Observation of the anisotropic exceptional point in cavity magnonics system

Zhang Gao-Jian; Wang Yi-Pu

doi:10.7498/aps.69.20191632

Abstract

We theoretically and experimentally demonstrate the anisotropic exceptional points (EPs) in the cavity magnonics system where magnons in a one millimeter-diameter yttrium iron garnet (YIG) sphere are coherently coupled with the microwave photons in a three-dimensional microwave cavity. The damping nature makes the cavity magnonics system inherently non-Hermitian. By solving the eigenvalues and eigenvectors of non-Hermitian Hamiltonian, a series of interesting and essential characteristics of the system can be obtained. Therefore, non-Hermitian physics has received more and more attention in both theory and experiment communities. Among them, exceptional points correspond to the non-Hermitian system’s degenerate states where the eigenvalues of the non-Hermitian system are identical, and the eigenvectors are parallel. The coupled cavity photon-magnon system has high tunability of coupling strength and cavity external damping rate, which is very suitable for studying EPs -related physics. Exceptional points (EPs) are crucial in all kinds of non-Hermitian physical systems, which have both fundamental and applicational importance. For instance, it can be used for sensitive detection by monitoring spectrum splitting of degenerate modes when a perturbation to be sensed occurs. The EPs can be anisotropic, which means that it has a different function relation when the system approaches the EPs along different parameter paths of the system. In this paper, by carefully designing the parameter space, we observe the anisotropic exceptional point in the coupled cavity photon-magnon system. It shows the linear and square-root behavior when the EPs are approached from different directions in the parameter space. One of the parameters is the position of the YIG sphere in the cavity, which determines the coherent coupling strength between the cavity mode and the magnon mode. Another parameter is the number of the gasket between the cavity signal loading port and the cavity external surface, which determines the external damping rate of the cavity mode. Both of these parameters can be easily and accurately adjusted experimentally. Our study paves the way for exploring anisotropic EPs based sensing technologies and more non-Hermitian related physics in the cavity magnonics system.

Keywords:

Authors and contacts

Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China

Corresponding author: Wang Yi-Pu, alex13@csrc.ac.cn

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References

[1]	Xiang Z L, Ashhab S, You J Q, Nori F 2013 Rev. Mod. Phys. 85 623 Google Scholar
[2]	Kurizki G, Bertet P, Kubo Y, Mølmer K, Petrosyan D, Rabl P, Schmiedmayer J 2015 Proc. Natl. Acad. Sci. U.S.A. 112 3866 Google Scholar
[3]	Wallquist M, Hammerer K, Rabl P, Lukin M, Zoller P 2009 Phys. Scr. T137 014001 Google Scholar
[4]	Kimble H J 2008 Nature (London) 453 1023 Google Scholar
[5]	Huebl H, Zollitsch C W, Lotze J, Hocke F, Greifenstein M, Marx A, Gross R, Goennenwein S T B 2013 Phys. Rev. Lett. 111 127003 Google Scholar
[6]	Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K, Nakamura Y 2014 Phys. Rev. Lett. 113 083603 Google Scholar
[7]	Zhang X, Zou C L, Jiang L, Tang H X 2014 Phys. Rev. Lett. 113 156401 Google Scholar
[8]	Goryachev M, Farr W G, Creedon D L, Fan Y, Kostylev M, Tobar M E 2014 Phys. Rev. Appl. 2 054002 Google Scholar
[9]	Bai L, Harder M, Chen Y P, Fan X, Xiao J Q, Hu C M 2015 Phys. Rev. Lett. 114 227201 Google Scholar
[10]	Zhang D, Wang X M, Li T F, Luo X Q, Wu W, Nori F, You J Q 2015 npj Quantum Inf. 1 15014 Google Scholar
[11]	Soykal Ö O, Flatté M E 2010 Phys. Rev. Lett. 104 077202 Google Scholar
[12]	Rameshti B Z, Cao Y, Bauer G E W 2015 Phys. Rev. B 91 214430 Google Scholar
[13]	Cao Y, Yan P, Huebl H, Goennenwein S T B, Bauer G E W 2015 Phys. Rev. B 91 094423 Google Scholar
[14]	Bourhill J, Kostylev N, Goryachev M, Creedon D L, Tobar M E 2016 Phys. Rev. B 93 144420 Google Scholar
[15]	Kostylev N, Goryachev M, Tobar M E 2016 Appl. Phys. Lett. 108 062402 Google Scholar
[16]	Yao B M, Gui Y S, Xiao Y, Guo H, Chen X S, Lu W, Chien C L, Hu C M 2015 Phys. Rev. B 92 184407 Google Scholar
[17]	Zhang X, Zou C L, Jiang L, Tang H X 2016 Sci. Adv. 2 e1501286 Google Scholar
[18]	Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Usami K, Nakamura Y 2015 Science 349 405 Google Scholar
[19]	Quirion D L, Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Nakamura Y 2017 Sci. Adv. 3 e1603150 Google Scholar
[20]	Osada A, Hisatomi R, Noguchi A, Tabuchi Y, Yamazaki R, Usami K, Sadgrove M, Yalla R, Nomura M, Nakamura Y 2016 Phys. Rev. Lett. 116 223601 Google Scholar
[21]	Hisatomi R, Osada A, Tabuchi Y, Ishikawa T, Noguchi A, Yamazaki R, Usami K, Nakamura Y 2016 Phys. Rev. B 93 174427 Google Scholar
[22]	Haigh J A, Langenfeld S, Lambert N J, Baumberg J J, Ramsay A J, Nunnenkamp A, Ferguson A J 2015 Phys. Rev. A 92 063845 Google Scholar
[23]	Zhang X, Zhu N, Zou C L, Tang H X 2016 Phys. Rev. Lett. 117 123605 Google Scholar
[24]	Liu T, Zhang X, Tang H X, Flatté M E 2016 Phys. Rev. B 94 060405 Google Scholar
[25]	Wang Y P, Zhang G Q, Zhang D, Luo X Q, Xiong W, Wang S P, Li T F, Hu C M, You J Q 2016 Phys. Rev. B 94 224410 Google Scholar
[26]	Sharma S, Blanter Y M, Bauer G E W 2017 Phys. Rev. B 96 094412 Google Scholar
[27]	Sharma S, Blanter Y M, Bauer G E W 2018 Phys. Rev. Lett. 121 087205 Google Scholar
[28]	Osada A, Gloppe A, Hisatomi R, Noguchi A, Yamazaki R, Nomura M, Nakamura Y, Usami K 2018 Phys. Rev. Lett. 120 133602 Google Scholar
[29]	Liu Z X, Wang B, Xiong H, Wu Y 2018 Opt. Lett. 43 3698 Google Scholar
[30]	Zhang X, Zou C L, Zhu N, Marquardt F, Jiang L, Tang H X 2015 Nat. Commun. 6 8914 Google Scholar
[31]	Lambert N J, Haigh J A, Langenfeld S, Doherty A C, Ferguson A J 2016 Phys. Rev. A 93 021803 Google Scholar
[32]	Bai L, Harder M, Hyde P, Zhang Z, Hu C M, Chen Y P, Xiao J Q 2017 Phys. Rev. Lett. 118 217201 Google Scholar
