Guiding ultraslow weak-light bullets with Airy beams in a coherent atomic system

Chao Hang and Guoxiang Huang

Phys. Rev. A 89, 013821 – Published 17 January 2014

Abstract

We investigate the possibility of guiding stable ultraslow weak-light bullets by using Airy beams in a cold, lifetime-broadened four-level atomic system via electromagnetically induced transparency (EIT). We show that under EIT condition the light bullet with ultraslow propagating velocity and extremely low generation power formed by the balance between diffraction and nonlinearity in the probe field can be not only stabilized but also steered by the assisted field. In particular, when the assisted field is taken to be an Airy beam, the light bullet can be trapped into the main lobe of the Airy beam, propagate ultraslowly in longitudinal direction, accelerate in transverse directions, and move along a parabolic trajectory. We further show that the light bullet can bypass an obstacle when guided by two sequential Airy beams. A technique for generating ultraslow helical weak-light bullets is also proposed.

Received 7 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.013821

Authors & Affiliations

Chao Hang^* and Guoxiang Huang^†

State Key Laboratory of Precision Spectroscopy and Department of Physics, East China Normal University, Shanghai 200062, China

^*chang@phy.ecnu.edu.cn
^†gxhuang@phy.ecnu.edu.cn

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Vol. 89, Iss. 1 — January 2014

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Images

Figure 1
(a) Energy-level diagram and excitation scheme of the lifetime-broadened four-state atomic system interacting with a weak pulsed probe field (with half Rabi frequency $Ω_{p}$ ), a strong cw control field (with half Rabi frequency $Ω_{c}$ ), and a weak cw assisted field (with half Rabi frequency $Ω_{a}$ ). $Δ_{3},$ $Δ_{2}$ , and $Δ_{4}$ are one-photon, two-photon, and three-photon detunings, respectively. The energy levels are taken from the $D_{2}$ line of $^{87}$ Rb atoms, with $| 1 〉 = | 5 S_{1 / 2}, F = 1, m_{F} = - 1 〉$ , $| 2 〉 = | 5 S_{1 / 2}, F = 2, m_{F} = 0 〉$ , $| 3 〉 = | 5 P_{3 / 2}, F = 2, m_{F} = - 1 〉$ , and $| 4 〉 = | 5 P_{3 / 2}, F = 1, m_{F} = 1 〉$ . $f_{i j} = {| p_{i j} / D_{} |}^{2} \times 120$ is the relative transition strength, with $D = 3.58 \times 10^{- 27}$ cm C and $p_{i j}$ being the dipole transition matrix element between the state $| i 〉$ and the state $| j 〉$ . (b) The geometry of the system. The lower part shows the intensity pattern of the assisted field, chosen as an Airy beam, in the $x$ - $y$ plane.
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Figure 2
The imaginary part Im $K (ω)$ [panel (a)] and the real part Re $K (ω)$ [panel (b)] of the linear dispersion relation $K (ω)$ of the probe field as functions of $ω$ . In both panels, the dashed and solid lines correspond to the presence ( $Ω_{c} = 1.0 \times 10^{7}$ s $^{- 1}$ ) and the absence ( $Ω_{c} = 0$ ) of the control field, respectively. The other parameters are given in the text. A transparency window is opened for the large control field [the dashed line in panel (a)]. The steep slope of the dashed-dotted line for the large control field [the dashed line in panel (b)] results in an ultraslow group velocity.
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Figure 3
Guiding a linear light bullet with Airy beam ( $U_{0} = 1.8 \times 10^{- 6}$ s $^{- 1}$ ). (a)–(c) Intensity patterns of the linear light bullet in the $x$ - $z$ plane at $τ / τ_{0} = 0$ , 2, and 4, respectively. Dashed lines denote the trajectory of the main lobe of the Airy potential. (d)–(f) Intensity patterns of the linear light bullet in the $x$ - $y$ plane at $τ / τ_{0} = 0$ , 2, and 4, respectively. A significant diffraction can be observed in (c) and (f).
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Figure 4
Guiding nonlinear light bullet with Airy beam. (a)–(c) Intensity patterns of the nonlinear light bullet in the $x$ - $z$ plane at $τ / τ_{0} = 0$ , 2, and 4, respectively. (d)–(f) Intensity patterns of the nonlinear light bullet in the $x$ - $y$ plane at $τ / τ_{0} = 0$ , 2, and 4. (g) Schematic diagram of the propagation of the nonlinear light bullet in three-dimensional space. $V_{x}$ ( $V_{y}$ ) is the velocity in the $x$ ( $y$ ) direction; $V_{r} = \sqrt{V_{x}^{2} + V_{y}^{2}}$ is the radial velocity in the $x$ - $y$ plane; $V_{g}$ is the group velocity in the $z$ direction; $V_{LB} = \sqrt{V_{r}^{2} + V_{g}^{2}}$ is the total velocity; $R$ is the transverse displacement after the nonlinear light bullet passing through the atomic medium; $θ$ is the angle between $V_{r}$ and $V_{g}$ ; the (red) solid spheres represent the nonlinear light bullet. (h) $R$ and $θ$ as functions of $z$ . The solid and dashed lines are analytical results; the “ $\times$ ” symbols are results by numerical simulations.
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Figure 5
Nonlinear light bullet bypasses an obstacle. (a)–(e) Intensity patterns of the nonlinear light bullet in the $x$ - $z$ plane for $τ / τ_{0} = 0$ , 2, 4, 6, and 8, respectively. The black polygons represent an obstacle. (f) The propagation of the nonlinear light bullet [represented by the (red) solid spheres] in the 3D space. The “ $Λ$ ” shape trajectory of the nonlinear light bullet enables it to bypass the obstacle (represented by the black polygon).
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Figure 6
Generating a nonlinear helical light bullet with Airy beam. (a)–(d) Intensity patterns of the nonlinear light bullet in the $x$ - $y$ plane at $t / τ_{0} = 0$ , 2, 4, and 6, respectively, for $τ_{1} = 4$ and $b = 0.05$ . (e) The propagation of the helical light bullet in 3D space. The solid line with arrow denotes the motion trajectory of the nonlinear light bullet. The radius of the ring $R \approx 5.66 R_{⊥}$ .
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