Two-Photon Dynamics in Coherent Rydberg Atomic Ensemble

Bing He, A. V. Sharypov, Jiteng Sheng, Christoph Simon, and Min Xiao

Phys. Rev. Lett. 112, 133606 – Published 4 April 2014

Abstract

We study the interaction of two photons in a Rydberg atomic ensemble under the condition of electromagnetically induced transparency, combining a semiclassical approach for pulse propagation and a complete quantum treatment for quantum state evolution. We find that the blockade regime is not suitable for implementing photon-photon cross-phase modulation due to pulse absorption and dispersion. However, approximately ideal cross-phase modulation can be realized based on relatively weak interactions, with counterpropagating and transversely separated pulses.

Received 19 December 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.133606

Authors & Affiliations

Bing He^1,2, A. V. Sharypov^3,4, Jiteng Sheng², Christoph Simon¹, and Min Xiao^2,5,*

¹Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2 N 1N4, Canada
²Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
³Kirensky Institute of Physics, 50 Akademgorodok, Krasnoyarsk 660036, Russia
⁴Siberian Federal University, 79 Svobodny Avenue, Krasnoyarsk 660041, Russia
⁵National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China

^*mxiao@uark.edu

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Vol. 112, Iss. 13 — 4 April 2014

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Figure 1
(a) Two single-photon pulses propagate in a Rydberg atomic ensemble. (b) Two pulses propagate in two parallel waveguides filled with Rydberg atoms. For the insignificant diffraction or the propagation in (b), the pulse profiles for the numerical simulations in Figs. 3 and 4 can be approximated as one dimensional. (c) Atomic level scheme for the system. Without pulse interaction there is $Δ_{1} + Δ_{2} = 0$ under the EIT condition. Here, $Δ_{1} = ω_{e g} - ω_{p}$ and $Δ_{2} = ω_{r e} - ω_{c}$ ( $ω_{p}$ is the input pulses’ central frequency, and $ω_{c}$ is the frequency of the pump beam). The Rydberg level is shifted by $Δ_{R}$ due to the interaction with another pulse.
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Figure 2
(a) Shift of refractive curves with increasing $Δ_{R} < 0$ . Here, $n - 1 = 1 / 2 Re {χ^{(1)}}$ [25] is obtained from the approximated solution to Eqs. (4a) and (4b) in a slow light regime, where $Δ_{R}$ changes slightly during the decay time on the order of $1 / γ$ . We take the photon detuning $Δ_{1} = 2 γ$ and the pump detuning $Δ_{2} = - 2 γ$ under the EIT condition with $Δ_{R} = 0$ . The pulses’ initial group velocity under the EIT condition is set as $v_{g} = 10 m / s$ ( $v_{g} = c / n_{g}$ with $n_{g} = n + ω_{p} \partial n / \partial ω_{p}$ ), while the pump Rabi frequency is $Ω_{c} = 2 γ$ . The excited level $| e ⟩$ is $5 P_{1 / 2}$ of ${}^{87}{Rb}$ . The dashed curve is that for the two-level system of the corresponding parameters. The minus signs of the horizontal axis labels come from our definition of $Δ_{1}$ . (b) Group velocity $v_{g}$ vs $Δ_{R}$ with the same pulse and pump detuning as in (a). The thicker solid curve is for $Ω_{c} = 2 γ$ , whereas the thinner curve is about $Ω_{c} = 4 γ$ . The dashed line is the group velocity of the corresponding two-level system.
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Figure 3
Numerical simulation for pulse motion. Here, we adopt the relative distance coordinate $Z$ as a substitute for the time scale. We use $| g ⟩ = 5 S_{1 / 2}$ , $| e ⟩ = 5 P_{1 / 2}$ , and $| r ⟩ = 82 D_{3 / 2}$ of ${}^{87}{Rb}$ , with the VdW coefficient $| C_{6} | = 8500 GHz μ m^{6}$ [26] and $γ = 2 π \times 5.75 MHz$ . The system parameters are chosen as $N g^{2} / Ω_{c}^{2} = 0.75 \times 10^{7}$ [ $v_{g} (- L) = 10 m / s$ ], $Ω_{c} = 2 γ$ , and $Δ_{1} = - Δ_{2} = 2 γ$ . The input Gaussian shaped pulses with $f (z) = (1 / \sqrt{π} σ)^{1 / 2} e^{- (1 / 2) z^{2} / σ^{2}}$ have the initial size $σ = 11.1 μ m$ , with the corresponding bandwidth well fitted into the EIT window. The dashed curves are about the transverse separation $a = 0.58 σ$ , the thicker solid ones for $a = σ$ , and the thinner solid ones for $a = 1.5 σ$ . The iterative step size for the numerical simulation is $0.005 σ$ . (a) Interaction-induced $Δ_{R} (Z)$ at pulse centers. The insertion is the refined plot for $a = 1.5 σ$ . (b) Transmission rate $T (Z) = \exp (- k_{p} \int_{- L}^{Z} d y Im {χ^{(1)} (y)})$ . (c) Group velocity $v_{g} (Z) = c / n_{g} (Z)$ at pulse centers. The most transversely adjacent situation shows a velocity platform near the Rydberg blockade. (d) Group velocity deviation ratio $δ_{v} (Z)$ . For the same $Z$ , more extending pulses have higher $δ_{v}$ due to more spatially inhomogeneous interaction.
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Figure 4
(a) Fidelity and cross phase of photon-photon XPM for two counterpropagating pulses with the transverse separation $a = 1.5 σ$ in Fig. 3. $L$ is the medium size. The system parameters are the same as in Fig. 3. The inset describes an imagined situation by reducing the initial pulse velocity to $10^{- 2} m / s$ . (b) Fidelity and cross phase for two pulses propagating together along two tracks separated by $a = 1.5 σ$ . Because of pulse absorption, their group velocity is not stable in such copropagation (for example, it drops from 11.007 to 11.002 m/s from $L = 2 σ$ to $5 σ$ ).
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