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Tunable unconventional phonon blockade in Fabry-Perot cavity and optical parametric amplifier composite system

Li Hong Zhang Si-Qi Guo Ming Li Mei-Xuan Song Li-Jun

Tunable unconventional phonon blockade in Fabry-Perot cavity and optical parametric amplifier composite system

Li Hong, Zhang Si-Qi, Guo Ming, Li Mei-Xuan, Song Li-Jun
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  • In this paper, we present a scheme to realize an unconventional photon blockade effect in a Fabry-Perot cavity and optical parametric amplifier (OPA) composite system. The system includes a tunable phase of complex driving strength, the second-order correlation function is used to describe the photon statistical properties. The numerical simulation of the photon blockade effect is conducted with different parameters. Our calculations show that the unconventional photon blockade effect can be controlled by the tunable phase of complex driving strength. Under the weak driving condition, the exact optimal conditions for strong photon anti-bunching are analytically derived (i.e. the optimal nonlinear gain of optical parametric amplifier and the phase of the field driving for the strong photon anti-bunching are obtained), and obtain the analytic calculations of the second-order correlation function. Under the optimal conditions, we perform a numerical simulation with different parameters. The optimal conditions for strong photon anti-bunching are found by analytic calculations, which are in good agreement with the numerical results. The results provide a platform for coherently operating the photon blockade and have potential applications in quantum information processing and quantum optical devices.
      Corresponding author: Song Li-Jun, songlj@jlenu.edu.cn
    [1]

    Imamoglu A, Schmidt H, Woods G, Deutsch M 1997 Phys. Rev. Lett. 79 1467

    [2]

    Liew T C H, Savona V 2010 Phys. Rev. Lett. 104 183601

    [3]

    Cao C, Mi S C, Wang T, Zhang R, Wang C 2016 IEEE J. Quantum Electron. 52 7000205

    [4]

    Cao C, Mi S C, Gao Y P, He L Y, Yang D, Wang T J, Zhang R, Wang C 2016 Sci. Rep. 6 22920

    [5]

    Cao Cong, Chen Xi, Duan Y W, Fan L, Zhang R, Wang T J, Wang C 2017 Optik 130 659

    [6]

    张秀龙, 鲍倩倩, 杨明珠, 田雪松 2018 物理学报 67 104203

    Zhang X L, Bao Q Q, Yang M Z, Tian X S 2018 Acta Phys. Sin. 67 104203

    [7]

    廖庆洪, 叶杨, 李红珍, 周南润 2018 物理学报 67 40302

    Liao Q H, Ye Y, Li H Z, Zhou N R 2018 Acta Phys. Sin. 67 40302

    [8]

    Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E, Kimble H J 2005 Nature 436 87

    [9]

    Greentree A D, Tahan C, Cole J H, Hollenberg L C L 2006 Nat. Phys. 2 856

    [10]

    Angelakis D G, Santos M F, Bose S 2007 Phys. Rev. A 76 031805

    [11]

    Shen H Z, Zhou Y H, Yi X X 2015 Phys. Rev. A 91 063808

    [12]

    Shen H Z, Zhou Y H, Yi X X 2014 Phys. Rev. A 90 023849

    [13]

    Irvine W T M, Hennessy K, Bouwmeester D 2006 Phys. Rev. Lett. 96 057405

    [14]

    Zhou Y H, Shen H Z, Yi X X 2015 Phys. Rev. A 92 023838

    [15]

    Shen H Z, Zhou Y H, Liu H D, Wang G C, Yi X X 2015 Opt. Express 23 32835

    [16]

    Zhou Y H, Zhang S S, Shen H Z, Yi X X 2017 Opt. Lett. 42 1289

    [17]

    Shen H Z, Shang C, Zhou Y H, Yi X X 2018 Phys. Rev. A 98 023856

    [18]

    Shen H Z, Xu S, Zhou Y H, Wang G C, Yi X X 2018 J. Phys. B 51 035503

    [19]

    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819

    [20]

    Su S L, Tian Y Z, Shen H Z, Zang H P, Liang E J, Zhang S 2017 Phys. Rev. A 96 042335

    [21]

    Su S L, Gao Y, Liang E J, Zhang S 2017 Phys. Rev. A 95 022319

    [22]

    Su S L, Liang E J, Zhang S, Wen J J, Sun L L, Jin Z, Zhu A D 2016 Phys. Rev. A 93 012306

    [23]

