Research progress of topological photonics

Wang Hong-Fei; Xie Bi-Ye; Zhan Peng; Lu Ming-Hui; Chen Yan-Feng

doi:10.7498/aps.68.20191437

Abstract

Inspired by topological phases and phase transitions in condensed matter, a new research field based on topological band theory, topological photonics, has emerged. It breaks through the traditional idea of light regulation by optical superposition principle of real space and energy band theory of solids of reciprocal space, providing a novel mechanism of optical regulation and rich properties of transport and light manipulation. Such as transmission properties of against backscattering and rubout to defects and disorders, selective transports dependent on spin-orbit coupling, and high dimensional manipulation of light. This review paper classifies different topological photonic systems by dimensions, briefly introducing the topological model, the novel physical phenomena, and the corresponding physical picture, such as SSH models, photonic quantum Hall effects, photonic quantum spin Hall effects, photonic Floquet topological insulator, and photonic three-dimensional topological insulator; other advanced platforms such as higher-order, non-Hermitian, and nonlinear topological platforms are also involved; a summary and outlook about the current development, advantages, and challenges of this field are present in the end.

Keywords:

topological phases in condensed matter /
topological photonics /
higher-order topological insulators /
non-Hermitian photonic systems

Authors and contacts

1.
National Laboratory of Solid State Microstructures, Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China
2.
School of Physics, Nanjing University, Nanjing 210093, China
3.
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
4.
Jiangsu Key Laboratory of Artificial Functional Materials, Nanjing 210093, China

Corresponding author: Lu Ming-Hui, luminghui@nju.edu.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2018YFA0306200, 2017YFA0303702), the National Natural Science Foundation of China (Grant Nos. 11474158, 51732006, 11890700), and the National Science Fund for Distinguished Young Scholars of China (Grant No.11625418)