[33]	Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M, You J Q 2018 Phys. Rev. Lett. 120 057202 Google Scholar
[34]	Hyde P, Yao B M, Gui Y S, Zhang G Q, You J Q, Hu C M 2018 Phys. Rev. B 98 174423 Google Scholar
[35]	Harder M, Yang Y, Yao B M, Yu C H, Rao J W, Gui Y S, Stamps R L, Hu C M 2018 Phys. Rev. Lett. 21 137203 Google Scholar
[36]	Grigoryan V L, Shen K, Xia K 2018 Phys. Rev. B 98 024406 Google Scholar
[37]	Bhoi B, Kim B, Jang S H, Kim J, Yang J, Cho Y J, Kim S K 2019 Phys. Rev. B 99 134426 Google Scholar
[38]	Yang Y, Rao J W, Gui Y S, Yao B M, Lu W, Hu C M 2019 Phys. Rev. Applied 11 054023 Google Scholar
[39]	Rao J W, Yu C H, Zhao Y T, Gui Y S, Fan X L, Xue D S, Hu C M 2019 New J. Phys. 21 065001 Google Scholar
[40]	Proskurin I, Macedo R, Stamps R L 2019 New J. Phys. 21 095003 Google Scholar
[41]	Yao B m, Yu T, Zhang X, Lu W, Gui Y S, Hu C M, Blanter Y M 2019 arXiv: 1906.12142 v2[cond-mat.mes-hall]
[42]	Yu W C, Wang J J, Yuan H Y, Xiao J 2019 Phys. Rev. Lett. 123 227201 Google Scholar
[43]	Wang Y P, Rao J W, Yang Y, Xu P C, Gui Y S, Yao B M, You J Q, Hu C M 2019 Phys. Rev. Lett. 123 127202 Google Scholar
[44]	Li J, Zhu S Y, Agarwal G S 2018 Phys. Rev. Lett. 121 203601 Google Scholar
[45]	Li J, Zhu S Y, Agarwal G S 2019 Phys. Rev. A 99 021801(R)Google Scholar
[46]	Bender C M 2007 Rep. Prog. Phys. 70 947 Google Scholar
[47]	陈增军, 宁西京 2003 物理学报 52 2683 Google Scholar Chen Z J, Ning X J 2003 Acta Phys. Sin. 52 2683 Google Scholar
[48]	Gilbert T L 2004 IEEE Trans. Magn. 40 3443 Google Scholar
[49]	Stancil D D, Prabhakar A 2009 Spin Waves Theory and Applications (New York: Springer) pp263–296
[50]	Zhang D, Luo X Q, Wang Y P, Li T F, You J Q 2017 Nat. Commun. 8 1368 Google Scholar
[51]	Dembowski C, Gräf H D, Harney H L, Heine A, Heiss W D, Rehfeld H, Richter A 2001 Phys. Rev. Lett. 86 787 Google Scholar
[52]	Dembowski C, Dietz B, Gräf H D, Harney H L, Heine A, Heiss W D, Richter A 2003 Phys. Rev. Lett. 90 034101 Google Scholar
[53]	Gao T, Estrecho E, Bliokh K Y, Liew T C H, Fraser M D, Brodbeck S, Kamp M, Schneider C, Hfling S, Yamamoto Y, Nori F, Kivshar Y S, Truscott A G, Dall R G, Ostrovskaya E A 2015 Nature (London) 526 554 Google Scholar
[54]	Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik A, Libisch F, Milburn T J, Rabl P, Moiseyev N, Rotter S 2016 Nature (London) 537 76 Google Scholar
[55]	Heiss W 2012 J. Phys. A 45 444016 Google Scholar
[56]	Ingrid R 2009 Journal Phys. A 42 153001
[57]	Eleuch H, Rotter I 2014 Eur. Phys. J. D 68 74 Google Scholar
[58]	Ding K, Ma G, Xiao M, Zhang Z Q, Chan C T 2016 Phys. Rev. X 6 021007 Google Scholar
[59]	Lu L, Joannopoulos J D, Soljacic M 2014 Nat. Photonics 8 821 Google Scholar
[60]	Zhou H, Peng C, Yoon Y, Hsu C W, Nelson K A, Fu L, Joannopoulos J D, Soljacic M, Zhen B 2018 Science 359 1009 Google Scholar
[61]	Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402 Google Scholar
[62]	Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature (London) 548 187 Google Scholar
[63]	Lin Z, Pick A, Loncar M, Rodriguez A W 2016 Phys. Rev. Lett. 117 107402 Google Scholar
[64]	Jing H, Özdemir S K, Lu H, Nori F 2017 Sci. Rep. 7 3386 Google Scholar
[65]	Nada M Y, Othman M A K, Capolino F 2017 Phys. Rev. B 96 184304 Google Scholar
[66]	Zhang G Q, You J Q 2019 Phys. Rev. B 99 054404 Google Scholar
[67]	Kato T 1966 Perturbation Theory for Linear Operators (Berlin: Springer-Verlag) p516
[68]	Ding K, Ma G, Zhang Z Q, Chan C T 2018 Phys. Rev. Lett. 121 085702 Google Scholar
[69]	Rotter I 2009 J. Phys. A 42 153001 Google Scholar
[70]	Eleuch H, Rotter I 2015 Eur. Phys. J. D 69 229 Google Scholar
[71]	Kittel C 1948 Phys. Rev. 73 155 Google Scholar