    Zhou Y H, Shen H Z, Shao X Q, Yi X X 2016 Opt. Express 24 17332

    [24]

    Tang J, Geng W, Xu X 2015 Sci. Rep. 5 9252

    [25]

    Majumdar A, Bajcsy M, Rundquist A, Vuckovic J 2012 Phys. Rev. Lett. 108 183601

    [26]

    Zhang W, Yu Z, Liu Y, Peng Y 2014 Phys. Rev. A 8 043832

    [27]

    Flayac H, Savona V 2016 Phys. Rev. A 94 013815

    [28]

    Gerace D, Savona V 2014 Phys. Rev. A 89 031803

    [29]

    Lemonde M A, Didier N, Clerk A A 2014 Phys. Rev. A 90 063824

    [30]

    Xu X W, Li Y J 2013 J. Phys. B 46 035502

    [31]

    Wicz A, Li H R, Miranoao J Q, Nori F, Jing H 2018 Phys. Rev. Lett. 121 153601

    [32]

    石海泉, 谢智强, 徐勋卫, 刘念华 2018 物理学报 67 044203

    Shi H Q, Xie Z Q, Xu X W, Liu N H 2018 Acta Phys. Sin. 67 044203

    [33]

    Sarma B, Sarma A K 2017 Phys. Rev. A 96 053827

  • 图 1  (a) 用激光抽运OPA, 在腔内产生参量放大的腔结构示意图; (b) 量子干涉系统的跃迁路径

    Figure 1.  (a) Schematic diagram of the cavity setup with an OPA which is pumped by a laser to produce parametric amplification in the cavity; (b) transition paths of the system for quantum interference.

    图 3  等时二阶关联函数$\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$数值结果随OPA非线性增益G和相位$\theta $的等高线图${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $\phi = 0.5\;{\rm{rad}}$

    Figure 3.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the nonlinear gain G of the OPA and phase $\theta $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $\phi = 0.5\;{\rm{rad}}$.

    图 2  等时二阶关联函数${g^{\left(2\right)}}(0)$随OPA非线性增益G的变化${\varOmega / {\kappa = 0.01}}$, $\theta = - 0.0341{\text{π}}$, ${\varDelta _a} = 1$

    Figure 2.  Variation curves of the zero-time-delay second-order correlation function ${g^{\left(2\right)}}(0)$ with the nonlinear gain G of the OPA. Other parameters are ${\varOmega / {\kappa = 0.01}}$, $\theta = - 0.0341{\text{π}}$, ${\varDelta _a} = 1$.

    图 4  等时二阶关联函数$\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$数值结果随相位$\theta $$\phi $的等高线图${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$

    Figure 4.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the phase $\theta $ and $\phi $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$.

    图 5  等时二阶关联函数$\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$数值结果随OPA非线性增益$G$和相位$\phi $变化的等高线图${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$

    Figure 5.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the nonlinear gain of the optical parametric amplifier $G$ and phase $\phi $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$.

    图 6  (a) 二阶关联函数${g^{\left(2\right)}}(0)$随OPA非线性增益$G$的变化, 其中蓝色实线由数值求解方程(3)得出, 红色菱形由(13)和(15)式解析得出; 其他参数为${\varDelta _a} = 1, \; \varOmega /\kappa = 0.01,$ $\phi = 0.5\;{\rm{rad}}, \; \theta = {\theta _{\rm opt}}$; (b) 二阶关联函数${g^{\left(2\right)}}(0)$随相位$\theta $的变化, 蓝色实线由数值求解方程(3)得出, 红色圆形由(13)和(15)式解析得出; 其他参数为${\varDelta _a} = 1, \;\varOmega /\kappa = 0.01,$ $\phi = 0.5\;{\rm{rad}}, \;G = {G_{\rm opt}}$

    Figure 6.  (a) The second-order correlation functions ${g^{\left(2\right)}}(0)$ vs. the nonlinear gain of the optical parametric amplifier $G$; the blue solid line indicates the numerical results by numerically solving Eq. (3) and the red diamond corresponds to the analytical results of Eq. (13) and Eq. (15); other parameters are ${\varDelta _a} = 1, \; \varOmega /\kappa = 0.01, \; \phi = 0.5\;{\rm{rad}}, \; \theta = {\theta _{\rm opt}}$; (b) the second-order correlation functions ${g^{\left(2\right)}}(0)$ vs. the phase $\theta $; the blue solid line indicates the numerical results by numerically solving Eq. (3) and the red diamond corresponds to the analytical results of Eq. (13) and Eq. (15). Other parameters are ${\varDelta _a} = 1, \;\varOmega /\kappa = 0.01, \;\phi = 0.5\;{\rm{rad}},$ G = Gopt.

