一维非互易声学晶体的非厄米趋肤态操控

黄泽鑫; 圣宗强; 程乐乐; 曹三祝; 陈华俊; 吴宏伟

doi:10.7498/aps.73.20241087

摘要

Abstract

作者及机构信息

Authors and contacts

文章全文

1. 引　言
2. 非互易耦合的实现与调控
3. 趋肤界面态的调控与实验观测
4. 结　论

参考文献

摘要

近年来, 基于非厄米拓扑理论, 研究者们通过调制声学晶体中的非互易耦合, 揭示了体态向界面塌陷的趋肤效应. 本工作实验设计了具有不同绕组数域之间的拓扑趋肤界面, 以操纵能量聚焦到非厄米一维声腔链的中间或两端. 首先, 通过电声耦合的方法实现了两个声学腔之间的非互易耦合, 并研究其特性. 其次, 将非互易耦合腔扩展成链状, 通过调制非互易电-声耦合来构建趋肤界面的位置. 实验结果表明, 对于不同的非互易耦合分布, 声音可以集中在中间界面或两端界面, 并且通过改变非互易耦合方向, 可以将趋肤界面从中间切换到两端. 本研究结果为设计控制声音传播的先进拓扑声学装置提供了一个新平台.

关键词:

Abstract

Topological insulators possess strong topological protection properties and can manipulate the wave propagation to combat disorder and defects. And now they have grown into a large research field in photonic and phononic crystals. However, the conventional topological band theory is used to describe a closed photonic/phononic crystal that is assumed to be a Hermitian system. In fact, actual physical systems often couple with external environment, and generate non-Hermitian Hamiltonians with complex eigenvalues. Recently, many novel topological properties have been induced by the interaction between non-Hermitian phase and topological phase. A prominent example is non-Hermitian skin effect that all eigenstates are localized to the boundary in open system, which is different from the conventional topological edge state. This unique physical phenomenon has inspired various applications, such as wave funneling, enhanced sensing, and topological lasing. In this work, we describe the non-Hermitian skin effect by using winding number domains. The sign of the winding number domain determines the rotation direction of the loops in the complex frequency plane, whose sign can be controlled by the nonreciprocal coupling direction. In this work, we design different topological skin interfaces between different domains with opposite winding numbers to manipulate the energy focusing on middle or two-end of non-Hermitian one-dimensional acoustic cavity chain. In experiment, we use an electroacoustic coupling method, in which a unidirectional coupler composed of microphones, speakers, phase shifters, and amplifiers is used to introduce positive and negative non-reciprocal couplings between the two acoustic cavities, and study the characteristics of these non-reciprocal couplings. Then, the non-reciprocal coupling cavities are extended into a chain structure, and the magnitudes and signs of the non-reciprocal couplings are flexibly controlled by using phase shifters and amplifiers. Through this method, we successfully construct the interfaces between different winding number domains, achieving a one-dimensional non-Hermitian skin effect at various interfaces. The experimental results indicate that the sound can be focused on the middle interface or two-end interfaces for different nonreciprocal coupling distributions, and the skin interface can also be switched from middle to two-end by exchanging the nonreciprocal coupling direction of the domains. Our research results provide greater flexibility for designing acoustic devices and also a new platform for exploring advanced topological acoustic systems for controlling sound propagation.

Keywords:

PACS:

0365
0250

作者及机构信息

1.
安徽理工大学力学与光电物理学院, 淮南　232001

2.
安徽理工大学基础物理研究中心, 淮南　232001

3.
南京大学, 固体微结构物理国家重点实验室, 南京　210093

通信作者: 圣宗强, zqsheng@aust.edu.cn ; 吴宏伟, hwwu@aust.edu.cn

基金项目: 安徽省高校自然科学研究项目(批准号: 2022AH040114)和安徽省高校协同创新项目(批准号: GXXT-2022-015)资助的课题.

Authors and contacts

1.
School of Mechanics and Photoelectric Physics, Anhui University of Science and Technology, Huainan 232001, China

2.
Center for Fundamental Physics, Anhui University of Science and Technology, Huainan 232001, China

3.
National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

Corresponding author: Sheng Zong-Qiang, zqsheng@aust.edu.cn ; Wu Hong-Wei, hwwu@aust.edu.cn

Funds: Project supported by the Natural Science Foundation of the Higher Education Institutions of Anhui Province, China (Grant No. 2022AH040114) and the University Synergy Innovation Program of Anhui Province, China (Grant No. GXXT-2022-015).