Cited By

图 1 (a)归一化耦合常数$ g/g_0 $和归一化衰减系数差$ \Delta\kappa/(2 g_0) $在参数空间$ (\alpha, \beta) $ 中分别用黄色和绿色表示, 红色的实线表示的是奇异点连成的线; (b)系统本征频率在参数空间中的虚部, 黑色实线交叉的地方就是各向异性奇异点的位置; (c)表示本征频率的虚部分别沿着$ \alpha (\beta = 0) $(实线)和$ \beta(\alpha = 0) $ (虚线)方向; (d)表示的是态分别沿着$ \alpha (\beta = 0) $(实线)和$ \beta(\alpha = 0) $(虚线)方向的相位信息; 上面的参数为: $ \omega_0 = 1, \gamma_{\rm m} = 0, g_0 = 0.5 $

Figure 1. (a) The normalized coupling constant $ g/g_0 $ and normalized loss difference $ \Delta\kappa/(2 g_0) $ in the parameter space $ (\alpha, \beta) $ are shown by the yellow and green surfaces, respectively. The solid red line corresponds to a line of EPs; (b) the imaginary part of the eigenfrequencies in the parameter space as a function of $ \beta $ and $ \alpha $, where the black solid line crossing is the position of the anisotropic EP; (c) imaginary part of the eigenfrequencies along $ \alpha (\beta = 0) $ adjusting direction (solid line) and $ \beta(\alpha = 0) $ adjusting direction (dotted line), respectively; (d) phase rigidity of the corresponding states along $ \alpha (\beta = 0) $ adjusting direction (solid line) and $ \beta(\alpha = 0) $ adjusting direction (dotted line), respectively. The parameters used are $ \omega_0 = 1, \gamma_{\rm m} = 0 $ and $ g_0 = 0.5 $.

DownLoad: Full-Size Img PowerPoint

图 2 (a) 腔-自旋波量子系统的实验装置图, 把包含YIG 小球的3D腔放入一个室温可高精度调节的双轭双调型电磁铁产生的磁场中, 通过网络分析仪测出腔的透射谱; (b) 表示的是YIG小球黏在一根玻璃钢棒上, 通过一个4 mm的孔, 在腔的一侧插入到3D矩形腔内. 端口1和端口2用来得到腔模的传输谱; (c) 3D腔的透射谱, 横轴为电磁铁线圈的电流(既偏置磁场的大小), 纵轴为探测场的频率. 腔模和Kittle模共振时, 两子系统间的耦合强度最大为$ g_0/2{\text{π}}$ = 8.45 MHz

Figure 2. Schematic of the experiment setup. The 3D cavity containing the YIG sphere is placed in the static magnetic field generated by a double-yoke double-tuned electromagnet which can be accurately adjusted at room temperature, and the transmission spectrum of the cavity is measured by network analyzer; (b) the YIG sphere is attached to a fiberglass reinforced plastic rod and inserted into a 3D rectangular cavity through a 4 mm hole on one side of the cavity. Ports 1 and 2 are used to obtain the transmission spectrum of cavity modes; (c) transmission spectrum of coupled system as a function of the current of the electromagnet coil(the magnitude of the biased magnetic field) and the probe field frequency. When the cavity mode and Kittle mode are resonant, the normal mode splitting equals to $ 2 g_0/2{\text{π}} $ = 16.9 MHz.

DownLoad: Full-Size Img PowerPoint

图 3 (a) 系统的耦合强度g和YIG小球在腔中位置x的二次函数关系, 三角形代表实验得到的结果. 黑色实抛物线线是理论拟合实验数据的结果. 其中深红、黄色和绿色分别表示耦合强度为 8.17, 5.54, 3.47 MHz; (b) 3D腔的衰减$ \kappa $和垫片数量y呈线性函数关系, 品红色原点代表实验测量的结果. 蓝色实线代表理论拟合的结果

Figure 3. (a) The coherent coupling strength as a function of the position of the YIG sphere in the cavity, and the triangle dots the experimental results. The black solid curve is the result of theoretical fitting of experimental data. Among them, crimson, yellow and green dots indicate the coupling strengthes equal to: 8.17, 5.54 and 3.47 MHz, respectively; (b) the damping rate $ \kappa $ of the 3D cavity as a function of the number of gaskets y between the cavity and SMA connector, and the magenta dots represent the measured results. The solid blue line is the theoretical fitting curve.