  • [1]

    Imamoglu A, Schmidt H, Woods G, Deutsch M 1997 Phys. Rev. Lett. 79 1467

    [2]

    Liew T C H, Savona V 2010 Phys. Rev. Lett. 104 183601

    [3]

    Cao C, Mi S C, Wang T, Zhang R, Wang C 2016 IEEE J. Quantum Electron. 52 7000205

    [4]

    Cao C, Mi S C, Gao Y P, He L Y, Yang D, Wang T J, Zhang R, Wang C 2016 Sci. Rep. 6 22920

    [5]

    Cao Cong, Chen Xi, Duan Y W, Fan L, Zhang R, Wang T J, Wang C 2017 Optik 130 659

    [6]

    张秀龙, 鲍倩倩, 杨明珠, 田雪松 2018 物理学报 67 104203

    Zhang X L, Bao Q Q, Yang M Z, Tian X S 2018 Acta Phys. Sin. 67 104203

    [7]

    廖庆洪, 叶杨, 李红珍, 周南润 2018 物理学报 67 40302

    Liao Q H, Ye Y, Li H Z, Zhou N R 2018 Acta Phys. Sin. 67 40302

    [8]

    Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E, Kimble H J 2005 Nature 436 87

    [9]

    Greentree A D, Tahan C, Cole J H, Hollenberg L C L 2006 Nat. Phys. 2 856

    [10]

    Angelakis D G, Santos M F, Bose S 2007 Phys. Rev. A 76 031805

    [11]

    Shen H Z, Zhou Y H, Yi X X 2015 Phys. Rev. A 91 063808

    [12]

    Shen H Z, Zhou Y H, Yi X X 2014 Phys. Rev. A 90 023849

    [13]

    Irvine W T M, Hennessy K, Bouwmeester D 2006 Phys. Rev. Lett. 96 057405

    [14]

    Zhou Y H, Shen H Z, Yi X X 2015 Phys. Rev. A 92 023838

    [15]

    Shen H Z, Zhou Y H, Liu H D, Wang G C, Yi X X 2015 Opt. Express 23 32835

    [16]

    Zhou Y H, Zhang S S, Shen H Z, Yi X X 2017 Opt. Lett. 42 1289

    [17]

    Shen H Z, Shang C, Zhou Y H, Yi X X 2018 Phys. Rev. A 98 023856

    [18]

    Shen H Z, Xu S, Zhou Y H, Wang G C, Yi X X 2018 J. Phys. B 51 035503

    [19]

    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819

    [20]

    Su S L, Tian Y Z, Shen H Z, Zang H P, Liang E J, Zhang S 2017 Phys. Rev. A 96 042335

    [21]

    Su S L, Gao Y, Liang E J, Zhang S 2017 Phys. Rev. A 95 022319

    [22]

    Su S L, Liang E J, Zhang S, Wen J J, Sun L L, Jin Z, Zhu A D 2016 Phys. Rev. A 93 012306

    [23]

    Zhou Y H, Shen H Z, Shao X Q, Yi X X 2016 Opt. Express 24 17332

    [24]

    Tang J, Geng W, Xu X 2015 Sci. Rep. 5 9252

    [25]

    Majumdar A, Bajcsy M, Rundquist A, Vuckovic J 2012 Phys. Rev. Lett. 108 183601

    [26]

    Zhang W, Yu Z, Liu Y, Peng Y 2014 Phys. Rev. A 8 043832

    [27]

    Flayac H, Savona V 2016 Phys. Rev. A 94 013815

    [28]

    Gerace D, Savona V 2014 Phys. Rev. A 89 031803

    [29]

    Lemonde M A, Didier N, Clerk A A 2014 Phys. Rev. A 90 063824

    [30]

    Xu X W, Li Y J 2013 J. Phys. B 46 035502

    [31]

    Wicz A, Li H R, Miranoao J Q, Nori F, Jing H 2018 Phys. Rev. Lett. 121 153601

    [32]

    石海泉, 谢智强, 徐勋卫, 刘念华 2018 物理学报 67 044203

    Shi H Q, Xie Z Q, Xu X W, Liu N H 2018 Acta Phys. Sin. 67 044203

    [33]

