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光学非互易性是指系统的输入端和输出端交换位置后所表现出的具有不同光学响应的性质, 一般指促进光场的正向传输并抑制反方向传输这一特性. 例如光隔离器[1,2]、环形器[3]和定向放大器[4]等非互易器件不仅是光学系统中重要的组成部分, 而且在构建量子网络和实现量子通信上也具有重要应用. 实现光学非互易传输首先要打破洛伦兹互易定理, 目前主要实现方法是基于法拉第磁致旋光效应通过磁光材料来改变信号的偏振状态[5], 但该方法工艺复杂且需要强磁场, 使得光学非互易器件难以实现片上集成. 此外, 其他实现非互易特性的方法有手性量子光学[6-8], 利用非线性光学介质[9]、系统参数的时空调制[10], 原子系综的光诱导磁化[11] 、热原子碰撞[12], 谐振模式下损耗的诱导[13]和量子压缩[14], 以及光机械相互作用[15,16]等. 手性量子光学[6-8]是实现光学非互易的重要方法之一, 通过环形腔与量子发射器的手性耦合来实现光非互易性, 并且这种方法有望实现片上集成.
光力耦合作用实质上是通过辐射压力实现的, 光学腔和力学振子耦合构成的腔光力学系统是近些年研究的热点. 该系统是观察宏观量子效应和验证量子力学基本规律的重要平台, 能够产生很多量子效应, 如量子纠缠[17]和力学振子的基态冷却[18,19]等; 同时可观察到很多量子光学现象, 如光力诱导透明[20]、光力诱导吸收和放大[21]、Fano共振[22]; 还广泛应用于光信息存储[23]、快光和慢光效应[24]等. 此外, 随着纳米技术的发展, 该系统中力学振子的空间尺度可达到纳米级别, 这为实现非互易器件的片上集成提供了可能. Xu和Li[25]在三模光力系统中实现了光学非互易性, 通过调控光力耦合率间的相位差来启动非互易响应, 这诱导了系统的时间反演对称性破缺. 最有趣的是, 该三模光力系统不仅可用于两光学模和机械模间的三端口光力循环器, 而且可用于单光子级别并集成到芯片, 还可能将三端口光力循环器最终提供量子信息处理和量子模拟应用的基础. 基于此我们提出在双光力系统中, 通过两种大小不同的光力耦合作用(即系统包含两个不同的力学振子)来实现更加新奇的光非互易特性. 理论上提出的双腔光力系统实验上可通过芯片上的微波循环器来实现, 主要采用频率可调的硅上绝缘体的电磁系统来实现芯片上的微波循环器. 具体为将两个独立的光力变频器组合在一起, 通过同时耦合两个电磁腔模式到两个不同的振动模式的机械模[15,16].
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研究由两个光学腔与两个力学振子组成的双光力系统如图1所示,
${a_i}$ 和${b_j}$ 分别表示光学腔和力学振子的湮灭算符$\left( {i, j = 1, 2} \right)$ ,${\omega _i}$ 与${\omega _m}$ 分别表示对应的本征频率. 两光学腔作为系统的输入端或输出端分别与两力学振子相耦合, 耦合常数${g_{ij}}$ 表示光学腔${a_i}$ 与力学振子${b_j}$ 之间的光子耦合强度. 两光学腔之间存在直接耦合, 耦合强度为$J({a_1^{\dagger}} {a_2} + {a_2^{\dagger} {a_1}})$ ,$J$ 表示腔模间线性耦合强度. 频率为${\omega _{\text{d}}}$ 的驱动场和频率为${\omega _{\text{p}}}$ 的探测场从两侧对光学腔进行驱动,${\varepsilon _{{\text{d}}i}}$ 和${\varepsilon _{{\text{p}}i}}$ 分别是驱动场和探测场的振幅, 它们和功率之间的关系是${\varepsilon _{{\text{d}}i, {\text{p}}i}} = \sqrt {2{P_{{\text{d}}i, {\text{p}}i}}/\left( {\hbar {\omega _{{\text{d, p}}}}} \right)} $ .
图 1 双腔双光力系统示意图, 其中光学腔
${a_i}$ 通过光力耦合相互作用${g_{ij}}$ 与力学振子${b_j}$ 相耦合, 同时两光学腔之间存在腔模线性耦合相互作用, 腔模两侧存在振幅为${\varepsilon _{{\text{d}}i}}$ 的驱动场和${\varepsilon _{{\text{p}}i}}$ 的探测场Figure 1. Diagram of double-cavity dual-optomechanical system. The optical cavity
${a_i}$ is coupled to the mechanical oscillator${b_j}$ by the optomechanical coupling interaction$ {g}_{ij}$ . And there is a cavity mode linear coupling interaction between two optical cavities, and there are the coupling fields with amplitude${\varepsilon _{{\text{d}}i}}$ and the probe fields with amplitude${\varepsilon _{{\text{p}}i}}$ on both sides of the cavity modes.系统的哈密顿量是
$ \begin{split} H =\;& \mathop \sum \nolimits_i^2 \hbar {\omega _i}a_i^\dagger {a_i} + \mathop \sum \nolimits_j^2 \hbar {\omega _m}b_j^\dagger {b_j} \\ &+ \hbar J\left( {a_1^\dagger {a_2} + a_2^\dagger {a_1}} \right) + \mathop \sum \nolimits_{i,j}^2 \hbar {g_{ij}}a_i^\dagger {a_i}\left( {b_j^\dagger + {b_j}} \right) \\ &+ {\text{i}}\hbar \mathop \sum \nolimits_i^2 \left( {a_i^\dagger {\varepsilon _{{\text{d}}i}}{{\text{e}}^{ - {\text{i}}{\omega _{\text{d}}}t}} - {a_i}\varepsilon _{{\text{d}}i}^*{{\text{e}}^{{\text{i}}{\omega _{\text{d}}}t}}} \right)\\ & + {\text{i}}\hbar \mathop \sum \nolimits_i^2 \left( {a_i^\dagger {\varepsilon _{{\text{p}}i}}{{\text{e}}^{ - {\text{i}}{\omega _{\text{p}}}t}} - {a_i}\varepsilon _{{\text{p}}i}^*{{\text{e}}^{{\text{i}}{\omega _{\text{p}}}t}}} \right).\\[-16pt] \end{split} $ 系统相对驱动场频率
${\omega _{\text{d}}}$ 做旋转后, 根据海森伯运动方程及对易关系$ [{a}_{i}, {a}_{i}^{†}]=1 $ ,$ [{b}_{i}, {b}_{i}^{†}]=1 $ , 并且唯象地引入耗散速率, 系统中光学腔与力学振子的量子朗之万方程被得到$\begin{split} &{{\dot a}}_{i}=-\text{i}{\varDelta }_{i}{a}_{i}-\text{i}J{a}_{3-i}-\text{i}\displaystyle \sum\nolimits_{j}^{2}{g}_{ij}{a}_{i}\left({b}_{j}^{†}+{b}_{j}\right)\\ & \quad\quad +{\varepsilon }_{\text{d}i}+{\varepsilon }_{\text{p}i}{\text{e}}^{-\text{i}\varDelta t}-{\kappa }_{i}{a}_{i} ,\\ & {\dot b}_{j}=-\text{i}{\omega }_{m}{b}_{j}-\text{i}\displaystyle \sum\nolimits_{i}^{2}{g}_{ij}{a}_{i}^{†}{a}_{i}-{\gamma }_{j}{b}_{j} ,\end{split} $ 其中
$\varDelta = {\omega _{\text{p}}} - {\omega _{\text{d}}}$ 表示探测场与驱动场之间的失谐,${\varDelta _i} = {\omega _i} - {\omega _{\text{d}}}$ 表示光学腔ai与驱动场之间的失谐,${\kappa _i}$ 和${\gamma _j}$ 分别是${a_i}$ 和${b_j}$ 的耗散速率.假设没有探测场时, 令方程(2)的时间求导项为0, 即可得到稳态平均值:
$\begin{split} & {a_{{\text{1s}}}} = \frac{{ - {\text{i}}J{\varepsilon _{{\text{d2}}}} + \left( {{\kappa _2} + {\text{i}}\varDelta'_{2}} \right){\varepsilon _{{\text{d1}}}}}}{{\left( {{\kappa _1} + {\text{i}}\varDelta'_{1}} \right)\left( {{\kappa _2} + {\text{i}}\varDelta'_{2}} \right) + {J^2}}}, \\ &{a_{{\text{2s}}}} = \frac{{ - {\text{i}}J{\varepsilon _{{\text{d1}}}} + \left( {{\kappa _1} + {\text{i}}\varDelta'_{1}} \right){\varepsilon _{{\text{d2}}}}}}{{\left( {{\kappa _1} + {\text{i}}\varDelta'_{1}} \right)\left( {{\kappa _2} + {\text{i}}\varDelta'_{2}} \right) + {J^2}}},\\ & {b}_{j\text{s}}=\frac{-\text{i}\displaystyle \sum\nolimits_{i}^{2}{g}_{ij}{\left|{a}_{i\text{s}}\right|}^{2}}{{\gamma }_{j}+\text{i}{\omega }_{m}},\end{split} $ 其中,