文章全文

1. 引　言

拓扑绝缘体因其强大的拓扑保护特性和在无序和缺陷环境中控制波传播的能力^[1–9], 已经成为光子晶体和声子晶体中的重要研究领域. 然而, 传统的拓扑能带理论是基于其厄米特性, 主要用于描述封闭的光子晶体或声子晶体. 实际上, 物理系统总是与外部环境发生耦合, 所以推导出的哈密顿量通常具有复特征值, 并且是非厄米的. 近年来, 非厄米相与拓扑相之间的相互作用诱导出了许多新的拓扑性质^[10–18], 一个新奇的现象被命名为非厄米趋肤效应, 在开放边界条件下系统所有本征态都局域于边界^[19–23], 这与传统的拓扑边界态不同. 这种独特的物理现象激发了各种应用, 如波漏斗^[24]、增强传感^[25,26]和拓扑激光^[27,28]等.

非厄米趋肤效应由Hatano-Nelson模型描述的非互易耦合(non-reciprocal hopping)所控制^[29,30]. 最近, 该模型已经在多个有源实验平台上进行了验证, 如光纤环路光子晶格^[24]、量子行走^[31]、电路^[32]和电声调制声子晶体^[33]. 在非厄米系统中, 趋肤效应可以用点能隙拓扑来描述, 在周期边界条件下, 复本征频谱在复平面上形成闭环. 在开放边界条件下, 本征频谱在复平面上呈直线变化, 本征态呈现向边界坍缩的趋肤效应. 人们在该领域开展了一系列研究, 如在光子谐振腔中引入了频率合成维度, 以实现在复平面上获得任意绕组数, 而不局限于单个环路^[34]. 通过引入远程非互易耦合, 在声学晶体中实验实现了扭曲绕组^[34]. 除主动声平台之外, 复频率激励也使得在无增益介质的被动声晶体中实现趋肤效应^[35]. 近年来, 随着非厄米系统的发展, 在赝自旋二维非厄米Su-Schrieffer-Heeger模型(SSH模型)中观察到二阶趋肤效应, 在电路和声学晶体中所有体态都坍缩到角态. 除了经典的趋肤效应, 还提出了许多新颖的非厄米趋肤现象, 例如几何依赖的趋肤效应^[36,37]和混合趋肤拓扑效应^[38–40].

本文通过精心设计的双边域声电耦合, 在声腔链的不同界面上成功实现了一维非厄米趋肤态. 为了诱导非互易耦合现象, 首先, 通过添加电子元器件来研究两个腔体之间的耦合特性, 实现了可调谐的非互易耦合. 进一步地, 通过调制非互易电声耦合, 在声腔链中构造了不同绕组数域之间的界面. 实验结果表明, 声音可以集中在中间界面或两端界面, 并且可以通过交换非互易耦合方向来切换趋肤位置. 这些发现可能为开发先进的拓扑声学器件, 如声漏斗、增强传感和声天线等, 提供新的思路.

2. 非互易耦合的实现与调控

本文的目标是通过电声元器件调控腔体之间的非互易耦合, 进而实现非厄米趋肤效应及其趋肤界面调控的研究. 因此, 在设计电声耦合调控之前, 首先介绍两个声学腔体耦合的固有模式特征. 展示了两个通过对称交叉管道连接的声学腔体示意图, 此时系统具有互易耦合特性. 声学谐振腔之间的间距 $s=24\;{\mathrm{m}}{\mathrm{m}}$ , 每个谐振腔的尺寸为 $L=92\;{\mathrm{m}}{\mathrm{m}},\; w=72\;{\mathrm{m}}{\mathrm{m}}$ , $h=112\;{\mathrm{m}}{\mathrm{m}}$ . 交叉的窄管产生的耦合系数为 ${t}_{0}$ , 窄管宽度为 ${d}_{1}=20\;{\mathrm{m}}{\mathrm{m}}$ .

$图 1 (a) 通过交叉管连接的相同尺寸的谐振腔模型; (b) 单个谐振腔的本征模式和两个由交叉管连接的谐振腔的本征模式\r\nFig. 1. (a) Resonator models of the same size connected by cross tubes; (b) the eigenmodes of a single resonator and the eigenmode of two resonators connected by cross tubes.$

图 1 (a) 通过交叉管连接的相同尺寸的谐振腔模型; (b) 单个谐振腔的本征模式和两个由交叉管连接的谐振腔的本征模式

Fig. 1. (a) Resonator models of the same size connected by cross tubes; (b) the eigenmodes of a single resonator and the eigenmode of two resonators connected by cross tubes.

