图 1  原子与参量驱动泵浦腔的相互作用示意图. 腔中包含$ \chi^{(2)} $非线性介质和二能级原子, 其中非线性介质受到振幅$ \Omega_{p} $, 频率为$ \omega_{p} $, 相位为$ \theta_{p} $的外部驱动场泵浦, 同时原子被单光子激发至激发态. 为了消除光学参量放大带来的额外耗散, 腔耦合了一个压缩参数为$ r_{e} $, 参考相位为$ \theta_{e} $的压缩真空库

Fig. 1.  The schematic of our proposed method for investigating the interaction between an atom and a parametrically driven cavity. The optical cavity contains a $ \chi^{(2)} $ nonlinear medium and one two-level atom, where the nonlinear medium is pumped by a driving field of amplitude $ \Omega_{p} $, frequency $ \omega_{p} $ and phase $ \theta_{p} $, and the atom is excited to the excited state by a single photon. In order to eliminate the additional dissipation caused by optical parametric amplification, the cavity couples to a squeezed-vacuum reservoir with the squeezing parameter $ r_{e} $ and a reference phase $ \theta_{e} $.

图 2  原子的单光子辐射谱, 黑色虚线对应的驱动场强度$ \Omega_{p} = 0 $, 红色实线对应的驱动场强度$ \Omega_{p} = 0.799\gamma $, 其它参数取值为$ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $

Fig. 2.  The radiation spectrum of the atom. The black dashed line and the red solid line are plotted with the driving field driving intensities of $ \Omega_{p} = 0 $ and $ \Omega_{p} = 0.799\gamma $, respectively. Other parameters are $ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $.

图 3  腔的辐射谱, 黑色虚线对应的驱动场强度$ \Omega_{p} = 0 $, 红色实线对应的驱动场强度$ \Omega_{p} = 0.799\gamma $, 其他参数的取值为$ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $

Fig. 3.  The transmission spectrum of the cavity. The black dashed line and red solid line are plotted with the driving intensities of $ \Omega_{p} = 0 $ and $ \Omega_{p} = 0.799\gamma $, respectively. Other parameters are $ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $.

图 4  腔的辐射谱, 图4(a)红色实线和黑色虚线分别对应$ \Omega_{p} = 0 $时耦合强度为$ g = 0.5\gamma $和$ g = 1.62\gamma $时的辐射谱, 图4(b)红色实线对应$ \Omega_{p} = 0.799\gamma $, $ g = 0.5\gamma $时的辐射谱, 黑色虚线对应$ \Omega_{p} = 0 $, $ g = 1.62\gamma $时的透射谱. 其他参数的取值为$ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ \kappa = \gamma $

Fig. 4.  The transmission spectrum of the cavity. Fig. 4(a) is plotted with parametric pump field intensity $ \Omega_{p} = 0 $, where the red solid line and black dashed line are corresponding to the coupling strength of $ g = 0.5\gamma $ and $ g = 1.62\gamma $, respectively. In Fig. 4(b) the red solid line is plotted with $ \Omega_{p} = 0.799\gamma $ and $ g = 0.5\gamma $, while the black dashed line is plotted with $ \Omega_{p} = 0 $ and $ g = 1.62\gamma $. Other parameters are $ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ \kappa = \gamma $.

图 5  原子和腔模的辐射强度随驱动场强度的变化曲线, 嵌入图为根据公式(20)得到的放大因子m随驱动场强度的变化曲线, 其他参数为$ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $

Fig. 5.  The intensity of the atomic and cavity mode spectra as a function of the driving field intensity. The inset illustrates the dependence of the amplification factor m on the driving field intensity, which is derived from Eq. (20). Other parameters are $ \Delta_{c} = 0.8\gamma $, $ \Delta_{a} = 0 $, $ g = 0.5\gamma $, $ \kappa = \gamma $.