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

•The concept of the time delay in quantum scattering is briefly reviewed.•Statistical properties of the Wigner time delay in 1D disordered quantum mechanics are described.•In the presence of a chiral symmetry in 1D, the existence of a limit law for the time delay is related to a chiral zero mode.•We review results obtained within random matrix theory for the distribution, moments and correlation functions of the Wigner–Smith time delay matrix.•These concepts are applied for non-linear and AC transport in coherent conductors.

The concepts of Wigner time delay and Wigner–Smith matrix allow us to characterise temporal aspects of a quantum scattering process. The paper reviews the statistical properties of the Wigner time delay for disordered systems; the case of disorder in 1D with a chiral symmetry is discussed and the relation with exponential functionals of the Brownian motion is underlined. Another approach for the analysis of time delay statistics is the random matrix approach, from which we review few results. As a practical illustration, we briefly outline a theory of non-linear transport and AC transport developed by Büttiker and coworkers, where the concept of Wigner–Smith time delay matrix is a central piece allowing us to describe screening properties in out-of-equilibrium coherent conductors.