Coherence in a cold-atom photon switch

We consider an all-optical photon switch in an atomic gas which is gated via a so-called Rydberg spin wave, i.e., a single Rydberg excitation that is coherently shared by the whole ensemble. A current challenge is to preserve the coherence of the Rydberg spin wave during the operation of the switch, which would enable, for example, its coherent optical readout. By employing a simple model description we investigate how the coherence of the Rydberg spin wave is affected during the operation of the switch. The coherence becomes increasingly protected with growing interatomic interaction strength. For the strongly interacting limit we derive analytical expressions for the spin-wave fidelity as a function of the optical depth and bandwidth of the incoming photon.

Figure 1

(a) EIT level scheme. The ground state $|1\rangle$, excited state $|2\rangle$ (decay rate $\gamma$), and Rydberg state $|3\rangle$ are resonantly coupled by a single-photon field $\mathcal{E}(z,t)$ (with collective coupling strength $g$) and a classical field of Rabi frequency $\mathrm{\Omega }$. Initially a gate photon is stored as a spin wave in the Rydberg state $|4\rangle$ (indicated by the green circle). (b,c) Polarization profiles $P_j(z,t)$ for a spin wave consisting of two possible gate atom positions $Z_j$ ($j=1,2$) and their dependence on the blockade radius $R_{\mathrm{b}}$ and the system length $L$. (b) For $L>R_{\text{b}}$ and $|Z_2-Z_1|>2R_{\text{b}}$, the polarization profiles associated with the two gate atom positions are distinguishable. (c) When $L\lesssim R_{\text{b}}$ the polarization profile is independent of the gate atom position, which leads to enhanced coherence of the stored spin wave.

Figure 2

(a) Photon transmission $T$ as a function of the pulse duration $\tau$ for $g=1000\gamma$ (squares) and $g=500\gamma$ (circles). The medium becomes transparent when $\gamma \tau \ll 1$, i.e., the bandwidth $\mathrm{\Delta }\omega$ of the pulse is large. (b) Photon transmission as a function of the coupling constant $g$ for $R_{\text{b}}=L$ (squares) and $R_{\text{b}}=L/2$ (circles). The solid curve is the analytical result obtained from Eq. (7). The dashed curve is plotted as a guide to the eye. Note that $T\approx 0$ when $g=1000\gamma$ for both $R_{\text{b}}=L$ and $R_{\text{b}}=L/2$. The data is calculated for rubidium atoms with the parameters $L=20\mu \mathrm{m}, \mathrm{\Omega }=2\gamma$, and $\gamma \approx 2\pi \times 5.7$ MHz. The blockade radius can be changed through selecting different Rydberg states. For example, $R_{\text{b}}=L=20\mu \mathrm{m}$ when $|3\rangle =|127S\rangle$ and $|4\rangle =|130S\rangle$, where $C_6\approx 4.2\times {10}^6\text{GHz}\mu {\text{m}}^6$.

Figure 3

(a,b) Squared modulus of the polarization ${\left|P(z,t)\right|}^2$ and (c,d) the time-integrated intensity $I_{\text{p}}\left(z\right)$ for $R_{\text{b}}=L/2$ and $g=1000\gamma$ for two different positions of the gate atom. The gate atom (green circle) is located at $Z_i=0$ in panels (a) and (b) and at $Z_i=L$ in panels (c) and (d). The dashed line marks the blockade radius with respect to the gate atom position.

Figure 4

(a) Fidelity (circles) and transmission (squares) as a function of the blockade radius $R_{\text{b}}$ with $g=1000\gamma$. The transmission is already negligible when $R_{\text{b}}/L>0.4$, but the fidelity approaches unity only when $R_{\text{b}}\sim L$. (b) Intensity $I_{\text{p}}\left(z\right)$ in the strong blockade regime ($R_{\text{b}}=L$) for $g=1000\gamma$ (square) and $g=512\gamma$ (circle). The squares and circles are numerical data. The solid and dotted curves are the analytical results obtained from Eq. (6).