(and both approach $1$) for an ideal QPG in which the number of photons is conserved and all the atoms remain in state $|3\rangle$. This ideal condition is verified in the limit of large detunings $\delta_j \gg \gamma_{kj}$ so that spontaneous emission is significantly suppressed and can be neglected, and very small couplings $g_j\sqrt{N_a}\ll \Omega_j$. In this limit, each component of the initial state of Eq. (\[iniz\]) practically coincides with the dark state of the four independent Hamiltonian dynamics discussed in Sec. \[sec:model\]. The system with the initial state containing zero probe and trigger photons does not evolve, i.e. stays in the initial state $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |0_p\rangle \otimes  |0_t\rangle$ all the time.

![image](CPS500)![image](Flong.eps)

The subsystem containing one probe and zero trigger photons as the initial state evolves according to a reduced three-dimensional Hamiltonian, which in the basis formed by the states $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |1_p\rangle \otimes  |0_t\rangle$, $|e_2^{(0,0)}\rangle $ and $|e_1^{(0,0)}\rangle $, is given by

\[eq:HamiltonianMatrices\] $$\label{eq:Hp}
H_{p} = \left(\begin{array}{ccc}0 & g_p\sqrt{N_{a}} & 0 \\g_p\sqrt{N_{a}} & \delta_{2} & \Omega_{1} \\0 & \Omega_{1} & \epsilon_{12}\end{array}\right).$$ Similarly, the subsystem containing one trigger and zero probe photons as the initial state evolves according to a reduced three-dimensional Hamiltonian, which in the basis formed by 