$\varDelta' _i = {\varDelta _i} +\displaystyle\sum\nolimits_j^2 {g_{ij}}\left( {{b_{j{\text{s}}}} + b_{j{\text{s}}}^*} \right)$ .将系统算符用平均值和各自的相对涨落值来表示, 即
${a_i} = {a_{i{\text{s}}}} + \text{δ} {a_i}$ ,$ {b}_{j}= {b}_{j\text{s}}+\text{δ} {b}_{j}$ , 并且保留其中的线性项, 得到线性化的海森伯-朗之万方程:$\begin{split}& \text{δ} {\dot{a}}_{i}= -\text{i}{\varDelta' _i}\text{δ} {a}_{i}-\text{i}J{\text{δ}} {a}_{3-i}\\&\quad\;\;\quad -{\text{ie}}^{\text{i}{\alpha }_{i}}\displaystyle\sum\nolimits_{j}^{2}{G}_{ij}\left({\text{δ}} {b}_{j}^{†}+{\text{δ}} {b}_{j}\right) +{\varepsilon }_{\text{p}i}{\text{e}}^{-\text{i}\varDelta t}-{\kappa }_{i}{\text{δ}} {a}_{i},\\& {\text{δ}} {\dot{b}}_{j}=-\text{i}{\omega }_{m}{\text{δ}} {b}_{j}-\text{i}\displaystyle\sum\nolimits_{i}^{2}{G}_{ij}\left({\text{δ}} {a}_{i}{\text{e}}^{-\text{i}{\alpha }_{i}}+{\text{δ}} {a}_{i}^{†}{\text{e}}^{\text{i}{\alpha }_{i}}\right)\\& \quad\;\;\quad -{\gamma}_{j}{\text{δ}} {b}_{j} ,\\[-10pt]\end{split} $ 其中,
${G_{ij}} = {g_{ij}}\left| {{a_{i{\text{s}}}}} \right|$ 表示光腔模$i$ 和力学振子$j$ 之间的有效光力耦合强度, 满足${G_{ij}}{{\text{e}}^{{\text{i}}{\alpha _i}}} = {g_{ij}}{a_{i{\text{s}}}}$ . 采用相互作用表象${\text{δ}} {a_i} \to {\text{δ}} {a_i}{{\text{e}}^{ - {\text{i}}\varDelta'_i t}}$ ,${\text{δ}} {b_j} \to {\text{δ}} {b_j}{{\text{e}}^{ - {\text{i}}{\omega _m}t}}$ , 得到$\begin{split} &{\text{δ}} {\dot{a}}_{i}=-\text{i}J{\text{δ}} {a}_{3-i}{\text{e}}^{-\text{i}\left(\varDelta'_{3-i}-\varDelta'_{i}\right)t} -{\text{ie}}^{\text{i}{\alpha }_{i}} \\ & \quad\;\;\quad \times \displaystyle\sum\nolimits_{j}^{2}{G}_{ij} \big[{\text{δ}} {b}_{j}^{†}{\text{e}}^{\text{i}({\omega }_{m}+\varDelta'_{i})t} + {\text{δ}} {b}_{j}{\text{e}}^{-\text{i} ({\omega }_{m}-\varDelta'_{i} )t} \big]\\ &\quad\;\;\quad +{\varepsilon }_{\text{p}i}{\text{e}}^{-\text{i} (\varDelta -\varDelta'_{i})t}-{\kappa }_{i}{\text{δ}} {a}_{i},\\ &{\text{δ}} {\dot{b}}_{j}=-\text{i}\displaystyle\sum\nolimits_{i}^{2}{G}_{ij}\big[{\text{e}}^{-\text{i}{\alpha }_{i}}{\text{δ}} {a}_{i}{\text{e}}^{-\text{i}\left(\varDelta'_{i}-{\omega }_{m}\right)t} \\ &\quad\;\;\quad +{\text{e}}^{\text{i}{\alpha }_{i}}{\text{δ}} {a}_{i}^{†}{\text{e}}^{\text{i}\left(\varDelta'_{i}+{\omega }_{m}\right)t}\big]-{\gamma }_{j}{\text{δ}} {b}_{j} .\\[-12pt] \end{split} $ 接着采用红失谐驱动, 即
$x = \varDelta - {\omega _m}$ ,$\varDelta'_{1} = \varDelta'_{2} = {\omega _m}$ , 并且忽略高频振荡项得到$\begin{split} & {\text{δ}} {\dot{a}}_{i} = -\text{i}J{\text{δ}} {a}_{3-i}-{\text{ie}}^{\text{i}{\alpha }_{i}}\displaystyle\sum\nolimits_{j}^{2}{G}_{ij}{\text{δ}} {b}_{j} + {\varepsilon }_{\text{p}i}{\text{e}}^{-\text{i}xt}-{\kappa }_{i}{\text{δ}} {a}_{i} ,\\ & {\text{δ}} {\dot{b}}_{j}=-\text{i}\displaystyle\sum\nolimits_{i}^{2}{G}_{ij}{\text{e}}^{-\text{i}{\alpha }_{i}}{\text{δ}} {a}_{i}-{\gamma }_{j}{\text{δ}} {b}_{j} .\\[-15pt]\end{split}$ 先讨论相等的光力耦合, 使
${G_{ij}}$ 简化为$ {G_{11}} = {G_{21}} = {G_1} $ ,$ {G_{12}} = {G_{22}} = {G_2} $ . 这里的${G_1}$ 和${G_2}$ 分别表示力学振子${b_1}$ 和${b_2}$ 对一个光学腔的有效光力耦合强度. 首先可以得到以下矩阵形式:$ \begin{split}&\dot{\boldsymbol{u}} = {\boldsymbol{Mu}} + {{\boldsymbol{u}}_{{\text{in}}}} ,\\ & {\boldsymbol{u}} = {\left\{{{\text{δ}} {a_1}}\quad {{\text{δ}} {a_2}}\quad {{\text{δ}} {b_1}}\quad {{\text{δ}} {b_2}} \right\}^{\text{T}}},\\ & {{\boldsymbol{u}}_{{\text{in}}}} = {\left\{ {{\varepsilon _{{\text{p1}}}}{{\text{e}}^{ - {\text{i}}xt}}} \quad {{\varepsilon _{{\text{p2}}}}{{\text{e}}^{ - {\text{i}}xt}}}\quad 0\quad 0\right\}^{\text{T}}} ,\end{split} $ T代表转置. 系数矩阵为
$ {\boldsymbol{M}} = \begin{bmatrix} { - {\kappa _1}} & { - {\text{i}}J} & { - {\text{i}}{G_1}{{\text{e}}^{{\text{i}}{\alpha _1}}}} & { - {\text{i}}{G_2}{{\text{e}}^{{\text{i}}{\alpha _1}}}} \\ { - {\text{i}}J} & { - {\kappa _2}} & { - {\text{i}}{G_1}{{\text{e}}^{{\text{i}}{\alpha _2}}}} & { - {\text{i}}{G_2}{{\text{e}}^{{\text{i}}{\alpha _2}}}} \\ { - {\text{i}}{G_1}{{\text{e}}^{ - {\text{i}}{\alpha _1}}}} & { - {\text{i}}{G_1}{{\text{e}}^{ - {\text{i}}{\alpha _2}}}} & {-{\gamma _1}} & 0 \\ { - {\text{i}}{G_2}{{\text{e}}^{ - {\text{i}}{\alpha _1}}}} & { - {\text{i}}{G_2}{{\text{e}}^{ - {\text{i}}{\alpha _2}}}} & 0 & { - {\gamma _2}} \end{bmatrix}. $ 只有当系数矩阵M所有的本征值都有负实部时, 系统才稳定, 本文中系统一直是稳定的. 具体的稳定性条件可根据Routh-Hurwitz稳态判据[26]给出. 假设(7)式中解的形式为
