states of the form of Eq. (\[eq:gen\_state\]). This limit allows a drastic simplification of the effective time evolution. Following Duan [*et al.*]{} [@Cirac02], we can introduce Fourier transforms of the individual atomic operators $s^\mu_{kl} = \sum_{j=0}^{N_a-1}\sigma_{kl}^je^{ij\mu/N_a}/\sqrt{N_a}$, where $s^0_{kl}=\hat{S}_{kl}$ are the collective operators defined in Eq. (\[eq:collective1\]). The sum over the atoms in Eq. (\[eq:mastercoll\]) then transforms to the sum over the collective atomic modes with index $\mu$. In the weak excitation limit, the operators $s^\mu_{kl}$ approximately commute with each other. This means that they represent independent collective atomic modes, and one can trace over the $\mu \neq 0$ modes, so that the spontaneous emission term in the master equation becomes $$\sum_{kl}\frac{\gamma_{kl}}{2}\left( 2\hat{S}_{kl}\rho\hat{S}_{kl}^\dagger
- \hat{S}_{kl}^\dagger\hat{S}_{kl}\rho - \rho\hat{S}_{kl}^\dagger\hat{S}_{kl} \right),
\label{eq:mastereff}$$ where the sum is now over the six “collective” spontaneous decay channels only, each characterized by the *single-atom* decay rate $\gamma_{kl}$. A similar argument applies to the dephasing term in the master equation (\[eq:mastercoll\]). In fact, if we restrict to the subspace of the symmetric collective states of Eq. (\[eq:gen\_state\]) involving only single atomic excitations, we can approximate in the dephasing terms of the master equation, $$\label{eq:mastereff2}
\sum_{k}\gamma_{kk}\sum_{j=1}^{N_a}
\sigma_{kk}^j\rho\sigma_{kk}^j \simeq \sum_{k}\gamma_{kk} \hat{S}_{kk}\rho\hat{S}_{kk},$$ where $ \hat{S}_{kk}$ is given by Eq. (\[eq:collective2\]). Using Eqs. 