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图 4 在腔的三种不同损耗下测得的传输谱. 所有实线表示应用图3中得到的系统的耦合强度和腔模的衰减系数实验值代入到输入- 输出理论中得到的, 腔模的衰减系数分别为15.64, 18.19和21.32 MHz

Figure 4. (a)Transmission spectra measured under three different damping rates of the cavity. All the solid lines are calculated using the input-output theory. The damping rates of the cavity mode are 15.64, 18.19 and 21.32 MHz, respectively.

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图 5 (a) 在不同腔模损耗下本征频率虚部关于YIG小球在腔中位置x的函数关系. 实线是利用前面实验得到的耦合强度和腔模的损耗(参见图3)计算得到的, 圆点是实验测得的数据; (b)本征频率虚部在垫片数量分别为6, 4, 2的时候并分别用点划线、靛青和蓝色实线表示, 点划线显示了线性交叉行为; (c) 当系统耦合强度$ g_0/2{\text{π}}$ = 8.45 MHz 时, 本征频率虚部关于腔的端口处垫片数量y(腔模损耗)的关系. 实线是理论计算的结果, 品红色方块是实验结果

Figure 5. (a) Imaginary part of the eigenfrequencies as a function of the position x of the YIG sphere in the cavity with different number of gaskets. The solid curves are calculated using the coupling strengthes and the damping rates of the cavity modes obtained from the previous experiments (see Fig. 3), and the dots are obtained from the experimental data shown in Fig. 4; (b) imaginary part of the eigenfrequencies is plotted by black dotted line, indigo line and blue line when the number of gaskets are 6, 4 and 2, respectively. The black dotted line shows the linear crossing behavior; (c) when the coupling strength of the system is $ g_0/2{\text{π}}$ = 8.45 MHz, imaginary part of the eigenfrequencies are plotted as function of the gaskets y. The solid line is the result of theoretical calculation and the magenta square dots are the experimental results.

