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通过解析求解与数值模拟相结合的方法, 研究了由克尔介质单模腔与光学参量放大器组成的混合量子系统中光子阻塞效应的调控机制. 建立了包含腔场衰减的有效哈密顿量主方程, 采用Fock态基矢展开至双光子截断近似, 解析求解稳态薛定谔方程获得了光子阻塞最佳条件. 通过对比数值模拟结果与解析结果, 解析结果与等时二阶关联函数的数值模拟高度一致, 验证了理论的正确性. 研究结果表明, 在参数适当的条件下, 系统中可以存在光子阻塞. 系统处于共振时, 平均光子数显著增加, 这对实现高亮度的单光子源十分必要. 进一步的驱动相位变化可导致最佳阻塞区域在驱动力强度与光学参量放大器非线性系数F-G参数二维平面发生位移甚至最佳光子阻塞区域形成的抛物线开口方向发生反转, 数值结果和理论结果均证实了驱动力相位对光子阻塞效应的调控作用. 值得一提的是, 在克尔非线性强度在宽参数范围内, 系统始终存在显著的光子阻塞效应, 展现出典型的普适光子阻塞特征. 物理机制分析表明, 光子阻塞源于系统两条光子跃迁路径在特定参数下的量子干涉相消, 有效地抑制了双光子激发. 克尔非线性虽调制系统能级但不影响量子干涉路径, 使光子阻塞效应在宽参数范围内保持稳定.
By combining analytical solutions and numerical simulations, we investigate the control mechanism of photon blockade effects in a hybrid quantum system consisting of a Kerr-medium single-mode cavity coupled with an optical parametric amplifier (OPA). To study photon blockade in the system, the dynamics are described by a master equation derived from the effective Hamiltonian, which considers single-mode cavity decay. In order to obtain analytical solutions under optimal photon blockade conditions, the quantum state of the system is expanded to the two-photon level based on the Fock state, and the steady-state probability amplitudes are derived by solving the Schrödinger equation, thereby yielding analytical expressions for the optimal photon blockade regime. The results demonstrate that photon blockade can be achieved in the system at appropriate parameters. Comparative analysis shows excellent agreement between the analytical results and numerical simulations of the equal-time second-order correlation function, validating both the correctness of the analytical solutions and the effectiveness of photon blockade in the system. The numerical results show that the average photon number significantly increases under resonant conditions, providing theoretical support for optimizing single-photon source brightness, which is essential for achieving high-brightness single-photon sources. Furthermore, variations in the driving phase can cause the optimal photon blockade region to shift in the two-dimensional parameter space of driving strength and OPA nonlinear coefficient, and even reverse the opening direction of the parabolic-shaped optimal blockade region. Both numerical and theoretical results confirm the regulatory effect of the driving phase on photon blockade. Additionally, the influence of Kerr nonlinearity is examined. The results show that photon blockade persists robustly over a broad range of Kerr nonlinear strengths, exhibiting universal characteristics. Physical mechanism analysis indicates that the photon blockade effect originates from destructive quantum interference between two photon transition pathways in the system under specific parameters, effectively suppressing two-photon excitation. Although Kerr nonlinearity modulates the energy levels of the system, it does not affect the quantum interference pathways, thus keeping the photon blocking effect stable over a wide parameter range. -
图 1 等时二阶关联函数$ {g^{\left( 2 \right)}}\left( 0 \right) $的对数随不同物理量之间的变化图像 (a) 等时二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right)$的对数值随驱动力强度${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa }$和光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$的变化, 其他参数设置为$\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} {12}}} \right. } {12}}$和${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$; (b) ${g^{\left( 2 \right)}}\left( 0 \right)$的对数值光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$和驱动力相位${\phi _a}$的变化关系图像, 其他参数设置为${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa } = 0.1$和${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$. 两幅图中的白色虚线, 由方程(13)给出, 表示光子阻塞最佳条件的解析结果
Fig. 1. Logarithmic value of the equal-time second-order correlation function $ {g^{\left( 2 \right)}}\left( 0 \right) $ versus different physical parameters are presented: (a) Logarithmic value of $ {g^{\left( 2 \right)}}\left( 0 \right) $ as a function of the driving strength $ {F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa } $ and the nonlinear coefficient $ {G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa } $ of the optical parametric amplifier, where c and ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$; (b) logarithmic value of $ {g^{\left( 2 \right)}}\left( 0 \right) $ as a function of the nonlinear coefficient $ {G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa } $ and the driving phase $ \phi $, where ${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa } = 0.1$ and ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$. In both figures, the white dashed lines, derived from Eq. (13), indicate the analytical solutions corresponding to the optimal conditions for photon blockade.
