Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering

Abstract

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.

Energy-dependent correlations in the S-matrix of chaotic systems

“…(7), except for the fact that the usual Weingarten function is replaced by the generalization W (2) M,ǫ . For example, we now have…”

“…This situation can be realized in microwave scattering in metallic cavities [1,2,3,4,5,6] and in electron scattering in condensed matter systems, [7,8,9,10,11] among other possibilities. We assume there is a well defined decay rate for the ray dynamics, Γ; this means that the total energy inside the system decays exponentially in time as e −Γt .…”

“…[2,25,26,27] This has been used to compute correlations involving up to four matrix elements, [28,29,30,31] and leads to exact results which are not quite explicit, since some difficult integrals remain to be evaluated. Asymptotic expansions in the parameter 1/M are available for the simplest case of only two elements.…”

Open problems in applying random-matrix theory to nuclear reactions

“…Because of the technical difficulties in the calculation of such averages, the predictive power of the GOE for nuclear reactions is actually much more limited than for spectral fluctuations. General analytical results exist only for the correlation function involving a pair of S-matrix elements [23], for select values of the correlation function involving three or four S-matrix elements [24,25], and for the probability distribution of single S-matrix elements [26,27,28,29]. The complete joint probability distribution of all S-matrix elements is known [30,31,32] only in the Ericson regime (strongly overlapping resonances).…”

Current Status of Nuclear Physics Research

“…Microwave billiards provide very useful systems to study quantum chaos, because they already contain a degree of chaoticity in their classical dynamics. In fact, the eigenvalues and wave functions of quantum microwave billiards have been studied by Achim Richter's group [227][228][229] with a high precision insight into quantum chaos phenomena.…”