by ultrafast photoexcitations [@Akba13; @zachmannNJP13; @Schny11; @Papen07; @Papen08; @Papen09]. In particular, it has been shown that excitation pulses with pulse duration $\tau_{\textrm{p}}$, see Eq. , shorter than the inverse BCS gap energy $| \Delta |$, i.e., $\tau_{\textrm{p}} \ll h / ( 2 | \Delta | )$, generate coherent oscillations in the order parameter amplitude and in the quasiparticle occupations with frequency $\omega_{\Delta_{\infty}}  \approx  2 |\Delta | / \hbar $ [@volkov1974; @amin2004; @barankov2004; @Yuz05; @Yuz06]. Moreover, in the presence of an optical-phonon branch with energy of the order of $\hbar \omega_{\textrm{ph}}  \approx  2 | \Delta | $, coherent quasipersistent phonon oscillations can be generated [@Schny11]. While the creation of these coherent oscillations is well understood theoretically, it has remained an open question whether and how this nonadiabatic BCS state dynamics could be observed in experimental pump-probe spectra [@Papen08].

In this paper, we numerically simulate the pump-probe response of nonequilibrium BCS superconductors coupled to optical-phonon modes and determine the signatures of the coherent Cooper pair and phonon dynamics in the pump-probe conductivity. Our aim is to provide systematic calculations of the pump-probe response, such that experimental measurements can more easily be interpreted and compared to the theoretical predictions. Using density-matrix theory [@Ross02], 