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, related to the energy (or frequency) derivative of the scattering matrix S. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix Q s = −i S −1/2 ∂ ε S S −1/2 , and obtain the joint distribution of S and Q s for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) 0 < T 1. We derive two representations of the distribution of Q s in terms of matrix integrals 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 τ W = (1/N ) tr Q s , which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution P N,β (τ ) of the Wigner time delay in the weak coupling limit N T 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 P N,β (τ ) ∼ τ −βN 2 /2−3/2 exp − βN T /(8τ ) for τ T .(ii) The distribution shows the universal τ −3/2 behaviour in some intermediate range T τ 1/(T N 2 ). (iii) It has a power law decay P N,β (τ ) ∼ T 2 N 3 (T N 2 τ ) −2−βN/2 for large τ 1/(T N 2 ).

Wigner time delay and related concepts: Application to transport in coherent conductors

“…where ∆ N (γ) = i<j (γ i − γ j ) denotes the Vandermonde determinant (here, I have slightly simplified the form given in [101]). This form reduces to (85) for κ → 1, as we now check.…”

“…(118) is precisely the BFB result (85) for β = 2, as it should. Although complicate, the form (115) was used in order to derive the distribution of the Wigner time delay [101]. Extracting the marginal distribution of the proper time or correlations seems however difficult.…”

“…The marginal distribution of the partial time delays is known exactly for any symmetry class [88,86,175]. A convenient integral representation was derived in Appendix C of [101] by extending the method of Ref. [95] to arbitrary number of channels.…”

Morphology of wetting-layer states in a simple quantum-dot wetting-layer model

“…It has been shown that for weakly open systems (see, e.g. [56]) the lifetime statistics is given by a χ 2 distribution…”

Reflection Time Difference as a Probe of S-Matrix Zeroes in Chaotic Resonance Scattering

“…Statistics of Wigner time delays and related quantities in wave-chaotic scattering attracted a considerable interest for about three decades and is still an active research topic, see [53,54,55,56,57,58,59,60,61,62,63,64] and references therein. Interestingly, τ W (λ) naturally appears in characterization of systems with spatially-uniform absorption.…”