recent pump-probe experiments on Nb$_{1-x}$Ti$_x$N thin films [@matsunagaPRL13].

![ \[FigM2\] \[plot:sigma-adi\] (Color online) (a) Real part of the pump-probe conductivity, $\mathrm{Re}[ \sigma(\delta t,\omega)]$, versus $\omega$ and $\delta t >0$ for the adiabatic regime \[$\tau_{\textrm{p}}=20$ ps, $|\textbf{A}_{\textrm{p}} |=0.5\cdot10^{-8}$ Js/(Cm)\] in the absence of phonons. Panel (b) shows a detail of the pump-probe response for frequencies just below the pump frequency $ \omega_{\textrm{p}} = 1.21  \omega_{\Delta_{\infty}}$. The small dip near $\omega  \approx 1.15 \omega_{\Delta_{\infty}}$ is due to Pauli blocking. (c) Time dependence of $| \Delta (t) |$ for the same parameters as in panel (a). ](Fig_2){width="\columnwidth"}

### Adiabatic regime, $\tau_{\textrm{p}} \gg \tau_{\Delta}$

It is interesting to contrast the pump-probe response induced by ultrashort pump pulses \[Fig. \[FigM1\](a)\] with the one in the adiabatic regime, where the pump pulse duration $\tau_{\textrm{p}}$ is much longer than the dynamical time scale of the superconductor $\tau_{\Delta}$. The absorption spectrum $\textrm{Re} [ \sigma(\delta t,\omega)]$ as a function of delay time $\delta t$ and frequency $\omega$ for a pump pulse with $\tau_{\textrm{p}} \gg \tau_{\Delta}$ is shown in Fig. \[FigM2\](a). In this adiabatic regime, the pump pulse excites only the normal quasiparticle densities, $\langle\alpha^{\dagger}_{\textbf{k}}\alpha^{\phantom{\dagger}}_{\textbf{k}' } \rangle$ and $\langle\beta^{\dagger}_{\textbf{k}}\beta^{\phantom{\dagger}}_{\textbf{k}' } \rangle$, whereas the anomalous ones, $\langle\alpha^{\dagger}_{\textbf{k}}\beta^{\dagger}_{\textbf{k}' } \rangle$ and $\langle\alpha^{\phantom{\dagger}}_{\textbf{k}}\beta^{\phantom{\dagger}}_{\textbf{k}' 