$ {\text{δ}} v = {\text{δ}} {v_ + }{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}} {v_ - }{{\text{e}}^{ {\text{i}}xt}},{\text{ }}v = {a_i},{b_j}\text{, } $ 最终得出各自的相对涨落值:
$\begin{split} &{\text{δ}} {a_{1 + }} = \left( {{\kappa _{2x}} + {C_x}} \right)\frac{{{\varepsilon _{{\text{p1}}}}}}{{Z\left( x \right)}} + \left( { - {\text{i}}J - {{\text{e}}^{{\text{i}}\alpha }}{C_x}} \right)\frac{{{\varepsilon _{{\text{p2}}}}}}{{Z\left( x \right)}},\\ & {\text{δ}} {a_{2 + }} = \left( {{\kappa _{1x}} + {C_x}} \right)\frac{{{\varepsilon _{{\text{p2}}}}}}{{Z\left( x \right)}} + \left( { - {\text{i}}J - {{\text{e}}^{ - {\text{i}}\alpha }}{C_x}} \right)\frac{{{\varepsilon _{{\text{p1}}}}}}{{Z\left( x \right)}},\\ & {\text{δ}} {b_{1 + }} = \frac{{{G_1}}}{{{\gamma _{1x}}}}\left[\left( { - J - {\text{i}}{{\text{e}}^{ - {\text{i}}\alpha }}{\kappa _{2x}}} \right)\frac{{{\varepsilon _{{\text{p1}}}}}}{{Z\left( x \right)}}\right. \\ & \quad\quad\quad\left.+ \left( { - J{{\text{e}}^{ - {\text{i}}\alpha }} - {\text{i}}{\kappa _{1x}}} \right)\frac{{{\varepsilon _{{\text{p2}}}}}}{{Z\left( x \right)}} \right],\\ &{\text{δ}} {b_{2 + }} = \frac{{{G_2}}}{{{\gamma _{2x}}}}\left[\left( { - J - {\text{i}}{{\text{e}}^{ - {\text{i}}\alpha }}{\kappa _{2x}}} \right)\frac{{{\varepsilon _{{\text{p1}}}}}}{{Z\left( x \right)}}\right. \\ & \quad\quad\quad \left. + \left( { - J{{\text{e}}^{ - {\text{i}}\alpha }} - {\text{i}}{\kappa _{1x}}} \right)\frac{{{\varepsilon _{{\text{p2}}}}}}{{Z\left( x \right)}}\right],~~~~{\text{δ}} {v_ - } = 0.\\[-18pt]\end{split}$ 这里,
$\begin{split} & Z\left( x \right) = \left( { - 2{\text{i}}J\cos \alpha + {\kappa _{1x}} + {\kappa _{2x}}} \right){C_x} + {J^2} + {\kappa _{1x}}{\kappa _{2x}},\\ &{C_x} = {G_1^2}\big/ {{\gamma _{1x}}} +{G_2^2} \big/ {\gamma _{2x}},\\ & {\kappa _{ix}} = {\kappa _i} - {\text{i}}x ,~~{\gamma _{jx}} = {\gamma _j} - {\text{i}}x.\\[-10pt]\end{split} $ 由结果可知, 具有物理效应的是相位差
$\alpha = {\alpha _1} - {\alpha _2}$ . 取简化条件$\varDelta'_{1} = \varDelta'_{2} = {\omega _m}$ ,${\kappa _1} ={\kappa _2} = \kappa $ , (3)式简化为$\begin{split} & {a_{{\text{1s}}}} = \frac{{\kappa {\varepsilon _{{\text{d1}}}} + {\text{i}}({\omega _m}{\varepsilon _{{\text{d1}}}} - J{\varepsilon _{{\text{d2}}}})}}{{{J^2} + {\kappa ^2} - \omega _m^2 + 2{\text{i}}\kappa {\omega _m}}},\\ & {a_{{\text{2s}}}} = \frac{{\kappa {\varepsilon _{{\text{d2}}}} + {\text{i}}({\omega _m}{\varepsilon _{{\text{d2}}}} - J{\varepsilon _{{\text{d1}}}})}}{{{J^2} + {\kappa ^2} - \omega _m^2 + 2{\text{i}}\kappa {\omega _m}}}. \end{split} $ 基于(11)式得到相位:
$\begin{split} & {\alpha _1} = \arctan \dfrac{{({J^2} + {\kappa ^2} - \omega _m^2)\left({\omega _m} - J\dfrac{{{\varepsilon _{{\text{d2}}}}}}{{{\varepsilon _{{\text{d1}}}}}}\right) - 2{\kappa ^2}{\omega _m}}}{{\kappa ({J^2} + {\kappa ^2} - \omega _m^2) + 2\kappa {\omega _m}\left({\omega _m} - J\dfrac{{{\varepsilon _{{\text{d2}}}}}}{{{\varepsilon _{{\text{d1}}}}}}\right)}},\\ & {\alpha _2} = \arctan \dfrac{{({J^2} + {\kappa ^2} - \omega_m^2)\left({\omega _m} - J\dfrac{{{\varepsilon _{{\text{d1}}}}}}{{{\varepsilon _{{\text{d2}}}}}}\right) - 2{\kappa ^2}{\omega _m}}}{{\kappa ({J^2} + {\kappa ^2} - \omega _m^2) + 2\kappa {\omega _m}\left({\omega _m} - J\dfrac{{{\varepsilon _{{\text{d1}}}}}}{{{\varepsilon _{{\text{d2}}}}}}\right)}} .\end{split} $ 根据(12)式中的相位, 可以看出: 通过两驱动场振幅的比值
$ {{{\varepsilon _{{\text{d2}}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{{\text{d2}}}}} {{\varepsilon _{{\text{d1}}}}}}} \right. } {{\varepsilon _{{\text{d1}}}}}} $ 进而达到调控相位差$ \alpha = {\alpha _1} - {\alpha _2} $ . 例如:$ {{{\varepsilon _{{{\text{d}}_2}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{{{\text{d}}_2}}}} {{\varepsilon _{{{\text{d}}_1}}}}}} \right. } {{\varepsilon _{{{\text{d}}_1}}}}} = 1 $ 时, 相位差$ \alpha = 0 $ ;$\begin{split} \;& \frac{{{\varepsilon _{{\text{d2}}}}}}{{{\varepsilon _{{\text{d1}}}}}} = \frac{1}{{2 J{\omega _m}}}\big[\left( {{J^2} + \omega _m^2 + {\kappa ^2}} \right) \\ ~~~~~~~~ & \pm \sqrt {{J^2} + \omega _m^4 + {\kappa ^4} + 2{J^2}{\kappa ^2} - 2{J^2}\omega _m^2 + 2{\kappa ^2}\omega _m^2} \big]\end{split} $ 时, 相位差
$ \alpha = {\pi /2}$ .系统的输出场可通过以下标准的输入-输出关系[27]得出
$ \varepsilon _{{\text{p}}i}^{{\text{out}}} + \varepsilon _{{\text{p}}i}^{{\text{in}}}{{\text{e}}^{ - {\text{i}}xt}} = \sqrt {2{\kappa _i}} {\text{δ}} {a_i}\text{, } $ 其中