    Sarma B, Sarma A K 2017 Phys. Rev. A 96 053827

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  • Received Date:  27 January 2019
  • Accepted Date:  04 April 2019
  • Available Online:  16 August 2019
  • Published Online:  01 June 2019

Tunable unconventional phonon blockade in Fabry-Perot cavity and optical parametric amplifier composite system

    Corresponding author: Song Li-Jun, songlj@jlenu.edu.cn
  • 1. Institute for Interdisciplinary Quantum Information Technology, Jilin Engineering Normal University, Changchun 130052, China
  • 2. Jilin Engineering Laboratory for Quantum Information Technology, Changchun 130052, China

Abstract: In this paper, we present a scheme to realize an unconventional photon blockade effect in a Fabry-Perot cavity and optical parametric amplifier (OPA) composite system. The system includes a tunable phase of complex driving strength, the second-order correlation function is used to describe the photon statistical properties. The numerical simulation of the photon blockade effect is conducted with different parameters. Our calculations show that the unconventional photon blockade effect can be controlled by the tunable phase of complex driving strength. Under the weak driving condition, the exact optimal conditions for strong photon anti-bunching are analytically derived (i.e. the optimal nonlinear gain of optical parametric amplifier and the phase of the field driving for the strong photon anti-bunching are obtained), and obtain the analytic calculations of the second-order correlation function. Under the optimal conditions, we perform a numerical simulation with different parameters. The optimal conditions for strong photon anti-bunching are found by analytic calculations, which are in good agreement with the numerical results. The results provide a platform for coherently operating the photon blockade and have potential applications in quantum information processing and quantum optical devices.

    • 单光子源在量子光学领域至关重要, 理想的单光子源在量子信息和量子通信领域意义重大. 1997年, Imamoglu等[1]首次提出光子阻塞的概念, 光子阻塞效应是实现单光子源的方法之一. 2010年, Liew和Savona[2]在弱Kerr非线性条件下仍获得强的反聚束效应, 首次提出非常规光子阻塞的概念. 近年来, 腔光力系统在理论和实验上都取得了快速的发展[3-7], 为量子现象和光学效应研究提供了新途径, 尤其在非线性方面为光子阻塞机制实现单光子源奠定了基础. Birnbaum等[8]首次在腔量子电动力学(quantum electro-dynamics, QED)系统中成功观测到光子阻塞效应以来, 人们分别在耦合空腔阵列[9,10]、一维光波导[11]、量子腔耦合系统[12]、光学系统[13-22]和电路QED系统[23]中观测到光子阻塞现象. 在实现非常规光子阻塞方面同样进展迅速, 如耦合单量子点腔系统[24]、有损双模纳米谐振腔系统[25,26]、三阶非线性耦合的单模腔[11,15,27]、二阶非线性耦合腔[10,28]、高斯压缩态[29]和耦合光机械系统[30]等均观测到了非常规光子阻塞现象. 2018年, Wicz等[31]利用非互易量子光学首次揭示了依赖方向的光子阻塞效应. 同年, 石海泉等[32]在多模光力系统中研究了非传统声子阻塞效应.

      本文研究基于包含简并光学参量放大(optical parametric amplifier, OPA)的Fabry-Perot腔内实现非常规光子阻塞的可能性. 尽管相关系统与内容已经有人研究, 但本文包含复合型驱动强度的相位, 并解析给出包含此相位的最优化条件. 通过数值模拟, 验证了优化条件的有效性. 研究发现复合型驱动的相位对非常规光子阻塞效应有着显著的影响.

    2.   物理模型
    • 考虑典型的光力系统: 在Fabry-Perot腔中包含简并的OPA装置, 如图1(a)所示, 系统的哈密顿量可以写为[33]

      其中$a({a^† })$是光模的降(升)算符, ${\omega _a}$是腔场的共振频率, ${\omega _{\rm{l}}}$是驱动激光频率, $\varOmega {{\rm{e}}^{{\rm{i}}\phi }}$是复合型驱动强度, $G$是OPA的非线性增益, $\theta $是外场驱动OPA的相位.

      假设驱动强度$\varOmega $非常小, 通过$U = \exp \left[ {{\rm{i}}{\omega _{\rm{l}}}t{a^† }a} \right]$旋转变换, 有效哈密顿量变为

      其中${\varDelta _a} = {\omega _a} - {\omega _{\rm{l}}}$表示腔的失谐量. 在下面的研究中, 主要探索相位$\phi $对光子阻塞的影响.