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References

[1]	Haldane F D, Raghu S 2008 Phys. Rev. Lett. 100 013904 Google Scholar
[2]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905 Google Scholar
[3]	Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772 Google Scholar
[4]	Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljacic M 2015 Phys. Rev. Lett. 115 253901 Google Scholar
[5]	Cheng X, Jouvaud C, Ni X, Mousavi S H, Genack A Z, Khanikaev A B 2016 Nat. Mater. 15 542 Google Scholar
[6]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901 Google Scholar
[7]	Nalitov A V, Malpuech G, Tercas H, Solnyshkov D D 2015 Phys. Rev. Lett. 114 026803 Google Scholar
[8]	Yang Y, Xu Y F, Xu T, Wang H X, Jiang J H, Hu X, Hang Z H 2018 Phys. Rev. Lett. 120 217401 Google Scholar
[9]	Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233 Google Scholar
[10]	Nathan F, Abanin D, Berg E, Lindner N H, Rudner M S 2019 Phys. Rev. B 99 195133 Google Scholar
[11]	Leykam D, Rechtsman M C, Chong Y D 2016 Phys. Rev. Lett. 117 013902 Google Scholar
[12]	Titum P, Lindner N H, Rechtsman M C, Refael G 2015 Phys. Rev. Lett. 114 056801 Google Scholar
[13]	Leykam D, Chong Y D 2016 Phys. Rev. Lett. 117 143901 Google Scholar
[14]	Mukherjee S, Spracklen A, Valiente M, Andersson E, Ohberg P, Goldman N, Thomson R R 2017 Nat. Commun. 8 13918 Google Scholar
[15]	Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196 Google Scholar
[16]	Fang K, Yu Z, Fan S 2012 Nat. Photonics 6 782 Google Scholar
[17]	Lumer Y, Plotnik Y, Rechtsman M C, Segev M 2013 Phys. Rev. Lett. 111 243905 Google Scholar
[18]	Mukherjee S, Chandrasekharan H K, Ohberg P, Goldman N, Thomson R R 2018 Nat. Commun. 9 4209 Google Scholar
[19]	Zhu B, Zhong H, Ke Y, Qin X, Sukhorukov A A, Kivshar Y S, Lee C 2018 Phys. Rev. A 98 013855 Google Scholar
[20]	Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782 Google Scholar
[21]	Chen Y, Chen H, Cai G 2018 Appl. Phys. Lett. 112 013504 Google Scholar
[22]	Hafezi M, Lukin M D, Taylor J M 2013 New J. Phys. 15 063001 Google Scholar
[23]	Harder G, Bartley T J, Lita A E, Nam S W, Gerrits T, Silberhorn C 2016 Phys. Rev. Lett. 116 143601 Google Scholar
[24]	Barik S, Karasahin A, Flower C, Cai T, Miyake H, DeGottardi W, Hafezi M, Waks E 2018 Science 359 666 Google Scholar
[25]	Klitzing K v, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[26]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[27]	Kohmoto M 1985 Ann. Phys. 160 343 Google Scholar
[28]	Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106802 Google Scholar
[29]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[30]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[31]	Vocke D, Roger T, Marino F, Wright E M, Carusotto I, Clerici M, Faccio D 2015 Optica 2 484 Google Scholar
[32]	Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639 Google Scholar
[33]	Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907 Google Scholar
[34]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402 Google Scholar
[35]	Lee T E 2016 Phys. Rev. Lett. 116 133903 Google Scholar
[36]	Kawabata K, Shiozaki K, Ueda M 2018 Phys. Rev. B 98 165148 Google Scholar
[37]	Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808 Google Scholar
[38]	Wang H F, Gupta S K, Zhu X Y, Lu M H, Liu X P, Chen Y F 2018 Phys. Rev. B 98 214101 Google Scholar
[39]	Qi B, Zhang L, Ge L 2018 Phys. Rev. Lett. 120 093901 Google Scholar
[40]	Yao S, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802 Google Scholar
[41]	Feng L, El-Ganainy R, Ge L 2017 Nat. Photonics 11 752 Google Scholar
[42]	Midya B, Zhao H, Feng L 2018 Nat. Commun. 9 2674 Google Scholar
[43]	El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Nat. Phys. 14 11 Google Scholar
[44]	Zeuner J M, Rechtsman M C, Plotnik Y, Lumer Y, Nolte S, Rudner M S, Segev M, Szameit A 2015 Phys. Rev. Lett. 115 040402 Google Scholar
[45]	Gupta S K, Zou Y, Zhu X Y, Lu M H, Zhang L, Liu X P, Chen Y F 2018 arXiv preprint arXiv: 1803.00794
[46]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167 Google Scholar
