Time delay correlations in chaotic scattering: random matrix approach

Abstract

We study the correlations of time delays in a model of chaotic resonance scattering based on the random matrix approach. Analytical formulae which are valid for arbitrary number of open channels and arbitrary coupling strength between resonances and channels are obtained by the supersymmetry method. We demonstrate that the time delay correlation function, though being not a Lorentzian, is characterized, similar to that of the scattering matrix, by the gap between the cloud of complex poles of the S-matrix and the real energy axis.

Correlation Functions of Impedance and Scattering Matrix Elements in Chaotic Absorbing Cavities

“…One has the Gaussian orthogonal ensemble (GOE, the Dyson symmetry index β = 1 and H symmetric) for chaotic systems with preserved TRS and the Gaussian unitary ensemble (GUE, β = 2 and H Hermitian) for those with fully broken TRS. For similar reasons the approach is independent of particular statistical assumptions on coupling amplitudes V a n (as long as M N [36,37]), which may be chosen as fixed [29] or random [30] variables. They enter final expressions only by means of M transmission coefficients (also so-called sticking probabilities):…”

“…For orthogonal symmetry it was done in [6], where the overall decaying factor e −γτ due to absorption was also confirmed by comparison to the experimental result for the form factor measured in microwave cavities. It is useful for the qualitative description to note that P ab (τ ) and 2J ac (τ ) are quite similar to the "norm leakage" decay function [54] and the form factor of the Wigner time delays [37], respectively (they would coincide exactly at γ = 0, if we put T a,b,c = 0 appearing explicitly in denominators of (13a) and thereafter). Then one can follow analysis performed in these papers, see also [53], to find qualitatively …”

Semiclassical expansion of parametric correlation functions of the quantum time delay

“…For the GOE case, the correlation function can be found for the case without parametric variations in [20], and for the case with parametric variations in [22] The parametric correlation function, which we consider here, is expressed by a triple integral…”

“…In this article we consider two-point correlation functions of the Wigner time delay in open systems, for which there are very comprehensive results from RMT available [20,21,22]. We evaluate a semiclassical periodic orbit expansion for the Fourier transform of the two-point correlation function, the form factor K(τ, x, y, M).…”

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

“…[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.…”