下载: 全尺寸图片幻灯片

我们计算了单个腔体的偶极模式, 利用COMSOL多物理场声压模块的特征频率求解器对仿真结果进行求解. 在模拟中, 腔体和窄管区域设置为空气, 空气密度 ${\rho }_{0}=1.21\;{\mathrm{k}}{\mathrm{g}}/{{\mathrm{m}}}^{3}$ , 声速 $v= 343\;{\mathrm{m}}/{\mathrm{s}}$ . 如图1(b), 当腔体尺寸与上述相同时, 单腔的偶极模式频率为1542.4 Hz, 当腔体增大时, 谐振频率会下降. 因此, 可以通过调整谐振腔大小来设置我们期望的谐振频率.

接下来, 将两个谐振腔通过两个交叉的窄管连接, 使其发生耦合, 如图1(b)右侧所示, 通过对特征模式的求解, 当频率为1558.1 Hz与1523.4 Hz时, 特征模式会产生对称模式与反对称模式. 在高频处, 两腔处于同相模式, 表明耦合效应为正.

通过打破上述结构中的互易性来产生非互易现象, 如所示. 利用树脂材料进行3D打印, 制作与仿真模型对应的结构. 腔体上的开孔是用于插入声源、探测器和单向耦合器. 未使用时, 这些孔是密闭的. 在两腔之间使用单向耦合器, 实现非互易耦合 $\kappa =\rho {{\mathrm{e}}}^{{\mathrm{i}}\theta }$ , 其中 $\rho$ 和 $\theta$ 分别由放大器和移相器控制. 麦克风在腔1接收到声信号, 信号由放大器(Type LM386)放大, 并通过移相器(Type MCP41010)调整相位, 最后通过扬声器将声信号输出到腔2. 这种单向放大引入了非互易性, 紧束缚模型如图2(b)可以得到如下哈密顿量^[34]:

$图 2 (a) 两个相同尺寸的谐振腔1和2, 其中腔体通过交叉管连接, 通过单向耦合器实现非互易耦合; (b) 紧束缚模型示意图; (c) 当放大器关闭时, 对透射谱进行实验测量和数值拟合; (d), (e)当放大器打开时, 实验测量了负耦合(d)和正耦合(e), 并对透射谱进行了数值拟合, 其中红色圆圈与蓝色正方形为实验结果, 红色实线与蓝色实线为数值拟合结果\r\nFig. 2. (a) Two identical size resonators 1 and 2. The cavity is connected by cross pipe, and the non-reciprocal coupling is realized by unidirectional coupler. (b) Schematic diagram of a tight-binding model. (c) When the amplifier is turned off, the transmission spectrum is experimentally measured and numerically fitted. (d), (e) When the amplifier is turned on, the negative coupling (d) and positive coupling (e) are experimentally measured and numerically fitted to the transmission spectra. The red circle and blue square are experimental results, and the red solid line and blue solid line are numerical fitting results.$

图 2 (a) 两个相同尺寸的谐振腔1和2, 其中腔体通过交叉管连接, 通过单向耦合器实现非互易耦合; (b) 紧束缚模型示意图; (c) 当放大器关闭时, 对透射谱进行实验测量和数值拟合; (d), (e)当放大器打开时, 实验测量了负耦合(d)和正耦合(e), 并对透射谱进行了数值拟合, 其中红色圆圈与蓝色正方形为实验结果, 红色实线与蓝色实线为数值拟合结果

Fig. 2. (a) Two identical size resonators 1 and 2. The cavity is connected by cross pipe, and the non-reciprocal coupling is realized by unidirectional coupler. (b) Schematic diagram of a tight-binding model. (c) When the amplifier is turned off, the transmission spectrum is experimentally measured and numerically fitted. (d), (e) When the amplifier is turned on, the negative coupling (d) and positive coupling (e) are experimentally measured and numerically fitted to the transmission spectra. The red circle and blue square are experimental results, and the red solid line and blue solid line are numerical fitting results.

下载: 全尺寸图片幻灯片

$H=\left[\begin{array}{cc}{\omega }_{0}-{\mathrm{i}}\left({\gamma }_{0}+{\gamma }_{1}\right)& {t}_{0}\\ {t}_{0}+\kappa & {\omega }_{0}-{\mathrm{i}}\left({\gamma }_{0}+{\gamma }_{1}\right)\end{array}\right] ,$

(1)

${\mathrm{式}}{\mathrm{中}}{\gamma }_{0}$ 为空腔损耗; 声源和探测器具有相同的尺寸, 插入损耗为 ${\gamma }_{1}$ ; ω₀为腔体的谐振频率.

