state $|3\rangle$. Since we consider only the polarization mode interacting with the medium, both for the probe and the trigger, the initial state can be written as $$\begin{aligned}
|\psi_{in}\rangle &=& \bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i \otimes \left(c_{00}|0_p\rangle \otimes |0_t\rangle+c_{01}|0_p\rangle \otimes|1_t\rangle \right.
\nonumber \\
&& \left. +c_{10}|1_p\rangle \otimes |0_t\rangle
+c_{11}|1_p\rangle \otimes |1_t\rangle \right). \label{iniz}\end{aligned}$$ Due to the above assumptions, the passage of the two pulses through the atomic medium of length $L$ corresponds to the time evolution of this state, for a time $t_{int}=L/v_g$, according to the master equation [@FleischhauerRMP] $$\begin{aligned}
 \label{eq:mastercoll}
\dot{\rho} &=& -\frac{i}{\hbar}\left[ H,\, \rho\right] +\sum_{k}\frac{\gamma_{kk}}{2}\sum_{j=1}^{N_a}
\left( 2\sigma_{kk}^j\rho\sigma_{kk}^j - \sigma_{kk}^j\rho
- \rho\sigma_{kk}^j \right),\nonumber \\
&& + \sum_{kl}\frac{\gamma_{kl}}{2}\sum_{j=1}^{N_a}\left( 2\sigma_{kl}^j\rho\sigma_{kl}^{j \dagger}
- \sigma_{kl}^{j \dagger} \sigma_{kl}^{j}\rho
- \rho\sigma_{kl}^{j \dagger} \sigma_{kl}^{j}\right),\end{aligned}$$ including not only the coherent interaction described by the Hamiltonian of Eq. (\[eq:coll\_ham\]), but also the spontaneous emission from the excited states $l=2,4$ to the ground states $k=1,3,5$ ($\gamma_{kl}$ denotes the corresponding decay rate) and the dephasing of levels $|k\rangle $, $k=1,2,4,5$, with dephasing rate $\gamma_{kk}$. Typically the dephasing rates are much smaller than the decay rates, $\gamma_{kl} \gg \gamma_{kk}$, $\forall \;k,l$.

Since the initial state of Eq. (\[iniz\]) contains at most two excitations, the coherent time evolution driven by Eq. (\[eq:coll\_ham\]) is simple and 