图 1  两二能级巨原子耦合两个一维波导构成的量子路由示意图. 二能级巨原子a (b)与波导m (n)存在两次耦合, 位置坐标分别为$ x_\mathrm{m1} = 0 $ ($ x_\mathrm{n1} = 0 $) 和$ x_\mathrm{m2}=l $ ($ x_\mathrm{n2}= l $), 耦合强度均为$ g_\mathrm{a} $ ($ g_\mathrm{b} $), 局域耦合相位分别为$ \theta_{1} $ ($ \theta_{3} $), $ \theta_{2} $ ($ \theta_{4} $). λ 表示巨原子之间的偶极-偶极相互作用强度

Figure 1.  Schematic configuration of routing single photons in two channels made of two one-dimensional waveguides. The giant two-level atom a (b) interacts with waveguide m (n) at $ x_\mathrm{m1} = 0 $ ($ x_\mathrm{n1} = 0 $) and $ x_\mathrm{m2}=l $ ($ x_\mathrm{n2}=l $), characterized by coupling strengths $ g_\mathrm{a} $ ($ g_\mathrm{b} $) and local coupling phases $ \theta_{1} $ ($ \theta_{3} $), $ \theta_{2} $ ($ \theta_{4} $). The dipole-dipole interaction strength between the two giant atoms is denoted by λ.

图 2  散射率$ T_{{\mathrm{a}}} $(黑色实线)、$ R_{{\mathrm{a}}} $(红色虚线)、$ T_{{\mathrm{b}}} $(蓝色点线)和$ R_{{\mathrm{b}}} $(绿色点划线)在不同的累积相位ϕ、局域耦合相位差$ \theta_{21} $和$ \theta_{43} $以及偶极相互作用强度λ下随失谐Δ的演化行为　(a) $ \phi=\pi $, $ \theta_{21}=\pi/4 $, $ \theta_{43} = 0 $, $ \lambda/\varGamma = 4 $; (b) $ \phi=\theta_{21}=\theta_{43}=\pi/2 $, $ \lambda/\varGamma = 4 $; (c) $ \phi=\theta_{21}=\pi/2 $, $ \theta_{43} = 3\pi/2 $, $ \lambda/\varGamma = 4 $; (d) $ \phi=\pi/3 $, $ \theta_{21} = 2\pi/3 $, $ \theta_{43}=\pi/3 $, $ \lambda/\varGamma=\sqrt{15} $. 其中图(d)插图为$ T_{{\mathrm{b}}} $和$ R_{{\mathrm{b}}} $的最大散射率及两者之和随$ \theta_{43} $的调控情况

Figure 2.  Scattering rates $ T_{{\mathrm{a}}} $ (solid black line), $ R_{{\mathrm{a}}} $ (dashed red line), $ T_{{\mathrm{b}}} $ (dotted blue line), and $ R_{{\mathrm{b}}} $ (dash-dotted green line) versus the detuning Δ with different accumulated phases ϕ, local coupling phase differences $ \theta_{21} $, $ \theta_{43} $, and dipole interaction strengths λ: (a) $ \phi = \pi $, $ \theta_{21} = \pi/4 $, $ \theta_{43} = 0 $, $ \lambda/\varGamma = 4 $; (b) $ \phi = \theta_{21} = \theta_{43} = \pi/2 $, $ \lambda/\varGamma = 4 $; (c) $ \phi = \theta_{21} = \pi/2 $, $ \theta_{43} = 3\pi/2 $, $ \lambda/\varGamma = 4 $; (d) $ \phi = \pi/3 $, $ \theta_{21} = 2\pi/3 $, $ \theta_{43} = \pi/3 $, $ \lambda/\varGamma = \sqrt{15} $. The inset in panel (d) shows the maximum scattering rates of $ T_{{\mathrm{b}}} $ and $ R_{{\mathrm{b}}} $ and their sum as functions of $ \theta_{43} $.

图 3  散射率 (a) $ T_{{\mathrm{a}}} $ 和 (b) $ T_{{\mathrm{b}}} $ 随失谐Δ和原子偶极相互作用强度λ的变化. (c) 和 (d) 分别为在给定的几个λ 的取值下与图(a)和图(b) 对应的曲线图. 参数取值为 (a), (c) $ \phi=\pi $, $ \theta_{21}=\pi/4 $, $ \theta_{43} = 0 $; (b), (d)$ \phi=\pi/6 $, $ \theta_{21}= \theta_{43} = 5\pi/6 $

Figure 3.  Scattering rates (a) $ T_{{\mathrm{a}}} $ and (b) $ T_{{\mathrm{b}}} $ as functions of the detuning Δ and the dipole interaction strength λ. (c) and (d) show the corresponding curves for several values of λ in panels (a) and (b), respectively. The parameter values are: (a), (c)$ \phi = \pi $, $ \theta_{21} = \pi/4 $, $ \theta_{43} = 0 $; (b), (d) $ \phi = \pi/6 $, $ \theta_{21} = \theta_{43} = 5\pi/6 $.