      Figure 1.  (a) Schematic diagram of the cavity setup with an OPA which is pumped by a laser to produce parametric amplification in the cavity; (b) transition paths of the system for quantum interference.

      用等时二阶关联函数分析光子阻塞效应, 其表达式如下:

      二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right) > 1$表示光子存在聚束效应, 会极大提高腔内双光子存在的概率, 反之${g^{\left( 2 \right)}}\left( 0 \right) < 1$表示光子存在反聚束效应, 会有效抑制腔内双光子存在的概率. 如果二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right) \to 0$, 表示系统处于完全光子阻塞机制下, 腔内同时出现两个光子的概率趋近于零.

      考虑到系统的耗散, 系统动力学演化过程可以由如下主方程描述:

      其中$\kappa $表示腔的耗散率. 通过数值求解主方程得到稳态解

    3.   数值结果
    • 本文所有计算结果都是基于弱驱动条件, 令${\varOmega / {\kappa = 0.01}}$, 数值模拟结果如图2图3所示. 为方便起见, 将耗散率$\kappa $归一化. 在图2中, 展示了关联函数数值计算结果随相位$\phi $$G$的变化, 结果表明对于不同的$G$存在光子反聚束效应. 相位$\phi $分别取0.5, 0.8和1.2 rad, 数值结果表明$\phi = 0.5\;{\rm{rad}}$时对应优化的强反聚束效应. 为了得到对应反聚束效应的优化参数, 图3展示了$\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$随OPA非线性增益$G$和相位$\theta $的等高线图, 其他参数为$\phi = 0.5\;{\rm{rad}}$, ${\varDelta _a} = 1$. 由图3可见, 在一小区域内${g^{\left(2\right)}}(0)\ll 1$, 这区域内选定参数可实现强的反聚束效应.

      Figure 3.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the nonlinear gain G of the OPA and phase $\theta $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $\phi = 0.5\;{\rm{rad}}$.

      Figure 2.  Variation curves of the zero-time-delay second-order correlation function ${g^{\left(2\right)}}(0)$ with the nonlinear gain G of the OPA. Other parameters are ${\varOmega / {\kappa = 0.01}}$, $\theta = - 0.0341{\text{π}}$, ${\varDelta _a} = 1$.

    4.   光子阻塞最优条件
    • 系统演化过程用Fock态表示, 假设初始时刻处在$\left| 0 \right\rangle $态上, 系统含时演化的态$\left| \varPsi \right\rangle $

      其中${C_m}\;(m = 0, 1, 2)$表示量子态的概率幅, 通过求解薛定谔方程可得到${C_m}$, 考虑到系统的耗散情况, 此时薛定谔方程为

      有效非厄米哈密顿量

      其中${\varDelta '_a} = {\varDelta _a} - {{{\rm{i}}\kappa } / 2}$. 把态$\left| \varPsi \right\rangle $和哈密顿量$H'$代入到(7)式中, 得到概率幅所满足的方程:

      通过对系数耦合方程组的求解, 可以得到稳态解. 当态$\left| 2 \right\rangle $等于0时, 系统可达到完全光子阻塞效应, 在此条件下可以解出光子阻塞的最优化条件. 因此, 在(9)式中, 令${C_2} = 0$, 在弱驱动条件下, 方程(9)中第一个式子总是近似满足, 所以只需考虑最后两个方程进行计算:

      为了保证${C_0}$${C_1}$有非奇异解, 可解析给出优化条件

      其中“opt”表示$G$$\theta $的优化解. 值得一提的是, 这些条件取决于腔失谐、驱动激光振幅和复合驱动强度的相位. 由于最优条件与驱动OPA的抽运场参数相对应, 所以这些参数可以通过调节OPA抽运场来控制. 当最优条件(11)和(12)式同时满足时, 可获得较强的反聚束效应.