$ \varepsilon _{{\text{p}}i}^{{\text{in}}} = {{{\varepsilon _{{\text{p}}i}}} \mathord{\left/ {\vphantom {{{\varepsilon _{{\text{p}}i}}} {\sqrt {2{\kappa _i}} }}} \right. } {\sqrt {2{\kappa _i}} }} $ . 考虑只有单侧探测场时, 可以得出系统散射矩阵元的一般关系式:$\begin{split} & {T_{{a_1} \to {a_1}}} = \left| {{{\varepsilon _{\rm{p1^+}}^{\rm{out}}} \big/ {\varepsilon _{{\text{p1}}}^{{\text{in}}}} } } \right|,~~ {T_{{a_2} \to {a_1}}} = \left| {{{\varepsilon _{\rm{p1^+}}^{\rm{out}}} \big/ {\varepsilon _{{\text{p2}}}^{{\text{in}}}} } } \right|,\\ & {T_{{a_1} \to {a_2}}} = \left| {{{\varepsilon _{\rm{p2^+}}^{\rm{out}}}\big/{\varepsilon _{{\rm{p1}}}^{{\text{in}}}} } } \right| ,~~ {T_{{a_2} \to {a_2}}} = \left| {{{\varepsilon _{\rm{p2^+}}^{\rm{out}}}\big/{\varepsilon _{{\text{p2}}}^{{\text{in}}}} } } \right|,\\ &{T_{{a_1} \to {b_1}}} = \left| {{{\varepsilon _{\rm{b1^+}}^{\rm{out}}}\big/{\varepsilon _{{\rm{p1}}}^{{\text{in}}}} } } \right|,~~ {T_{{a_2} \to {b_1}}} = \left| {{{\varepsilon _{\rm{b1^+}}^{\rm{out}}}\big/ {\varepsilon _{{\rm{p2}}}^{{\text{in}}}} } } \right|,\\ &{T_{{a_1} \to {b_2}}} = \left| {{{\varepsilon _{\rm{b2^+}}^{\rm{out}}}\big/ {\varepsilon _{{\rm{p1}}}^{{\text{in}}}} } } \right|,~~{T_{{a_2} \to {b_2}}} = \left| {{{\varepsilon _{\rm{b2^+}}^{\rm{out}}} \big/ {\varepsilon _{{\rm{p2}}}^{{\text{in}}}} } } \right|.\end{split} $ 上述散射振幅角标以
$ {T_{{a_1} \to {a_2}}} $ 为例, 表示信号从光学腔${a_1}$ 传输到光学腔${a_2}$ 的透射振幅, 下面把$ {T_{{a_1} \to {a_2}}} $ 简单表示为$ {T_{{a_1}{a_2}}} $ , 其他同理. 注意(14)式振子输出场表示被振子吸收的探测场部分. -
首先令耗散速率
${\kappa _1} = {\kappa _2} = \kappa $ , 且当系统失谐量$x = 0$ 时, (14)式中的透射振幅可简化为$\begin{split} & {T_{{a_2}{a_1}}} = \left| {\frac{{2\kappa \left( { - {\text{i}}J - {{\text{e}}^{{\text{i}}\alpha }}\kappa C} \right)}}{{Z\left( 0 \right)}}} \right| ,\\ & {T_{{a_1}{a_2}}} = \left| {\frac{{2\kappa \left( { - {\text{i}}J - {{\text{e}}^{ - {\text{i}}\alpha }}\kappa C} \right)}}{{Z\left( 0 \right)}}} \right|, \end{split}$ 其中, 光力协同性
$C = \displaystyle\sum\nolimits_j^2 {{G_j^2} \big/{\kappa {\gamma _j}}}$ . 根据(10)式, 得到$ Z\left( 0 \right) = \left( { - 2{\text{i}}J\kappa \cos \alpha + 2{\kappa ^2}} \right)C + {J^2} + {\kappa ^2}. $ 这里重点讨论两个不同方向的透射振幅
$ {T_{{a_1}{a_2}}} $ 和$ {T_{{a_2}{a_1}}} $ , 主要由光腔模之间的线性耦合$J$ 和光力耦合部分$C$ 相互影响决定. 从(15)式可以看出,$ {T_{{a_1}{a_2}}} $ 和$ {T_{{a_2}{a_1}}} $ 在相位差$\alpha = n{\text{π }}$ ($n$ 为整数)时相等, 系统表现为互易性, 此时无法实现探测光的单向隔离.为了实现系统的隔离性, 要求系统从一侧输入的探测场到另一侧输出时会被抑制, 而不影响反方向传输的探测场. 考虑完美光非互易性情况, 即
$ {T_{{a_1}{a_2}}} = 1, {\text{ }}{T_{{a_2}{a_1}}} = 0 $ 或${T_{{a_2}{a_1}}} = 1, {T_{{a_1}{a_2}}} = 0$ , 可得$\begin{split} & \alpha = \pm {\text{π }}/2 + 2 n{\text{π }}(n为整数),\\ & {J / \kappa } = C, J = \kappa .\end{split}$ 图2显示传输振幅随相位差
$ \alpha $ 的周期变化. 透射振幅$ {T_{{a_1}{a_2}}} $ 和${T_{{a_2}{a_1}}}$ 变化周期为2π, 但$ {T_{{a_1}{a_2}}} $ 和${T_{{a_2}{a_1}}}$ 具有π的相位差, 且变化趋势相反. 图2表明当相位差改变π后, 系统两端将会呈现出与原来方向相反的光学响应, 如在$\alpha = {{\text{π }}/2}$ 时, 探测场从${a_2} \to {a_1}$ 能够完美透过, 即${T_{{a_2}{a_1}}} = 1$ , 而${a_1} \to {a_2}$ 被完全抑制, 即${T_{{a_1}{a_2}}} = 0$ ; 随着相位差变至$\alpha = {{3{\text{π }}}/ 2}$ , 探测场从${a_2} \to {a_1}$ 被完全抑制, 即${T_{{a_2}{a_1}}} = 0$ , 从${a_1} \to {a_2}$ 则能完美透过, 即${T_{{a_1}{a_2}}} = 1$ . 可见, π相位的改变导致透射振幅${T_{{a_1}{a_2}}}$ 和${T_{{a_2}{a_1}}}$ 的完全抑制和完美透射发生反转. 反射振幅${T_{{a_1}{a_1}}}$ 和${T_{{a_2}{a_2}}}$ 曲线的变化周期为π, 二者完全重合, 并在$\alpha = n{\text{π }}$ ($n$ 为整数)时达到极大. 而在$\alpha = {\text{π }}/2 + n{\text{π }}$ ($n$ 为整数)时反射部分消失, 而透射振幅${T_{{a_1}{a_2}}}$ 最小(或最大),${T_{{a_2}{a_1}}}$ 最大(或最小), 此时光非互易现象最明显. 下面讨论选择相位差α = π/2.
图 2
${x / \kappa } = 0$ 时, 传输振幅T随相位差α的变化 (其他参数:${J/\kappa } = C = 1$ )Figure 2. Transmission amplitudes T at
${x / \kappa } = 0$ are plotted against the phase difference α (Other parameters:${J/\kappa } = $ $ C = 1$ ).图3讨论了标准光力系统中力学振子对信号的非互易传输所产生的影响, 这时系统为b1模存在而b2模消失(
$ {G}_{1}\ne 0, {G}_{2}=0 $ )的三模光力系统. 当力学振子为一个输入端口时, 系统出现三端口环形器的一些功能[25]. 图3显示的传输振幅随失谐量的变化与文献[1]结果一致, 且系统中光学腔与力学振子的耗散速率对谱线宽度存在影响. 所有传输振幅关于失谐量为0的位置, 即共振频率($x/ \kappa = 0 $ )对称, 此时探测场与驱动场之间的拍频与力学振子频率相等. 图3(a)实现对探测场${\varepsilon _{{\text{p}}1}}$ 的完全隔离, 因为探测场${\varepsilon _{{\text{p}}1}}$ 在共振频率处被力学振子完全吸收, 而未通过反射或透射部分传输出该系统. 总之, 探测光子之间的相消干涉实现了对探测场${\varepsilon _{{\text{p}}1}}$ 的完全隔离. 图3(b)实现探测场${\varepsilon _{{\text{p}}2}}$ 的完美透射, 因为探测场${\varepsilon _{{\text{p}}2}}$ 在共振频率处能够从系统中全部透射出去, 而没有能量反射出来或被力学振子所吸收. 探测场${\varepsilon _{{\text{p}}1}}$ 的完全隔离和${\varepsilon _{{\text{p}}2}}$ 的完美透射说明标准光力系统具有光非互易性质, 能够实现光的单向隔离性, 有助于实现光隔离器和光开关[28]. 但在远离共振频率时, 探测场的非互易性质会明显减弱, 表现为不同方向的透射振幅逐渐重合, 系统趋于互易. 换言之, 该过程中信号被力学振子吸收的部分变弱而反射部分快速增强. 可见, 标准光力系统只能实现对特定频率(共振频率)的光隔离器和光开关.
