give rise to an exponential damping of the coherent phonon and order parameter oscillations. Recent pump-probe experiments on Nb$_{1-x}$Ti$_x$N films [@matsunagaPRL12; @matsunagaPRL13] have shown that these relaxation processes occur on time scales of tens of picoseconds, which is much larger than the period of the order parameter oscillations.

By factorizing higher-order density matrices according to Eq. , a closed set of differential equations for the quasiparticle density matrices and the mean phonon amplitudes can be derived using Heisenberg’s equation of motion. A derivation of these differential equations and some other technical details are given in Appendix \[sec:eom\]. In Sects. \[sec:result\] and \[sec:Results\_zwei\], we numerically solve this set of differential equations to obtain the temporal evolution of the order parameter amplitude $| \Delta (t) |$, the lattice displacement $U ({\bf r}, t)$, and the current density ${\bf j}_{{\bf q}_{\textrm{pr}}} (t)$. From the Fourier transform of the latter quantity, the pump-probe conductivity $\sigma ( \delta t, \omega )$ is readily obtained using Eq. . In order to study different time-scale regimes, we adjust in the following the pump-pulse width $\tau_{\textrm{p}}$, the phonon energy $\hbar \omega_{\textrm{ph}}$, and the integrated pump-pulse intensity $|\textbf{A}_{\textrm{p}} |^2 \tau_{\textrm{p}}$. For each regime we determine the signatures of the 