corresponding reduced Hamiltonian, with $\lambda_{H_{pt}}
\leftrightarrow \phi_{11}$, $\lambda_{H_{p}} \leftrightarrow \phi_{10}$, $\lambda_{H_{t}} \leftrightarrow \phi_{01}$ and $\phi_{00} = 0$, in agreement the with general definition of Eq. (\[eq:def\_cps\]). Following this procedure results in the following CPS $$\begin{aligned}
\label{eq:phipert}
\phi &=& \frac{g_{p}^{2}g_{t}^{2}N_a^2 t_{int}}{(\epsilon_{34}\delta_{3} - \Omega_{4}^{2})(\epsilon_{12}\delta_{2} - \Omega_{1}^{2})}
\times \nonumber \\ &&\times\left[\frac{\epsilon_{34}(\epsilon_{12}^{2}+\Omega_{1}^{2})}{(\epsilon_{12}\delta_{2} - \Omega_{1}^{2})}
+ \frac{\epsilon_{12}(\epsilon_{34}^{2} + \Omega_{4}^{2})}{(\epsilon_{34}\delta_{3} - \Omega_{4}^{2})}\right].\end{aligned}$$ This prediction is verified by the numerical solution of Eq. (\[eq:mastercoll2\]) in the limit of large detunings and small couplings. However the resulting CPS is too small, even for very long interaction times (i.e., long gas cells): for example, for $g_{p,t}\sqrt{N_a}=0.5$ MHz, $\epsilon_{12,34}=1.9$ MHz, $\Omega_{1,4}=65$ MHz and $\delta_{2,3}=1.9$ GHz, we obtain a tiny CPS of only $3\times 10^{-4}$ radians when $t_{int}=10^{-4}$ s, which corresponds to $L\simeq 30$ km. This is not surprising because this limit corresponds to a dispersive regime far from EIT. In this regime, transparency is achieved by means of a strong coupling field, producing a well-separated Autler-Townes doublet [@FleischhauerRMP]. At the same time, the size of nonlinearity is small due to the extremely weak coupling of the quantized fields to their respective transitions. The results are, therefore, not different from those expected from XPM in a standard optical fiber [@AgarwalBook]. Therefore, 