图 3 单个力学振子b1影响下, 不同方向输入场
${\varepsilon _{{\text{p}}1}}$ (a)和${\varepsilon _{{\text{p}}2}}$ (b)的传输振幅T随标准化失谐$x/\kappa $ 的变化 (其他参数:$\alpha = {{\text{π }} \mathord{\left/ {\vphantom {{\text{π }} 2}} \right. } 2}$ ,${J \mathord{\left/ {\vphantom {J \kappa }} \right. } \kappa } = 1$ ,${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma _1}} \kappa }} \right. } \kappa } = 0.25$ ,${G_2} = 0$ ,${{{G_1}}/ \kappa } = 0.5$ )Figure 3. Transmission amplitude T of input fields
${\varepsilon _{{\text{p}}1}}$ (a) and${\varepsilon _{{\text{p}}2}}$ (b) in different directions as a function of normalized detuning$x/\kappa $ for only a single mechanical oscillator b1 (Other parameters:$\alpha = {{\text{π }}/2}$ ,${J/ \kappa } = 1$ ,${{{\gamma _1}} / \kappa } = 0.25$ ,${G_2} = 0$ ,${{{G_1}} / \kappa } = 0.5$ )隔离率是衡量光隔离器等非互易器件性能的主要参数, 因此引入单向隔离率
$I = \left( {{T_{{a_2}{a_1}}} - {T_{{a_1}{a_2}}}} \right)\big/ \left( {{T_{{a_2}{a_1}}} + {T_{{a_1}{a_2}}}} \right)$ 表征系统以光腔模${a_1}$ 作为输入端的隔离程度和系统实现光非互易现象的程度[29]. 图4显示标准光力系统(力学振子b1单独作用)在x = 0处单向隔离率I随有效光力耦合强度${G_1}$ 的变化, 发现随着有效光力耦合的出现, 系统出现了光非互易现象, 并在${{{G_1}}/ \kappa } = 0.5$ 处使隔离率达到最大值, 且${G_1}$ 再增大时隔离率下降. 可见, 恰当地选择有效光力耦合强度可以制作光隔离器.
图 4 x = 0时, 单向隔离率I 随有效光力耦合强度
${G_1}$ 的变化 (给定的参数:$\alpha = {{\text{π }}/ 2}$ ,${J/ \kappa } = 1$ ,${G_2} = 0$ ,${{{\gamma _1}}/ \kappa } = $ $ 0.25$ )Figure 4. Unidirectional isolation rate I at x = 0 versus effective optomechanical coupling strengths
${G_1}$ (The given parameters:$\alpha = {{\text{π }}/ 2}$ ,${J / \kappa } = 1$ ,${G_2} = 0$ ,${{{\gamma _1}} / \kappa } = 0.25$ )图5显示双腔双光力系统在x = 0处的单向隔离率I随有效光力耦合强度
${G_1}$ 和${G_2}$ 变化, 发现腔模间耦合强度${J/\kappa } = 1$ 时, 由(10)式可知, 只有满足总的光力协同性$C = {J/ \kappa }$ 系统才能具备较大的单向隔离率, 如图5红色区域. 当单个力学振子作用时, 只有有效的光力耦合强度${G_1}$ 和${G_2}$ 分别靠近$ \sqrt{\kappa {\gamma }_{1}} $ 和$ \sqrt{\kappa {\gamma }_{2}} $ 时, 系统才能达到高隔离率. 在保证隔离程度一定时, 符合条件的有效光力耦合强度的范围与其自身力学振子的耗散速率呈正相关, 如隔离率I ≥ 0.95区域内, 因耗散速率${{{\gamma _2}} / \kappa } \gt {{{\gamma _1}}/ \kappa }$ , 所以相对于力学振子b1来说, b2对光学腔的有效光力耦合强度${G_2}$ 的可选择范围更大, 这一特性使系统拥有更高的容错率.
图 5 x = 0时, 单向隔离率I随有效光力耦合强度
${G_1}$ 和${G_2}$ 的变化 (给定的参数:$\alpha = {{\text{π }}/ 2}$ ,${J/\kappa } = 1$ ,${{{\gamma _1}}/ \kappa } = 0.25$ ,${{{\gamma _2}}/ \kappa } = 9$ )Figure 5. Unidirectional isolation rate I at x = 0 versus the effective optomechanical coupling strengths G1 and G2 (The given parameters:
$\alpha = {{\text{π }}/ 2}$ ,${J/ \kappa } = 1$ ,${{{\gamma _1}}/\kappa } = 0.25$ ,${{{\gamma _2}}/\kappa } = $ $ 9$ ).图6讨论了透射振幅T和对应的隔离率I随失谐量
$x/\kappa $ 的变化, 图6(a1), (a2)和图6(b1), (b2)分别对应图5中${G_2} = 0$ (a点和b点)和${{{G_1}}/\kappa } $ = 0.3 (c点和d点). 注意这里力学振子${b_1}$ 和${b_2}$ 是可分辨的, 通过不同的耗散速率体现, 不失一般性地设定力学振子对应的两耗散速率具有不同的数量级, 取${\gamma _2} \gt {\gamma _1}$ (注意${\gamma _2} \lt {\gamma _1}$ 的情况与${\gamma _2} \gt {\gamma _1}$ 类似; 而${\gamma _2} = {\gamma _1}$ 时两力学振子相同, 对系统的影响等价于单个力学振子时的情形).
图 6 透射振幅T和对应的隔离率I随标准化失谐
$x/\kappa $ 的变化 (其他参数与图5相同) (a1) G2 = 0, G1/κ = 0.5; (a2) G2 = 0, G1/κ = 0.3; (b1) G1/κ = 0.3, G2/κ = 2.4; (b2) G1/κ = 0.3, G2/κ = 3.0Figure 6. Transmission amplitudes T and the corresponding isolation rate I versus normalized detuning
$x/\kappa $ (The given parameters are the same as the ones in Fig. 5): (a1) G1/κ = 0.5; (a2) G1/κ = 0.3; (b1) G2/κ = 2.4; (b2) G2/κ = 3.0.为了能够得到更加明显的双峰谱线, 图7讨论透射振幅T和对应的隔离率I随标准化失谐
$x/\kappa $ 的变化. 图7(a1)表明力学振子${b_2}$ 单独作用可使系统实现完美光非互易传输, 对应于图5中的e点. 而图7(a2)表明当力学振子${b_1}$ 加入后, 在共振频率处打开了一个反向透射窗口(红实线), 此时系统出现具有明显双峰现象的隔离率曲线. 该结果不仅能隔离相应频率的噪声, 且共振频率处的反向透射振幅达到很高. 相较于理想条件, 即力学振子${b_1}$ 耗散速率为0时, 即${{{\gamma _1}}/ \kappa } = 0$ , 共振频率处的信号能够实现完美的反向透过(图7(b1)). 图7(b2)表明当光力耦合强度从${{{G_1}}/\kappa } = 0.3$ 增加至${{{G_1}} / \kappa } = 0.8$ 时, 双峰最大隔离率会降低, 同时最大隔离率对应的频率范围也发生了改变. 可见, 通过适当地调控参数, 该系统可以有选择性地实现不同频率的信号、不同程度的反向通过和隔离, 这些结果极大地丰富了一些光学非互易器件的功用性.