[47]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[48]	Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402 Google Scholar
[49]	Lieu S 2018 Phys. Rev. B 97 045106 Google Scholar
[50]	Malzard S, Poli C, Schomerus H 2015 Phys. Rev. Lett. 115 200402 Google Scholar
[51]	Soljačić M, Joannopoulos J D 2004 Nat. Mater. 3 211 Google Scholar
[52]	Dobrykh D A, Yulin A V, Slobozhanyuk A P, Poddubny A N, Kivshar Y S 2018 Phys. Rev. Lett. 121 163901 Google Scholar
[53]	Haddad L H, Weaver C M, Carr L D 2015 New J. Phys. 17 063033 Google Scholar
[54]	Soljačić M, Luo C, Joannopoulos J D, Fan S 2003 Opt. Lett. 28 637 Google Scholar
[55]	Berger V 1998 Phys. Rev. Lett. 81 4136 Google Scholar
[56]	Adair R, Chase L L, Payne S A 1989 Phys. Rev. B 39 3337 Google Scholar
[57]	Fleischer J W, Segev M, Efremidis N K, Christodoulides D N 2003 Nature 422 147 Google Scholar
[58]	Zhou X, Wang Y, Leykam D, Chong Y D 2017 New J. Phys. 19 095002 Google Scholar
[59]	Rajesh C, Georgios T 2019 arXiv: 1904.09466 v1
[60]	Hadad Y, Khanikaev A B, Alù A 2016 Phys. Rev. B 93 155112 Google Scholar
[61]	Mingaleev S F, Kivshar Y S 2001 Phys. Rev. Lett. 86 5474 Google Scholar
[62]	Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W 2018 arXiv: 1812.08326
[63]	Hu H, Huang B, Zhao E, Liu W V 2019 arXiv: 1905.03727 v1
[64]	Ezawa M 2018 Phys. Rev. B 98 201402 Google Scholar
[65]	Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S, Bernevig B A, Neupert T 2018 Sci. Adv. 4 eaat0346 Google Scholar
[66]	Ezawa M 2018 Phys. Rev. Lett. 120 026801 Google Scholar
[67]	Khalaf E 2018 Phys. Rev. B 97 205136 Google Scholar
[68]	van Miert G, Ortix C 2018 Phys. Rev. B 98 081110 Google Scholar
[69]	Călugăru D, Juričić V, Roy B 2019 Phys. Rev. B 99 041301 Google Scholar
[70]	Kunst F K, van Miert G, Bergholtz E J 2018 Phys. Rev. B 97 241405 Google Scholar
[71]	Ezawa M 2018 Phys. Rev. B 97 155305 Google Scholar
[72]	Ezawa M 2018 Phys. Rev. B 98 045125 Google Scholar
[73]	Yasutomo O, Feng L, Ryota K, Katsuyuki W, Katsunori W, Yasuhiko A, Satoshi I 2018 arXiv: 1812.10171
[74]	Peterson C W, Benalcazar W A, Hughes T L, Bahl G 2018 Nature 555 346 Google Scholar
[75]	Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147 Google Scholar
[76]	Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903 Google Scholar
[77]	Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698 Google Scholar
[78]	Zak J 1989 Phys. Rev. Lett. 62 2747 Google Scholar
[79]	Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Opt. Lett. 34 1633 Google Scholar
[80]	Poli C, Bellec M, Kuhl U, Mortessagne F, Schomerus H 2015 Nat. Commun. 6 6710 Google Scholar
[81]	Keil R, Zeuner J M, Dreisow F, Heinrich M, Tunnermann A, Nolte S, Szameit A 2013 Nat. Commun. 4 1368
[82]	Schomerus H 2013 Opt. Lett. 38 1912 Google Scholar
[83]	Meier E J, An F A, Gadway B 2016 Nat. Commun. 7 13986 Google Scholar
[84]	Ling C W, Xiao M, Chan C T, Yu S F, Fung K H 2015 Opt. Express 23 2021 Google Scholar
[85]	Slobozhanyuk A P, Poddubny A N, Miroshnichenko A E, Belov P A, Kivshar Y S 2015 Phys. Rev. Lett. 114 123901 Google Scholar
[86]	Parto M, Wittek S, Hodaei H, Harari G, Bandres M A, Ren J, Rechtsman M C, Segev M, Christodoulides D N, Khajavikhan M 2018 Phys. Rev. Lett. 120 113901 Google Scholar
[87]	Zhao H, Miao P, Teimourpour M H, Malzard S, El-Ganainy R, Schomerus H, Feng L 2018 Nat. Commun. 9 981 Google Scholar
[88]	St-Jean P, Goblot V, Galopin E, Lemaître A, Ozawa T, Le Gratiet L, Sagnes I, Bloch J, Amo A 2017 Nat. Photonics 11 651 Google Scholar
[89]	Kitagawa T, Broome M A, Fedrizzi A, Rudner M S, Berg E, Kassal I, Aspuru-Guzik A, Demler E, White A G 2012 Nat. Commun. 3 882 Google Scholar
[90]	Tarasinski B, Asbóth J K, Dahlhaus J P 2014 Phys. Rev. A 89 042327 Google Scholar
[91]	Barkhofen S, Nitsche T, Elster F, Lorz L, Gábris A, Jex I, Silberhorn C 2017 Phys. Rev. A 96 033846 Google Scholar
[92]	Cardano F, D'Errico A, Dauphin A, Maffei M, Piccirillo B, de Lisio C, De Filippis G, Cataudella V, Santamato E, Marrucci L, Lewenstein M, Massignan P 2017 Nat. Commun. 8 15516 Google Scholar
[93]	Yannopapas V 2011 Phys. Rev. B 84 195126 Google Scholar
