electron-phonon coupling strength is taken to be $g_{\text{ph}}=0.1$ meV. (b), (c) Time dependence of the order parameter amplitude $| \Delta (t) |$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (d) Spectral distribution of the coherent phonon oscillations. ](Fig_7){width="\columnwidth"}

![ \[FigM7\] \[plot:sigma-neg-phonon\] (Color online) (a) Real part of the pump-probe signal, $\textrm{Re} [ \sigma(\delta t,\omega)]$, versus $\omega$ and $\delta t <0$ for the nonadiabatic regime \[$\tau_{\textrm{p}}=0.5$ ps, $|\textbf{A}_{\textrm{p}}|=13\cdot10^{-8}$ Js/(Cm)\] in the presence of a phonon mode with energy $\hbar \omega_{\textrm{ph}} = 2$ meV $ \approx 1.19 \, \hbar \omega_{\Delta_{\infty}}$ and coupling strength $g_{\text{ph}}=0.1$ meV. (b) Fourier transform of the data in panel (a), i.e., $\textrm{Re} ( \sigma  )$ as a function of Fourier frequency $\omega_{\delta t}$ and absorption frequency $\omega$. (c), (d) Time dependence of the order parameter amplitude $| \Delta (t) |$ and the lattice displacement $U (0, t)$, respectively, for the same parameters as in panel (a). (e) Spectral distribution of the order parameter and coherent phonon oscillations (dashed blue and solid red lines, respectively). ](Fig_8){width="\columnwidth"}

In this section we analyze the pump-probe response of a superconductor coupled to an optical phonon. To that end, we numerically solve the set of kinetic 