图 7 透射振幅T和对应的隔离率I随标准化失谐
$x/\kappa $ 的变化 (其他参数:$\alpha = {{\text{π }} / 2}$ ,${J /\kappa } = 1$ ,${{{\gamma _2}}/ \kappa } = 9$ ,${{{G_2}} / \kappa } = 3$ ) (a1) γ1/κ = 0.01, G1/κ = 0; (a2) γ1/κ = 0.01, G1/κ = 0.3; (b1) γ1 = 0, G1/κ = 0.3; (b2) γ1 = 0, G1/κ = 0.8Figure 7. Transmission amplitudes T and the corresponding isolation rate I versus normalized detuning
$x/\kappa $ (The other parameters are$\alpha = {{\text{π }}/ 2}$ ,${J / \kappa } = 1$ ,${{{\gamma _2}}/ \kappa } = 9$ ,${{{G_2}} / \kappa } = 3$ ): (a1) γ1/κ = 0.01, G1/κ = 0; (a2) γ1/κ = 0.01, G1/κ = 0.3; (b1) γ1 = 0, G1/κ = 0.3; (b2) γ1 = 0, G1/κ = 0.8.对比图6(a1)和图7(a1), 均显示出完美光非互易性, 且力学振子
${b_2}$ 作用时的频谱宽度(图7(a1)和图7(a2))大于只有力学振子${b_1}$ 作用时的频谱宽度(图6(a1)和图6(a2)), 可见频谱宽度与力学振子耗散速率(${\gamma _2} \gt {\gamma _1}$ )呈正相关[30]. 图6(a1)和图6(a2)表明单个力学振子${b_1}$ 作用时, 隔离率在$x = 0$ 附近随失谐量的变化很敏感, 并随失谐量的增大快速衰减至0; 图6(a1)显示单个力学振子${b_1}$ 作用下系统达到了完美的光非互易性, 但随着光力耦合强度${G_1}$ 从${{{G_1}}/\kappa } = 0.5$ (图6(a1))降至${{{G_1}}/\kappa } = 0.3$ (图6(a2)), 光非互易程度减弱.图6(b1)和图6(b2)显示两力学振子
${b_1}$ 和${b_2}$ 共同作用下, 系统在共振频率处可以再次达到完美光非互易性(图6(b1)), 并且光非互易范围比图6(a1)还要宽, 但窄于只有力学振子${b_2}$ 作用时的情形(图7(a1)). 此时隔离率谱线可看作是力学振子${b_1}$ 和${b_2}$ 单独作用时谱线的 “结合”, 共振频率附近力学振子${b_1}$ 的影响占主导, 远离共振频率处力学振子${b_2}$ 的影响则占主导. 该结论可用于理解图6和图7中当${\gamma _1}$ 越小时, 隔离率谱线的峰或谷越尖锐; 而当${\gamma _2}$ 越大时, 谱线两边越平缓. 当力学振子${b_2}$ 的光力耦合强度增加至${{{G_2}} \mathord{\left/ {\vphantom {{{G_2}} \kappa }} \right. } \kappa } = 3$ (图6(b2)), 隔离率曲线在共振频率两侧对称地出现两个峰值, 计算发现隔离率曲线出现双峰须有效光力耦合强度满足如下条件:$\begin{split} & G_2^4G_1^2{\gamma _1}{\left( {{\gamma _2} - {\gamma _1}} \right)^2} + G_1^4G_2^2{\gamma _2}{\left( {{\gamma _2} - {\gamma _1}} \right)^2} \\ & ~ - {\kappa ^2}G_2^2\gamma _1^4{\gamma _2} - {\kappa ^2}G_1^2\gamma _2^4{\gamma _1} \gt 0, ~~( {{G_1},{G_2} \ne 0}).\end{split} $ 为了详细讨论隔离率曲线出现双峰的情况, 图8绘制了隔离率随两力学振子光力耦合强度的变化. 注意图8单峰区和双峰区的分割线与坐标轴无交点, 这正体现出只有两个力学振子共同作用时, 才能出现具有双峰的隔离率谱线. 这是因为两力学振子共同作用的双腔双光力系统与三腔光力系统相比多了一条光力耦合路径, 在此路径的干涉作用下在共振频率处本该抑制的信号变得可以透过, 这样共振频率两端的抑制程度变得更加突出. 由此可见, 通过调整参数在该系统中可有选择地实现两种特定频率光的较大隔离率(两个峰值), 同时有望实现中间共振频率信号的反向通过.
图 8 隔离率I随有效光力耦合强度
${{{G_1}}/\kappa }$ 和${{{G_2}} / \kappa }$ 的单峰区及双峰区 (其他参数与图5相同)Figure 8. Single peak region and double peak region of the isolation rate I versus the effective optomechanical coupling strengths
${{{G_1}} / \kappa }$ and${{{G_2}} / \kappa }$ (Other parameters are the same as the ones in Fig. 5).总之, 两力学振子共同作用的双腔双光力系统能够得到独特的隔离率谱线, 是多条路径间量子干涉的结果. 以探测场
${\varepsilon _{{\text{p}}1}}$ 传输方向为例, 两条光力耦合路径${a_1} \to {b_1} \to {a_2}$ 和${a_1} \to {b_2} \to {a_2}$ 与腔间路径${a_1} \to {a_2}$ 进行干涉, 当相位差$\alpha = {{\text{π }} \mathord{\left/ {\vphantom {{\text{π }} 2}} \right. } 2}$ 时, 两种路径之间的相长干涉导致探测场的正向通过, 反之相消干涉会抑制探测场的反向通过. 与双腔光力学系统相比, 通过力学振子${b_2}$ 增加的光力耦合路径${a_1} \to {b_2} \to {a_2}$ 不但能使系统实现完美的光非互易传输, 而且还能得到一些新的光非互易特征. -
主要研究了双腔双光力系统的光非互易性质, 首先给出了双腔光力学系统和双腔双光力系统在共振频率处实现完美光非互易性的条件, 然后讨论了相位差
$\alpha = {{\text{π }}/ 2}$ 时系统的光非互易性随有效光力耦合强度和耗散速率的变化, 主要通过透射振幅和隔离率来刻画. 结果发现双腔双光力系统能够得到更加丰富多变的隔离率谱线, 通过两力学振子的共同作用系统不仅可以实现完美光非互易性, 而且隔离率谱线具备每个力学振子单独作用时的特性. 双腔双光力系统不仅可实现对两种特定频率探测场的隔离, 而且还可实现对共振频率处信号的反向传输. 此外, 该系统的这些光非互易特性不仅可以用于量子开关的制备, 而且能广泛应用于量子通信和量子器件等方面.
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光学非互易器件是光学系统中重要的组成部分, 例如光隔离器. 为了提高这类器件的隔离特性, 提出一种双腔双光力系统, 系统以两种大小不同的光力耦合强度分别与两光学腔相耦合. 该系统在红失谐场的驱动下, 通过调控相位差既可以实现光非互易现象, 又能决定光在系统中透过与隔离的方向, 这种性质来源于光力耦合相互作用和光学腔模耦合之间的量子干涉效应. 研究过程中把系统中相关算符用其平均值加各自的相对涨落值来表示, 然后由涨落值利用输入-输出关系得到系统的传输振幅, 进而得到系统隔离率. 讨论了系统隔离率随光力耦合强度变化的分布情况, 发现两个力学振子的共同作用能使系统有更高的容错率. 该系统通过选用一个品质因数更高的力学振子, 能够实现两种特定频率光的较大隔离率, 并且同时能够保证共振频率信号的反向传输 .