图 2 系统的平均光子数的对数$ \lg \left[ N \right] $随不同参数的变化 (a) 不同驱动力强度${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa }$下, $ \lg \left[ N \right] $随失谐量${\Delta \mathord{\left/ {\vphantom {\Delta \kappa }} \right. } \kappa }$的变化; (b) 不同光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$下, $ \lg \left[ N \right] $随驱动力相位$\phi $的变化; (c) 不同光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$下, $ \lg \left[ N \right] $随失谐量${\Delta \mathord{\left/ {\vphantom {\Delta \kappa }} \right. } \kappa }$的变化; (d) 不同克尔非线性强${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa }$下, $ \lg \left[ N \right] $随失谐量${\Delta \mathord{\left/ {\vphantom {\Delta \kappa }} \right. } \kappa }$的变化
Fig. 2. Logarithmic value of the average photon number $ N $ versus different parameters: (a) $ \lg \left[ N \right] $ as a function of detuning ${\Delta \mathord{\left/ {\vphantom {\Delta \kappa }} \right. } \kappa }$ at different driving strengths ${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa }$; (b) phase dependence of $ \lg \left[ N \right] $ under varying of the optical parametric amplifier nonlinear coefficients ${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$; (c) detuning dependence of $ \lg \left[ N \right] $ for different optical parametric amplifier nonlinear coefficients ${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$; (d) $ \lg \left[ N \right] $ versus detuning ${\Delta \mathord{\left/ {\vphantom {\Delta \kappa }} \right. } \kappa }$ at distinct Kerr nonlinearity strengths ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa }$.
图 3 不同驱动力相位$\phi $情况下, 等时二阶关联函数$ {g^{\left( 2 \right)}}\left( 0 \right) $的对数随驱动力强度${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa }$和光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$的变化 (a) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} {12}}} \right. } {12}}$; (b) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 6}} \right. } 6}$; (c) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 4}} \right. } 4}$; (d) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 3}} \right. } 3}$; (e) $\phi = {{{{5\pi }}} \mathord{\left/ {\vphantom {{{{5\pi }}} {12}}} \right. } {12}}$; (f) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 2}} \right. } 2}$. 图中的白色虚线由方程(13)给出, 表示光子阻塞最佳条件的解析结果. 克尔非线性强度均设置为${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$
Fig. 3. Logarithmic value of $ {g^{\left( 2 \right)}}\left( 0 \right) $ as a function of the driving strength $ {F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa } $ and the nonlinear coefficient $ {G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa } $ of the optical parametric amplifier under different driving phases $\phi $: (a) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} {12}}} \right. } {12}}$; (b) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 6}} \right. } 6}$; (c) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 4}} \right. } 4}$; (d) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 3}} \right. } 3}$; (e) $\phi = {{{{5\pi }}} \mathord{\left/ {\vphantom {{{{5\pi }}} {12}}} \right. } {12}}$; (f) $\phi = {{\pi} \mathord{\left/ {\vphantom {{\pi} 2}} \right. } 2}$. In all panels, the white dashed lines, derived from Eq. (13), represent the analytical solutions for the optimal photon blockade conditions. The Kerr nonlinearity strength was consistently set to ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.5$ in the numerical simulations.
图 4 不同克尔非线性强度${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa }$情况下, 等时二阶关联函数$ {g^{\left( 2 \right)}}\left( 0 \right) $的对数随驱动力强度${F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa }$和光学参量放大器非线性系数${G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa }$的变化图像 (a) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.1$; (b) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 1.0$; (c) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 2.0$; (d) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 5.0$. 图中的虚线由方程(13)给出, 表示光子阻塞最佳条件的解析结果
Fig. 4. Logarithmic value of $ {g^{\left( 2 \right)}}\left( 0 \right) $ as a function of the driving strength $ {F \mathord{\left/ {\vphantom {F \kappa }} \right. } \kappa } $ and the nonlinear coefficient $ {G \mathord{\left/ {\vphantom {G \kappa }} \right. } \kappa } $ of the optical parametric amplifier under different Kerr nonlinearity strength ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa }$: (a) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 0.1$; (b) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 1.0$; (c) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 2.0$; (d) ${U \mathord{\left/ {\vphantom {U \kappa }} \right. } \kappa } = 5.0$. In all panels, the white dashed lines, derived from Eq. (13), represent the analytical solutions for the optimal photon blockade conditions.
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