DownLoad: Full-Size Img PowerPoint

[1]	Xiang Z L, Ashhab S, You J Q, Nori F 2013 Rev. Mod. Phys. 85 623 Google Scholar
[2]	Kurizki G, Bertet P, Kubo Y, Mølmer K, Petrosyan D, Rabl P, Schmiedmayer J 2015 Proc. Natl. Acad. Sci. U.S.A. 112 3866 Google Scholar
[3]	Wallquist M, Hammerer K, Rabl P, Lukin M, Zoller P 2009 Phys. Scr. T137 014001 Google Scholar
[4]	Kimble H J 2008 Nature (London) 453 1023 Google Scholar
[5]	Huebl H, Zollitsch C W, Lotze J, Hocke F, Greifenstein M, Marx A, Gross R, Goennenwein S T B 2013 Phys. Rev. Lett. 111 127003 Google Scholar
[6]	Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K, Nakamura Y 2014 Phys. Rev. Lett. 113 083603 Google Scholar
[7]	Zhang X, Zou C L, Jiang L, Tang H X 2014 Phys. Rev. Lett. 113 156401 Google Scholar
[8]	Goryachev M, Farr W G, Creedon D L, Fan Y, Kostylev M, Tobar M E 2014 Phys. Rev. Appl. 2 054002 Google Scholar
[9]	Bai L, Harder M, Chen Y P, Fan X, Xiao J Q, Hu C M 2015 Phys. Rev. Lett. 114 227201 Google Scholar
[10]	Zhang D, Wang X M, Li T F, Luo X Q, Wu W, Nori F, You J Q 2015 npj Quantum Inf. 1 15014 Google Scholar
[11]	Soykal Ö O, Flatté M E 2010 Phys. Rev. Lett. 104 077202 Google Scholar
[12]	Rameshti B Z, Cao Y, Bauer G E W 2015 Phys. Rev. B 91 214430 Google Scholar
[13]	Cao Y, Yan P, Huebl H, Goennenwein S T B, Bauer G E W 2015 Phys. Rev. B 91 094423 Google Scholar
[14]	Bourhill J, Kostylev N, Goryachev M, Creedon D L, Tobar M E 2016 Phys. Rev. B 93 144420 Google Scholar
[15]	Kostylev N, Goryachev M, Tobar M E 2016 Appl. Phys. Lett. 108 062402 Google Scholar
[16]	Yao B M, Gui Y S, Xiao Y, Guo H, Chen X S, Lu W, Chien C L, Hu C M 2015 Phys. Rev. B 92 184407 Google Scholar
[17]	Zhang X, Zou C L, Jiang L, Tang H X 2016 Sci. Adv. 2 e1501286 Google Scholar
[18]	Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Usami K, Nakamura Y 2015 Science 349 405 Google Scholar
[19]	Quirion D L, Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Nakamura Y 2017 Sci. Adv. 3 e1603150 Google Scholar
[20]	Osada A, Hisatomi R, Noguchi A, Tabuchi Y, Yamazaki R, Usami K, Sadgrove M, Yalla R, Nomura M, Nakamura Y 2016 Phys. Rev. Lett. 116 223601 Google Scholar
[21]	Hisatomi R, Osada A, Tabuchi Y, Ishikawa T, Noguchi A, Yamazaki R, Usami K, Nakamura Y 2016 Phys. Rev. B 93 174427 Google Scholar
[22]	Haigh J A, Langenfeld S, Lambert N J, Baumberg J J, Ramsay A J, Nunnenkamp A, Ferguson A J 2015 Phys. Rev. A 92 063845 Google Scholar
[23]	Zhang X, Zhu N, Zou C L, Tang H X 2016 Phys. Rev. Lett. 117 123605 Google Scholar
[24]	Liu T, Zhang X, Tang H X, Flatté M E 2016 Phys. Rev. B 94 060405 Google Scholar
[25]	Wang Y P, Zhang G Q, Zhang D, Luo X Q, Xiong W, Wang S P, Li T F, Hu C M, You J Q 2016 Phys. Rev. B 94 224410 Google Scholar
[26]	Sharma S, Blanter Y M, Bauer G E W 2017 Phys. Rev. B 96 094412 Google Scholar
[27]	Sharma S, Blanter Y M, Bauer G E W 2018 Phys. Rev. Lett. 121 087205 Google Scholar
[28]	Osada A, Gloppe A, Hisatomi R, Noguchi A, Yamazaki R, Nomura M, Nakamura Y, Usami K 2018 Phys. Rev. Lett. 120 133602 Google Scholar
[29]	Liu Z X, Wang B, Xiong H, Wu Y 2018 Opt. Lett. 43 3698 Google Scholar
[30]	Zhang X, Zou C L, Zhu N, Marquardt F, Jiang L, Tang H X 2015 Nat. Commun. 6 8914 Google Scholar
[31]	Lambert N J, Haigh J A, Langenfeld S, Doherty A C, Ferguson A J 2016 Phys. Rev. A 93 021803 Google Scholar
[32]	Bai L, Harder M, Hyde P, Zhang Z, Hu C M, Chen Y P, Xiao J Q 2017 Phys. Rev. Lett. 118 217201 Google Scholar
[33]	Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M, You J Q 2018 Phys. Rev. Lett. 120 057202 Google Scholar
[34]	Hyde P, Yao B M, Gui Y S, Zhang G Q, You J Q, Hu C M 2018 Phys. Rev. B 98 174423 Google Scholar