为了获取系统的固有参数并实现可控的非互易耦合, 通过实验和数值模拟计算了在关闭和打开单向耦合器的情况下的透射谱. 为了在实验中检索耦合, 定义 ${S}_{ij}$ , 其中 $i$ 是探测器的位置, $j$ 是声源的位置. 此时, 系统的动力学方程为^[34]

$\begin{split} &{S}_{21}\left(\omega \right) = \frac{-2{\mathrm{i}}{\gamma }_{1}\left({t}_{0}+\kappa \right)}{\left[{\mathrm{i}}\left(\omega -{\omega }_{0}\right)-{\gamma }_{0}-{\gamma }_{1}\right]\left[{\mathrm{i}}\left(\omega -{\omega }_{0}\right)-{\gamma }_{0}-{\gamma }_{1}\right]+(t_0 + \kappa) t_0}, \end{split}$

(2a)

$\begin{split}&{S}_{12} (\omega) = \frac{-2{\mathrm{i}}{\gamma }_{1}{t}_{0}}{\left[{\mathrm{i}}\left(\omega -{\omega }_{0}\right)-{\gamma }_{0}-{\gamma }_{1}\right]\left[{\mathrm{i}}\left(\omega -{\omega }_{0}\right)-{\gamma }_{0}-{\gamma }_{1}\right]+(t_0 + \kappa) t_0}. \end{split}$

(2b)

实验中, 我们使用频率范围为1490—1590 Hz的声波对系统进行扫频测量. 将测量得到的各个频率点的声压强度绘制成图像, 如所示. 当单向耦合器关闭时, 系统是互易的( $\kappa =0$ ), 此时产生的 $\left|{S}_{12}\right|$ 和 $\left|{S}_{21}\right|$ 的透射谱几乎是重叠的. 利用已建立的动力学方程拟合可以得到参数: ${\omega }_{0}= 1535.2{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ , ${t}_{0}=16{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ , ${\gamma }_{0}=0.08{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ , ${\gamma }_{1}=3.3{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ . 当打开单向耦合器时, 传输响应 $\left|{S}_{12}\right|$ 和 $\left|{S}_{21}\right|$ 在图像上会产生明显的不同, 此时系统是非互易的. 当系统的单向耦合值为负时(), $\left|{S}_{12}\right|$ 的传输响应会明显高于 $\left|{S}_{21}\right|$ , 并且共振峰距离变窄, 通过数值拟合得到单向耦合值 $\kappa =-6.5{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ . 当系统的单向耦合值为正时(), 情况与负值相反. $\left|{S}_{21}\right|$ 的传输响应会明显高于 $\left|{S}_{12}\right|$ , 并且共振峰距离变宽, 此时通过数值拟合得到的单向耦合值 $\kappa =+5-0.7{\mathrm{i}}{\mathrm{ }}\;{\mathrm{H}}{\mathrm{z}}$ . 从图2(d), (e)可以看出, 实验结果与数值拟合结果非常一致. 通过不断重复这个过程, 在声腔链上实现大范围的非互易耦合. 单向耦合值的虚部由相位延迟所致, 并导致透射谱上左右峰值不对称, 即对称模式和反对称模式的响应强度不同. 在实验中, 虽然可以通过移相器调整电路中的相位, 有时仍无法完全消除相位延迟. 较小的虚部对实验影响不大. 注意, 单向耦合器在工作超过一定增益范围时, 可能会经历自激效应. 通过合理的选择功率放大器和系统的工作频率, 最大限度地缓解了这一问题.

3. 趋肤界面态的调控与实验观测

我们使用绕组数来描述非厄米趋肤效应. 以10个腔体组成的声腔链为理论模型, 在非互易耦合的作用下, 系统中的能量会局域于边界处, 局域的方向由单向耦合 $\kappa$ 的正负决定. 当 $\kappa$ = 0时, 系统是厄米的, 特征频谱在周期边界和开放边界条件下的复频谱发生重叠, 此时系统不具有趋肤效应. 当 $\kappa \ne 0$ 时, 本征频谱的实部和虚部如, 所示, 当非互易耦合的正负不同时, 本征频谱的实部(蓝色曲线)保持不变, 而虚部(红色曲线)则相反, 并且虚部关于 $\kappa =0$ 不对称分布, 表明系统具有非互易性. 在不同的边界条件下, 本征频谱在复频率平面上呈现不同的形状, 在周期边界条件下, 本征频谱在复频率平面上形成旋转的闭环, 在开放边界条件下, 则在复频率平面上形成一条直线. 对于非厄米系统, 可以计算绕组数^[34]

