Optimal light storage in atomic vapor

We study procedures for the optimization of efficiency of light storage and retrieval based on the dynamic form of electromagnetically induced transparency in hot Rb vapor. We present a detailed analysis of two recently demonstrated optimization protocols: a time-reversal-based iteration procedure, which finds the optimal input signal pulse shape for any given control field, and a procedure based on the calculation of an optimal control field for any given signal pulse shape. We verify that the two procedures are mutually consistent and that they both independently achieve the maximum memory efficiency for any given optical depth. We observe good agreement with theoretical predictions for moderate optical depths $(<25)$, while at higher optical depths the experimental efficiency falls below the theoretically predicted values. We identify possible effects responsible for this reduction in memory efficiency.

Figure 1

(Color online) The three-level $\Lambda$ scheme used in theoretical calculations (a), the schematic (b), and the example control (c) and signal (d) fields during light storage. At the writing stage $(t<0)$, an input signal pulse ${\mathcal{E}}_{\mathrm{in}}\left(t\right)$ propagates through the atomic medium with low group velocity $v_g$ in the presence of a control field envelope $\Omega \left(t\right)$. While compressed inside the cell, the pulse is mapped onto a spin wave $S\left(z\right)$ by turning the control field off at time $t=0$. After a storage period $\tau$, the spin wave is mapped back into an output signal pulse ${\mathcal{E}}_{\mathrm{out}}\left(t\right)$ using the retrieval control field envelope $\Omega \left(t\right)$ $(t>\tau )$.

Figure 2

(Color online) Experimental apparatus (see text for abbreviations). Inset: Schematic of the ${}^{87}\mathrm{Rb}$ $D_1$ line level structure and relevant $\Lambda$ systems formed by control and signal fields.

Figure 3

(Color online) Iterative signal pulse optimization. The experimental data (solid black lines) are taken at $60.5°\mathrm{C}$ $(\alpha L=24)$ using $16\mathrm{mW}$ constant control field during writing and retrieval (solid red line in the top panel) with a $\tau =100\mu \mathrm{s}$ storage interval. Numerical simulations are shown with blue dashed lines. Left: Input pulses for each iteration. Right: Signal field after the cell, showing leakage of the initial pulse for $t<0$ and the retrieved signal field ${\mathcal{E}}_{\mathrm{out}}$ for $t>100\mu \mathrm{s}$. Here and throughout the paper, all pulses are shown in the same scale, and all input pulses are normalized to have the same area ${\int }_{-\mathrm{T}}^0{\mid {\mathcal{E}}_{\mathrm{in}}\left(t\right)\mid }^{2}dt=1$, where $t$ is time in microseconds.

Figure 4

(Color online) (a) Experimental (solid) and theoretical (dashed) optimized signal pulses obtained after five steps of the iteration procedure for three different powers of the constant control fields during writing and retrieval stages. (b) Corresponding memory efficiencies determined for each iteration step. Theoretically predicted optimal efficiency value is shown by the dashed line. The temperature of the cell was $60.5°\mathrm{C}$ $(\alpha L=24)$.

Figure 5

(Color online) Storage of three signal pulses $\left({\mathrm{a}}^{\prime }\right),\left({\mathrm{b}}^{\prime }\right),\left({\mathrm{c}}^{\prime }\right)$ using calculated optimal storage $(t<0)$ control fields (a), (b), (c). Input signal pulse shapes are shown in black dotted lines. The same graphs also show the leakage of the pulses (solid black lines for $t<0$) and retrieved signal pulses $(t>100\mu \mathrm{s})$ using flat control fields at the retrieval stage (dashed red lines), or using time-reversed control fields (solid red lines). Graphs $\left({\mathrm{a}}^{″}\right),\left({\mathrm{b}}^{″}\right),\left({\mathrm{c}}^{″}\right)$ show the results of numerical calculations of $\left({\mathrm{a}}^{\prime }\right),\left({\mathrm{b}}^{\prime }\right),\left({\mathrm{c}}^{\prime }\right)$. The temperature of the cell was $60.5°\mathrm{C}$ $(\alpha L=24)$.

Figure 6

(Color online) (a) Eight randomly selected signal pulse shapes (black lines) and their corresponding optimal control fields (red lines). (b) Memory efficiency for the eight signal pulse shapes using calculated optimized control fields at the writing stage, and flat control fields (open red diamonds) or inverted writing control fields (solid black circles) at the retrieval stage. Theoretically predicted optimal memory efficiency is shown by a dashed line. The temperature of the cell was $60.5°\mathrm{C}$ $(\alpha L=24)$.

Figure 7

(Color online) Memory efficiency as a function of optical depth obtained by carrying out iterative signal optimization until convergence. (a) At each optical depth, we considered constant control fields at four different power levels (indicated on the graph) during writing and retrieval stages. Note that many experimental data points overlap since the converged efficiencies are often the same for different control fields. Dashed lines are to guide the eye. Thin and thick black solid lines show the theoretically predicted maximum efficiency assuming no spin-wave decay and assuming an efficiency reduction by a factor of 0.82 during the $100\mu \mathrm{s}$ storage period, respectively. (b) Thin and thick black lines are the same as in (a), while the three lines with markers are calculated efficiencies for three different control fields (indicated on the graph) assuming spin-wave decay with a $500\mu \mathrm{s}$ time constant during all three stages of the storage process (writing, storage, retrieval).

Figure 8

(Color online) Results of the optimization procedures for different optical depths $\alpha L=24$ (red), 40 (black), and 50 (green). The top panel [(a), (b)] shows storage and retrieval (b) of the optimized input signal pulses (a) obtained by running iterative optimization until convergence for a constant control field of power $8\mathrm{mW}$ [dash-dotted line in (b)]. Solid lines correspond to experimental results, while dashed lines show the results of numerical simulations. In the bottom panel [(c), (d)], (c) shows the calculated optimal writing control fields $(t<0)$ for a steplike signal pulse [dotted line in (d)] and the time reverses of these control fields used during retrieval $(t>100\mu \mathrm{s})$, while (d) shows the resulting storage followed by retrieval.

Figure 9

(Color online) (a) Level diagram illustrating Stokes field $\left(E_S\right)$ generation due to resonant four-wave mixing. (b) Memory efficiency for retrieval in the signal channel [same as the red filled diamonds in Fig. 7a], Stokes channel, and the total for both channels. The efficiencies are obtained by carrying out iterative optimization until convergence for constant writing and retrieval control fields of $16\mathrm{mW}$ power. Dashed lines are to guide the eye. The solid line (same as the thick black line in Fig. 7) shows the theoretically predicted maximum efficiency assuming an efficiency reduction by a factor of 0.82 during the $100\mu \mathrm{s}$ storage period.