Three coupled ultraslow temporal solitons in a five-level tripod atomic system

Wen-Xing Yang, Ai-Xi Chen, Liu-Gang Si, Kaijun Jiang, Xiaoxue Yang, and Ray-Kuang Lee

Phys. Rev. A 81, 023814 – Published 16 February 2010

Abstract

We propose a scheme to generate three coupled ultraslow optical solitons in a five-level tripod atomic system. We show that the detrimental distortions of the three weak probe fields due to dispersion effects under weak driving conditions can be well balanced by self- and cross-phase-modulation effects, which leads to the three coupled ultraslow temporal optical solitons with the temporal, group velocity, and amplitude values nearly matched. In contrast to other schemes, our model uses optical fibers, including multichromatic optical solitons in the stimulated Raman scattering, and produces three coupled ultraslow optical solitons in a small propagation distance of less than $1$ cm with $\frac{1}{2}$ Rabi frequency of the driving field, typically less than 100 MHz.

Received 14 December 2008

DOI:https://doi.org/10.1103/PhysRevA.81.023814

Authors & Affiliations

Wen-Xing Yang^1,2,*, Ai-Xi Chen³, Liu-Gang Si⁴, Kaijun Jiang⁵, Xiaoxue Yang⁴, and Ray-Kuang Lee²

¹Department of Physics, Southeast University, Nanjing 210096, People's Republic of China
²Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan
³Department of Applied Physics, School of Basic Science, East China Jiaotong University, Nanchang 330013, People's Republic of China
⁴School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China
⁵Laboratoire Kastler-Brossel, École Normale Supérieure (ENS), 24 rue Lhomond, 75005 Paris, France

^*wenxingyang2@126.com

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Vol. 81, Iss. 2 — February 2010

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Figure 1
(a) Five-level tripod atomic system interacts with a cw coherent control laser (frequency $ω_{d}$ and $\frac{1}{2}$ Rabi frequency $Ω_{d}$ ) and three pulsed probe fields (frequency $ω_{a, b, c}$ and $\frac{1}{2}$ Rabi frequency $Ω_{a, b, c}$ ). All the atoms are assumed to be initially prepared in the states $| 1 〉$ , $| 2 〉$ , and $| 3 〉$ with near amplitude as $ρ_{11} ≃ ρ_{22} ≃ ρ_{33} ≃ 1 / 3$ . (b) Possible arrangement of experimental apparatus. The arrowhead represents the propagation direction of the fields; $σ^{-}$ , $π$ , and $σ^{+}$ represent the different polarized directions of the three weak probe fields; and $ω_{d}$ represents the control field. The three weak fields are sent into the atomic medium to form three coupled ultraslow temporal optical solitons.Reuse & Permissions
Figure 2
Absorption coefficients $α_{a}$ (a), $α_{b}$ (b), and $α_{c}$ (c) of the three pulsed probe fields versus the dimensionless $\frac{1}{2}$ Rabi frequency $Ω_{d} τ_{0}$ for several different two-photon detunings $Δ_{2} τ_{0}$ with the parameters $(γ_{41} + γ_{42} + γ_{43}) τ_{0} = 0.053$ , $γ_{31} τ_{0} = γ_{32} τ_{0} = γ_{21} τ_{0} = 6 \times 10^{- 5}$ , $γ_{54} τ_{0} = 0.69 \times 10^{- 2}$ , $κ_{41} τ_{0} = κ_{42} τ_{0} = κ_{43} τ_{0} = 10$ cm $^{- 1}$ , $Δ_{1} τ_{0} = 3$ , $Δ_{3} τ_{0} = - 1 \times 10^{- 3}$ , and $Δ_{4} τ_{0} = - 5 \times 10^{- 4}$ .Reuse & Permissions
Figure 3
Relative group velocities $V_{ga} / c$ (a), $V_{gb} / c$ (b), and $V_{gc} / c$ (c) of the three pulsed probe fields versus the dimensionless $\frac{1}{2}$ Rabi frequency $Ω_{d} τ_{0}$ for $τ_{0} = 1 \times 10^{- 8}$ s, with the other parameters being the same as in Fig. 2.Reuse & Permissions
Figure 4
$| Ω_{j} / U_{0} |$ [(a) $| Ω_{a} / U_{0} |$ , (b) $| Ω_{b} / U_{0} |$ , (c) $| Ω_{c} / U_{0} |$ ] as the function of the dimensionless time $t / τ_{0}$ obtained by numerically integrating Eqs. (62) and (63). All the parameters are present in the main text.Reuse & Permissions
Figure 5
Comparison of analytical solutions of Eqs. (63)–(62) (dashed lines) and the full numerical solutions (solid lines) obtained by directly integrating Eqs. (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) without using any approximations. The dotted lines are numerical solutions without including nonlinear terms (i.e., without SPM and CPM terms). Each curve contains three indistinguishable traces, that is, $| Ω_{a} / U_{0} |$ , $| Ω_{b} / U_{0} |$ , and $| Ω_{c} / U_{0} |$ . The parameters ( $Δ_{j}$ , $γ_{ij}$ , $Ω_{d}$ , $U_{0}$ , $τ$ , and $κ_{41, 42, 43}$ ) are the same as in Fig. 4.Reuse & Permissions
Figure 6
$| Ω_{a} / U_{0} |$ , $| Ω_{b} / U_{0} |$ , and $| Ω_{c} / U_{0} |$ (dashed line) at $z = 0.8$ cm obtained by numerically integrating Eqs. (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) using three matched Gaussian input pulses. The central peak has a hyperbolic secant shape (dash-dotted line) and has an apparent advancement relative to that of the signal pulses propagating through a vacuum (dotted line). Each curve contains three indistinguishable traces.Reuse & Permissions

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