图 1  一维(1D)腔光力耦合晶格系统模型图, $ a_n $和$ b_n $分别表示腔场和机械振子的模式, 其中包含$ N+1 $个腔模式和$ N $个机械模式

Fig. 1.  Schematic of the 1D optomechanical coupling lattice system, $ a_n $ and $ b_n $represent the modes of the cavity field and the mechanical oscillator, respectively, and including $ N+1 $ cavity modes and $ N $mechanical modes.

图 2  系统能谱E与参数$ \phi $的物理图像　(a) $ M_{1} = M_{2} = 0 $; (b) $ M_{1} = 0, M_{2} = 0.1 $. 参数$ \phi $取值范围为(0, $2\text{π}$), $N = 15$

Fig. 2.  The energy spectrum E of the system vs the parameter $ \phi $: (a) $ M_{1} = M_{2} = 0 $; (b) $ M_{1} = 0, M_{2} = 0.1 $. The range of parameter $ \phi $ is (0, $2\text{π}$), $ N = 15 $

图 3  系统能谱E与参数$ \phi $的物理图像　(a) $ \psi = 0 $; (b) $ \psi = 0.1\pi $; (c) $ \psi = 0.9\pi $; (d) $ \psi = \pi $. 参数 $ \phi $ 取值范围为($0, 2\pi $), $N = 15$

Fig. 3.  The energy spectrum E of the system vs the parameter $ \phi $: (a) $ \psi=0 $; (b) $ \psi=0.1\pi $; (c) $ \psi=0.9\pi $; (d) $ \psi=\pi $. The range of parameter $ \phi $ is (0, $ 2\pi $), $ N = 15 $

图 4  边缘态分布与晶格数的物理图像　(a), (c) $ \phi = 2\pi/3 $, $ \psi = 3\pi/2, \pi/2, \pi $; (b), (d) $ \phi = \pi/3 $, $ \psi = 4\pi/3, 2\pi/3, \pi $

Fig. 4.  The state distribution vs the lattice number: (a), (c) $ \phi = 2\pi/3 $, $ \psi = 3\pi/2, \pi/2, \pi $; (b), (d) $ \phi = \pi/3 $, $ \psi = 4\pi/3, 2\pi/3, \pi $

图 5  沿晶格方向腔场稳态的平均光子数N与晶格数的物理图像　(a)晶格数为10, 驱动频率为$ \varDelta _{a, n} = 0.25 t $, $ \psi = 0.25\pi $; (b)晶格数为4, 驱动频率是$ \varDelta _{a, n} = - 0.01 t $, $ \psi = 0.25\pi $, 黑色虚线$ \kappa = 0.1 $, 红色点线$ \kappa = 0.15 $, 蓝色点虚线$ \kappa = 0.2 $; (c)$ \psi = $$ 0.2\pi, 0.23\pi, 0.26\pi $; (d)在腔的最左边(黑色线圆)、中间(红色虚线正方形)和最右边(蓝色点虚线菱形)驱动晶格.其他参数为$ \varOmega _{1 a} = 0.1 t $, $ \phi = 0.8\pi $

Fig. 5.  The average photon number N vs the lattice number: (a) The lattice size 10, $ \varDelta _{a, n} = 0.25 t $, $ \psi = 0.25\pi $; (b) the lattice size 4, $ \varDelta _{a, n} = - 0.01 t $, $ \psi = 0.25\pi $, black dotted line $ \kappa = 0.1 $, the red dotted line $ \kappa = 0.15 $, the blue dotted line $ \kappa = 0.2 $; (c) $ \psi = 0.2\pi, 0.23\pi, 0.26\pi $; (d) lattice site driven at the far left (black circles), middle (red dashed squares) and rightmost (blue dotted diamonds). Other parameters are Ω_1a = 0.1t and ϕ=0.8π

图 6  反射系数相的环绕数随参数$ \phi $的变化　(a1), (a2), (a3)蓝色线条为$ \psi = \pi/2 $, 红色线条为$ \psi = - \pi/2 $; (b) $\psi = $$ \pi/2$, $ \kappa = 0.1, 0.5, 1.5, 2 $; (c) $ \psi = - \pi/2 $, $ \kappa=0.1, 0.5, 1.5, 2 $. 其他参数为$\varOmega _{1 a} = 0.75 t$, 晶格数为10

Fig. 6.  The winding number of the reflection coefficient phase varies with parameter$ \phi $: (a1), (a2), (a3) The blue line is $ \psi = \pi/2 $, the red line is $ \psi = - \pi/2 $; (b) $ \psi = \pi/2 $, $ \kappa=0.1, 0.5, 1.5, 2 $; (c) $ \psi = - \pi/2 $, $ \kappa = 0.1, 0.5, 1.5, 2 $. Other parameters are $\varOmega _{1 a} = 0.75 t$ and lattice size 10.

