## 5 Intervalley Scattering and Franck-Condon Transition
### Model and Method  
The model and method introduced in [12] are reviewed. We start from the following Hamiltonian (\(\equiv H\)),  
\[H=H_{0}+H_{\rm I},H_{0}=H_{\rm e}+H_{\rm P},H_{\rm I}=H_{\rm ee}+H_{\rm ep}, \tag{13}\]  
\[H_{\rm e}=\sum_{\mathbf{k},\sigma}E(\mathbf{k})a_{\mathbf{k},\sigma}^{\dagger}a_{\mathbf{k}, \sigma},H_{\rm P}=\sum_{\mathbf{q}}\omega_{\mathbf{q}}b_{\mathbf{q}}^{\dagger}b_{\mathbf{q}}, \tag{14}\]  
\[H_{\rm ee}=U\sum_{I}n_{I,\uparrow}n_{I,\downarrow}, \tag{15}\]  
\[H_{\rm ep}=(2N)^{\frac{1}{2}}\sum_{\mathbf{q},\mathbf{k},\sigma}S_{\mathbf{q}}(b_{\mathbf{q}}^{ \dagger}+b_{-\mathbf{q}})a_{\mathbf{k}-\sigma}^{\dagger}a_{\mathbf{k},\sigma}. \tag{16}\]  
Here, \(E_{i}(\mathbf{k})\) is the one electron energy in the conduction band, where \(i=\Gamma\) or L correspond to each valley state. The others are the same notation as those in Section 4.2. The unit of length is the lattice constant along the [111] direction of the GaAs crystal. \(U\) is the on-site Coulombic repulsion energy. The time constants for the e-e process and the elastic e-ph one are estimated as inverse of the corresponding transition rate, which is derived by the second order perturbation theory. Thus, we can follow the same manner in Section 4.2 and obtain the rate equation as:  
\[\frac{\partial\langle n_{\mathbf{k},\sigma}(t)\rangle}{\partial t}=\left\{\Gamma_{ \rm ee,\mathbf{k},\sigma}^{+}+\Gamma_{\rm ep,\mathbf{k},\sigma}^{+}(T_{\rm P})\right\} -\left\{\Gamma_{\rm ee,\mathbf{k},\sigma}^{-}+\Gamma_{\rm ep,\mathbf{k},\sigma}^{-}(T _{\rm P})\right\}, \tag{17}\]  
where \(\Gamma_{\rm ee,\mathbf{k},\sigma}^{\pm}\) and \(\Gamma_{\rm ep,\mathbf{k},\sigma}^{\pm}(T_{\rm P})\) are the transition rate for the e-e and e-ph processes, respectively. Since we treat only the \(\Gamma\)-L process soon after the photo-excitation, we need transition rates only for the decay channel of the \(\Gamma\) valley electrons to the L valley. For the e-e scattering and the elastic e-ph one, the decay transition rates are given by:  
\[\Gamma_{\rm ee,\mathbf{k},\sigma}^{-}=\frac{2\pi U^{2}}{N^{2}}\sum_{\mathbf{k}^{ \prime},\mathbf{q}}\delta(E_{L}(\mathbf{k}+\mathbf{q})+E_{L}(\mathbf{k}^{\prime}-\mathbf{q})-E_{ \Gamma}(\mathbf{k})-E_{\Gamma}(\mathbf{k}^{\prime})), \tag{18}\]  
and:  
\[\Gamma_{\rm ep,\mathbf{k},\sigma}^{-}(T_{\rm P})=\frac{2\pi}{N}\sum_{\mathbf{q}}S_{ \mathbf{q}}^{2}(\langle\langle n_{\mathbf{q}}(T_{\rm P})\rangle\rangle+\frac{1}{2}) \delta(E_{L}(\mathbf{k}+\mathbf{q})-E_{\Gamma}(\mathbf{k})), \tag{19}\]  
respectively, with the phonon number \(\langle\langle n_{\mathbf{q}}(T_{\rm P})\rangle\rangle\). \(E_{\Gamma(L)}\) represents the band energy in the \(\Gamma\)(L)-valley. In the derivation of Equations (18) and (19), it is assumed that the initial \(\Gamma\) valley is fully occupied and the L valley is completely unoccupied. Although this assumption may restrict the situation rather strongly, we can know the trend and characteristics of each scattering process to argue the possible ultrafast IVS. Figure 8: A schematic explanation of (**a**) elastic e-e scattering (Coulomb burst) and (**b**) elastic e-phonon (ph) scattering is shown. Adapted with the permission from [12], copyright Journal of the Physical Society of Japan, 2016.