coherent Cooper pair and phonon dynamics in the pump-probe response.

![image](Fig_1a){width="\columnwidth"} ![image](Fig_1b){width="\columnwidth"}

Pump-probe response in the absence of phonons {#sec:result}
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Before studying the evolution of the coupled Bogoliubov quasiparticle-phonon system in Sect. \[sec:Results\_zwei\], it is instructive to first consider the pump-probe response of a BCS superconductor in the absence of phonons. Hence, we first solve the set of equations of motion  for $g_{\textrm{ph}} = 0$ and compute $\sigma ( \delta t, \omega )$ for positive and negative pump-probe delay times $\delta t$.

Positive pump-probe delay time
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We start by discussing the case where the probe pulse follows the pump pulse after a positive delay time $\delta t$. Both the nonadiabatic and the adiabatic regimes are considered, corresponding to $\tau_{\textrm{p}} \ll \tau_{\Delta}$ and $\tau_{\textrm{p}} \gg \tau_{\Delta}$, respectively; see Figs. \[FigM1\] and \[FigM2\].

### Nonadiabatic regime, $\tau_{\textrm{p}}  \ll \tau_{\Delta}$

In Fig. \[FigM1\](a) we plot the real part of the pump-probe signal, $\textrm{Re} [ \sigma(\delta t,\omega)]$, versus delay time $\delta t$ and frequency $\omega$ for the regime $\tau_{\textrm{p}} \ll \tau_{\Delta}$, where the Bogoliubov quasiparticle densities build up coherently. The imaginary part of $\sigma(\delta t,\omega)$, which shows similar features as the real part, is presented in Fig. \[FigM11\] of Appendix \[sec:example\_im\]. In 