that, due to the combined action of the cross-decay channels and of the Hamiltonian (\[eq:coll\_ham\]), the states $|e_1^{(1,0)}\rangle$, $|e_2^{(1,0)}\rangle $, $|e_5^{(0,1)}\rangle $ and $|e_4^{(0,1)}\rangle $ are coupled also to *doubly excited* atomic collective states without photons which are neglected by our treatment. However, as we shall see below in the paper, a good QPG performance is possible only when spontaneous emission events are rare. Under this condition, the probability to populate these doubly excited atomic collective states during the atom-field interaction is completely negligible, and therefore our model based on the effective single five-level atom description provided by Eq. (\[eq:mastercoll2\]) is essentially correct.

In summary, the model outlined above relies on the single-photon nature of the excitations. In this case, the collective operators (\[eq:collectiveop\]) effectively switch between the states making the superposition ${\left|\psi_{in}\right\rangle}$ (\[iniz\]), and the symmetric collective states ${\left|e_r^{(n_p,n_t)}\right\rangle}$ (\[eq:gen\_state\]). This is central to the reasoning leading to the effective master equation (\[eq:mastercoll2\]).

To characterize the QPG operation, we calculate the CPS $\phi$ of Eq. (\[eq:def\_cps\]) and the [*fidelity*]{} of the gate. The accumulated CPS $\phi$ as a function of the interaction time $t_{int}$ is obtained by using the fact that the phase shifts $\phi_{ij}$ of Eq. (\[eq:def\_cps\]) are 