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Cavity quantum electrodynamics (QED) provides a fundamental platform for implementing light-matter interactions at the single-particle level, having been extensively investigated in fundamental physics and quantum information. Recent advances in parametric squeezing techniques have demonstrated remarkable capabilities for exponentially enhancing coherent coupling between an atom and a cavity. However, the full extent of manipulating quantum optical phenomena using these techniques still requires further exploration. This work systematically investigates the effects of optical parametric amplification on single-photon excited atom-cavity systems within a parametric driven cavity. In the proposed model, optical parametric amplification converts the driving photons into a squeezed cavity mode, which can enhance the atom-cavity interaction to the strong coupling region. Through analytical derivation of atomic and cavity radiation spectra, we demonstrate that the optical parametric amplification can lead to the splitting of atomic radiation spectra, but produces negligible effects on spectral intensity. Conversely, the cavity transmission spectrum exhibits both pronounced splitting and nonlinear intensity amplification. Notably, when driving field intensity approaches critical region, the intensity of the cavity radiation spectrum can be significantly enhanced. The underlying mechanism originates from parametric driving amplification, which converts the driving light into a squeezed cavity mode. When this squeezed mode is mapped back to the original mode of the cavity through Bogoliubov squeezing transformation, the pump photons in the squeezed cavity mode are converted into the radiation spectrum of the cavity, which leads to the amplification of the cavity radiation spectrum. This parametric enhancement protocol not only deepens fundamental understanding of engineered light-matter interactions but also establishes a practical framework for improving single-photon detection sensitivity in cavity-based quantum systems. These findings hold promising implications for quantum sensing and information processing applications.
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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
Figure 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 $
Figure 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 $
Figure 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 $
Figure 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 $
Figure 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 $.
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