$图 3 (a), (c) 当非互易耦合为正(a)或负(c)时, 周期边界条件下本征谱的实部和虚部; (b), (d) 红色区域表示在周期边界条件下, 复本征频谱在复频率平面上形成闭环; 绿色虚线对应于开放边界条件下的本征频谱; 单向耦合为正, 圈数$ W=-1 $ (b), 闭环顺时针旋转; 当单向耦合为负时, 圈数$ W=+1 $ (d), 闭环逆时针旋转\r\nFig. 3. (a), (c) When the non-reciprocal coupling is positive (a) or negative (c), the real and imaginary parts of the characteristic spectrum under periodic boundary conditions. (b), (d) The red region indicates that the complex eigenfrequency spectrum forms a closed loop in the complex frequency plane under periodic boundary conditions. The green dashed lines correspond to the eigenfrequency spectrum under open boundary conditions. The unidirectional coupling is positive, the winding number $ W=-1 $ (b), and the closed loop rotates clockwise. When the unidirectional coupling is negative, the winding number $ W=+1 $ (d), and the closed loop rotates counterclockwise.$

图 3 (a), (c) 当非互易耦合为正(a)或负(c)时, 周期边界条件下本征谱的实部和虚部; (b), (d) 红色区域表示在周期边界条件下, 复本征频谱在复频率平面上形成闭环; 绿色虚线对应于开放边界条件下的本征频谱; 单向耦合为正, 圈数

$W=-1$

(b), 闭环顺时针旋转; 当单向耦合为负时, 圈数

$W=+1$

(d), 闭环逆时针旋转

Fig. 3. (a), (c) When the non-reciprocal coupling is positive (a) or negative (c), the real and imaginary parts of the characteristic spectrum under periodic boundary conditions. (b), (d) The red region indicates that the complex eigenfrequency spectrum forms a closed loop in the complex frequency plane under periodic boundary conditions. The green dashed lines correspond to the eigenfrequency spectrum under open boundary conditions. The unidirectional coupling is positive, the winding number

$W=-1$

(b), and the closed loop rotates clockwise. When the unidirectional coupling is negative, the winding number

$W=+1$

(d), and the closed loop rotates counterclockwise.

下载: 全尺寸图片幻灯片

$W=\frac{1}{2{\text{π}}{\mathrm{i}}}\int_{-{\text{π}}}^{{{\text{π}}}}\frac{\partial \omega \left(k\right)/\partial k}{\omega \left(k\right)-{\omega }_{{\mathrm{b}}}}{\mathrm{d}}k ,$

(3)

其中 ${\omega }_{{\mathrm{b}}}$ 为参考频率. 当 $\kappa > 0$ 时(), 复频率平面内的闭环顺时针旋转, 此时 $W=-1$ , 所有特征态会向右坍缩. 当 $\kappa < 0$ 时(), 复频率平面内的闭环逆时针旋转, 此时 $W=+1$ , 所有特征态会向左坍缩. 当 $\kappa$ 存在虚部时, 图像会发生倾斜, 倾斜的方向与大小与 $\kappa$ 虚部的正负和大小有关.

上文已经证明可以通过在两腔耦合中加入单向耦合器实现非互易耦合, 接下来为了在实验上实现声音集中在中间界面和两端界面, 我们将这种非互易耦合拓展到一维声腔链, 整个系统包括10个腔体, 如图4(a), 相邻声腔之间用交叉管道和单向耦合器连接. 以5腔为中间界面, 5腔左侧非互易耦合为负, 右侧非互易耦合为正, 紧束缚模型如图4(b)所示. 由于趋肤效应的存在, 声音会聚集在边界处, 当非互易耦合为负时, 声音会向左坍缩, 非互易耦合为正时, 声音会向右坍缩, 从而在一维声腔链上实现同时趋向于两侧的趋肤效应. 实验中, 使用LabVIEW控制数据采集卡(USB-5511, Smacq)产生输入信号, 信号经过放大器放大后, 通过扬声器作为点声源发出, 并放置在样品背面的孔中. 测量时, 在5腔中放入频率为1532 Hz声信号, 作为激发源, 将探测器(378B02, PCB压电)依次插入第1—10个腔体测量振幅并记录. 如图4(c), 可以看到1腔和10腔内的声压几乎相同并且高于5腔, 将实验测量的数据与数值拟合的结果进行对比(图4(d)), 红色实线为数值拟合结果, 蓝色虚线为实验测量数据, 两者拟合很好, 由于声源位于5腔, 测量时5腔内声压偏高.

