Distributions of off-diagonal scattering matrix elements: Exact results

Abstract

The recently derived distributions for the scattering-matrix elements in quantum chaotic systems are not accessible in the majority of experiments, whereas the cross sections are. We analytically compute distributions for the off-diagonal cross sections in the Heidelberg approach, which is applicable to a wide range of quantum chaotic systems. We thus eventually fully solve a problem which already arose more than half a century ago in compound-nucleus scattering. We compare our results with data from microwave and compound-nucleus experiments, particularly addressing the transition from isolated resonances towards the Ericson regime of strongly overlapping ones.

Distribution of Off-Diagonal Cross Sections in Quantum Chaotic Scattering: Exact Results and Data Comparison

“…Rozhkov, Fyodorov, and Weaver [49,50] computed a related quantity, namely the statistics of transmitted power. Putting forward a new variant of the supersymmetry method, we recently calculated the distributions of the real and the imaginary parts of the off-diagonal S matrix [51,52]. In a related study, Fyodorov and Nock obtained the distributions of off-diagonal elements of the Wigner K matrix [53].…”

“…At first sight, one might expect that this task leads to doubling the size of the supersymmetric non-linear sigma model as compared to Refs. [48,51,52], rendering further evaluation forbiddingly complicated. However, we recently discovered that a simple, yet farreaching modification and generalization of our supersymmetry technique in Refs.…”

“…However, we recently discovered that a simple, yet farreaching modification and generalization of our supersymmetry technique in Refs. [51,52] yields P (x 1 , x 2 ) without enlarging this size.…”

“…In Refs. [51,52] we proceeded similarly, but for marginal distributions and thus univariate characteristic functions that depend either on k 1 or on k 2 . Absorbing them as complex variable k into the definition of A preserves its Hermiticity.…”

Joint probability distributions for projection probabilities of random orthonormal states

“…Ensembles of scattering matrices may be derived from the unitary group, also [19]. In this area, there are connections to the distribution of transmission eigenvalues [20], and the probability distributions of individual scattering matrix elements in the so called Heidelberg model [21,22].…”