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Optical nonreciprocity with blue-detuned driving in two-cavity optomechanics

Zhang Li-Wei Li Xian-Li Yang Liu

Optical nonreciprocity with blue-detuned driving in two-cavity optomechanics

Zhang Li-Wei, Li Xian-Li, Yang Liu
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  • Radiation pressure in an optomechanical system can be used to generate various quantum phenomena. Recently, one paid more attention to the study of optical nonreciprocity in an optomechanical system, and nonreciprocal devices are indispensable for building quantum networks and ubiquitous in modern communication technology. Here in this work, we study how to realize the perfect optical nonreciprocity in a two-cavity optomechanical system with blue-detuned driving. Our calculations show that the optical nonreciprocity comes from the quantum interference of signal transmission between two possible paths corresponding to the two interactions in this system, i.e. optomechanical interaction and linearly-coupled interaction. According to the standard input-output relation of optical field in cavity optomechanics, we obtain the expression of output optical field, from which we can derive the essential conditions to achieve the perfect optical nonreciprocity, and find there are two sets of coupling strengths both of which can realize the perfect optical nonreciprocal transmission. Because the system is driven by blue-detuned driving, the system is stable only under some conditions which we can obtain according to the Routh-Hurwitz criterion. Due to the blue-detuned driving, there will be transmission gain (transmission amplitude is greater than 1) in the nonreciprocal transmission spectrum. We also find that the bandwidth of nonreciprocal transmission spectrum is in proportion to mechanical decay rate if mechanical decay rate is much less than the cavity decay rate. In other words, in a realistic optomechanical parameter regime, where mechanical decay rate is much less than cavity decay rate, the bandwidth of nonreciprocal transmission spectrum is very narrow. Our results can also be applied to other parametrically coupled three-mode bosonic systems and may be used to realize the state transfer process and optical nonreciprocal transmission in an optomechanical system.
      Corresponding author: Li Xian-Li, lxl7158@163.com ; Yang Liu, lyang@hrbeu.edu.cn
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    Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201

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    Zhang B, Liang J Q, Lü J G, Liang Z Z, Feng S Y 2018 Acta Phys. Sin. 67 070701

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

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    Huang R, Miranowicz A, Liao J Q, Nori F, Jing H 2018 Phys. Rev. Lett. 121 153601

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    Jayich A M, Sankey J C, Zwickl B M, Yang C, Thompson J D, Girvin S M, Clerk A A, Marquardt F, Harris J G E 2008 New J. Phys. 10 095008

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    Agarwal G S, Huang S 2014 New J. Phys. 16 033023

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  • 图 1  双腔光力学系统示意图, 两光学腔通过光力相互作用与一个力学振子相耦合, 振幅为$\varepsilon_{\rm c}$$\varepsilon_{\rm d}$($\varepsilon_{\rm L}$$\varepsilon_{\rm R}$)的强耦合场 (探测场)分别从左右两侧驱动腔模$c_{1}$$c_{2}$, 同时两腔模之间存在线性耦合相互作用J

    Figure 1.  A two-cavity optomechanical system with a mechanical resonator interacted with two cavities. Two strong coupling fields (probe fields) with amplitudes $\varepsilon_{\rm c}$ and $\varepsilon_{\rm d}$ ($\varepsilon_{\rm L}$ and $\varepsilon _{\rm R}$) are used to drive cavity $c_{1}$ and $c_{2}$ respectively. Meanwhile, the two cavities are linearly coupled to each other with coupling strength J

    图 2  传输振幅$T_{\rm {LR}}$(红线)和$T_{\rm {RL}}$(黑线)在不同力学振子耗散速率下随着标准化失谐$x/\kappa$的变化曲线 (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=2; 其他参数: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{+}$$J=J_{+}$(见(16)式)

    Figure 2.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\kappa$ for different cavity damping rate: (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=2. Other parameters: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{+}$ and $J=J_{+}$ according to Eq. (16)

    图 3  传输振幅$T_{\rm {LR}}$(红线)和$T_{\rm {RL}}$(黑线)在不同力学振子耗散速率下随着标准化失谐$x/\kappa$的变化曲线 (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=10. 其他参数: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{-}$$J=J_{-}$(见(16)式)

    Figure 3.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\kappa$ for different mechancial damping rate: (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=10. Other parameters: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{-}$ and $J=J_{-}$ according to Eq. (16)