      把方程(6)代入方程(3), 在弱耦合条件下, 态的概率幅满足${C_0} \gg {C_1} \gg C{}_2$, 此时得到等时二阶关联函数

      在弱抽运极限下, 基态布居数近似为1, 其他能级布居数微乎其微, 可忽略不计, 在这种情况下, 方程(9)变为

      求解(14)式得到

      通过数值求解方程(3)模拟了$\lg\left[ {{g^{\left(2\right)}}\left(0\right)} \right]$随外场驱动OPA的相位$\theta $和复合型驱动相位$\phi $变化的等高线图, 结果如图4所示. 其中$G$满足(11)式, 即$G = {G_{\rm opt}}$. 其中红色虚线由(11)式画出, 研究发现非常规光子阻塞发生的地方正好是$G$取最优值的地方. 同样地, 在图5中, 模拟了$\lg\left[ {{g^{\left(2\right)}}\left(0\right)} \right]$随非线性增益$G$和复合型驱动相位$\phi $变化的等高线图, 其中$\theta $满足(12)式, 即$\theta = {\theta _{\rm opt}}$. 其中红色虚线部分由(12)式画出, 发现非常规光子阻塞发生的地方正好是$\theta $取最优值的地方. 由此可知, 光子的统计性质可以通过改变复合型驱动强度的相位$\phi $、OPA的非线性增益$G$和外场驱动OPA的相位$\theta $来调节. 系统的跃迁路径如图1(b)所示, 双光子激发共有2条跃迁路径, 一条是由驱动场激发从$\left| 0 \right\rangle $态到$\left| 1 \right\rangle $态, 然后从$\left| 1 \right\rangle $态再到$\left| 2 \right\rangle $态; 另一条是OPA作用直接从$\left| 0 \right\rangle $态到$\left| 2 \right\rangle $态. 当满足阻塞优化条件时, 这两条路径上的光子发生量子干涉相消, 干涉的结果为光子不能占据$\left| 2 \right\rangle $态, 所以发生强反聚束效应, 即产生光子阻塞效应.

      Figure 4.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the phase $\theta $ and $\phi $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$.

      Figure 5.  Contour plot of the second-order correlation functions $\lg \left[ {{g^{\left(2\right)}}\left(0\right)} \right]$ vs. the nonlinear gain of the optical parametric amplifier $G$ and phase $\phi $. Other parameters are ${\varOmega / {\kappa = 0.01}}$, ${\varDelta _a} = 1$, $G = {G_{\rm opt}}$.

      图6中, 分别数值模拟了二阶关联函数${g^{\left(2\right)}}(0)$随非线性增益$G$和相位$\theta $的变化. 图6(a)为模拟得到的${g^{\left(2\right)}}(0)$$G$的变化曲线, 蓝色实线是由数值求解方程(3)得到的, 红色菱形是由(13)和(15)式得到的解析结果, 其中$\theta = {\theta _{\rm opt}}$$\phi = 0.5\;{\rm{rad}}$; 图6(b)为模拟得到的${g^{\left(2\right)}}(0)$$\theta $的变化曲线, 蓝色实线是由数值求解方程(3)得到的, 红色圆形是由(13)和(15)式得到的解析结果, 其中$G = {G_{\rm opt}}$$\phi = 0.5\;{\rm{rad}}$. 可以看出数值模拟与解析结果相符合, 说明了解析结果的正确性.

      Figure 6.  (a) The second-order correlation functions ${g^{\left(2\right)}}(0)$ vs. the nonlinear gain of the optical parametric amplifier $G$; the blue solid line indicates the numerical results by numerically solving Eq. (3) and the red diamond corresponds to the analytical results of Eq. (13) and Eq. (15); other parameters are ${\varDelta _a} = 1, \; \varOmega /\kappa = 0.01, \; \phi = 0.5\;{\rm{rad}}, \; \theta = {\theta _{\rm opt}}$; (b) the second-order correlation functions ${g^{\left(2\right)}}(0)$ vs. the phase $\theta $; the blue solid line indicates the numerical results by numerically solving Eq. (3) and the red diamond corresponds to the analytical results of Eq. (13) and Eq. (15). Other parameters are ${\varDelta _a} = 1, \;\varOmega /\kappa = 0.01, \;\phi = 0.5\;{\rm{rad}},$ G = Gopt.

    5.   结 论
    • 本文研究了在Fabry-Perot腔和OPA复合系统中实现非常规光子阻塞效应, 给出了光子阻塞出现的最优化条件. 研究发现, 可以通过调整复合驱动强度中的相位来实现光子反聚束效应. 本文考虑了弱驱动和弱非线性条件, 通过选择最优解, 从数值和解析两方面论证了系统的强反聚束现象, 发现数值模拟与解析结果是一致的. 本研究为精确控制光子阻塞提供了方案, 并为制备优良单光子源提供了理论基础.

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