$图 4 (a) 实验设备图片, 10个谐振腔通过交叉管和单向耦合器连接; (b) 声音集中在两端界面时的紧束缚模型示意图. 第1—5个空腔之间非互易耦合为负, 第5—10个空腔之间非互易耦合为正; (c) 时域信号传输的测量, 其中结果分别在1, 5, 10腔中测量; (d) 声音集中在两端界面时的声场分布(结果被归一化), 声源频率为1532 Hz\r\nFig. 4. (a) Experimental equipment picture, 10 resonators connected by cross tubes and unidirectional couplers. (b) Schematic diagram of a tight-binding model when sound is concentrated at the two-end interfaces. The non-reciprocal coupling between 1 to 5 cavities is negative, and the non-reciprocal coupling between 5 to 10 cavities is positive. (c) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (d) Field distribution of sound locality at the two-end interfaces and the frequency of the sound source is 1532 Hz. The results are normalized.$

图 4 (a) 实验设备图片, 10个谐振腔通过交叉管和单向耦合器连接; (b) 声音集中在两端界面时的紧束缚模型示意图. 第1—5个空腔之间非互易耦合为负, 第5—10个空腔之间非互易耦合为正; (c) 时域信号传输的测量, 其中结果分别在1, 5, 10腔中测量; (d) 声音集中在两端界面时的声场分布(结果被归一化), 声源频率为1532 Hz

Fig. 4. (a) Experimental equipment picture, 10 resonators connected by cross tubes and unidirectional couplers. (b) Schematic diagram of a tight-binding model when sound is concentrated at the two-end interfaces. The non-reciprocal coupling between 1 to 5 cavities is negative, and the non-reciprocal coupling between 5 to 10 cavities is positive. (c) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (d) Field distribution of sound locality at the two-end interfaces and the frequency of the sound source is 1532 Hz. The results are normalized.

下载: 全尺寸图片幻灯片

接下来通过改变非互易耦合方向将趋肤界面从两端切换到中间. 以5腔为中心, 交换左右两侧非互易耦合方向, 紧束缚模型如图5(a)所示. 当以频率为1532 Hz的声波在5腔中激发时, 声音集中在5腔中, 不向两侧扩散, 如图5(b)所示, 可以看到1腔和10腔中的信号明显小于5腔, 这是因为5腔左侧非互易耦合为正, 声音向右侧坍缩, 5腔右侧非互易耦合为负, 能量向左坍缩, 导致声音会被限制在5腔内部, 无法传播. 将实验测量的数据和数值拟合的结果进行对比, 可以看到实验和理论结果一致(如图5(c)左侧). 为了证明声音在中间界面处的聚集并非由于声源位于该界面引起的, 我们将声源切换到3腔(如图5(c)右侧), 结果发现声音仍然聚集在中间界面, 进一步证实了非厄米趋肤效应的存在.

$图 5 (a) 声音集中在中间界面时的紧束缚模型示意图, 第1—5个空腔之间非互易耦合为正, 第5—10个空腔之间非互易耦合为负; (b) 时域信号传输测量, 其中结果分别在1, 5, 10腔中测量; (c) 声音集中在中间界面时的声场分布(结果被归一化), 声源的频率为1532 Hz\r\nFig. 5. (a) Schematic diagram of a tight-binding model when sound is concentrated at the intermediate interface. The non-reciprocal coupling between 1 to 5 cavities is positive, and the non-reciprocal coupling between 5 to 10 cavities is negative. (b) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (c) Field distribution of sound locality at the middle interface. The excitation frequency of the sound source is 1532 Hz. The results are normalized.$

图 5 (a) 声音集中在中间界面时的紧束缚模型示意图, 第1—5个空腔之间非互易耦合为正, 第5—10个空腔之间非互易耦合为负; (b) 时域信号传输测量, 其中结果分别在1, 5, 10腔中测量; (c) 声音集中在中间界面时的声场分布(结果被归一化), 声源的频率为1532 Hz

Fig. 5. (a) Schematic diagram of a tight-binding model when sound is concentrated at the intermediate interface. The non-reciprocal coupling between 1 to 5 cavities is positive, and the non-reciprocal coupling between 5 to 10 cavities is negative. (b) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (c) Field distribution of sound locality at the middle interface. The excitation frequency of the sound source is 1532 Hz. The results are normalized.

下载: 全尺寸图片幻灯片

4. 结　论

综上所述, 本文利用单向耦合器在声腔链上实现不对称耦合, 并且在声腔链上制造了不同的界面, 实验实现了声音集中在不同的界面处. 首先, 通过设计两个声腔结构, 在两腔之间实现了非互易耦合. 并通过观察本征频谱和计算绕组数, 证明了非互易性的存在. 本研究还将结构扩展为声腔链, 通过移相器灵活控制非互易耦合的正负, 构造了不同圈数域之间的界面, 实现了不同界面上的一维非厄米趋肤效应. 通过实验和数值计算证明了声音可以集中在声腔链的中间界面和两端界面. 这些例子表明, 利用工程界面可以有效控制非互易声学晶体的非厄米趋肤效应, 为控制声音传播提供方便的工具. 这些研究结果将为设计高灵敏度传感器、单向波导和能量收集器方面提供理论基础和应用价值.