Optical non-reciprocal devices such as the isolators are quite important components in optical systems. To realize the non-reciprocal transmission of the light, the Lorenz reciprocity theorem must be broken first and the main method is that Faraday magnetic rotation effect is used to change the polarization state of the signal through magneto-optical materials. However, this method is difficult to achieve on-chip integration. So using optomechanical system is presented to overcome the difficulty. In order to improve the isolation characteristics of the device, a double-cavity double-optomechanical system, which is coupled to two optical modes by two mechanical oscillators with two different optomechanical coupling strengths, is proposed. Driven by the red detuning field in such a system, the non-reciprocal phenomenon can be realized by regulating the phase difference, and the direction of light transmission and isolation can be determined as well. This property is determined by the quantum interference effect between the optomechanical coupling strengths and the couplings of the optical cavity modes. The method is that the relative operators are represented by their average value plus their relative fluctuations, and then according to the input-output relationship the transmission amplitude and the isolation rate are obtained. We mainly discuss the distribution of the isolation rate as a function of the optomechanical coupling strength. The results are that the combined action of two mechanical modes can make the system have higher fault tolerance rate. The other mechanical mode can make the system achieve a large isolation rate at two specific frequencies and the reverse transmission in the resonant frequency signals at the same time. -
Keywords:
- double-cavity double-optomechanical system /
- nonreciprocal property /
- transmission amplitude /
- isolation rate
[1] Xia C C, Yan X B, Tian X D, Gao F 2019 Opt. Commun. 451 197
Google Scholar
[2] 朱雪松, 刘星雨, 张岩 2022 物理学报 71 150701
Google Scholar
Zhu X S, Liu X Y, Zhang Y 2022 Acta Phys. Sin. 71 150701
Google Scholar
[3] Jing Y W 2022 Sci. Rep. 12 5844
Google Scholar
[4] Jiang C, Song L N, Li Y 2018 Phys. Rev. A 97 053812
Google Scholar
[5] Asadchy V S, Mirmoosa M S, Diaz-Rubio A, Fan S, Tretyakov S A 2020 Proc. IEEE 108 1684
Google Scholar
[6] Xia K Y, Lu G W, Lin G W, Cheng Y Q, Niu Y P, Gong S Q, Twamley J 2014 Phys. Rev. A 90 043802
Google Scholar
[7] Scheucher M, Hilico A, Will E, Volz J, Rauschenbeutel A 2016 Science 354 1577
Google Scholar
[8] Tang J S, Nie W, Tang L, Chen M Y, Su X, Lu Y Q, Nori F, Xia K Y 2022 Phys. Rev. Lett. 128 203602
Google Scholar
[9] Hu Y Q, Qi Y H, You Y, Zhang S C, Lin G W, Li X L, Gong J B, Gong S Q, Niu Y P 2021 Phys. Rev. Appl. 16 014046
Google Scholar
[10] Peterson C W, Benalcazar W A, Lin M, Hughes T L, Bahl G 2019 Phys. Rev. Lett. 123 063901
Google Scholar
[11] Hu X X, Wang Z B, Zhang P F, Chen G J, Zhang Y L, Li G, Zou X B, Zhang T C, Tang H X, Dong C H, Guo G C, Zou C L 2021 Nat. Commun. 12 2389
Google Scholar
[12] Liang C, Liu B, Xu A N, Wen X, Lu C C, Xia K Y, Tey M K, Liu Y C, You L 2020 Phys. Rev. Lett. 125 123901
Google Scholar
[13] Huang X Y, Lu C C, Liang C, Tao H G, Liu Y C 2021 Light Sci. Appl. 10 30
Google Scholar
[14] Tang L, Tang J S, Chen M Y, Nori F, Xiao M, Xia K Y 2022 Phys. Rev. Lett. 128 083604
Google Scholar
[15] Barzanjeh S, Wulf M, Peruzzo M, Kalaee M, Dieterle P B, Painter O, Fink J M 2017 Nat. Commun. 8 953
Google Scholar
[16] Peterson G A, Lecocq F, Cicak K, Simmonds R W, Aumentado J, Teufel J D 2017 Phys. Rev. X 7 031001
Google Scholar
[17] Tang J D, Cai Q Z, Cheng Z D, Xu N, Peng G Y, Chen P Q, Wang D G, Xia Z W, Wang Y, Song H Z, Zhou Q, Deng G W 2022 Phys. Lett. A 429 127966
Google Scholar
[18] 刘妮, 王建芬, 梁九卿 2020 物理学报 69 064202
Google Scholar
Liu N, Wang J F, Liang J Q 2020 Acta Phys. Sin. 69 064202
Google Scholar
[19] Jiang C, Liu Y L, Sillanpaa M A 2021 Phys. Rev. A 104 013502
Google Scholar
[20] Xiao R J, Pan G X, Xiu X M 2021 Chin. Phys. B 30 034209
Google Scholar
[21] Yan K X, Zhang Y C, Cui Y S, Jiang C 2020 Opt. Commun. 475 126249
Google Scholar
[22] Qu K N, Agarwal G S 2013 Phys. Rev. A 87 63813
Google Scholar
[23] He Y 2016 Phys. Rev. A 94 063804
Google Scholar
[24] Singh S K, Parvez M, Abbas T, Peng J X, Mazaheri M, Asjad M 2022 Phys. Lett. A 442 128181
Google Scholar
[25] Xu X W, Li Y 2015 Phys. Rev. A 91 053854
Google Scholar
[26] DeJesus E X, Kaufman C 1987 Phys. Rev. A 35 5288
Google Scholar
[27] Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520
Google Scholar
[28] Ullah K 2019 Chin. Phys. B 28 114209
Google Scholar
[29] Chen Y T, Du L, Liu Y M, Zhang Y 2020 Opt. Express 28 7095
Google Scholar
[30] 张利巍, 李贤丽, 杨柳 2019 物理学报 68 170701
Google Scholar
Zhang L W, Li X L, Yang L 2019 Acta Phys. Sin. 68 170701
Google Scholar
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图 1 双腔双光力系统示意图, 其中光学腔
${a_i}$ 通过光力耦合相互作用${g_{ij}}$ 与力学振子${b_j}$ 相耦合, 同时两光学腔之间存在腔模线性耦合相互作用, 腔模两侧存在振幅为${\varepsilon _{{\text{d}}i}}$ 的驱动场和${\varepsilon _{{\text{p}}i}}$ 的探测场Fig. 1. Diagram of double-cavity dual-optomechanical system. The optical cavity
${a_i}$ is coupled to the mechanical oscillator${b_j}$ by the optomechanical coupling interaction$ {g}_{ij}$ . And there is a cavity mode linear coupling interaction between two optical cavities, and there are the coupling fields with amplitude${\varepsilon _{{\text{d}}i}}$ and the probe fields with amplitude${\varepsilon _{{\text{p}}i}}$ on both sides of the cavity modes.
图 2
${x / \kappa } = 0$ 时, 传输振幅T随相位差α的变化 (其他参数:${J/\kappa } = C = 1$ )Fig. 2. Transmission amplitudes T at
${x / \kappa } = 0$ are plotted against the phase difference α (Other parameters:${J/\kappa } = $ $ C = 1$ ).
图 3 单个力学振子b1影响下, 不同方向输入场
${\varepsilon _{{\text{p}}1}}$ (a)和${\varepsilon _{{\text{p}}2}}$ (b)的传输振幅T随标准化失谐$x/\kappa $ 的变化 (其他参数:$\alpha = {{\text{π }} \mathord{\left/ {\vphantom {{\text{π }} 2}} \right. } 2}$ ,${J \mathord{\left/ {\vphantom {J \kappa }} \right. } \kappa } = 1$ ,${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma _1}} \kappa }} \right. } \kappa } = 0.25$ ,${G_2} = 0$ ,${{{G_1}}/ \kappa } = 0.5$ )Fig. 3. Transmission amplitude T of input fields
${\varepsilon _{{\text{p}}1}}$ (a) and${\varepsilon _{{\text{p}}2}}$ (b) in different directions as a function of normalized detuning$x/\kappa $ for only a single mechanical oscillator b1 (Other parameters:$\alpha = {{\text{π }}/2}$ ,${J/ \kappa } = 1$ ,${{{\gamma _1}} / \kappa } = 0.25$ ,${G_2} = 0$ ,${{{G_1}} / \kappa } = 0.5$ )
图 4 x = 0时, 单向隔离率I 随有效光力耦合强度
${G_1}$ 的变化 (给定的参数:$\alpha = {{\text{π }}/ 2}$ ,${J/ \kappa } = 1$ ,${G_2} = 0$ ,${{{\gamma _1}}/ \kappa } = $ $ 0.25$ )Fig. 4. Unidirectional isolation rate I at x = 0 versus effective optomechanical coupling strengths
${G_1}$ (The given parameters:$\alpha = {{\text{π }}/ 2}$ ,${J / \kappa } = 1$ ,${G_2} = 0$ ,${{{\gamma _1}} / \kappa } = 0.25$ )
图 5 x = 0时, 单向隔离率I随有效光力耦合强度
${G_1}$ 和${G_2}$ 的变化 (给定的参数:$\alpha = {{\text{π }}/ 2}$ ,${J/\kappa } = 1$ ,${{{\gamma _1}}/ \kappa } = 0.25$ ,${{{\gamma _2}}/ \kappa } = 9$ )Fig. 5. Unidirectional isolation rate I at x = 0 versus the effective optomechanical coupling strengths G1 and G2 (The given parameters:
$\alpha = {{\text{π }}/ 2}$ ,${J/ \kappa } = 1$ ,${{{\gamma _1}}/\kappa } = 0.25$ ,${{{\gamma _2}}/\kappa } = $ $ 9$ ).
