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This work is to investigate the single-photon scattering in a waveguide quantum electrodynamics system consisting of two dipole-coupled giant atoms, each interacting with a separate one-dimensional infinite waveguide at two distinct coupling points. Our primary objective is to establish a theoretical framework for manipulating photon propagation paths via quantum interference induced by multiple coupling points and local phase engineering. Unlike traditional chiral coupling schemes, an innovative method, in which the coupling phases are designed locally at each atom-waveguide interface, is used to achieve effective chiral coupling, thereby introducing novel quantum interference mechanisms. Using a real-space approach, we derive analytical expressions for four-port scattering amplitudes. We establish the conditions for achieving perfect directional routing to any output port and demonstrate the coherent control mechanisms implemented by geometric and local coupled phases. Continuous frequency tunability is primarily achieved through dipole-dipole interaction, and finely tuned through the cumulative phase and local coupling phases. Local phase differences precisely regulate port-specific probability distributions within the waveguides while preserving total routing efficiency. Furthermore, we elucidate the mechanisms of nonreciprocal transport and chiral scattering. The analysis reveals different governing principles: perfect nonreciprocity arises from the interplay of the cumulated phase, local coupling phases, photon-atom detuning, and dipole-dipole interaction. In contrast, perfect chiral scattering depends entirely on the cumulated phase and local coupling phases, and is independent of detuning. Notably, under the phase-matching conditions, the system achieves both perfect chiral and directional routing, and realizes frequency-selective path-asymmetric photon control. These findings provide a comprehensive framework for manipulating quantum interference in multi-atom waveguide systems, highlighting applications in quantum information processing, including tunable single-photon routers, isolators, and chiral quantum nodes. By implementing superconducting circuits, the local phase can be dynamically adjusted, thus proving the feasibility of the experiment.
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Keywords:
- giant atom /
- directional routing /
- nonreciprocity /
- chiral scattering
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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, 其位置坐标为$ 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} $、$ \theta_{43} $和偶极相互作用强度λ下随失谐Δ的演化行为. 参数取值情况为: (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} $. 其中插图为(d)图中$ T_{{\mathrm{b}}} $和$ R_{{\mathrm{b}}} $的最大散射率及两者和值随$ \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. The parameter values are as follows: (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 (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) 图分别为在给定的几个λ 的取值下与 (a) 和 (b) 对应的曲线图. 参数取值为: (a) 和 (c) 图中 $ \phi=\pi $、$ \theta_{21}=\pi/4 $、$ \theta_{43} = 0 $; (b) 和 (d) 图中 $ \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. Panels (c) and (d) show the corresponding curves for several values of λ in (a) and (b), respectively. The parameter values are: $ \phi = \pi $, $ \theta_{21} = \pi/4 $, $ \theta_{43} = 0 $ for panels (a) and (c); $ \phi = \pi/6 $, $ \theta_{21} = \theta_{43} = 5\pi/6 $ for panels (b) and (d).
图 4 (a) 在局域耦合相位差$ \theta_{21}=\theta_{43}=\pi-\phi $条件下$ T_{{\mathrm{b}}} $随失谐Δ和累积相位ϕ 的变化; (b) 几个典型的累积相位ϕ取值下与(a)图对应的单光子散射曲线图, 参数取值为: 黑色实线$ \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)为双峰间距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 (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 (a), showing the detunings of the double peaks $ \varDelta_{\pm} $ as functions 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}}} $随累积相位ϕ和失谐Δ的演化情况; (b)和(c)图分别给出了$ \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 $ 的$ T'_{{\mathrm{a}}} $和$ T_{{\mathrm{a}}} $的散射曲线(蓝色虚线). 其它参数为: $ \theta_{21}=\theta_{43}=\pi/2 $, $ \lambda/\varGamma = 4 $
Figure 5. The 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} $的取值下手性度C与$ \theta_{43} $ 的变化关系; (c)—(h)图为完美手性散射下的$ 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. The influence 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 $.
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