    图 4  传输振幅$T_{\rm {LR}}$(红线)和$T_{\rm {RL}}$(黑线)在不同非互易相位差θ和耦合强度G时随着标准化失谐$x/\gamma$的变化曲线 (a) $\theta=-\dfrac{{\text{π}}}{4}$$G=G_{-}$; (b) $\theta=-\dfrac{{\text{π}}}{4}$$G=G_{+}$; (c) $\theta=-\dfrac{3{\text{π}}}{4}$$G=G_{-}$; (d) $\theta=-\dfrac{3{\text{π}}}{4}$$G=G_{+}$; 其他参数: $\gamma/\kappa=10^{-3}$, $J=J_{\pm}$$G=G_{\pm}$(见(19)式)

    Figure 4.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\gamma$ for different nonreciprocal phase θ and coupling strength G: (a) $\theta=-\dfrac{{\text{π}}}{4}$ and $G=G_{-}$; (b) $\theta=-\dfrac{{\text{π}}}{4}$ and $G=G_{+}$; (c) $\theta=-\dfrac{3{\text{π}}}{4}$ and $G=G_{-}$; (d) $\theta=-\dfrac{3{\text{π}}}{4}$ and $G=G_{+}$. Other parameters: $\gamma/\kappa=10^{-3}$, coupling strengths $J=J_{\pm}$ and $G=G_{\pm}$ according to Eq. (19)

  • [1]

    Jalas D, Petrov A, Eich M, et al. 2013 Nat. Photonics 7 579

    [2]

    Aplet L J, Carson J W 1964 Appl. Opt. 3 544

    [3]

    Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391

    [4]

    陈雪, 刘晓威, 张可烨, 袁春华, 张卫平 2015 物理学报 64 164211

    Chen X, Liu X W, Zhang K Y, Yuan C H, Zhang W P 2015 Acta Phys. Sin. 64 164211

    [5]

    陈华俊, 方贤文, 陈昌兆, 李洋 2016 物理学报 65 194205

    Chen H J, Fang X W, Chen C Z, Li Y 2016 Acta Phys. Sin. 65 194205

    [6]

    严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201

    Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201

    [7]

    班章, 梁静秋, 吕金光, 梁中翥, 冯思悦 2018 物理学报 67 070701

    Zhang B, Liang J Q, Lü J G, Liang Z Z, Feng S Y 2018 Acta Phys. Sin. 67 070701

    [8]

    Chen R X, Shen L T, Yang Z B, Wu H Z, Zheng S B 2014 Phys. Rev. A 89 023843

    [9]

    Liao J Q, Wu Q Q, Nori F 2014 Phys. Rev. A 89 014302

    [10]

    Yan X B 2017 Phys. Rev. A 96 053831

    [11]

    He Q Y, Ficek Z 2014 Phys. Rev. A 89 022332

    [12]

    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

    [13]

    Kiesewetter S, He Q Y, Drummond P D, Reid M D 2014 Phys. Rev. A 90 043805

    [14]

    Lin Q, He B, Ghobadi R, Simon C 2014 Phys. Rev. A 90 022309

    [15]

    He Q Y, Reid M D 2013 Phys. Rev. A 88 052121

    [16]

    Yan X B, Deng Z J, Tian X D, Wu J H 2019 Opt. Express 27 024393

    [17]

    He B, Yang L, Lin Q, Xiao M 2017 Phys. Rev. Lett. 118 233604

    [18]

    Li Y, Wu L A, Wang Z D 2011 Phys. Rev. A 83 043804

    [19]

    Deng Z J, Li Y, Gao M, Wu C W 2012 Phys. Rev. A 85 025804

    [20]

    Liu Y C, Wen H Y, Wei W C, Xiao Y F 2013 Chin. Phys. B 22 114213

    [21]

    Huang S M, Agarwal G S 2009 Phys. Rev. A 79 013821

    [22]

    Agarwal G S, Huang S M 2010 Phys. Rev. A 81 041803(R)

    [23]

    Shu J 2011 Chin. Phys. Lett. 28 104203

    [24]

    陈华俊, 米贤武 2011 物理学报 60 124206

    Chen H J, Mi X W 2011 Acta Phys. Sin. 60 124206

    [25]

    Han Y, Cheng J, Zhou L 2011 J. Phys. B 44 165505

    [26]

    Zhang J Q, Li Y, Feng M, Xu Y 2012 Phys. Rev. A 86 053806

    [27]