[94]	Minkov M, Savona V 2016 Optica 3 200 Google Scholar
[95]	Liu K, Shen L, He S 2012 Opt. Lett. 37 4110 Google Scholar
[96]	Umucalılar R O, Carusotto I 2011 Phys. Rev. A 84 043804 Google Scholar
[97]	He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. U S A 113 4924 Google Scholar
[98]	Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photonics 7 1001 Google Scholar
[99]	Pasek M, Chong Y D 2014 Phys. Rev. B 89 075113 Google Scholar
[100]	Lu L, Fu L, Joannopoulos J D, Soljačić M 2013 Nat. Photonics 7 294 Google Scholar
[101]	Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622 Google Scholar
[102]	Yang B, Guo Q, Tremain B, Liu R, Barr L E, Yan Q, Gao W, Liu H, Xiang Y, Chen J 2018 Science 359 1013 Google Scholar
[103]	Dubcek T, Kennedy C J, Lu L, Ketterle W, Soljacic M, Buljan H 2015 Phys. Rev. Lett. 114 225301 Google Scholar
[104]	Roy S, Kolodrubetz M, Goldman N, Grushin A G 2018 2D Mater. 5 024001
[105]	Gao W, Yang B, Lawrence M, Fang F, Beri B, Zhang S 2016 Nat. Commun. 7 12435 Google Scholar
[106]	Xiao M, Lin Q, Fan S 2016 Phys. Rev. Lett. 117 057401 Google Scholar
[107]	Lin Q, Xiao M, Yuan L, Fan S 2016 Nat. Commun. 7 13731 Google Scholar
[108]	Kawakami T, Hu X 2017 Phys. Rev. B 96 235307 Google Scholar
[109]	Yan Q, Liu R, Yan Z, Liu B, Chen H, Wang Z, Lu L 2018 Nat. Phys. 14 461 Google Scholar
[110]	Yan Z, Bi R, Shen H, Lu L, Zhang S C, Wang Z 2017 Phys. Rev. B 96 041103 Google Scholar
[111]	Bi R, Yan Z, Lu L, Wang Z 2017 Phys. Rev. B 96 201305 Google Scholar
[112]	Lu L, Gao H, Wang Z 2018 Nat. Commun. 9 5384 Google Scholar
[113]	Slobozhanyuk A P, Khanikaev A B, Filonov D S, Smirnova D A, Miroshnichenko A E, Kivshar Y S 2016 Sci. Rep. 6 22270 Google Scholar
[114]	Slobozhanyuk A, Mousavi S H, Ni X, Smirnova D, Kivshar Y S, Khanikaev A B 2016 Nat. Photonics 11 130
[115]	Yang Y, Gao Z, Xue H, Zhang L, He M, Yang Z, Singh R, Chong Y, Zhang B, Chen H 2019 Nature 565 622 Google Scholar
[116]	Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljačić M 2016 Nat. Phys. 12 337 Google Scholar
[117]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[118]	Ochiai T 2017 Phys. Rev. A 96 043842 Google Scholar
[119]	Xu Y, Wang S T, Duan L M 2016 Phys. Rev. Lett. 118 045701
[120]	Zhen B, Hsu C W, Igarashi Y, Lu L, Kaminer I, Pick A, Chua S L, Joannopoulos J D, Soljacic M 2015 Nature 525 354 Google Scholar
[121]	Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C, Szameit A 2017 Nat. Mater. 16 433 Google Scholar
[122]	Zhao H, Qiao X, Wu T, Midya B, Longhi S, Feng L 2019 Science 365 1163 Google Scholar
[123]	Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079
[124]	Martinez Alvarez V M, Barrios Vargas J E, Berdakin M, Foa Torres L E F 2018 Eur. Phys. J. Spec. Top. 227 1295 Google Scholar
[125]	Cerjan A, Raman A, Fan S 2016 Phys. Rev. Lett. 116 203902 Google Scholar
[126]	Pan M, Zhao H, Miao P, Longhi S, Feng L 2018 Nat. Commun. 9 1308 Google Scholar
[127]	Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803 Google Scholar
[128]	Kleinman D A 1962 Phys. Rev. 126 1977 Google Scholar
[129]	Adler E 1964 Phys. Rev. 134 A728 Google Scholar
[130]	Lumer Y, Plotnik Y, Rechtsman M C, Segev M 2013 Phys. Rev. Lett. 111 263901 Google Scholar
[131]	Jia N, Schine N, Georgakopoulos A, Ryou A, Clark L W, Sommer A, Simon J 2018 Nat. Phys. 14 550 Google Scholar
[132]	Roushan P, Neill C, Megrant A, Chen Y, Babbush R, Barends R, Campbell B, Chen Z, Chiaro B, Dunsworth A, Fowler A, Jeffrey E, Kelly J, Lucero E, Mutus J, O’Malley P J J, Neeley M, Quintana C, Sank D, Vainsencher A, Wenner J, White T, Kapit E, Neven H, Martinis J 2016 Nat. Phys. 13 146
[133]	Reinhard A, Volz T, Winger M, Badolato A, Hennessy K J, Hu E L, Imamoğlu A 2011 Nat. Photonics 6 93
[134]	Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Dan B, Preiss P M, Grusdt F, Kaufman A M, Greiner M 2017 Nature 546 519 Google Scholar
[135]	Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W 2019 Phys. Rev. Lett. 122 233902 Google Scholar
[136]	Ota Y, Liu F, Katsumi R, Watanabe K, Wakabayashi K, Arakawa Y, Iwamoto S 2019 Optica 6 786 Google Scholar

