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

Abstract

We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $\omega$ is described by the scattering matrix $\mathcal{S}(\omega )$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $\mathcal{Q}=-\mathrm{i}{\mathcal{S}}^{†}{\partial }_{\omega }\mathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $\mathcal{S}={\mathrm{e}}^{2\mathrm{i}kL}{\mathcal{U}}_L{\mathcal{U}}_R$ (with ${\mathcal{U}}_L={\mathcal{U}}_R^{\mathrm{T}}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $\tilde{\mathcal{Q}}={\mathcal{U}}_R\mathcal{Q}{\mathcal{U}}_R^{†}=(2L/v)1_N-\mathrm{i}{\mathcal{U}}_L^{†}{\partial }_{\omega }\left({\mathcal{U}}_L{\mathcal{U}}_R\right){\mathcal{U}}_R^{†}$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $\tilde{\mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $L\to \infty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $\mathcal{Q}$'s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $⟨\mathrm{tr}(\mathcal{Q})⟩$, $⟨\mathrm{tr}({\mathcal{Q}}^2)⟩$ and $⟨[\mathrm{tr}(\mathcal{Q}){]}^2⟩$. Finally we derive a partial differential equation for the resolvent $g(z;L)={\lim }_{N\to \infty }(1/N)\mathrm{tr}\left\{\right(z{1}_N-N\mathcal{Q}{)}^{-1}\}$ in the large $N$ limit.

arXiv comment30 pages, LaTeX

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