Distribution of the Wigner–Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

Abstract

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner-Smith time delay matrix Q for a chaotic cavity with uniform absorption, coupled via N perfect channels. In the unitary class β = 2 we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large N properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an application of our results, we study the statistical properties of the Wigner time delay τ W = tr[Q]/N in the presence of absorption.

Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

“…Based on this distribution, many results have been obtained for ideal contacts: cumulants [19] and distribution [20] of its trace tr {Q}, or other correlations [21][22][23][24][25] (see the updated preprint version of reference [10] for an exhaustive review). Several generalizations of BFB's distribution have been obtained more recently: the case of non-ideal contacts has been studied [26,27], BdG symmetry classes [26] and the effect of absorption (for ideal contacts) [28].…”

Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

“…[10] for an exhaustive review). Several generalizations of BFB's distribution have been obtained more recently : the case of non-ideal contacts has been studied [23,24], BdG symmetry classes [23] and the effect of absorption (for ideal contacts) [25].…”

Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities

“…The paper thus was among the first promoting interest in statistics of Wigner time delays in wave-chaotic scattering, which after three decades still remains an active research topic, see e.g. [20][21][22][23][24][25][26][27][28] as well as [29][30][31][32][33][34] and references therein. Measuring the phaseshifts δ(ω) independently allowed to test experimentally the relation between R(ω) and the Wigner time-delay, and overall good agreement has been reported in [13], with discrepancies close to the deepest minima attributed to inacuracies in numerical differentiation.…”