    Lü X Y, Zhang W M, Ashhab S, Wu Y, Nori F 2013 Sci. Rep. 3 2943

    [28]

    Zhou L, Cheng J, Han Y, Zhang W P 2013 Phys. Rev. A 88 063854

    [29]

    He B 2012 Phys. Rev. A 85 063820

    [30]

    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

    [31]

    Cao C, Mi S C, Wang T J, Zhang R, Wang C 2016 IEEE J. Quantum Electron. 52 1

    [32]

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

    [33]

    Xiong X R, Gao Y P, Liu X F, Cao C, Wang T J, Wang C 2018 Sci. China-Phys. Mech. Astron. 61 090322

    [34]

    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

    [35]

    Manipatruni S, Robinson J T, Lipson M 2009 Phys. Rev. Lett. 102 213903

    [36]

    Hafezi M, Rabl P 2012 Opt. Express 20 7672

    [37]

    Wang Z, Shi L, Liu Y, Xu X, Zhang X 2015 Sci. Rep. 5 8657

    [38]

    Shen Z, Zhang Y L, Chen Y, Zou C L, Xiao Y F, Zou X B, Sun F W, Guo G C, Dong C H 2016 Nat. Photonics 10 657

    [39]

    Peterson G A, Lecocq F, Cicak K, Simmonds R W, Aumentado J, Teufel J D 2017 Phys. Rev. X 7 031001

    [40]

    Barzanjeh S, Wulf M, Peruzzo M, Kalaee M, Dieterle P B, Painter O, Fink J M 2017 Nat. Commun. 8 953

    [41]

    Fang K, Luo J, Metelmann A, Matheny M H, Marquardt F, Clerk A A, Painter O 2017 Nat. Phys. 13 465

    [42]

    Maayani S, Dahan R, Kligerman Y, Moses E, Hassan A U, Jing H, Nori F, Christodoulides D N, Carmon T 2018 Nature (London) 558 569

    [43]

    Xu X W, Li Y 2015 Phys. Rev. A 91 053854

    [44]

    Xu X W, Li Y, Chen A X, Liu Y 2016 Phys. Rev. A 93 023827

    [45]

    Tian L, Li Z 2017 Phys. Rev. A 96 013808

    [46]

    Malz D, Tóth L D, Bernier N R, Feofanov A K, Kippenberg T J, Nunnenkamp A 2018 Phys. Rev. Lett. 120 023601

    [47]

    Jiang Y, Maayani S, Carmon T, Nori F, Jing H 2018 Phys. Rev. Applied 10 064037

    [48]

    Li Y, Huang Y Y, Zhang X Z, Tian L 2017 Opt. Express 25 18907

    [49]

    Huang R, Miranowicz A, Liao J Q, Nori F, Jing H 2018 Phys. Rev. Lett. 121 153601

    [50]

    Xu X W, Zhao Y J, Wang H, Jing H, Chen A X 2018 arXiv: 1809.07596

    [51]

    Lü H, Jiang Y, Wang Y Z, Jing H 2017 Photonics Res. 5 367

    [52]

    Habraken S J M, Stannigel K, Lukin M D, Zoller P, Rabl P 2012 New J. Phys. 14 115004

    [53]

    Seif A, DeGottardi W, Esfarjani K, Hafezi M 2018 Nat. Commun. 9 1207

    [54]

    Thompson J D, Zwickl B M, Jayich A M, Marquardt F, Girvin S M, Harris J G E 2008 Nature (London) 452 72

    [55]

    Jayich A M, Sankey J C, Zwickl B M, Yang C, Thompson J D, Girvin S M, Clerk A A, Marquardt F, Harris J G E 2008 New J. Phys. 10 095008

    [56]

    Sankey J C, Yang C, Zwickl B M, Jayich A M, Harris J G E 2010 Nat. Phys. 6 707

    [57]

    Agarwal G S, Huang S 2014 New J. Phys. 16 033023

    [58]

    Yan X B, Cui C L, Gu K H, Tian X D, Fu C B, Wu J H 2014 Opt. Express 22 4886

    [59]

    DeJesus E X, Kaufman C 1987 Phys. Rev. A 35 5288

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  • Received Date:  16 February 2019
  • Accepted Date:  19 May 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

Optical nonreciprocity with blue-detuned driving in two-cavity optomechanics

    Corresponding author: Li Xian-Li, lxl7158@163.com
    Corresponding author: Yang Liu, lyang@hrbeu.edu.cn
  • 1. College of Electronic Science, Northeast Petroleum University, Daqing 163318, China
  • 2. College of Automation, Harbin Engineering University, Harbin 150001, China

