Photon storage in $\Lambda$-type optically dense atomic media. II. Free-space model

In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we presented a universal physical picture for describing a wide range of techniques for storage and retrieval of photon wave packets in $\Lambda$-type atomic media in free space, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo based techniques. This universal picture produced an optimal control strategy for photon storage and retrieval applicable to all approaches and yielded identical maximum efficiencies for all of them. In the present paper, we present the full details of this analysis as well some of its extensions, including the discussion of the effects of non-degeneracy of the two lower levels of the $\Lambda$ system. The analysis in the present paper is based on the intuition obtained from the study of photon storage in the cavity model in the preceding paper [Gorshkov et al., previous paper, Phys. Rev. A. 76, 033804 (2007)].

Figure 1

(Color online) (a) Storage, (b) forward retrieval, and (c) backward retrieval setup. The smooth solid curve is the generic control field shape $\left(\Omega \right)$ for adiabatic storage or retrieval; the dotted square pulse indicates a $\pi$-pulse control field for fast storage or retrieval. The dashed line indicates the quantum field $\mathcal{E}$ and the spin-wave mode $S$. During storage, $\mathcal{E}(L,t)$ is the “leak,” whereas it is the retrieved field during retrieval.

Figure 2

Optimal modes ${\tilde{S}}_d\left(\tilde{z}\right)$ to retrieve from (in the forward direction) at indicated values of $d$. The flipped versions of these modes ${\tilde{S}}_d(1-\tilde{z})$ are the optimal modes for backward retrieval and are also the optimal (normalized) spin waves $S(\tilde{z},T)∕\sqrt{{\eta }_{\mathrm{s}}^{\max }}$ for adiabatic and fast storage if it is optimized for storage alone or for storage followed by backward retrieval (${\eta }_{\mathrm{s}}^{\max }$ is the maximum storage efficiency).

Figure 3

Input mode ${\mathcal{E}}_{\mathrm{in}}\left(t\right)$ (dashed) defined in Eq. (40) and control fields $\Omega \left(t\right)$ (in units of $\sqrt{d\gamma ∕T}$) that maximize for this ${\mathcal{E}}_{\mathrm{in}}\left(t\right)$ the efficiency of resonant adiabatic storage (alone or followed by backward retrieval) at $d=1$, 10, 100, and $d\to \infty$.

Figure 4

${\eta }_{\mathrm{back}}^{\max }$ (solid) and ${\eta }_{\mathrm{forw}}^{\max }$ (dotted) are maximum total efficiency for adiabatic (or fast) storage followed by backward or forward retrieval, respectively. ${\eta }_{\text{square}}$ (dashed) is the total efficiency for resonant storage of ${\mathcal{E}}_{\mathrm{in}}\left(t\right)$ from Eq. (40) followed by backward retrieval, where the storage control field is a naive square pulse with the power set by $v_{g}T=L$.

Figure 5

For different values of $d$, the optimal spin-wave mode to be used during storage followed by forward retrieval.

Figure 6

Breakdown of optimal adiabatic storage in free space at $Td\gamma \lesssim 10$. In (a), the total efficiency of storage followed by backward retrieval is plotted for $\Delta =0$ and $d=1$, 10, 100, and 1000. The horizontal dashed lines are the maximal values. Dotted lines are obtained for the input from Fig. 3 using adiabatic equations to shape the storage control but then using exact equations to numerically compute the efficiency. In (b), the same plot is made for $d=10$ and $\Delta ∕\gamma =0,1,10,100,200$.

Figure 7

If the two metastable states are not degenerate, the efficiency of storage followed by backward retrieval will be lowered relative to the degenerate case, because the energy difference $\hslash {\omega }_{\mathrm{sg}}$ introduces a momentum difference $\Delta k={\omega }_{\mathrm{sg}}∕c$ between the quantum and classical fields. As a function of $d$, the figure shows the momentum $\Delta kL$, at which the optimal total efficiency of storage followed by backward retrieval falls to half of the $\Delta kL=0$ value (dashed), and at which it is decreased to the optimal efficiency with forward retrieval (solid).

Figure 8

(a) The magnitude and (b) the phase of the optimal mode for storage followed by backward retrieval at $d=20$ for the indicated values of $\Delta kL$. The phase of the optimal mode at $\Delta kL=0$ is 0. The phase is plotted for the forward-propagation slowly varying operators as defined in Eq. (A10).