q}_{\textrm{pr}} } \rangle 
- 
\langle\beta^{\dagger}_{\textbf{k}}\beta^{\phantom{\dagger}}_{{\bf k}+{\bf q}_{\textrm{pr}} } \rangle  
\big)
\Big]  . \nonumber\end{aligned}$$

Due to the interaction term $H_{\text{el-ph}}$ in Eq. , the kinetic equations for the single-particle density matrices $\langle\alpha^{\dagger}_{\textbf{k}}\alpha^{\phantom{\dagger}}_{\textbf{k}' } \rangle$, $\langle\beta^{\dagger}_{\textbf{k}}\beta^{\phantom{\dagger}}_{\textbf{k}' } \rangle$, etc., are not closed: Instead, $H_{\text{el-ph}}$ leads to an infinite hierarchy of equations of higher-order density matrices [@Ross02]. In order to study the generation of coherent phonons, it is sufficient to break this hierarchy at first order of the electron-phonon coupling strength $g_{\textrm{ph}}$. That is, phonon-assisted quantities, such as $\langle\alpha^{\dagger}_{\textbf{k}}\alpha^{\phantom{\dagger}}_{\textbf{k}+\textbf{q}}b^{\phantom{\dagger}}_{\textbf{p}}\rangle$, are factorized as follows [@Schny11; @Herb03]: $$\begin{aligned}
 \label{factorizeEq}
\langle\alpha^{\dagger}_{\textbf{k}}\alpha^{\phantom{\dagger}}_{\textbf{k}+\textbf{q}}b^{\phantom{\dagger}}_{\textbf{p}}\rangle
=
\langle\alpha^{\dagger}_{\textbf{k}}\alpha^{\phantom{\dagger}}_{\textbf{k}+\textbf{q}}\rangle
\langle b^{\phantom{\dagger}}_{\textbf{p}}\rangle.\end{aligned}$$ Finite expectation values of the mean phonon amplitudes $\langle b^{\phantom{\dagger}}_{\textbf{p}}\rangle$ and $\langle b^{\dagger}_{-\textbf{p}}\rangle$ correspond to a nonvanishing lattice displacement, $$\begin{aligned}
U(\textbf{r},t)=\sum_{\textbf{p}}\sqrt{\frac{\hbar}{2M\omega_{\text{ph}}N}}
\left(
 \langle b^{\phantom{\dagger}}_{\textbf{p}} \rangle + \langle b^{\dagger}_{-\textbf{p}} \rangle
\right)e^{+ i\textbf{p}\textbf{r}} ,\end{aligned}$$ where $M$ denotes the reduced mass of the lattice ions. We remark that breaking the hierarchy at first order in $g_{\textrm{ph}}$ amounts to neglecting all correlations among quasiparticles and phonons. In particular, relaxation processes due to quasiparticle-phonon and phonon-phonon scattering are not taken into account. Since we focus on time scales shorter than the coherent-phonon and quasiparticle life-times, we neglect all of these higher-order processes, which are expected to 