levels must be taken into account.

Photon Storage in a Four-level System {#sec:4L}
=====================================

We now extend the treatment of Sec. \[sec:3L\] to include another excited state that can couple to both the control field and the input quantum field as shown in Fig. \[fig:fig5\]. Because the maximum Rabi frequency of the control pulse can exceed the excited state level splitting $\Delta_e$, an atom can go from ${\ensuremath{|g\rangle}}$ to ${\ensuremath{|s\rangle}}$ via ${\ensuremath{|1\rangle}}$ or ${\ensuremath{|2\rangle}}$. Depending on the dipole matrix elements $\tilde{\mu}_{\alpha \beta}$, these paths can interfere constructively or destructively.

Following the same procedure as in Sec. \[sec:3L\] and adding another polarization $\hat{P}_2(z,t) = \sqrt{N}\hat{\sigma}_{g2}(z,t)$ results in the following equations of motion (see App. A for details):

$$\begin{aligned}
\partial_{\tilde{z}} \mathcal{E} &= i\sqrt{d \gamma}\left[\tilde{\mu}_{1g} P_1 + \tilde{\mu}_{2g} P_2\right] \label{eqn:4LEoME}\\
\partial_{\tau}P_1 &= \left(i \delta_g-\gamma \right) P_1+i \tilde{\mu}_{1s} \Omega(\tau) S +i \tilde{\mu}_{1g} \sqrt{d \gamma} \mathcal{E} \label{eqn:4LEoMP1}\\
\partial_{\tau}P_2 &= \left(i\delta_g -i\Delta_e-\gamma\right) P_2 + i \tilde{\mu}_{2s} \Omega(\tau) S \nonumber \\
&+ i \tilde{\mu}_{2g} \sqrt{d \gamma} \mathcal{E} \label{eqn:4LEoMP2}\\
\partial_{\tau}S &= i \left(\delta_g - \delta_s \right) S + i\Omega^*(\tau)\left[ \tilde{\mu}_{1s}P_1 + \tilde{\mu}_{2s}P_2 \right] \label{eqn:4LEoMS}\end{aligned}$$

where new relative dipole moments $\tilde{\mu}_{2g}$ and $\tilde{\mu}_{2s}$ have been introduced to represent the couplings to the additional excited state. As the spin wave 