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

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 $\mathcal{S}$. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix $\mathcal{Q}_s=-\mathrm{i}\hbar\,\mathcal{S}^{-1/2}\big(\partial_\varepsilon\mathcal{S}\big)\,\mathcal{S}^{-1/2}$, and obtain the joint distribution of $\mathcal{S}$ and $\mathcal{Q}_s$ for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) $0<T\leq1$. We derive two representations of the distribution of $\mathcal{Q}_s$ 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 $τ_\mathrm{W}=(1/N)\,\mathrm{tr}\big\{\mathcal{Q}_s\big\}$, which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution $\mathscr{P}_{N,β}(τ)$ of the Wigner time delay in the weak coupling limit $NT\ll1$ and identify three different asymptotic regimes.