Storage and retrieval of interacting photons in a Rydberg medium

Storing interacting photons in ensembles of highly excited Rydberg atoms makes it possible to realize photon-photon gates. The efficiency of stopping and retrieving pairs of single-photon pulses in any Rydberg medium is, however, significantly influenced by their mutual interaction. The corresponding dynamical process is rather complicated due to the interaction constantly changing with the relative positions of photon pulses and the associated dissipation. Applying the numerical calculations based on first principles, we reveal the detail of such a dynamical process by showing how the interacting pulses evolve in various situations. Other important issues, such as the use of nonadiabatic passage for stopping photons and the optimal way to regenerate photons, are also investigated.

Figure 1

(a) Energy-level scheme of a $\mathrm{\Xi }$-type or ladder-type atomic system. Here ${\mathrm{\Delta }}_p={\omega }_{eg}-{\omega }_p$ and ${\mathrm{\Delta }}_c={\omega }_{re}-{\omega }_c$, as the differences between a level gap and a field central frequency. (b) Coupling field ${\mathrm{\Omega }}_c\left(t\right)={\mathrm{\Omega }}_c^{M}tanh\{(t_{\text{off}}-t)/{\tau }_{\text{off}}\}\left(t⩽t_{\text{off}}\right)$; ${\mathrm{\Omega }}_c\left(t\right)=0\left(t_{\text{off}}<t<t_{\text{on}}\right)$; ${\mathrm{\Omega }}_c\left(t\right)={\mathrm{\Omega }}_c^{M}tanh\{(t_{\text{on}}-t)/{\tau }_{\text{on}}\}\left(t⩾t_{\text{on}}\right)$ switched off at $t_{\text{off}}$ and on at $t_{\text{on}}$ for realizing the storage and retrieval of a photon pulse. The solid, dot-dashed, and dotted curves indicate those with the gradually slowing speed of switch (corresponding to the increasing ${\tau }_{\text{off}}$ and ${\tau }_{\text{on}}$). (c) Geometry of the pulse propagations. The diameter of each ensemble is $d$ and the distance between the axes of the two ensembles is $a$. Two single-photon pulses propagate in two different ensembles either along the same direction like the red and blue one or from the opposite tips like the two red ones. The pulses are stopped in the ensembles as the yellow ones with their amplitudes diminished from the damping in the medium, and interact with each other through the long-range van der Waals potential.

Figure 2

Dynamical evolutions of photon pulses (a) and (c) and the induced spinwave in the unit $\mu {\mathrm{m}}^{-3/2}$ (b) and (d) throughout a whole storage-retrieval process. Two pulses counterpropagate in (a) and (b) and copropagate in (c) and (d). Two $400\text{−}\mu \mathrm{m}$-long ensembles fill with ${}^{87}\mathrm{Rb}$ atoms of the level scheme $\left|g\right.〉=5S_{1/2}$, $\left|e\right.〉=5P_{3/2}$, and $\left|r\right.〉=100S_{1/2}$, with $C_6=-2.3\times {10}^5$ GHz $\mu {\mathrm{m}}^6$, $\gamma =2\pi ·6.1$ MHz, and ${\gamma }^{\prime }=1.8$ kHz. The control field switch off (on) at $t_{\text{off}}=40\mu \mathrm{s}$ ($t_{\text{on}}=50\mu \mathrm{s}$) with ${\tau }_{\text{off}}={\tau }_{\text{on}}=5\mu \mathrm{s}$ and ${\mathrm{\Omega }}_c^M=2\pi ·2$ MHz. The parameters of the photon pulses are ${\mathrm{\Omega }}_p^M=0.01$ MHz, $t_p=12.0\mu \mathrm{s}$, and ${\tau }_p=7.0\mu \mathrm{s}$, and those of the ensembles are $N=2\times {10}^{13}{\mathrm{cm}}^{-3}$, $a=10\mu \mathrm{m}$, and $d=2\mu \mathrm{m}$.

Figure 3

(a1)–(d1) Two pulses counterpropagating are stopped in two 300-$\mu \mathrm{m}$-long ensembles given a control field being switched off with ${\tau }_c=10\mu \mathrm{s}$, $t_{\text{off}}=40\mu \mathrm{s}$. (a2)–(d2) The corresponding quantities for the copropagating pulses, while the control field is turned off at $t_{\text{off}}=24\mu \mathrm{s}$. Ensemble length for copropagation is flexible as long as the together pulses can be contained inside. (a1) and (a2) Spinwave profiles and (b1) and (b2) corresponding potentials $|V^{\text{eff}}\left(z\right)|$ as functions of $a$ (the distance between the two ensembles). Spinwave profiles obtained from stopping counterpropagating pulses (c1) and copropagating setup (c2) at different switch speeds ($a=10\mu \mathrm{m}$). Example of counterpropagating spinwave dynamics [(d1) $a=10\mu \mathrm{m}]$ and copropagating spinwave dynamics (d2). Here ${\mathrm{\Omega }}_c^M=2\pi ·2$ MHz and other parameters are the same as those in Fig. 2.

Figure 4

(a) Ratios of the Rabi frequency of a regenerated photon (at the exit) over the peak value of the input photon's Rabi frequency, given ${\mathrm{\Omega }}_c^{\text{out},M}=2\pi ·2$ MHz and $a=10\mu \mathrm{m}$. (b) Example of photon pulse evolution throughout a whole “write-in” and “read-out” process, for which we chose ${\mathrm{\Omega }}_c^{\text{out},M}=2\pi ·3$ MHz, ${\tau }_c=0.1\mu \mathrm{s}$, and $a=10\mu \mathrm{m}$. (c) Profiles of the photon pulses upon regeneration, given ${\mathrm{\Omega }}_c^{\text{out},M}=2\pi ·5$ MHz and ${\tau }_c=0.1\mu \mathrm{s}$. The dashed curve is the profile of the input photons for comparison. (d) Growth of the photon survival probability with the ratio of maximum read-out over maximum write-in Rabi frequency, with ${\tau }_c=0.1\mu \mathrm{s}$ and $a=10\mu \mathrm{m}$. The horizontal line is the corresponding probability achieved without the interaction between pulses. The inset shows the tendency for $a=8\mu \mathrm{m}$. The stopping of photons ahead of the regenerating processes here, except for those of $a\ne 10\mu \mathrm{m}$, is the same as the one in Fig. 3 with ${\mathrm{\Omega }}_c^{\text{in},M}=2\pi ·2$ MHz and ${\tau }_{\text{off}}=10\mu \mathrm{s}$.