Abstract: Radiation pressure in an optomechanical system can be used to generate various quantum phenomena. Recently, one paid more attention to the study of optical nonreciprocity in an optomechanical system, and nonreciprocal devices are indispensable for building quantum networks and ubiquitous in modern communication technology. Here in this work, we study how to realize the perfect optical nonreciprocity in a two-cavity optomechanical system with blue-detuned driving. Our calculations show that the optical nonreciprocity comes from the quantum interference of signal transmission between two possible paths corresponding to the two interactions in this system, i.e. optomechanical interaction and linearly-coupled interaction. According to the standard input-output relation of optical field in cavity optomechanics, we obtain the expression of output optical field, from which we can derive the essential conditions to achieve the perfect optical nonreciprocity, and find there are two sets of coupling strengths both of which can realize the perfect optical nonreciprocal transmission. Because the system is driven by blue-detuned driving, the system is stable only under some conditions which we can obtain according to the Routh-Hurwitz criterion. Due to the blue-detuned driving, there will be transmission gain (transmission amplitude is greater than 1) in the nonreciprocal transmission spectrum. We also find that the bandwidth of nonreciprocal transmission spectrum is in proportion to mechanical decay rate if mechanical decay rate is much less than the cavity decay rate. In other words, in a realistic optomechanical parameter regime, where mechanical decay rate is much less than cavity decay rate, the bandwidth of nonreciprocal transmission spectrum is very narrow. Our results can also be applied to other parametrically coupled three-mode bosonic systems and may be used to realize the state transfer process and optical nonreciprocal transmission in an optomechanical system.

    • 非互易光学器件在光源和接收器调换位置后可以使光信号表现出不同的传输特性. 由于其可以抑制多余的信号, 因此在量子信号处理和量子通信中有着重要的应用. 例如, 在量子超导电路中它们可以保护信号源不被读取器件发出的噪声干扰[1]. 为实现光学非互易性, 时间反演对称性破缺是必须的. 传统的非互易性光学器件都是依赖 强磁场去实现时间反演对称性破缺[2]. 然而由于需要较强磁场, 这些传统器件体积往往较大, 不便于微型化和集成化. 近年来由于纳米技术的进步, 使微纳系统中光学现象得到广泛研究[3-7]. 腔光力学系统中的光辐射压力可以使系统呈现出各种有趣的量子现象. 例如, 腔光力学系统中的量子纠缠[8-16], 力学振子的基态冷却[17-21], 光力诱导透明[22-26]以及非线性效应[26-33]和声子阻塞[34]等量子现象. 最近, 人们意识到光力耦合相互作用也可以产生光学非互易传输现象. 例如通过光力相互作用可以产生非互易光学反应在理论上被预言[35-37], 并在实验上得到证实[38-42]. 并在理论上指出如果采用适当的驱动场, 以力学模为中介的两个腔模之间的态转换可以是非互易的[43-45], 以及通过光力耦合相互作用可以实现信号非互易放大现象[46-48]. 在文献[49,50]中, 理论上给出了通过光力相互作用可以实现非互易光子阻塞效应, 以及在文献[51]中, 作者理论上指出通过光力相互作用可以实现非互易慢光. 另外, 在文献[52,53]中, 作者理论上预言了通过光力耦合可以实现声子环形器和热二极管. 然而在大部分文献当中人们常常采用红失谐的驱动场和$ \pm\dfrac{{\text{π}}}{2} $的非互易相位差去实现光学非互易性.

      本文研究了在蓝失谐驱动下, 在双腔光力系统(如图1所示)中如何实现光场的非互易传输. 在此模型中, Li等[48]利用力学驱动机制实现了光的非互易放大, 并在红失谐驱动下, 利用$ \pm\dfrac{{\text{π}}}{2} $的非互易相位差去实现光学非互易性[43]. 实际上, 此系统中的光学非互易性源于光力耦合和腔模线性耦合的共同作用, 使从不同路径传输的光信号之间产生干涉效应. 本文根据此物理机理并由腔光力学中标准的光场输入输出关系, 得到了实现完美的非互易光传输条件. 研究发现, 在系统中各耗散速率一定的情况下, 会有两套耦合强度可以实现光学非互易传输, 并且即使在非互易相位差不为$ \pm\dfrac{{\text{π}}}{2} $时系统依然可以实现完美光学非互易性. 最后根据劳斯-霍尔维茨(Routh-Hurwitz)稳态判据给出了系统在蓝失谐驱动下的稳定条件. 这些研究结果有望能应用于在光力系统中实现光频隔离器、非互易态转换等量子信息处理过程.