参考文献

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图 1 (a) 通过交叉管连接的相同尺寸的谐振腔模型; (b) 单个谐振腔的本征模式和两个由交叉管连接的谐振腔的本征模式

Fig. 1. (a) Resonator models of the same size connected by cross tubes; (b) the eigenmodes of a single resonator and the eigenmode of two resonators connected by cross tubes.

下载: 全尺寸图片幻灯片

图 2 (a) 两个相同尺寸的谐振腔1和2, 其中腔体通过交叉管连接, 通过单向耦合器实现非互易耦合; (b) 紧束缚模型示意图; (c) 当放大器关闭时, 对透射谱进行实验测量和数值拟合; (d), (e)当放大器打开时, 实验测量了负耦合(d)和正耦合(e), 并对透射谱进行了数值拟合, 其中红色圆圈与蓝色正方形为实验结果, 红色实线与蓝色实线为数值拟合结果

Fig. 2. (a) Two identical size resonators 1 and 2. The cavity is connected by cross pipe, and the non-reciprocal coupling is realized by unidirectional coupler. (b) Schematic diagram of a tight-binding model. (c) When the amplifier is turned off, the transmission spectrum is experimentally measured and numerically fitted. (d), (e) When the amplifier is turned on, the negative coupling (d) and positive coupling (e) are experimentally measured and numerically fitted to the transmission spectra. The red circle and blue square are experimental results, and the red solid line and blue solid line are numerical fitting results.

下载: 全尺寸图片幻灯片

(a), (c) 当非互易耦合为正(a)或负(c)时, 周期边界条件下本征谱的实部和虚部; (b), (d) 红色区域表示在周期边界条件下, 复本征频谱在复频率平面上形成闭环; 绿色虚线对应于开放边界条件下的本征频谱; 单向耦合为正, 圈数 $W=-1$ (b), 闭环顺时针旋转; 当单向耦合为负时, 圈数 $W=+1$ (d), 闭环逆时针旋转

(a), (c) When the non-reciprocal coupling is positive (a) or negative (c), the real and imaginary parts of the characteristic spectrum under periodic boundary conditions. (b), (d) The red region indicates that the complex eigenfrequency spectrum forms a closed loop in the complex frequency plane under periodic boundary conditions. The green dashed lines correspond to the eigenfrequency spectrum under open boundary conditions. The unidirectional coupling is positive, the winding number $W=-1$ (b), and the closed loop rotates clockwise. When the unidirectional coupling is negative, the winding number $W=+1$ (d), and the closed loop rotates counterclockwise.

下载: 全尺寸图片幻灯片

图 4 (a) 实验设备图片, 10个谐振腔通过交叉管和单向耦合器连接; (b) 声音集中在两端界面时的紧束缚模型示意图. 第1—5个空腔之间非互易耦合为负, 第5—10个空腔之间非互易耦合为正; (c) 时域信号传输的测量, 其中结果分别在1, 5, 10腔中测量; (d) 声音集中在两端界面时的声场分布(结果被归一化), 声源频率为1532 Hz

Fig. 4. (a) Experimental equipment picture, 10 resonators connected by cross tubes and unidirectional couplers. (b) Schematic diagram of a tight-binding model when sound is concentrated at the two-end interfaces. The non-reciprocal coupling between 1 to 5 cavities is negative, and the non-reciprocal coupling between 5 to 10 cavities is positive. (c) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (d) Field distribution of sound locality at the two-end interfaces and the frequency of the sound source is 1532 Hz. The results are normalized.

下载: 全尺寸图片幻灯片

图 5 (a) 声音集中在中间界面时的紧束缚模型示意图, 第1—5个空腔之间非互易耦合为正, 第5—10个空腔之间非互易耦合为负; (b) 时域信号传输测量, 其中结果分别在1, 5, 10腔中测量; (c) 声音集中在中间界面时的声场分布(结果被归一化), 声源的频率为1532 Hz

Fig. 5. (a) Schematic diagram of a tight-binding model when sound is concentrated at the intermediate interface. The non-reciprocal coupling between 1 to 5 cavities is positive, and the non-reciprocal coupling between 5 to 10 cavities is negative. (b) Measurement of signal transmission in time domain. The results are measured in cavity 1, 5 and 10 respectively. (c) Field distribution of sound locality at the middle interface. The excitation frequency of the sound source is 1532 Hz. The results are normalized.

