Resonances in a single-lead reflection from a disordered medium: σ\sigma-model approach

We develop a general non-perturbative characterisation of universal features of the density $\rho(\Gamma)$ of $S$-matrix poles (resonances) $E_n-i\Gamma_n$ describing waves incident and reflected from a disordered medium via a single $M$-channel waveguide/lead. Explicit expressions for $\rho(\Gamma)$ are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels ($M\sim 1$) the most salient features are tails $\rho(\Gamma)\sim 1/\Gamma$ for narrow resonances reflecting exponential localization and $\rho(\Gamma)\sim 1/\Gamma^2$ for broad resonances reflecting states located in the vicinity of the attached wire. For multimode wires with $M\gg 1$, intermediate asymptotics $\rho(\Gamma)\sim 1/\Gamma^{3/2}$ is shown to emerge reflecting diffusive nature of decay into wide enough contacts.Comment: Article+Supplemental Materia

Statistics of S-matrix poles for chaotic systems with broken time reversal invariance: a conjecture

In the framework of a random matrix description of chaotic quantum scattering the positions of $S-$matrix poles are given by complex eigenvalues $Z_i$ of an effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture on statistics of $Z_i$ for systems with broken time-reversal invariance and verify that it allows to reproduce statistical characteristics of Wigner time delays known from independent calculations. We analyze the ensuing two-point statistical measures as e.g. spectral form factor and the number variance. In addition we find the density of complex eigenvalues of real asymmetric matrices generalizing the recent result by Efetov\cite{Efnh}.Comment: 4 page

Resonances in a single-lead reflection from a disordered medium:σ-model approach

Using the framework of supersymmetric non-linear σ-model we develop a general non-perturbative characterization of universal features of the density ρ(Γ) of the imaginary parts (“width”) for S-matrix poles (“resonances”) describing waves incident and reflected from a disordered medium via M-channel waveguide/lead. Explicit expressions for ρ(Γ) are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels (M∼1) the most salient features are tails ρ(Γ)∼Γ−1 for narrow resonances reflecting exponential localization and ρ(Γ)∼Γ−2 for broad resonances reflecting states located in the vicinity of the attached wire. For multimode quasi 1D wires with M≫1, an intermediate asymptotics ρ(Γ)∼Γ−3/2 is shown to emerge reflecting diffusive nature of decay into wide enough contacts.</p