      Figure 1.  A two-cavity optomechanical system with a mechanical resonator interacted with two cavities. Two strong coupling fields (probe fields) with amplitudes $\varepsilon_{\rm c}$ and $\varepsilon_{\rm d}$ ($\varepsilon_{\rm L}$ and $\varepsilon _{\rm R}$) are used to drive cavity $c_{1}$ and $c_{2}$ respectively. Meanwhile, the two cavities are linearly coupled to each other with coupling strength J

    2.   理论模型与主要公式
    • 本文研究了一个双腔光力学系统, 如图1所示, 左右两个光学腔与中间一个力学振子通过光力相互作用耦合; $ {c}_{i} $($ \omega_{0} $)和$ b $($ \omega_{m} $)分别表示光学腔$ i $和力学振子的湮灭算符(本征频率); $ \kappa_{i} $$ \gamma $分别表示光腔i和力学振子的弛豫速率. 两个频率均为$ \omega_{\rm c} $($ \omega_{\rm p} $)、振幅分别为$ \varepsilon_{\rm c} $$ \varepsilon_{\rm d} $ ($ \varepsilon_{\rm L} $$ \varepsilon_{\rm R} $)的强驱动场(弱探测场) 分别从左右两侧射入并驱动腔模$ c_{1} $$ c_{2} $. 同时左右两腔之间由线性相互作用$ \hbar J(c_{1}^{†}c_{2}+c_{2}^{†}c_{1}) $相耦合, J为线性耦合强度. 在相对驱动场频率$ \omega_{\rm c} $做旋转后, 系统的哈密顿量($ \hbar = 1 $)可写为

      其中$ \varDelta_{\rm c} = \omega_{0}-\omega_{\rm c} $ ($ \varDelta = \omega_{\rm p}-\omega_{\rm c} $)为腔模(探测场)与驱动场之间的失谐, $ g_{i} $为光学腔i与力学振子之间的单光子耦合常数. 实际上, 此三模光力耦合系统在实验上是可行的, 如在法布里-珀罗(Fabry-Pérot)腔中加入力学膜的实验装置, 见文献[54-56].

      根据海森伯-郎之万方程, 由系统哈密顿量(1)式可得系统相关算符的运动方程为:

      在没有探测场时, 根据假设$ \langle bc_{i}\rangle = \langle b\rangle\langle c_{i}\rangle $, 可以得出各算符的稳态平均值为:

      其中, $ \varDelta_{1,2} = \varDelta_{\rm c}+g_{1,2}(b_{\rm s}+b_{\rm s}^{\ast}) $表示腔模和驱动场间的有效失谐. 当探测场不为零时, 可以把算符写成这个平均值和一个小的涨落值之和, 即$ b = b_{\rm s}+{\text{δ}} b $, $ c_{1} = c_{1{\rm s}}+{\text{δ}} c_{1} $, $ c_{2} = c_{2{\rm s}}+{\text{δ}} c_{2} $. 把算符的这种相加形式代入方程(2)中, 仅保留涨落的线性项, 并采用相互作用表象, 即$ {\text{δ}} b\rightarrow{\text{δ}} b{\rm e}^{-{\rm i}\omega_{\rm m}t} $, $ {\text{δ}} c_{1}\rightarrow{\text{δ}} c_{1}{\rm e}^{-{\rm i}\varDelta_{1}t} $, $ {\text{δ}} c_{2}\rightarrow{\text{δ}} c_{2}{\rm e}^{-{\rm i}\varDelta_{2}t} $, 则可以得到线性化的海森伯-郎之万方程为:

      其中$ G_{1} = g_{1}c_{1{\rm s}} $, $ G_{2} = g_{2}c_{2{\rm s}}{\rm e}^{-{\rm i}\theta} $. 由等式(3)可知, 通过调节驱动场$ \varepsilon_{\rm c} $$ \varepsilon_{\rm d} $可以有效调节光力耦合$ g_{1}c_{1{\rm s}} $$ g_{2}c_{2{\rm s}} $之间的非互易相位差θ (即调节驱动场$ \varepsilon_{\rm c} $$ \varepsilon_{\rm d} $的强度和相位可以使$ c_{1{\rm s}} $为实数, 而此时$ c_{2{\rm s}} $的辐角便是非互易相位差θ). 为简化, 本文只讨论相等耦合$ G_{1} = G_{2} = G $$ \kappa_{1} = \kappa_{2} = \kappa $, 并且设$ G(J) > 0 $. 本文讨论蓝失谐驱动, 即$ (\varDelta_{1}\approx\varDelta_{2}\approx-\omega_{\rm m}) $, 并假设力学振子频率$ \omega_{\rm m} $远大于耦合强度G, 则方程(4)可以化简为

      其中$ {{{v}}} \!=\! \{{\text{δ}} c_{1},{\text{δ}} c_{2},{\text{δ}} b^{†}\}^{\rm T}\!$, ${{{v}}}_{\rm {in}} \!=\! \{\varepsilon_{\rm L}{\rm e}^{-{\rm i}xt},\varepsilon_{\rm R}{\rm e}^{-{\rm i}xt},0\}^{\rm T} $ ($ x = \varDelta+\omega_{\rm m} $), 以及矩阵

      $ {\nu}_{\rm {in}} $的形式可以假设方程(5)的解具有${\text{δ}} s =$$ {\text{δ}} s_{+}{\rm e}^{-{\rm i}xt}+{\text{δ}} s_{-}{\rm e}^{{\rm i}xt} $ ($ s = b, c_{1}, c_{2} $)的形式, 经计算可得

      其中$ \gamma_{x} = \gamma-2{\rm i}x $, $ \kappa_{x} = \kappa-2{\rm i}x $, $ \delta s_{-} = 0 $.

      系统产生光学非互易性的物理根源是时间反演对称发生破缺, 这点也可从(5)式看出, 运动方程(5)式对应的系统等效哈密顿量为$ H_{\rm {eff}} = G{\text{δ}} c_{1}{\text{δ}} b+$$G{\rm e}^{-{\rm i}\theta}{\text{δ}} c_{2}{\text{δ}} b+J{\text{δ}} c_{1}{\text{δ}} c_{2}^{†}+{\rm {H.C.}} $. 当$ \theta\neq n{\text{π}} $($ n $为整数)时, 时间反演算符T与等效哈密顿算符$ H_{\rm {eff}} $不对易, 即$ [T,H]\neq0 $. 为研究系统的光学非互易性, 首先必须要求出系统左右两侧的输出光场$ \varepsilon^{\rm {out}}_{\rm L} $$ \varepsilon^{\rm {out}}_{\rm R} $. 输出场可由光力学中的输入输出关系[57,58]得出, 即

      其中$ \varepsilon_{\rm {L,R}}^{\rm {in}} = \varepsilon_{\rm {L,R}}/\sqrt{\kappa} $. 输出场也具有${\text{δ}} s = {\text{δ}} s_{+}{\rm e}^{-{\rm i}xt}+$$ {\text{δ}} s_{-}{\rm e}^{{\rm i}xt} $的形式, 由方程(7), (8)可得

      $ \varepsilon^{\rm {out}}_{\rm L-} = \varepsilon^{\rm {out}}_{\rm R-} = 0 $.

    3.   完美光学非互易性
    • 当系统呈现出完美的光学非互易性时, 传输振幅 $ T_{i\rightarrow j} $ ($ i,j =\rm{ L,R} $)应满足

      或者

      完美非互易性就意味着信号可以从系统的一侧完全传输到另一侧, 而另一侧的信号却一点也不可以传输过来. (10)式和(11)式代表光频隔离的两个不同的方向. 本文只讨论(10)式, 因为对(11)式的讨论是类似的. 下标$ \varepsilon_{\rm {R/L}}^{\rm {in}} = 0 $表示没有信号从右侧/左侧输入. 我们将忽略这些下标, 因为一般来说完美非互易性只讨论单侧输入的情况, 并且为方便将$ T_{i\rightarrow j} $简写为$ T_{ij} $.