图 4  (a)在局域耦合相位差$ \theta_{21}=\theta_{43}=\pi-\phi $条件下$ T_{{\mathrm{b}}} $随失谐Δ和累积相位ϕ 的变化; (b)几个典型的累积相位ϕ取值下与图(a)对应的单光子散射曲线图, 参数取值为: 黑色实线$ \phi=\pi/4 $, $ \theta_{21}=\theta_{43} = 3\pi/4 $, 红色点线$ \phi=\theta_{21}=\theta_{43}=\pi/2 $, 蓝色虚线$ \phi = 3\pi/4 $, $ \theta_{21}=\theta_{43}=\pi/4 $; (c)为图(a)中散射率最大值$ T_{{\mathrm{b}}} = 1 $的轮廓图, 即单光子谱线中双峰的峰位$ \varDelta_{\pm} $随ϕ的变化; (d)双峰间距d随ϕ 的变化. 其他参数为: $ \lambda/\varGamma = 4 $

Figure 4.  (a) Scattering rate $ T_{{\mathrm{b}}} $ as a function of the detuning Δ and the accumulated phase ϕ with $ \theta_{21} = \theta_{43} = \pi - \phi $. (b) The curves corresponding to panel (a) with some typical values of the accumulated phase ϕ. The parameter values are: solid black line $ \phi = \pi/4 $, $ \theta_{21} = \theta_{43} = 3\pi/4 $; dotted red line $ \phi = \theta_{21} = \theta_{43} = \pi/2 $; dashed blue line $ \phi = 3\pi/4 $, $ \theta_{21} = \theta_{43} = \pi/4 $. (c) Contour plot of the maximum scattering rate $ T_{{\mathrm{b}}} = 1 $ in panel (a), showing the detunings of the double peaks $ \varDelta_{\pm} $ as a function of ϕ. (d) The separation d of the double peaks versus ϕ. Other parameters are: $ \lambda/\varGamma = 4 $.

图 5  累积相位和原子耦合强度对光子非互易性的影响　(a)非互易度$ I_{{\mathrm{a}}} $随累积相位ϕ和失谐Δ的演化情况; (b), (c)分别给出了$ \phi = 3\pi/2 $和$ \pi/2 $下的两散射率$ T_{{\mathrm{a}}} $(黑色实线)和$ T'_{{\mathrm{a}}} $(红色点线)随Δ的变化曲线, 为了对比的需要, 两图还分别给出了$ \lambda/\varGamma = 6 $ 时的$ T'_{{\mathrm{a}}} $和$ T_{{\mathrm{a}}} $的散射曲线(蓝色虚线). 其他参数为$ \theta_{21}=\theta_{43}=\pi/2 $, $ \lambda/\varGamma = 4 $

Figure 5.  Influence of accumulated phase and atomic coupling strength on photon non-reciprocity: (a) Evolution of non-reciprocity degree $ I_{{\mathrm{a}}} $ with accumulated phase ϕ and detuning Δ; (b) and (c) show the scattering rates $ T_{{\mathrm{a}}} $ (solid black line) and $ T'_{{\mathrm{a}}} $ (dotted red line) as functions of Δ for $ \phi = 3\pi/2 $ and $ \phi = \pi/2 $, respectively. For comparison, the scattering curves of $ T'_{{\mathrm{a}}} $ and $ T_{{\mathrm{a}}} $ for $ \lambda/\varGamma = 6 $ (dashed blue line) are also shown in both panels. Other parameters are: $ \theta_{21} = \theta_{43} = \pi/2 $, $ \lambda/\varGamma = 4 $.

图 6  单光子手性散射受累积相位ϕ和局域耦合相位差$ \theta_{21} $, $ \theta_{43} $的影响　(a)累积相位$ \phi=\pi/2 $ 取值下手性度C随$ \theta_{43} $和$ \theta_{21} $的演化; (b)在给定的几个典型的$ \theta_{21} $取值下手性度C与$ \theta_{43} $之间的关系; (c)—(h)完美手性散射下的$ T_{{\mathrm{b}}} $和$ T'_{{\mathrm{b}}} $散射曲线图, 参数取值为(c) $ \phi=\theta_{21}=\pi/2 $, $ \theta_{43} = 0 $; (d) $ \phi=\pi/2 $, $ \theta_{21} = 3\pi/2 $, $ \theta_{43} = 0 $; (e) $ \phi=\theta_{21}=\theta_{43}=\pi/2 $; (f) $ \phi=\pi/2 $, $ \theta_{21}= \theta_{43} = $$ 3\pi/2 $; (g) $ \phi=\pi/12 $, $ \theta_{21}=\theta_{43} = 11\pi/12 $; (h) $ \phi=\pi/12 $, $ \theta_{21}=\theta_{43} = 13\pi/12 $

Figure 6.  Influences of accumulated phase ϕ and local coupling phase differences $ \theta_{21} $, $ \theta_{43} $ on single-photon chiral scattering. (a) Evolution of chirality C with $ \theta_{43} $ and $ \theta_{21} $ for $ \phi = \pi/2 $; (b) dependence of chirality C on $ \theta_{43} $ for several typical values of $ \theta_{21} $; (c)–(h) scattering curves of $ T_{{\mathrm{b}}} $ and $ T'_{{\mathrm{b}}} $ under perfect chiral scattering conditions. The parameter values are: (c) $ \phi = \theta_{21} = \pi/2 $, $ \theta_{43} = 0 $; (d) $ \phi = \pi/2 $, $ \theta_{21} = 3\pi/2 $, $ \theta_{43} = 0 $; (e) $ \phi = \theta_{21} = \theta_{43} = \pi/2 $; (f) $ \phi = \pi/2 $, $ \theta_{21} = \theta_{43} = 3\pi/2 $; (g) $ \phi = \pi/12 $, $ \theta_{21} = \theta_{43} = 11\pi/12 $; (h) $ \phi = \pi/12 $, $ \theta_{21} = \theta_{43} = 13\pi/12 $.