Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors*

Abstract

According to the Berry-Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE- like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré-Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.

Unidirectionality and Husimi functions in constant-width neutrino billiards

“…Furthermore, both the QB and the NB comprise eigenfunctions that are invariant under a discrete rotational symmetry, however, the first and second component of those of the NB transform differently under the respective rotation [16]. We demonstrated in [30] that, as a consequence, the circle NB and circle-sector NBs with inner angle 0 < φ 0 < 2π, constructed by cutting it along the symmetry lines φ = 0 and φ = φ 0 , do not have any common eigenstates and, as a consequence, the spectral properties of the latter may coincide with those of QBs with chaotic classical dynamics. The terms entering the BIE equation ( 24) can be expressed in terms of φ = φ − φ ′ ,…”

“…Furthermore, for billiards that possess a discrete rotational symmetry, for both the QB [11][12][13] and the NB the eigenstates can be separated according to the transformation properties under the corresponding rotation, however for the latter they differ for the two spinor eigenfunction components [29]. As a consequence, the circle NB and sectors of it do not have any common eigenstates [30]. Similarly, for billiards with a shape that generates a unidirectional dynamics, the unidirectionality of the current and the relative phase of the spinor components at the boundary resulting from the BC equation (58), which are clockwise for µ = +1 and counterclockwise for µ = −1, leads to the large splittings of the eigenvalues corresponding to clockwise and counterclockwise modes, respectively, and rules out dynamical tunneling; see also [35].…”

“…For this we extended the boundary integral equation (BIE) for massless NBs to massive ones [29] by proceeding as in [1]. Up to now, we only quoted the BIE in various publications [16,22,29,30], but did not provide a detailed derivation. Since this article is part of a special issue to celebrate Sir Michael Berry's 80th birthday, and the derivation relies on that given in [1] for the massless case, we decided to use the opportunity and to provide it in section 2.…”

Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

“…In sequence, we introduce a perturbation of the C-C 3 B, replacing the circumferences with ellipses. Recently, billiards with elliptical borders have been studied in other contexts, i.e., in singular potentials [30], in relativistic limits [31], and flows that move around chaotic cores [32]. We start the analysis by presenting the billiards, discussing their classical dynamics, showing some mixed phase spaces, and calculating the fraction of the chaotic sea on these phase spaces.…”

Properties of the eigenmodes and quantum-chaotic scattering in a superconducting microwave Dirac billiard with threefold rotational symmetry

“…It has been demonstrated in Refs. [51,52] that the spectral properties of GBs and NBs of corresponding shape do not coincide [1,8,24,28,[53][54][55][56][57][58][59]. These discrepancies were attributed to intervalley scattering at the boundary of GBs [55,58,59].…”