下载: 全尺寸图片幻灯片

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[2]	Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368 Google Scholar
[3]	贾鼎, 葛勇, 袁寿其, 孙宏祥 2019 物理学报 68 224301 Google Scholar Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301 Google Scholar
[4]	Cheng Q Q, Pan Y M, Wang H Q, Zhang C S, Yu D, Gover A, Zhang H J, Li T, Zhou L, Zhu S N 2019 Phys. Rev. Lett. 122 173901 Google Scholar
[5]	耿治国, 彭玉桂, 沈亚西, 赵德刚, 祝雪丰 2019 物理学报 68 227802 Google Scholar Geng Z G, Peng Y G, Shen Y X, Zhao D G, Zhu X F 2019 Acta Phys. Sin. 68 227802 Google Scholar
[6]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901 Google Scholar
[7]	王青海, 李锋, 黄学勤, 陆久阳, 刘正猷 2017 物理学报 66 204501 Google Scholar Wang Q H, Li F, Huang X Q, Lu J Y, Liu Z Y 2017 Acta Phys. Sin. 66 204501 Google Scholar
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[12]	Iwanow R, May-Arrioja D A, Christodoulides D N, Stegeman G I, Min Y, Sohler W 2005 Phys. Rev. Lett. 95 053902 Google Scholar
[13]	Shen Y X, Peng Y G, Zhao D G, Chen X C, Zhu J, Zhu X F 2019 Phys. Rev. Lett. 122 094501 Google Scholar
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[18]	Tang S, Wu J L, Lü C, Yao J B, Wang X S, Song J, Jiang Y Y 2023 New J. Phys. 25 033032 Google Scholar
[19]	Crespi A, Pepe F V, Facchi P, Sciarrino F, Mataloni P, Nakazato H, Pascazio S, Osellame R 2019 Phys. Rev. Lett. 122 130401 Google Scholar
[20]	Pinkse P W H, Fischer T, Maunz P, Rempe G 2000 Nature 404 365 Google Scholar
[21]	Schäfer F, Herrera I, Cherukattil S, Lovecchio C, Cataliotti F S, Caruso F, Smerzi A 2014 Nat. Commun. 5 3194 Google Scholar
[22]	Raimond J M, Sayrin C, Gleyzes S, Dotsenko I, Brune M, Haroche S, Facchi P, Pascazio S 2010 Phys. Rev. Lett. 105 213601 Google Scholar
[23]	Barontini G, Hohmann L, Haas F, Estève J, Reichel J 2015 Science 349 1317 Google Scholar
[24]	Weidemann S, Kremer1 M, Helbig T, Hofmann T, Stegmaier A, Greiter M, Thomale R, Szameit A 2020 Science 368 311 Google Scholar
[25]	Budich J C, Bergholtz E J 2020 Phys. Rev. Lett. 125 180403 Google Scholar
[26]	McDonald A, Clerk A A 2020 Nat. Commun. 11 5382 Google Scholar
[27]	Longhi S 2018 Ann. Phys. 530 1800023 Google Scholar
[28]	Zhu B f, Wang Q, Leykam D, Xue H, Wang Q J, Chong Y D 2022 Phys. Rev. Lett. 129 013903 Google Scholar
[29]	Hatano N, Nelson D R 1996 Phys. Rev. Lett. 77 570 Google Scholar
[30]	Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384 Google Scholar
[31]	Xiao L, Deng T S, Wang K K, Zhu G Y, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761 Google Scholar
[32]	Helbig T, Hofmann T, Imhof S, Abdelghany M, Kiessling T, Molenkamp L W, Lee C H, Szameit A, Greiter M, Thomale R 2020 Nat. Phys. 16 747 Google Scholar
[33]	Zhang Q C, Li Y T, Sun H F, Liu X, Zhao L K, Feng X L, Fan X Y, Qiu C Y 2023 Phys. Rev. Lett. 130 017201 Google Scholar
[34]	Zhang L, Yang Y H, Ge Y, Guan Y J, Chen Q L, Yan Q H, Chen F J, Xi R, Li Y Z, Jia D, Yuan S Q, Sun H X, Chen H S, Zhang B L 2021 Nat. Commun. 12 6297 Google Scholar
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[37]	Zhang K, Fang C, Yang Z S 2023 Phys. Rev. Lett. 131 036402 Google Scholar
[38]	Lee C H, Li L H, Gong J B 2019 Phys. Rev. Lett. 123 016805 Google Scholar
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