{#sec:transient}
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In Sec. \[sec:SSOperation\], we have found that the QPG operation of EIT-based nonlinear system in a steady-state is plagued by the trade-off between the phase shift size and the gate fidelity. In an attempt to find favorable conditions for the QPG operation, we consider the transient regime, when $\gamma t_{int}\lesssim 1$. As discussed above, in order to accumulate a significant CPS in such a short time one has to consider the strong coupling regime with a large ratio $g_j\sqrt{N_a}/\Omega_j$. Therefore, the trade-off between fidelity and a large nonlinear interaction is present also in the transient regime. However, when $g_j\sqrt{N_a}/\Omega_j$ is large, the transient dynamics is characterized by Rabi-like oscillations of the atomic populations and of the photon number, determining, as a consequence, coherent oscillations of the gate fidelity. In such a case one cannot exclude the existence of special values of the interaction time $t_{int}$ corresponding to a maximum of the gate fidelity close to one, and at the same time, to a value of the CPS close to $\pi$.

![image](cps-fidelity.eps)

We show that this fact is actually possible in Fig. \[fig:FidelityTransient\], where we see that a CPS of $\sim \pi$ radians is obtained in the transient regime 