switch only $\Omega_e$ and adiabatically eliminate the photonic modes we also obtain an irreversible transition from ${|e\rangle}\rightarrow{|g\rangle}$, but with a renormalization of the decay rates $\Gamma_\oned \rightarrow \Gamma_\oned | \frac{\Omega_e}{2 \delta} |^2$ and $\Gamma^* \rightarrow \Gamma^* | \frac{\Omega_e}{2\delta}|^2$. Hence, the Purcell factor $P_\oned = \Gamma_\oned / \Gamma^*$ is unchanged. In that situation our analysis is an alternative implementation to the one developed in Refs. [@beige00a; @beige00b; @tregenna02a].

Decoherence-Free Subspaces
--------------------------

In the case of equidistant spacing at positions commensurate with the wavelength of the guided mode, i.e. $z_n =n 2\pi/q(\omega_{{\rm a}})$, the effective interaction induced by the waveguide modes yields a pure Dicke model [@dicke54a] decay described by $${\cal L}_\DD \left[ \rho \right] 
    = \frac{\Gamma_\oned}{2} \left(S_{ge} \rho S_{eg} -  S_{eg} S_{ge} \rho \right) + \mathrm{H.c.},
    \label{Dicke}$$ where we have introduced the collective spin operator $S_{ge} = \sum_{n=1}^N \sigma^n_{ge}$. The states satisfying $S_{ge} {|\Psi\rangle}=0$ are *decoherence-free* with respect to the collective dissipation $\mathcal{L}_\DD$. These states can be easily described in the collective spin basis $\{| J, m_J, \alpha_J \rangle \}$, that is the eigenstates of the collective operators $S^2 =\sum_{i={x,y.z}}S_{i}^2$ and $S_z$ with $$\begin{aligned}
    S^2{|J,m_J,\alpha_J\rangle}&=J(J+1){|J,m_J,\alpha_J\rangle}\,,\\
    S_z{|J,m_J,\alpha_J\rangle}&=m_J {|J,m_J,\alpha_J\rangle}\,,\end{aligned}$$ where $J=N/2,N/2-1,\dots$, $m_J=-J, -J+1,\dots,J$. The index $\alpha_J$ is introduced because the states in the 