the states $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |0_p\rangle \otimes  |1_t\rangle$, $|e_4^{(0,0)}\rangle $ and $|e_5^{(0,0)}\rangle $, is given by $$\label{eq:Ht}
H_{t} = \left(\begin{array}{ccc}0 & g_t\sqrt{N_{a}} & 0 \\g_t\sqrt{N_{a}} & \delta_{3} & \Omega_{4} \\0 & \Omega_{4} & \epsilon_{34}\end{array}\right).$$ Finally, the subsystem containing initially one photon each in probe and trigger modes evolves according to a reduced five-dimensional Hamiltonian, which in the basis formed by the states $|e_1^{(0,1)}\rangle $, $|e_2^{(0,1)}\rangle $, $\bigotimes_{i=1}^{N_a} {\left|3\right\rangle}_i |1_p\rangle \otimes  |1_t\rangle$, $|e_4^{(1,0)}\rangle $, and $|e_5^{(1,0)}\rangle$, is given by $$\label{eq:Hpt}
H_{pt}=\left(\begin{array}{ccccc}\delta_{2} & \Omega_{1} & g_{p}\sqrt{N_{a}} & 0 & 0 \\\Omega_{1} & \epsilon_{12} & 0 & 0 & 0 \\g_{p}\sqrt{N_{a}}
& 0 & 0 & g_{t}\sqrt{N_{a}} & 0 \\0 & 0 & 0 & \delta_{3} & \Omega_{4} \\0 & 0 & g_{t}\sqrt{N_{a}} & \Omega_{4} & \epsilon_{34}\end{array}\right).$$

The phase accumulation experienced by the various components of the quantum state of the fields will be proportional to the eigenvalues of these matrices. The four phase shifts $\phi_{ij}$ can be evaluated as a fourth-order perturbation expansion of the eigenvalue corresponding to the dark state in each subspace, multiplied by the interaction time $t_{int}$. The CPS is then calculated as $$\label{eq:cps_eigs}
\phi = (\lambda_{H_{pt}} - \lambda_{H_{p}} - \lambda_{H_{t}}) t_{int},$$ where the $\lambda$’s denote the eigenvalues of the 