Density of scattering resonances in a disordered system

TL;DR

This work develops a universal nonlinear sigma-model framework to characterize the distribution of scattering-resonance widths in disordered and chaotic systems, valid for all three Wigner-Dyson symmetry classes and arbitrary lead coupling. By introducing a source-based generating function and performing a Fourier analysis on the sigma-model manifold, the authors reduce the problem to a tractable Cartan-variable integral and derive a central, broadly applicable formula for the resonance-width density ρ(η). They obtain explicit analytic results for small metallic grains in unitary, orthogonal, and symplectic classes, reproduce Moldauer-Simonius universal behavior for the mean decay rate, and reveal detailed asymptotics and crossover behavior as a function of channel number and coupling. Numerical simulations using random matrices show perfect agreement with the analytic predictions, underscoring the universality and practical relevance of the approach for mesoscopic scattering phenomena and related chaotic systems.

Abstract

Reflection of particles from a disordered or chaotic medium is characterized by a scattering matrix that can be represented as a superposition of resonances. Each resonance corresponds to an eigenstate inside the medium and has a width related to the decay time of this eigenstate. We develop a general approach to study the distribution function of these resonance widths based on the nonlinear sigma model. We derive an integral representation of the distribution function that works equally well for systems of any symmetry and for any type of coupling to the measuring device. From this integral representation we find explicit analytic expressions for the distribution function in the case of disordered metallic grains. We also compare the analytic results to large-scale numerical simulations and observe their perfect agreement.

Figures (8)

• Figure 1: Schematic depiction of a disordered sample attached to a ballistic measuring probe with $M$ conducting channels. The two parts of the system are coupled through a generic barrier characterized by a set of transmission probabilities $T_n$ in individual channels. Amplitudes of incident and reflected waves in the measuring lead are denoted by $\ket{\text{in}}$ and $\ket{\text{out}}$.
• Figure 2: An illustration of the sigma-model manifold by an ordinary two-dimensional sphere. The matrix $\Lambda$ corresponds to the north pole. Matrices $Q$ and $Q_a$ are represented as points on the sphere with the polar angles $\theta$ and $a$, respectively. Angular distance between $Q$ and $Q_a$ is $\tilde{\theta}$.
• Figure 3: The density of scattering poles $\rho(y)$ in a small metallic grain of the unitary class, Eq. (\ref{['rhounitM']}), with the attached measuring probe hosting $M = 1, \ldots 10$ conducting channels. Upper panel: perfect coupling with $g = 1$. Lower panel: each channel is coupled with $g = 3$ that corresponds to the transparency $T = 0.5$. We plot the distribution for the scaled variable $y/M$ to facilitate visual comparison of different curves.
• Figure 4: The density of scattering poles $\rho(y)$ in a small metallic grain of the orthogonal symmetry class, Eq. (\ref{['rhoorthM']}), attached to the measuring probe with $M$ perfectly conducting channels. Similar to Fig. \ref{['Fig:resA']}, we plot the distribution for the scaled variable $y/M$ to facilitate visual comparison of different curves.
• Figure 5: The density of scattering poles $\rho(y)$ in a small metallic grain of the symplectic symmetry class [Eq. (\ref{['rhosympA']}) expanded in a power series in $\alpha$] attached to the measuring probe with $M$ perfectly conducting channels. Similar to Figs. \ref{['Fig:resA']} and \ref{['Fig:resAIg1']}, we plot the distribution for the scaled variable $y/M$ to facilitate visual comparison of different curves.