[35]	Harder M, Yang Y, Yao B M, Yu C H, Rao J W, Gui Y S, Stamps R L, Hu C M 2018 Phys. Rev. Lett. 21 137203 Google Scholar
[36]	Grigoryan V L, Shen K, Xia K 2018 Phys. Rev. B 98 024406 Google Scholar
[37]	Bhoi B, Kim B, Jang S H, Kim J, Yang J, Cho Y J, Kim S K 2019 Phys. Rev. B 99 134426 Google Scholar
[38]	Yang Y, Rao J W, Gui Y S, Yao B M, Lu W, Hu C M 2019 Phys. Rev. Applied 11 054023 Google Scholar
[39]	Rao J W, Yu C H, Zhao Y T, Gui Y S, Fan X L, Xue D S, Hu C M 2019 New J. Phys. 21 065001 Google Scholar
[40]	Proskurin I, Macedo R, Stamps R L 2019 New J. Phys. 21 095003 Google Scholar
[41]	Yao B m, Yu T, Zhang X, Lu W, Gui Y S, Hu C M, Blanter Y M 2019 arXiv: 1906.12142 v2[cond-mat.mes-hall]
[42]	Yu W C, Wang J J, Yuan H Y, Xiao J 2019 Phys. Rev. Lett. 123 227201 Google Scholar
[43]	Wang Y P, Rao J W, Yang Y, Xu P C, Gui Y S, Yao B M, You J Q, Hu C M 2019 Phys. Rev. Lett. 123 127202 Google Scholar
[44]	Li J, Zhu S Y, Agarwal G S 2018 Phys. Rev. Lett. 121 203601 Google Scholar
[45]	Li J, Zhu S Y, Agarwal G S 2019 Phys. Rev. A 99 021801(R)Google Scholar
[46]	Bender C M 2007 Rep. Prog. Phys. 70 947 Google Scholar
[47]	陈增军, 宁西京 2003 物理学报 52 2683 Google Scholar Chen Z J, Ning X J 2003 Acta Phys. Sin. 52 2683 Google Scholar
[48]	Gilbert T L 2004 IEEE Trans. Magn. 40 3443 Google Scholar
[49]	Stancil D D, Prabhakar A 2009 Spin Waves Theory and Applications (New York: Springer) pp263–296
[50]	Zhang D, Luo X Q, Wang Y P, Li T F, You J Q 2017 Nat. Commun. 8 1368 Google Scholar
[51]	Dembowski C, Gräf H D, Harney H L, Heine A, Heiss W D, Rehfeld H, Richter A 2001 Phys. Rev. Lett. 86 787 Google Scholar
[52]	Dembowski C, Dietz B, Gräf H D, Harney H L, Heine A, Heiss W D, Richter A 2003 Phys. Rev. Lett. 90 034101 Google Scholar
[53]	Gao T, Estrecho E, Bliokh K Y, Liew T C H, Fraser M D, Brodbeck S, Kamp M, Schneider C, Hfling S, Yamamoto Y, Nori F, Kivshar Y S, Truscott A G, Dall R G, Ostrovskaya E A 2015 Nature (London) 526 554 Google Scholar
[54]	Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik A, Libisch F, Milburn T J, Rabl P, Moiseyev N, Rotter S 2016 Nature (London) 537 76 Google Scholar
[55]	Heiss W 2012 J. Phys. A 45 444016 Google Scholar
[56]	Ingrid R 2009 Journal Phys. A 42 153001
[57]	Eleuch H, Rotter I 2014 Eur. Phys. J. D 68 74 Google Scholar
[58]	Ding K, Ma G, Xiao M, Zhang Z Q, Chan C T 2016 Phys. Rev. X 6 021007 Google Scholar
[59]	Lu L, Joannopoulos J D, Soljacic M 2014 Nat. Photonics 8 821 Google Scholar
[60]	Zhou H, Peng C, Yoon Y, Hsu C W, Nelson K A, Fu L, Joannopoulos J D, Soljacic M, Zhen B 2018 Science 359 1009 Google Scholar
[61]	Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402 Google Scholar
[62]	Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature (London) 548 187 Google Scholar
[63]	Lin Z, Pick A, Loncar M, Rodriguez A W 2016 Phys. Rev. Lett. 117 107402 Google Scholar
[64]	Jing H, Özdemir S K, Lu H, Nori F 2017 Sci. Rep. 7 3386 Google Scholar
[65]	Nada M Y, Othman M A K, Capolino F 2017 Phys. Rev. B 96 184304 Google Scholar
[66]	Zhang G Q, You J Q 2019 Phys. Rev. B 99 054404 Google Scholar
[67]	Kato T 1966 Perturbation Theory for Linear Operators (Berlin: Springer-Verlag) p516
[68]	Ding K, Ma G, Zhang Z Q, Chan C T 2018 Phys. Rev. Lett. 121 085702 Google Scholar
[69]	Rotter I 2009 J. Phys. A 42 153001 Google Scholar
[70]	Eleuch H, Rotter I 2015 Eur. Phys. J. D 69 229 Google Scholar
[71]	Kittel C 1948 Phys. Rev. 73 155 Google Scholar

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Vol. 69, Issue 4 - 2020-02-20

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