      由(7)式和(9)式可得输出场为

      由(12)式可以看出, 当非互易相位差$ \theta = n{\text{π}} $(n为整数)时, 两输出场相等, 这说明光子传输是互易的. 而当$ \theta\neq n{\text{π}} $时, 两输出场不再相等, 即系统呈现出光学非互易性. 由(12)式可看出系统的非互易性来源于光力耦合相互作用G和腔模线性耦合相互作用J之间的量子相干效应. 由(12)式可得出(10)式成立时失谐x和线性耦合强度J必须满足的条件为:

      首先讨论$ \theta = -\dfrac{{\text{π}}}{2} $的情况, 此时输出场(12)式变为:

      把(13)式中的J代入(14)式中, 得到

      由(10)式和(15)式, 可以得出系统出现完美非互易性时, 耦合强度GJ必须满足:

      由于系统处于蓝失谐驱动下, 在某些条件下系统会不稳定. 系统要稳定, 矩阵M (见(6)式)的本征值一定具有负实部. 根据劳斯-霍尔维茨(Routh-Hurwitz)稳态判据[59], 可以得出具体的稳定条件如下:

      系统所有的参数必须满足(17)式. 由(17)式可知, 当耦合$ G = G_{-} $时, 系统始终都是稳定的, 而当$ G = G_{+} $时, 系统参数只有满足以下条件时系统才是稳定的:

      图2给出了在$ \theta = -\dfrac{{\text{π}}}{2} $, $ G = G_{+} $$ J = J_{+} $(见(16)式)时, 在力学振子弛豫速率$ \gamma $不同取值下, 传输振幅$ T_{\rm {LR}} $(红线)和$ T_{\rm {RL}} $(黑线)随着标准化失谐$ x/\kappa $的变化曲线. 从图2可知, $ \theta = -\dfrac{{\text{π}}}{2} $时传输谱线左右对称、有多个峰值并且完美光学非互易性出现在共振频率位置($ x = 0 $). 通过计算可知, 在$ \gamma/\kappa\ll1 $时非互易传输谱线线宽为$ \Delta\omega\propto\gamma $, 就是为什么当$ \gamma $很小时传输谱线非常狭窄的原因(见图2(a)和图2(b)). 而当力学振子耗散速率$ \gamma $增大时, 谱线宽度也随之增加, 并且会出现传输增幅增益(幅值大于1)的现象(见图2(c)图2(d)), 这是由于系统是处于蓝失谐驱动. 具体可由稳态条件(18)式看出, 当$ \gamma\gtrsim2.32\kappa $时, 系统会不稳定, 这就是为什么当$ \gamma $增大时系统会出现增益现象, 当$ \gamma/\kappa $增加到2时, 传输振幅$ T_{\rm {LR}} $$ T_{\rm {RL}} $的最大值均已远远大于1(见图2(d)).

      Figure 2.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\kappa$ for different cavity damping rate: (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=2. Other parameters: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{+}$ and $J=J_{+}$ according to Eq. (16)

      图3给出了在$ \theta = -\dfrac{{\text{π}}}{2} $, $ G = G_{-} $$ J = J_{-} $(见(16)式)时, 传输振幅$ T_{\rm {LR}} $(红线)和$ T_{\rm {RL}} $(黑线)在不同力学振子耗散速率下, 随着标准化失谐$ x/\kappa $的变化曲线. 从图3可知, 此时无论力学振子耗散速率取何值, 传输谱线$ T_{\rm {LR}} $都只有一个峰值, 并且峰值出现在共振频率的位置 ($ x = 0 $). 当$ \gamma/\kappa\ll1 $(见图3(a)图3(b))时非互易传输谱线非常狭窄, 这是因为此时谱线线宽为$ \Delta\omega\propto\gamma $. 随着力学振子耗散速率的增加, 传输谱线的宽度也随之不断增加(见图3(c)), 同时传输谱线$ T_{\rm {RL}} $(黑线)的最大值则不断降低. 但现在即使力学振子耗散速率$ \gamma\gg\kappa $(见图3(d))时也不再出现增益现象(因为此时系统不会出现非稳情况).

      Figure 3.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\kappa$ for different mechancial damping rate: (a) $\gamma/\kappa$=1/100; (b) $\gamma/\kappa$=1/10; (c) $\gamma/\kappa$=1; (d) $\gamma/\kappa$=10. Other parameters: $\theta=-\dfrac{{\text{π}}}{2}$, $G =G_{-}$ and $J=J_{-}$ according to Eq. (16)

