First Experimental Observation of Superscars in a Pseudointegrable Barrier Billiard

Abstract

With a perturbation body technique intensity distributions of the electric field strength in a flat microwave billiard with a barrier inside up to mode numbers as large as about 700 were measured. A method for the reconstruction of the amplitudes and phases of the electric field strength from those intensity distributions has been developed. Recently predicted superscars have been identified experimentally and -using the well known analogy between the electric field strength and the quantum mechanical wave function in a two-dimensional microwave billiard -their properties determined.PACS numbers: 05.45. Mt, 03.65.Sq, 42.25.Fx Planar polygonal billiards with angles α j = πm j /n j , where m j and n j are co-prime integers, have been studied both classically and quantum mechanically [1]. When m j = 1 the motion in phase space is not restricted to a torus like for integrable systems, but to a surface with a more complicated topology. Accordingly, such planar polygonal billiards are called pseudointegrable. It was established numerically (see [2,3] and refs. therein) that the statistical properties of the eigenvalues of the corresponding quantum systems are intermediate between those of a regular and a chaotic system. The properties of the wave functions of planar polygonal billiards are also intriguing, as they show a strong scarring behavior which can be related to families of periodic orbits [4]. In a plot of eigenfunctions in the barrier billiard scars are clearly distinguishable from non-scarred eigenfunctions. This pronounced scar structure does not disappear at large quantum numbers in contrast to that in chaotic systems [5,6,7]. To stress this difference it is proposed in [4] to call the scars in pseudointegrable systems superscars, an expression used by Heller in his early seminal paper [6] in a different context. The aim of this letter is to report on the experimental investigation of superscars in the barrier billiard. This is a rectangular billiard of area l x · l y which contains an infinitely thin barrier. In the experiment presented here the barrier is placed on the symmetry line x = l x /2 and its length equals l y /2, where l y is the length of the shorter side of the rectangle.In the present work the quantum barrier billiard is simulated by means of a microwave billiard. Microwave billiards are flat cylindrical resonators [8,9]. Below the critical frequency f c = c/2h, where c is the velocity of light and h is the height of the cavity, the electric field is the solution of the scalar Helmholtz equation with Dirichlet boundary conditions. This equation is mathematically equivalent to the Schrödinger equation for a quantum billiard of corresponding shape (see e.g. [8,9]). A is the area of the billiard. Hence, the number of experimentally accessible resonances increases with the area and decreases with the height of the billiard. The resonances can be well resolved, if their widths are small in comparison with the average spacing between adjacent resonances, i.e. a microwave cavity with a high qual...

Semiclassical Wave Functions and Energy Spectra in Polygon Billiards

“…The last condition is just the one which has justified the considerations of Bogomolny and Schmit on the superscar phenomenon [5,6].…”

“…POC's being not global periodic skeletons give rise however to the phenomenon known as superscars [5,6] which has been considered in our earlier paper [9].…”

“…In sec.8 periodic orbit channel (POC) skeletons of Bogommolny and Schmit [5,6] building global periodic skeletons are considered and their relations to these skeletons are discussed.…”

Inferring periodic orbits from spectra of simply shaped microlasers

“…One can argue that the origin of such families is analogous to the formation of superscar states in pseudointegrable billiards discussed in [22] and observed experimentally in microwave experiments in [23]. In general, periodic orbits of polygonal cavities form continuous families which can be considered as propagating inside straight channels obtained by unfolding classical motion (see Fig.…”

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…A deep insight into the structure of optical modes in polygonal microcavities has been provided by the superscar model (Lebental et al, 2007), which had been originally invented for polygonal billiards (Bogomolny and Schmit, 2004) and experimentally confirmed in microwave cavities (Bogomolny et al, 2006). In brief, the families of periodic ray trajectories in polygonal cavities are unfolded to fictitious straight rectangular channels passing through cavity corners.…”