interaction between electrons and phonons $$\begin{aligned}
 \label{froHam}
H_{\text{el-ph}}= \frac{1}{\sqrt{N}} \sum_{\textbf{p},\textbf{k},\sigma}g_{\text{ph}}\left(b^{\dagger}_{-\textbf{p}}+b^{\phantom{\dagger}}_{\textbf{p}}\right)c^{\dagger}_{\textbf{k}+\textbf{p},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma},\end{aligned}$$

where $N$ is the number of lattice sites. In Eq.  $c^{\phantom{\dagger}}_{\textbf{k},\sigma}$ ($c^{\dagger}_{\textbf{k},\sigma}$) represents the electron annihilation (creation) operator with momentum $\textbf{k}$ and spin $\sigma$, $\epsilon_{\textbf{k}}=\frac{\hbar^2\textbf{k}^2}{2m}-E_{\text{F}}$ is the electron dispersion relation, $m$ denotes the effective electron mass, and $E_{\text{F}}$ stands for the Fermi energy. The second sum in Eq.  is taken over the set $W$ of all $\textbf{k}$ vectors with $|\epsilon_{\textbf{k}}|\leq\hbar\omega_{\textrm{c}}$, $\omega_{\textrm{c}}$ being the frequency cutoff. The superconducting order parameter $\Delta (t) $ is assumed to be of $s$-wave symmetry, with $
\Delta (t) =\frac{W_0}{N}\sum_{\textbf{k}\in W} \langle c^{\phantom{\dagger}}_{-\textbf{k},\downarrow}c^{\phantom{\dagger}}_{\textbf{k},\uparrow} \rangle,
$ and $W_0$ an attractive interaction constant.

We emphasize that we are generally looking for signatures of the dynamics after the pump pulses and, in particular, in the dynamics of certain low-lying phonons. The phonon mode  considered here does not generate the superconductivity. The pairing interaction is assumed to be mediated by other bosons at higher energies, i.e., for instance by spin fluctuations or by phonons at high energies of the order of the Debye energy $\hbar \omega_{\mathrm{D}}\approx 30$meV. This energy scale is much larger than the energy scale of the Bogoliubov quasiparticles so that these other bosons influence the low-energy dynamics 