${\bf A}_{{\bf q}_{\textrm{pr}}} (\delta t, t)$ of the probe pulse, respectively. In the literature, there are also other ways discussed to determine the conductivity; see, for instance, Ref. . However, we think that in the present context of a pump and a probe pulse the above procedure suggests itself and is closest to what is experimentally done. Hence, the pump-probe conductivity is fully determined by the time evolution of the expectation values $\langle c^{\dagger}_{\textbf{k},\sigma}c^{\phantom{\dagger}}_{\textbf{k}+\textbf{q}_{\text{pr}},\sigma} \rangle (\delta t, t)$, which we numerically compute by integrating the corresponding equations of motion. To keep the notation light, we in the following omit the dependence on the delay time $\delta t$ because it is externally fixed by the timing of the pump and the probe pulse. Below we discuss how the dependence on the time $t$ is computed.

Density-Matrix Formalism {#sec:DMF}
------------------------

Using the density-matrix formalism, we derive equations of motion for the quasiparticle densities and the mean phonon amplitudes $\langle b^{\phantom{\dagger}}_{\textbf{p}}\rangle$ and $\langle b^{\dagger}_{-\textbf{p}}\rangle$ in this section. For this purpose, it is advantageous to perform a Bogoliubov transformation of the electron operators which diagonalizes the Hamiltonian $H_{\text{sc}}$ in the initial state. That is, we introduce new fermionic operators $\alpha_{\textbf{k}}$ and $\beta_{\textbf{k}}$, with $$\begin{aligned}
