restricted to a finite-dimensional Hilbert space involving few symmetric collective atomic states. In fact, each component of the initial state of Eq. (\[iniz\]) evolves independently in a different subspace. The component with no photon is an eigenstate of $H$ and does not evolve. The $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |0_p\rangle \otimes  |1_t\rangle$ component evolves in a three-dimensional Hilbert space which it spans together with the two states $|e_4^{(0,0)}\rangle $ and $|e_5^{(0,0)}\rangle$. Here, we have defined the symmetric collective states $$\label{eq:gen_state}
|e_r^{(n_p,n_t)}\rangle  = \frac{1}{\sqrt{N_{a}}}\sum_{i=1}^{N_{a}}{\left|3_{1},3_{2},\dots,r_{i},\dots,3_{N_{a}}\right\rangle} \otimes{\left|n_{p}\right\rangle}\otimes{\left|n_{t}\right\rangle},$$ where $r = 1,2,4,5$. In a similar fashion, the component with only one probe photon evolves in a three-dimensional Hilbert space spanned by the three states $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |1_p\rangle \otimes  |0_t\rangle$, $|e_1^{(0,0)}\rangle $ and $|e_2^{(0,0)}\rangle $. The component with one probe and one trigger photon evolves in the five dimensional subspace spanned by the four collective states $|e_1^{(0,1)}\rangle $, $|e_2^{(0,1)}\rangle $, $|e_4^{(1,0)}\rangle $, and $|e_5^{(1,0)}\rangle$ and the state $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |1_p\rangle \otimes  |1_t\rangle$.

Decoherence effects, and more specifically spontaneous emission from each atom complicates this dynamics. However, we are in the weak excitation limit where, for $l \neq 3$, $\langle \sigma_{ll}^j\rangle \simeq N_a^{-1} \ll 1$, as shown by the fact that the Hamiltonian dynamics involve only the symmetric atomic 