Cited By

图 1 SSH模型示意图, 每个元胞包含两个格点

Figure 1. Schematic of the SSH model, there are two sites in each unit cell.

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图 2 (a) 微纳加工(SSH 模型)的SEM图; (b) 单个柱子的模式; (c) 不同能带中的态及存在的边界态; (d) 利用波导环形阵列实现SSH模型

Figure 2. (a) SEM image of the coupled micropillars; (b) Modes of single micropillars; (c) Different modes of the micropillar array and edge states; (d) SSH microring array.

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图 3 (a) 旋磁光子晶体的示意图; (b) 向前向后的传输谱以及具有手性边界态的投影能带; (c) 大陈数光子晶体结构图; (d) 能带的带隙及其陈数

Figure 3. (a) Schematic of the gyromagnetic photonic crystal; (b) forward and backward spectra, and projected band structures with chiral edge states; (c) the diagram of large Chern number photonic crystals; (d) the band gap map and their Chern number.

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图 4 (a) Poincaré球上的LCP和RCP, 以及由PE和PM材料构成的光子晶体; (b) 没有赝自旋耦合以及具有赝自旋耦合的能带以及后者的投影能带; (c) 通过调节金属柱子实现赝自旋的耦合

Figure 4. (a) The polarization of LCP and RCP on the Poincaré sphere, and the photonic crystal consisting of PE and PM superlattices; (b) band structures without coupling between dseudospin states and with their coupling, and the projected band structures for the latter case; (c) photonic crystals consisting of metallic rods and collars at different positions, and their band strucutres.

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图 5 (a) 全介质光子晶体结构; (b) 收缩、高对称以及扩张晶格所对应的能带; (c) 赝自旋依赖的边界态的实验观测

Figure 5. (a) Schematic of all-dielectric photonic crystals; (b) band structures of shrinking and expanding lattices; (c) visualization of pseudospin-dependent edge states.

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图 6 (a) 谐振腔耦合单元; (b) 周期排布形成的耦合阵列

Figure 6. (a) Two coupled resonators in one unit cell; (b) a periodic array arranged by unit cells.

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图 7 (a) 光学谐振腔阵列的动态调制; (b) 激光直写波导系统的拓扑绝缘体构型; (c) 四种耦合组成的周期构型

Figure 7. (a) The resonator lattice with dynamic modulation; (b) floquet topological insulators using the femtosecond laser writing method; (c) four different bonds with different coupling.

