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In order to further investigate the non-reciprocity of light propagation in the defective atomic lattices, and due to its effective application in designing novel photonic devices, such as all-optical diodes and isolators, which are powerful tools for information processing and quantum simulation, we innovatively propose to use the Fibonacci sequence to modulate the arrangement of empty lattice cells that form a quasi periodic defective atomic lattices. In the electromagnetically induced transparency window, the probe light is almost not absorbed under the control of a strong coupling field (see
Fig. 1 ). The numerical simulation indicates that a wide nonreciprocal reflection band can be achieved by modulating the number of filled lattice cells, Fibonacci sequence, the period number in a single quasi period (seeFig. 2 ). These results provide more degrees of freedom for regulating nonreciprocal reflection with wide bandwidth and high contrast, and have potential applications in quantum computing and information processing.
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图 1 (a)三能级Lambda型相干原子系统; (b)一维准周期缺陷原子晶格与相干光场的作用; (c)一个满晶格周期中探测场平均极化率实部和虚部随失谐的变化及其与$ - 2\varDelta {\lambda _{{\mathrm{Lat}}}}/{\lambda _{{\mathrm{Lat}}}} \approx 0.0023 $的交点
Figure 1. (a) Three-level Lambda model coherent atomic system; (b) interaction between 1D quasi-periodic atomic lattice and coherent optical field; (c) the real and imaginary parts of average susceptibility in one filled lattice cell v.s. probe detuning, and the intersection with $ - 2\varDelta {\lambda _{{\text{Lat}}}}/{\lambda _{{\text{Lat}}}} \approx 0.0023 $.
图 2 (a), (b)展示了左右反射率$ {R_{\mathrm{l}}}_{, {\mathrm{r}}} $随失谐$ {\varDelta _{\mathrm{p}}} $的变化, 分别对应$ n \in \left[ {2, 16} \right] $和$ n \in \left[ {2, 21} \right] $; (c), (d)展示了左右反射率$ {R_{\mathrm{l}}}_{, {\mathrm{r}}} $随失谐$ {\varDelta _{\mathrm{p}}} $和斐波那契数列数量$ b(n) $中最大n值的变化. 其他参数: $ {{{N}}_0} = 7 \times {10^{11}}{\text{ }}{\mathrm{c{m}}^{ - 3}} $, $ \eta = 5 $, $ a = 40 $, $ c = 1 $, $ d = 1500 $, $ {\gamma _{31}} = $$ 6{\text{ MHz}} $, $ {\gamma _{21}} = 0.001{\text{ MHz}} $, $ {\varDelta _{\mathrm{c}}} = 15{\text{ MHz}} $, $ {\varOmega _{\mathrm{c}}} = 36{\text{ MHz}} $, $ {\lambda _{{\mathrm{Lat0}}}} = 781{\text{ nm}} $, $ {\lambda _{\mathrm{p}}} = 780.24{\text{ nm}} $, $ \varDelta {\lambda _{{\mathrm{Lat}}}} = - 0.9{\text{ nm}} $, $ {\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d} _{13}} = 1.0357 \times $$ {10^{ - 29}}{\text{ }}{\mathrm{C}}{ \cdot} {\mathrm{m}} $
Figure 2. (a) $ n \in \left[ {2, 16} \right] $ and (b) $ n \in \left[ {2, 21} \right] $ shows the reflectivities $ {R_{\mathrm{l}}}_{, {\mathrm{r}}} $ v.s. detuning $ {\varDelta _{\mathrm{p}}} $; (c), (d) the reflectivities $ {R_{\mathrm{l}}}_{, {\mathrm{r}}} $ v.s. the number of dissonance and Fibonacci series $ b(n) $ and detuning $ {\varDelta _{\mathrm{p}}} $, respectively. Other relevant parameters: $ {{{N}}_0} = 7 \times {10^{11}}{\text{ }}{\mathrm{c{m}}^{ - 3}} $, $ \eta = 5 $, $ a = 40 $, $ c = 1 $, $ d = 1500 $, $ {\gamma _{31}} = 6{\text{ MHz}} $, $ {\gamma _{21}} = 0.001{\text{ MHz}} $, $ {\varDelta _{\mathrm{c}}} = 15 \;{\mathrm{MHz}} $, $ {\varOmega _{\mathrm{c}}} = 36{\text{ MHz}} $, $ {\lambda _{{\mathrm{Lat0}}}} = 781{\text{ nm}} $, $ {\lambda _{\mathrm{p}}} = $$ 780.24{\text{ nm}} $, $ \varDelta {\lambda _{{\mathrm{Lat}}}} = - 0.9{\text{ nm}} $, ${\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d} _{13}}= 1.0357 \times {10^{ - 29}}{\text{ }}{\mathrm{C}} {\cdot} {\mathrm{m}} $.
