Unidirectionality and Husimi functions in constant-width neutrino billiards

Abstract

We investigate the spectral properties and Husimi functions of relativistic quantum billiards consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width by imposing boundary conditions on the spinor components. We consider those of neutrino billiards propoposed in [Berry and Mondragon, Proc. R. Soc. London A 412, 53 (1987)]. The classical dynamics of billiards of corresponding shape is predominantly chaotic. Constant-width billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic quantum billiard a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of constant-width neutrino billiards in the ultra-relativistic, i.e., the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the boundary conditions and the unidrectionality of the local current arising from them.

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

“…[16], we computed the symmetryprojected eigenstates of massive NBs using the method developed in Refs. [74,75]. For too small masses the eigenvalues corresponding to doublet partners are not degenerate, implying that we do only find agreement of the spectral properties of the NB with those of the DB, GB and HKB around the DPs for sufficiently large mass [74], even though these exhibit a selective excitation of the two sets of valley states [76][77][78][79][80].…”

Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles

“…Therefore, we employed an extension of the boundary-integral equations (BIEs) derived in Ref. [38] for massless NBs to massive ones [42,79].…”

“…(1) 1 (kρ) have a logarithmic and a 1/ρ singularity. The integral over these singularities leads to the sin θ β term on the left-hand side [79]. Accordingly, an interval [φ − δφ, φ + δφ], where δφ is arbitrarily small, is excluded from the integration range on the right-hand side.…”