\nonumber \\
& & \approx    e^{i(\tau\phi_{\vec{k}+\vec{q}} - \tau\phi_{\vec{k}}  - \tau'\phi_{\vec{k}'})/2} \frac{b_{v,s'}}{\sqrt{A}} \chi_{d}(\vec{k}'+\vec{q})\end{aligned}$$ Similarly, the electron capture process (Fig.\[fig:auger\_1\](a)) is described by the Hamiltonian, $$H_{ec} =  \frac{1}{A} \sum_{\vec{k},\vec{k}',\vec{q},s,s'} V(\vec{q}) L_{s,s'}(\vec{k},\vec{k}',\vec{q})  d^{\dagger}_{\sigma} b^{\dagger}_{\vec{k}',s'} b_{\vec{k}'+\vec{q},s'} c_{\vec{k},s}  +  h.c.$$ where overlap factor $L_{s,s'}(\vec{k},\vec{k}',\vec{q})$ equals, $$L_{s,s'}(\vec{k},\vec{k}',\vec{q}) \approx   e^{i(\tau'\phi_{\vec{k}'+\vec{q}} - \tau'\phi_{\vec{k}'}  - \tau \phi_{\vec{k}})/2} \frac{b^{*}_{c,s}}{\sqrt{A}} \chi^{*}_{d}(\vec{k}+\vec{q})$$ The potential of the defect does not appear in the Hamiltonian above. The reason for this is that it has already been taken into account in defining the non-interacting Hamiltonian, and its eigenstates, in Section (\[sec:nonintH\]).

Electron and Hole Capture Rates for Excitons
============================================

We assume an initial state described by the density operator $\rho_{i}$ in which the exciton occupation $n_{s,s',\alpha}(\vec{Q})$, defect occupation $f_{d}$, and conduction and valence band occupations are given by, $$\begin{aligned}
&& \langle d^{\dagger}_{\sigma}  d_{\sigma'}  \rangle    =  f_{d} \, \delta_{\sigma,\sigma'}  \nonumber \\ \label{eq:ex_avg}
&& \langle  c^{\dagger}_{\vec{k},s}  c_{\vec{k}',s'}\rangle  = f_{c,s}(\vec{k})  \delta_{s,s'}  \delta_{\vec{k},\vec{k}'} \nonumber \\
&& \langle   b^{\dagger}_{\vec{k},s}  b_{\vec{k}',s'}  \rangle  = f_{v,s}(\vec{k}) \delta_{s,s'}  \delta_{\vec{k},\vec{k}'} \nonumber \\
&& \langle B^{\dagger}_{s,s',\alpha}(\vec{Q})  B_{s,s',\alpha}(\vec{Q}) \rangle   =  n_{s,s',\alpha}(\vec{Q})  + \nonumber \\
&& \frac{1}{A}\sum_{\vec{k}} |\phi_{\alpha}(\vec{k})|^{2} f_{c,s}(\vec{k} + \frac{m_{e}}{m_{ex}} \vec{Q}) \left[ 1 - f_{v,s'}(\vec{k} -  \frac{m_{h}}{m_{ex}} \vec{Q}) \right] \nonumber \\ \end{aligned}$$ The angled brackets stand for ensemble averaging with respect to the the density operator $\rho_{i}$. 