图  1  (a), (b)平均场近似下, 不同晶格模型关于超流序参量 $ A = \braket{a} $ 的 $ J \text {-} g $ 相图　(a) JC晶格模型; (b) Rabi 晶格模型. 横坐标为格点之间的光子跃迁强度 $ J $ , 纵坐标为二能级原子和光子相互作用强度 $ g $ , 横纵坐标的单位为 $ \omega_0 $ , 颜色条表示超流序参量 $ A = \braket{a} $ 的大小. 深蓝色表示Mott绝缘相, 浅黄色表示超流体相. 其他参量取值为: $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , 光子截断数 $ N = 20 $ . (c), (d)对于不同的 $ J $ , 不同晶格模型的超流序参量 $ A $ 随 $ g $ 变化的图像　(c) JC晶格模型; (d) Rabi晶格模型

Figure  1.  (a), (b) Under the mean field approximation, the $ J \text {-} g $ phase diagram of different lattice models with respect to the superfluid order parameter $ A = \braket{a} $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the photon transition intensity $ J $ between the lattice, the ordinate is the two-level atom and photon interaction strength $ g $ , the unit of the abscissa and the ordinate is $ \omega_0 $ , and the color bar represents the value of the superfluid order parameter $ A = \braket{a} $ . Dark blue indicates Mott insulating phase, and light yellow indicates superfluid phase. Other parameters are taken as $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , and the number of the photon truncation $ N = 20 $ . (c), (d) For different $ J $ , the superfluid order parameter $ A $ of different lattice models varies with $ g $ : (c) JC lattice model; (d) Rabi lattice model.

图  2  平均场近似下, 不同晶格模型关于二阶关联函数 $ g^{2}(0) $ 的 $J\text {-} g$ 相图　(a) JC晶格模型; (b) Rabi晶格模型. 横坐标为格点之间的光子跃迁强度 $ J $ , 纵坐标为二能级原子和光子相互作用强度 $ g $ , 横纵坐标的单位为 $ \omega_0 $ , 颜色条表示二阶关联函数 $ g^{2}(0) $ 的值. 其他参量取值为: $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , 光子截断数 $ N = 20 $

Figure  2.  Under the mean field approximation, the $ J \text {-} g $ phase diagram of different lattice models with respect to the second-order correlation function $ g^{2}(0) $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the photon transition intensity $ J $ between the lattice, the ordinate is the two-level atom and photon interaction strength $ g $ , the unit of the abscissa and the ordinate is $ \omega_0 $ , the color bar is represented by the value of the second-order correlation function $ g^{2}(0) $ . $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , and the number of photon truncation $ N = 20 $ .

图  3  Kerr效应下不同晶格模型关于超流序参量 $ A = \braket{a} $ 的 $ \kappa \text {-} g $ 相图　(a) JC晶格模型; (b) Rabi 晶格模型. 横坐标为Kerr非线性强度 $ \kappa $ , 纵坐标为二能级原子和光子相互作用强度 $ g $ , 横纵坐标的单位为 $ \omega_0 $ , 颜色条表示超流序参量 $ A $ 的大小. 其他参量取值为: $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , $ J = 0.05 $ , 光子截断数 $ N = 20 $

Figure  3.  The $ \kappa \text {-} g $ phase diagram of different lattice models under the Kerr effect with respect to the superfluid order parameter $ A = \braket{a} $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the Kerr nonlinear intensity $ \kappa $ , the ordinate is the two-level atom and photon interaction strength $ g $ , the unit of the abscissa and the ordinate is $ \omega_0 $ , and the color bar represents the value of the superfluid order parameter $ A $ . Other parameters are taken as $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , $ J = 0.05 $ , and the number of photon truncation $ N = 20 $ .

图  4  Kerr效应下不同晶格模型关于二阶关联函数 $ g^{2}(0) $ 的 $ \kappa \text {-} g $ 相图　(a) JC晶格模型; (b) Rabi晶格模型. 横坐标为Kerr非线性强度 $ \kappa $ , 纵坐标为二能级原子和光子相互作用强度 $ g $ , 横纵坐标的单位为 $ \omega_0 $ , 颜色条表示二阶关联函数 $ g^{2}(0) $ . 其他参量取值为: $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , $ J = 0.05 $ , 光子截断数 $ N = 20 $

Figure  4.  The $ \kappa \text {-} g $ phase diagram of different lattice models under the Kerr effect with respect to the second-order correlation function $ g^{2}(0) $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the Kerr nonlinear intensity $ \kappa $ , the ordinate is the two-level atom and photon interaction strength $ g $ , the unit of the abscissa and the ordinate is $ \omega_0 $ , and the color bar represents the value of second-order correlation function $ g^{2}(0) $ . Other parameters are taken as $ \mathop{\omega_{0} = \omega_{1}} = 1 $ , $ J = 0.05 $ , and the number of photon truncation $ N = 20 $ .