图 6 透射振幅T和对应的隔离率I随标准化失谐
$x/\kappa $ 的变化 (其他参数与图5相同) (a1) G2 = 0, G1/κ = 0.5; (a2) G2 = 0, G1/κ = 0.3; (b1) G1/κ = 0.3, G2/κ = 2.4; (b2) G1/κ = 0.3, G2/κ = 3.0Fig. 6. Transmission amplitudes T and the corresponding isolation rate I versus normalized detuning
$x/\kappa $ (The given parameters are the same as the ones in Fig. 5): (a1) G1/κ = 0.5; (a2) G1/κ = 0.3; (b1) G2/κ = 2.4; (b2) G2/κ = 3.0.
图 7 透射振幅T和对应的隔离率I随标准化失谐
$x/\kappa $ 的变化 (其他参数:$\alpha = {{\text{π }} / 2}$ ,${J /\kappa } = 1$ ,${{{\gamma _2}}/ \kappa } = 9$ ,${{{G_2}} / \kappa } = 3$ ) (a1) γ1/κ = 0.01, G1/κ = 0; (a2) γ1/κ = 0.01, G1/κ = 0.3; (b1) γ1 = 0, G1/κ = 0.3; (b2) γ1 = 0, G1/κ = 0.8Fig. 7. Transmission amplitudes T and the corresponding isolation rate I versus normalized detuning
$x/\kappa $ (The other parameters are$\alpha = {{\text{π }}/ 2}$ ,${J / \kappa } = 1$ ,${{{\gamma _2}}/ \kappa } = 9$ ,${{{G_2}} / \kappa } = 3$ ): (a1) γ1/κ = 0.01, G1/κ = 0; (a2) γ1/κ = 0.01, G1/κ = 0.3; (b1) γ1 = 0, G1/κ = 0.3; (b2) γ1 = 0, G1/κ = 0.8.
图 8 隔离率I随有效光力耦合强度
${{{G_1}}/\kappa }$ 和${{{G_2}} / \kappa }$ 的单峰区及双峰区 (其他参数与图5相同)Fig. 8. Single peak region and double peak region of the isolation rate I versus the effective optomechanical coupling strengths
${{{G_1}} / \kappa }$ and${{{G_2}} / \kappa }$ (Other parameters are the same as the ones in Fig. 5). -
[1] Xia C C, Yan X B, Tian X D, Gao F 2019 Opt. Commun. 451 197
Google Scholar
[2] 朱雪松, 刘星雨, 张岩 2022 物理学报 71 150701
Google Scholar
Zhu X S, Liu X Y, Zhang Y 2022 Acta Phys. Sin. 71 150701
Google Scholar
[3] Jing Y W 2022 Sci. Rep. 12 5844
Google Scholar
[4] Jiang C, Song L N, Li Y 2018 Phys. Rev. A 97 053812
Google Scholar
[5] Asadchy V S, Mirmoosa M S, Diaz-Rubio A, Fan S, Tretyakov S A 2020 Proc. IEEE 108 1684
Google Scholar
[6] Xia K Y, Lu G W, Lin G W, Cheng Y Q, Niu Y P, Gong S Q, Twamley J 2014 Phys. Rev. A 90 043802
Google Scholar
[7] Scheucher M, Hilico A, Will E, Volz J, Rauschenbeutel A 2016 Science 354 1577
Google Scholar
[8] Tang J S, Nie W, Tang L, Chen M Y, Su X, Lu Y Q, Nori F, Xia K Y 2022 Phys. Rev. Lett. 128 203602
Google Scholar
[9] Hu Y Q, Qi Y H, You Y, Zhang S C, Lin G W, Li X L, Gong J B, Gong S Q, Niu Y P 2021 Phys. Rev. Appl. 16 014046
Google Scholar
[10] Peterson C W, Benalcazar W A, Lin M, Hughes T L, Bahl G 2019 Phys. Rev. Lett. 123 063901
Google Scholar
[11] Hu X X, Wang Z B, Zhang P F, Chen G J, Zhang Y L, Li G, Zou X B, Zhang T C, Tang H X, Dong C H, Guo G C, Zou C L 2021 Nat. Commun. 12 2389
Google Scholar
[12] Liang C, Liu B, Xu A N, Wen X, Lu C C, Xia K Y, Tey M K, Liu Y C, You L 2020 Phys. Rev. Lett. 125 123901
Google Scholar
[13] Huang X Y, Lu C C, Liang C, Tao H G, Liu Y C 2021 Light Sci. Appl. 10 30
Google Scholar
[14] Tang L, Tang J S, Chen M Y, Nori F, Xiao M, Xia K Y 2022 Phys. Rev. Lett. 128 083604
Google Scholar
[15] Barzanjeh S, Wulf M, Peruzzo M, Kalaee M, Dieterle P B, Painter O, Fink J M 2017 Nat. Commun. 8 953
Google Scholar
[16] Peterson G A, Lecocq F, Cicak K, Simmonds R W, Aumentado J, Teufel J D 2017 Phys. Rev. X 7 031001
Google Scholar
[17] Tang J D, Cai Q Z, Cheng Z D, Xu N, Peng G Y, Chen P Q, Wang D G, Xia Z W, Wang Y, Song H Z, Zhou Q, Deng G W 2022 Phys. Lett. A 429 127966
Google Scholar
[18] 刘妮, 王建芬, 梁九卿 2020 物理学报 69 064202
Google Scholar
Liu N, Wang J F, Liang J Q 2020 Acta Phys. Sin. 69 064202
Google Scholar
[19] Jiang C, Liu Y L, Sillanpaa M A 2021 Phys. Rev. A 104 013502
Google Scholar
[20] Xiao R J, Pan G X, Xiu X M 2021 Chin. Phys. B 30 034209
Google Scholar
[21] Yan K X, Zhang Y C, Cui Y S, Jiang C 2020 Opt. Commun. 475 126249
Google Scholar
[22] Qu K N, Agarwal G S 2013 Phys. Rev. A 87 63813
Google Scholar
[23] He Y 2016 Phys. Rev. A 94 063804
Google Scholar
[24] Singh S K, Parvez M, Abbas T, Peng J X, Mazaheri M, Asjad M 2022 Phys. Lett. A 442 128181
Google Scholar
[25] Xu X W, Li Y 2015 Phys. Rev. A 91 053854
Google Scholar
[26] DeJesus E X, Kaufman C 1987 Phys. Rev. A 35 5288
Google Scholar
[27] Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520
Google Scholar
[28] Ullah K 2019 Chin. Phys. B 28 114209
Google Scholar
[29] Chen Y T, Du L, Liu Y M, Zhang Y 2020 Opt. Express 28 7095
Google Scholar
[30] 张利巍, 李贤丽, 杨柳 2019 物理学报 68 170701
Google Scholar
Zhang L W, Li X L, Yang L 2019 Acta Phys. Sin. 68 170701
Google Scholar
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