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图 8 (a) 能够产生Weyl点以及节线的双螺旋光子晶体; (b) 具有Weyl点的金属夹杂的光子晶体

Figure 8. (a) Photonic crystals with two gyroid structures in one unit cell, and their band structures with Weyl points or nodal-line; (b) schematic of photonic crystals with the saddle-shaped metallic inclusion, and their Weyl points.

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图 9 (a) 三维全介质与双各向异性光子晶体; (b) 两种构型的光子晶体对应的能带; (c) 通过引入磁场破缺Dirac点的光子晶体构型

Figure 9. (a) 3 D all-dielectric and bianisotropic metacrystals; (b) band structures corresponding to two structures in (a); (c) photonic crystals with opened Dirac points when magnetization is applied on rods.

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图 10 (a) 动量空间中的奇异点以及具有增益损耗的紧束缚模型; (b) 具有增益损耗的波导阵列; (c) 具有奇异环的光子晶体板结构

Figure 10. (a) Exceptional points in momentum space, and the tight-binding model with gain and loss for α_i and β_i; (b) the waveguide array with gain and loss; (c) photonic crystal slabs with the ring of exceptional points.

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图 11 (a) 非线性SSH模型; (b) 与光强度相关的环绕数(贝利相位); (c) 将量子比特与它们的耦合器铺成二维格子的示意图; (d) 包含三个超导量子比特的超导回路

Figure 11. (a) The nonlinear SSH model; (b) the winding number (Berry phase) changed by intensity; (c) schematic diagram of qubits and their couplers in 2 D lattice; (d) the superconducting circuit including three qubits.

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图 12 (a) 介质柱构成的二维SSH模型的光子晶体; (b) 收缩、高对称与扩张晶格构型的能带结构; (c) 由收缩区域包围扩张区域构成的整体结构, 解的序号与本征频率的关系; (d) 实验中放于一个角的源激发的拐角态

Figure 12. (a) Photonic crystals of the 2D SSH model consisting of dielectric pillars; (b) band structures of shrinking, high symmetry and expending structures; (c) shrinking supercells contain expanding supercells, and the relationship between solution numbers and eigenfrequencies; (d) experimentally measured corner states when the source is placed at the corner.

