probe and trigger pulse interact for a sufficiently long time. If the two single photon pulses enter simultaneously the atomic medium, their interaction time $t_{int}$ is optimized when the group velocities of the two pulses are equal, so that $t_{int}=L/v_g$, where $v_g$ is the common group velocity of the pulses and $L$ is the length of the gas cell. The inherent *symmetry* of the scheme guarantees perfect group velocity matching for probe and trigger whenever $\delta_1=\delta_4$, $\delta_2=\delta_3$ and $g_{p}/\Omega_{1}=g_{t}/\Omega_{4}$, where $g_j=\mu_j \sqrt{\omega_j/2\hbar \epsilon_0 V_j}$ ($j=p,t$) is the coupling constant between the quantum mode with frequency $\omega_j$ and mode volume $V_j$, and the corresponding transition with electric dipole moment $\mu_j$.

The importance of group velocity matching for achieving a significant nonlinear phase shift was first pointed out in Ref. [@Lukin00], which suggested to use a mixture of two different atomic species to achieve this goal. The first kind of atoms generates XPM by means of a four-level *N* scheme [@Schmidt96], in which however only the probe field undergoes EIT and it is slowed down, while the second kind of atoms realizes a three-level $\Lambda$ scheme able to slow down the trigger pulse. Group velocity matching is achieved by means of an 