the discussion to ultracold ensembles. As was shown in Ref. [@ref:Gorshkov_PRA2], the important parameter governing the efficiency of the storage process is the optical depth, defined as $$d = \frac{g^2 N L}{\gamma c},$$ where $g$ is the single-photon coupling constant for the transition and $\gamma$ is the decay of the coherence of the excited state (for a purely radiative decay with no additional dephasing $2\gamma=\Gamma$, the spontaneous emission rate). Note that the $d$ used here and in Ref. [@ref:Gorshkov_PRA2] is equal to half of the optical depth as usually defined ($2d = d_{std} = \sigma \rho L$ where $\sigma$ is the resonant absorption cross-section of a single atom and $\rho$ is the number density). The classical control field is described by a Rabi frequency envelope $\Omega(z,t)=\Omega(z-t/c)$ centered at frequency $\nu_{es} = \omega_{es}+\delta_s$. As also noted in App. A, this Rabi frequency is defined to be 1/2 of the usually defined Rabi frequency. The quantum field is described by a slowly varying envelope operator $\mathcal{\hat{E}}(z,t)$ centered at frequency $\nu_{eg} = \omega_{eg}+\delta_g$. In addition to $\mathcal{\hat{E}}(z,t)$, there are two other operators required to describe the dynamics: the polarization operator $\hat{P}_1(z,t) = \sqrt{N}\hat{\sigma}_{g1}(z,t)$ and the spin-wave operator $\hat{S}(z,t) = \sqrt{N}\hat{\sigma}_{gs}(z,t)$ where $\hat{\sigma}_{\alpha \beta}(z,t)$ 