coherent Cooper pair dynamics as well as the finite density of coherent phonons produce oscillations in the pump-probe conductivity as a function of the delay time $\delta t$ between pump and probe pulses (see Figs. \[FigM1\]-\[FigM4\]). Interestingly, the pump-probe signal is resonantly enhanced and exhibits strong nondecaying oscillations when the frequency of the order-parameter oscillation is tuned to the phonon energy. This condition can be achieved by adjusting the integrated pump laser intensity (see Fig. \[FigM5\]).

The remainder of this paper is organized as follows. In Sect. \[Sec:ModHam\] we introduce the microscopic Hamiltonian of the model and specify the parameters for our numerical simulations. The equations of motion are derived in Sect. \[sec:method\] using density-matrix theory. Our numerical results are presented and analyzed in Sects. \[sec:result\] and \[sec:Results\_zwei\]. Finally, we summarize and conclude our findings in Sect. \[sec:conclusion\]. Some technical details and additional plots are presented in Appendixes \[sec:eom\] and \[sec:example\_im\], respectively.

Microscopic Hamiltonian {#Sec:ModHam}
=======================

We consider a single-band $s$-wave superconductor coupled to an optical-phonon mode described by the Hamiltonian $H=H_{\text{sc}}+H_{\text{ph}}+H_{\text{el-ph}}$, where $H_{\text{sc}}$ represents the mean-field BCS Hamiltonian

\[modHam\] $$\begin{aligned}
 \label{SChamm}
H_{\text{sc}}=\sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}}c^{\dagger}_{\textbf{k},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma}-\sum_{\textbf{k}\in W}\left(\Delta c^{\dagger}_{\textbf{k},\uparrow}c^{\dagger}_{-\textbf{k}\downarrow}+\Delta^{*} c^{\phantom{\dagger}}_{-\textbf{k},\downarrow}c^{\phantom{\dagger}}_{\textbf{k},\uparrow}\right),\end{aligned}$$ the free-phonon part $H_{\text{ph}}$ is given by $$\begin{aligned}
 \label{freePhon} 
H_{\text{ph}}=\sum_{\textbf{p}}\hbar\omega_{\text{ph}}\left(b^{\dagger}_{\textbf{p}}b^{\phantom{\dagger}}_{\textbf{p}}+\frac{1}{2}\right),\end{aligned}$$ and $H_{\text{el-ph}}$ denotes the 