Photon storage in $\Lambda$-type optically dense atomic media. I. Cavity model

In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we used a universal physical picture to optimize and demonstrate equivalence between 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. In the present paper, we perform the same analysis for the cavity model. In particular, we show that the retrieval efficiency is equal to $C∕(1+C)$ independent of the retrieval technique, where $C$ is the cooperativity parameter. We also derive the optimal strategy for storage and, in particular, demonstrate that at any detuning one can store, with the optimal efficiency of $C∕(1+C)$, any smooth input mode satisfying $TC\gamma ⪢1$ and a certain class of resonant input modes satisfying $TC\gamma \sim 1$, where $T$ is the duration of the input mode and $2\gamma$ is the transition linewidth. In the two subsequent papers of the series, we present the full analysis of the free-space model and discuss the effects of inhomogeneous broadening on photon storage.

Figure 1

(Color online) $\Lambda$-type medium coupled to a classical field (solid) with Rabi frequency $\Omega \left(t\right)$ and a quantum field (dashed). Due to collective enhancement [27], the quantum field couples to the spin-wave excitations with an effective coupling constant $g\sqrt{N}$, where $N$ is the number of atoms in the medium.

Figure 2

Breakdown of optimal adiabatic storage in a cavity at $TC\gamma \lesssim 10$. In (a), the total efficiency of storage followed by retrieval is plotted for $\Delta =0$ and $C=1$, 10, 100, and 1000. The horizontal dashed lines are the maximal values ${[C∕(1+C)]}^2$. Dotted lines are obtained for the input from Eq. (34) using adiabatic Eq. (26) to shape the storage control but then using exact Eqs. (10, 11) to numerically compute the efficiency. In (b), the same plot is made for $C=10$ and $\Delta ∕\gamma =0$, 1, 10, 100, and 1000.