Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems

Abstract

The $M$-dimensional scattering matrix $S(E)$ which connects incoming to outgoing waves in a chaotic systyem is always unitary, but shows complicated dependence on the energy. This is partly encoded in correlators constructed from traces of powers of $S(E+\epsilon )S^{†}(E-\epsilon )$, averaged over $E$, and by the statistical properties of the time delay operator, $Q(E)=-i\hslash S^{†}dS/dE$. Using a semiclassical approach for systems with broken time reversal symmetry, we derive two kind of expressions for the energy correlators: one as a power series in $1/M$ whose coefficients are rational functions of $\epsilon$, and another as a power series in $\epsilon$ whose coefficients are rational functions of $M$. From the latter we extract an explicit formula for $\mathrm{Tr}({\mathrm{Q}}^{\mathrm{n}})$ which is valid for all $n$ and is in agreement with random matrix theory predictions.

6 pages

References in corpus (5)

• Semiclassical Theory of Chaotic Conductors
• Semiclassical Approach to Chaotic Quantum Transport
• Efficient semiclassical approach for time delays
• Distribution of Off-Diagonal Cross Sections in Quantum Chaotic Scattering: Exact Results and Data Comparison
• Energy-dependent correlations in the $S$-matrix of chaotic systems