results are shown in Fig. \[fig:CPSshort\]. To obtain a CPS of the order of $\pi$ in a shorter interaction time ($t_{int}\sim 50/\gamma$), we have assumed a larger ratio $g_j\sqrt{N_a}/\Omega_j$. The trade-off between the amount of accumulated nonlinear phase shift and the gate fidelity is now even more pronounced: we find at best a CPS of $\sim\pi$ in correspondence with fidelities ${\mathcal F}(t_{int})$ and ${\mathcal F}^c(t_{int})$ equal to $65\%$ and $73\%$, respectively. As expected, having a stronger atom-field coupling enhances the processes lowering the fidelity. The system ends up with a large CPS faster, but this is achieved with a final state in which the probability of loosing the probe and trigger photons by spontaneous emission or within the atomic medium is no more negligible. We notice that in this short interaction time case dephasing does not have an appreciable effect, that is, the results without dephasing are indistinguishable from those with dephasing shown in Fig. \[fig:CPSshort\], due to fact that dephasing rates in a dilute gas are typically much smaller than decay rates. The only possible way to circumvent this trade-off is to explore the transient regime, which will be discussed in the following Section.

QPG Operation in Transient Regime 