      由(13)式可知, 当$ \theta\neq\pm\dfrac{{\text{π}}}{2} $时, 非互易性不会出现在$ x\neq0 $的位置, 也就是说, 当$ \theta\neq\pm\dfrac{{\text{π}}}{2} $时, 非互易谱线一定是非对称的图形. 下面以$ \theta = -\dfrac{{\text{π}}}{4} $$ -\dfrac{3{\text{π}}}{4} $为例讨论 $ \theta\neq\pm\dfrac{{\text{π}}}{2} $时传输谱线的性质. 由(12)式和(13)式可得, 当$ \theta = -\dfrac{{\text{π}}}{4} $$ -\dfrac{3{\text{π}}}{4} $时, 若要实现完美的非互易性, 耦合强度GJ必须满足

      由(19)式可知, 当$ \theta = -\dfrac{{\text{π}}}{4} $$ -\dfrac{3{\text{π}}}{4} $时, 要实现完美的非互易性, 则系统耗散必须满足$ \kappa\geqslant \gamma $. 图4给出了在力学振子耗散速率$ \gamma/\kappa = 10^{-3} $时, 传输振幅$ T_{\rm {LR}} $(红线)和$ T_{\rm {RL}} $(黑线)在不同非互易相位差θ和耦合强度G时 随着标准化失谐$ x/\gamma $的变化曲线. 其中耦合强度$ G_{\pm} $由(19)式给出. 由图4可知, 传输谱线$ T_{\rm {LR}} $(红线)和$ T_{\rm {RL}} $(黑线)在失谐$ x = \pm\dfrac{1}{2}\gamma $处确实会出现完美光学非互易性, 并且均会出现增益现象(增益是因为系统处于蓝失谐驱动). 虽然传输谱线$ T_{\rm {LR}} $$ T_{\rm {RL}} $自身不是左右对称的, 但对于耦合$ G = G_{+} $$ G = G_{-} $的谱线关于完美非互易位置轴对称(当$ \theta = -\dfrac{{\text{π}}}{4} $时, 对称轴$ x = \dfrac{1}{2}\gamma $, 如图4(a)图4(b)所示; 当$ \theta = -\dfrac{3{\text{π}}}{4} $时, 对称轴$ x = -\dfrac{1}{2}\gamma $, 见图4(c)图4(d)), 并且在整个非互易传输频谱上有$ T_{\rm {LR}} > T_{\rm {RL}} $.

      Figure 4.  Transmission amplitudes $T_{\rm {LR}}$ (red line) and $T_{\rm {RL}}$ (black line) are plotted vs normalized detuning $x/\gamma$ for different nonreciprocal phase θ and coupling strength G: (a) $\theta=-\dfrac{{\text{π}}}{4}$ and $G=G_{-}$; (b) $\theta=-\dfrac{{\text{π}}}{4}$ and $G=G_{+}$; (c) $\theta=-\dfrac{3{\text{π}}}{4}$ and $G=G_{-}$; (d) $\theta=-\dfrac{3{\text{π}}}{4}$ and $G=G_{+}$. Other parameters: $\gamma/\kappa=10^{-3}$, coupling strengths $J=J_{\pm}$ and $G=G_{\pm}$ according to Eq. (19)

    4.   结 论
    • 本文研究了双腔光力学系统在蓝失谐驱动下的光学非互易性. 由系统中的光力耦合相互作用G和腔模线性耦合相互作用J之间的量子相干效应, 在某些条件下, 可以使系统呈现出完美的光学非互易现象. 首先研究了非互易相位差$ \theta =$$ -\dfrac{{\text{π}}}{2} $的情况, 研究发现, 当系统中各耗散速率(力学耗散速率γ和腔模耗散速率κ)一定的情况下, 会有两组耦合强度($ G = G_{\pm} $$ J = J_{\pm} $)均可使系统出现完美非互易性. 由于系统处于蓝失谐驱动下, 会使系统出现非稳现象, 根据劳斯-霍尔维茨(Routh-Hurwitz)稳态判据我们给出了系统的稳定条件, 这种非稳现象也表现为非互易传输谱线会出现增益现象(谱线幅值大于1). 我们还发现当$ \gamma\ll\kappa $时, 非互易传输谱线的线宽$ \Delta\omega\propto\gamma $, 即当力学振子耗散速率很小时, 非互易传输谱线将会变得很狭窄. 最后研究了更一般的非互易相位差的情况$\left(\theta = -\dfrac{{\text{π}}}{4}\right.$$\left. -\dfrac{3{\text{π}}}{4}\right)$, 并给出了实现完美非互易传输的必要条件. 这些研究结果有望能应用于光力系统中量子态转换、非互易传输等量子信息处理过程.

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