Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts

Abstract

We consider wave propagation in a complex structure coupled to a finite number N of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, relatedtotheenergy(orfrequency)derivativeofthescatteringmatrixS. Wedeveloparandom matrixapproachtostudythestatisticalpropertiesofthesymmetrisedWigner-Smithtime-delay matrix Qs = −i~S−1/2∂εS S−1/2, and obtain the joint distribution of S and Qs for the systemwithnon-idealcontacts,characterisedbyafinitetransmissionprobability(perchannel) 0 < T 6 1. WederivetworepresentationsofthedistributionofQs intermsofmatrixintegrals specified by the Dyson symmetry index β = 1, 2, 4 (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay τW = (1/N)tr Qs , which is an important quantity providing the density of states of the open system. Using the obtainedresults,wedeterminethedistribution PN,β(τ) oftheWignertimedelayintheweak coupling limit NT 1 and identify the following three regimes. (i) The large deviations at small times (measured in units of the Heisenberg time) are characterised by the limiting behaviour PN,β(τ) ∼ τ−βN2/2−3/2 exp −βNT/(8τ) for τ . T. (ii) The distribution shows the universal τ−3/2 behaviour in some intermediate range T . τ . 1/(TN2). (iii) It has a power law decay PN,β(τ) ∼ T2N3(TN2τ)−2−βN/2 for large τ & 1/(TN2).