(\[eq:mastereff\]) and (\[eq:mastereff2\]), the master equation of Eq. (\[eq:mastercoll\]) in the weak excitation limit becomes $$\begin{aligned}
 \label{eq:mastercoll2}
\dot{\rho} &=& -\frac{i}{\hbar}\left[ H,\, \rho\right] +\sum_{k}\frac{\gamma_{kk}}{2} \left(2\hat{S}_{kk}\rho\hat{S}_{kk}-
\hat{S}_{kk}\rho - \rho\hat{S}_{kk}\right)
\nonumber \\
&& + \sum_{kl}\frac{\gamma_{kl}}{2}\left( 2\hat{S}_{kl}\rho\hat{S}_{kl}^\dagger
- \hat{S}_{kl}^\dagger\hat{S}_{kl}\rho - \rho\hat{S}_{kl}^\dagger\hat{S}_{kl} \right),\end{aligned}$$ that is, it involves only the operators of the collective atomic mode with index $\mu=0$. This actually means that the single photon probe and trigger pulses excite only a restricted number of collective atomic states, so that the atomic medium behaves as an effective *single* $5$-level atom, with a *collectively enhanced* coupling with the optical modes $g_j\sqrt{N_a}$, but with single-atom decay rates $\gamma_{kl}$, dephasing rates $\gamma_{kk}$, Rabi frequencies $\Omega_i$, and detunings $\delta_i$.

Spontaneous emission causes the four independent Hilbert subspaces corresponding to the four initial state components to become coupled. Moreover, the “cross” decay channels $|4\rangle \to |1\rangle $ and $|2 \rangle \to |5\rangle $ couple the above-mentioned collective states with six new states, $|e_1^{(1,0)}\rangle$, $|e_2^{(1,0)}\rangle $, $|e_3^{(2,0)}\rangle $ (populated if $\gamma_{41}\neq 0$), and $|e_5^{(0,1)}\rangle $, $|e_4^{(0,1)}\rangle $, $|e_3^{(0,2)}\rangle $ (populated if $\gamma_{25}\neq 0$). Therefore Eq. (\[eq:mastercoll2\]) actually describes dynamics in a Hilbert space of dimension $18$, which we have numerically solved in order to establish the performance of the QPG. Notice 