图 3 左右反射率$ {R_{\mathrm{l}}}_{, r} $随失谐$ {\varDelta _{\mathrm{p}}} $的变化(a) $ n \in \left[ {2, 7} \right] $; (b) $ n \in \left[ {8, 13} \right] $; (c) $ n \in \left[ {8, 17} \right] $. 相关参数: $ c = 10 $, 其他参数如图2所示
Figure 3. The reflectivities $ {R_{\mathrm{l}}}_{, r} $ v.s. detuning $ {\varDelta _{\mathrm{p}}} $: (a) $ n \in $$ \left[ {2, 7} \right] $; (b) $ n \in \left[ {8, 13} \right] $; (c) $ n \in \left[ {8, 17} \right] $. Here $ c = 10 $, other parameters are shown in Fig 2.
图 4 左右反射率$ {R_{\mathrm{l}}}_{, r} $和反射对比度$ {C_{{R}}} $随失谐$ {\varDelta _{\mathrm{p}}} $的变化 (a) $ a = 30 $; (b) $ a = 20 $; (c) $ a = 10 $. 相关参数: $ n \in $$ \left[ {8, 17} \right] $, $ c = 10 $, 其他参数如图2所示
Figure 4. The reflectivities $ {R_{\mathrm{l}}}_{, r} $ and the reflection contrast $ {C_{{R}}} $ v.s. detuning $ {\varDelta _{\mathrm{p}}} $: (a) $ a = 30 $; (b) $ a = 20 $; (c) $ a = $$ 10 $. Here $ n \in \left[ {8, 17} \right] $, $ c = 10 $, other parameters are shown in Fig 2.
图 5 左右反射率$ {R_{\mathrm{l}}}_{, r} $和反射对比度$ {C_{{R}}} $随失谐$ {\varDelta _{\mathrm{p}}} $的变化 (a) $ c = 20 $; (b) $ c = 25 $; (c) $ c = 30 $. 相关参数: $ n \in $$ \left[ {8, 17} \right] $, $ a = 20 $, 其他参数如图2所示
Figure 5. The reflectivities $ {R_{\mathrm{l}}}_{, r} $ and the reflection contrast $ {C_{{R}}} $ v.s. detuning $ {\varDelta _{\mathrm{p}}} $: (a) $ c = 20 $; (b) $ c = 25 $; (c) $ c = 30 $. Here $ n \in \left[ {8, 17} \right] $, $ a = 20 $, other parameters are shown in Fig 2.
图 6 (a)左反射率$ {R_{\mathrm{l}}} $和(b)右反射率$ {R_{\mathrm{r}}} $随探测场失谐$ {\varDelta _{\mathrm{p}}} $和强耦合场失谐$ {\varDelta _{\mathrm{c}}} $的变化. 相关参数: $ n \in \left[ {8, 17} \right] $, $ a = $$ 20 $, $ c = 30 $, 其他参数如图2所示
Figure 6. (a) The left reflectivity $ {R_{\mathrm{l}}} $ and (b) the right reflectivity $ {R_{\mathrm{r}}} $ v.s. the probe detuning $ {\varDelta _{\mathrm{p}}} $ and the strong coupling field detuning $ {\varDelta _{\mathrm{c}}} $. Here $ n \in \left[ {8, 17} \right] $, $ a = 20 $, $ c = 30 $, other parameters are shown in Fig 2.
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