (regular) case, where the average PR≃1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium-a chaotic billiard-level statistics described by a Brody distribution with parameter delta similar or equal to 0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)similar or equal to 1150 compared to the rectangular billiard (regular) case, where the average PR similar or equal to 1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
We study the low-energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of the self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through an appropriate truncation of the basis. We show that the signature of wave chaos in low-energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited. \textcopyright{} 1996 The American Physical Society.