图 7  最近邻相互作用之间的无序与系统能谱的物理图像　(a) $ \upsilon = 0.1 $; (b) $ \upsilon = 0.5 $; (c) $ \upsilon = 1 $; (d) $ \upsilon = 1.5 $; (e) $ \upsilon = 2 $; (f) $ \upsilon = 3 $. 其他参数为$ \psi = \pi/2 $, 晶格数为10

Fig. 7.  The effects of the disorder added into the nearest neighbor interaction on the energy spectra (a) $ \upsilon = 0.1 $; (b) $ \upsilon = 0.5 $; (c) $ \upsilon = 1 $; (d) $ \upsilon = 1.5 $; (e) $ \upsilon = 2 $; (f) $ \upsilon = 3 $. Other parameters are $ \psi = \pi/2 $ and lattice size 10.

图 8  在位无序与系统能谱的物理图像　(a) $ \mu = 0.1 $; (b) $ \mu = 0.5 $; (c) $ \mu = 1 $; (d) $ \mu = 1.5 $; (e) $ \mu = 2 $; (f) $ \mu = 3 $. 其他参数为$ \psi = \pi/2 $, 晶格数为10

Fig. 8.  The effects of the disorder added into the on-site potential energy on the energy spectra: (a) $ \mu = 0.1 $; (b) $ \mu = 0.5 $, (c) $ \mu = 1 $; (d) $ \mu = 1.5 $; (e) $ \mu = 2 $, (f) $ \mu = 3 $. Other parameters are $ \psi = \pi/2 $ and lattice size 10.

图 9  左边缘态的分布与缺陷强度$ \mu $, $ \upsilon $和晶格数的物理图像, 其他参数为$ \psi=\pi/2 $, 晶格数为10

Fig. 9.  The distribution of the left edge state is plotted with the defect intensity $ \mu $, $ \upsilon $, and lattice number, respectively, other parameters are $ \psi = \pi/2 $ and lattice size 10.

图 10  反射系数相的环绕数与无序缺陷的物理图像　(a) $ \upsilon = 0, 0.3, 0.6 $; (b) $ \upsilon = 1, 1.2, 1.4 $; (c) $ \mu = 0, 0.04, 0.08 $; (d) $\mu = 0.1, $$ 0.2, 0.3$. 其他参数为$ \psi = \pi/2 $, $ \kappa = 0.1 $, 晶格数为10

Fig. 10.  The winding number of the reflection coefficient phase varies with the disorder: (a) $ \upsilon = 0, 0.3, 0.6 $; (b) $ \upsilon = 1, 1.2, 1.4 $; (c) $ \mu = 0, 0.04, 0.08 $; (d) $ \mu = 0.1, 0.2, 0.3 $. Other parameters are $ \psi = \pi/2 $, $ \kappa = 0.1 $ and lattice size 10.

图 11  沿晶格方向腔场稳态的平均光子数与缺陷强度$ \mu $, $ \upsilon $和晶格数的物理图像　(a) $ \upsilon = 0, 1, 2, 3 $; (b) $ \mu = 0, 1, 2, 3 $. 其他参数为$\varOmega _{1 a} = 0.1 t$, $ \phi = 0.8\pi $, $\varDelta _{a, n} = - 0.5 t$, $ \kappa = 0.1 $, 晶格数为10

Fig. 11.  The average photon number $ N $ varies with the defect intensity $ \mu $, $ \upsilon $ and lattice number, respectively: (a) $ \upsilon = $$ 0, 1, 2, 3 $; (b) $ \mu = 0, 1, 2, 3 $. Other parameters are $\varOmega _{1 a} = 0.1 t$, $ \phi = 0.8\pi $, $\varDelta _{a, n} = - 0.5 t$, $ \kappa = 0.1 $ and lattice size 10.