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[1]	Haldane F D, Raghu S 2008 Phys. Rev. Lett. 100 013904 Google Scholar
[2]	Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905 Google Scholar
[3]	Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772 Google Scholar
[4]	Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljacic M 2015 Phys. Rev. Lett. 115 253901 Google Scholar
[5]	Cheng X, Jouvaud C, Ni X, Mousavi S H, Genack A Z, Khanikaev A B 2016 Nat. Mater. 15 542 Google Scholar
[6]	Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901 Google Scholar
[7]	Nalitov A V, Malpuech G, Tercas H, Solnyshkov D D 2015 Phys. Rev. Lett. 114 026803 Google Scholar
[8]	Yang Y, Xu Y F, Xu T, Wang H X, Jiang J H, Hu X, Hang Z H 2018 Phys. Rev. Lett. 120 217401 Google Scholar
[9]	Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233 Google Scholar
[10]	Nathan F, Abanin D, Berg E, Lindner N H, Rudner M S 2019 Phys. Rev. B 99 195133 Google Scholar
[11]	Leykam D, Rechtsman M C, Chong Y D 2016 Phys. Rev. Lett. 117 013902 Google Scholar
[12]	Titum P, Lindner N H, Rechtsman M C, Refael G 2015 Phys. Rev. Lett. 114 056801 Google Scholar
[13]	Leykam D, Chong Y D 2016 Phys. Rev. Lett. 117 143901 Google Scholar
[14]	Mukherjee S, Spracklen A, Valiente M, Andersson E, Ohberg P, Goldman N, Thomson R R 2017 Nat. Commun. 8 13918 Google Scholar
[15]	Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196 Google Scholar
[16]	Fang K, Yu Z, Fan S 2012 Nat. Photonics 6 782 Google Scholar
[17]	Lumer Y, Plotnik Y, Rechtsman M C, Segev M 2013 Phys. Rev. Lett. 111 243905 Google Scholar
[18]	Mukherjee S, Chandrasekharan H K, Ohberg P, Goldman N, Thomson R R 2018 Nat. Commun. 9 4209 Google Scholar
[19]	Zhu B, Zhong H, Ke Y, Qin X, Sukhorukov A A, Kivshar Y S, Lee C 2018 Phys. Rev. A 98 013855 Google Scholar
[20]	Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782 Google Scholar
[21]	Chen Y, Chen H, Cai G 2018 Appl. Phys. Lett. 112 013504 Google Scholar
[22]	Hafezi M, Lukin M D, Taylor J M 2013 New J. Phys. 15 063001 Google Scholar
[23]	Harder G, Bartley T J, Lita A E, Nam S W, Gerrits T, Silberhorn C 2016 Phys. Rev. Lett. 116 143601 Google Scholar
[24]	Barik S, Karasahin A, Flower C, Cai T, Miyake H, DeGottardi W, Hafezi M, Waks E 2018 Science 359 666 Google Scholar
[25]	Klitzing K v, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[26]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[27]	Kohmoto M 1985 Ann. Phys. 160 343 Google Scholar
[28]	Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106802 Google Scholar
[29]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[30]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[31]	Vocke D, Roger T, Marino F, Wright E M, Carusotto I, Clerici M, Faccio D 2015 Optica 2 484 Google Scholar
[32]	Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639 Google Scholar
[33]	Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907 Google Scholar
[34]	Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402 Google Scholar
[35]	Lee T E 2016 Phys. Rev. Lett. 116 133903 Google Scholar
[36]	Kawabata K, Shiozaki K, Ueda M 2018 Phys. Rev. B 98 165148 Google Scholar
[37]	Kunst F K, Edvardsson E, Budich J C, Bergholtz E J 2018 Phys. Rev. Lett. 121 026808 Google Scholar
[38]	Wang H F, Gupta S K, Zhu X Y, Lu M H, Liu X P, Chen Y F 2018 Phys. Rev. B 98 214101 Google Scholar
[39]	Qi B, Zhang L, Ge L 2018 Phys. Rev. Lett. 120 093901 Google Scholar
[40]	Yao S, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802 Google Scholar
[41]	Feng L, El-Ganainy R, Ge L 2017 Nat. Photonics 11 752 Google Scholar
[42]	Midya B, Zhao H, Feng L 2018 Nat. Commun. 9 2674 Google Scholar
[43]	El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Nat. Phys. 14 11 Google Scholar
[44]	Zeuner J M, Rechtsman M C, Plotnik Y, Lumer Y, Nolte S, Rudner M S, Segev M, Szameit A 2015 Phys. Rev. Lett. 115 040402 Google Scholar
[45]	Gupta S K, Zou Y, Zhu X Y, Lu M H, Zhang L, Liu X P, Chen Y F 2018 arXiv preprint arXiv: 1803.00794
[46]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167 Google Scholar
[47]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[48]	Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402 Google Scholar
[49]	Lieu S 2018 Phys. Rev. B 97 045106 Google Scholar
[50]	Malzard S, Poli C, Schomerus H 2015 Phys. Rev. Lett. 115 200402 Google Scholar
[51]	Soljačić M, Joannopoulos J D 2004 Nat. Mater. 3 211 Google Scholar
[52]	Dobrykh D A, Yulin A V, Slobozhanyuk A P, Poddubny A N, Kivshar Y S 2018 Phys. Rev. Lett. 121 163901 Google Scholar
[53]	Haddad L H, Weaver C M, Carr L D 2015 New J. Phys. 17 063033 Google Scholar
[54]	Soljačić M, Luo C, Joannopoulos J D, Fan S 2003 Opt. Lett. 28 637 Google Scholar
[55]	Berger V 1998 Phys. Rev. Lett. 81 4136 Google Scholar
[56]	Adair R, Chase L L, Payne S A 1989 Phys. Rev. B 39 3337 Google Scholar
[57]	Fleischer J W, Segev M, Efremidis N K, Christodoulides D N 2003 Nature 422 147 Google Scholar
[58]	Zhou X, Wang Y, Leykam D, Chong Y D 2017 New J. Phys. 19 095002 Google Scholar
[59]	Rajesh C, Georgios T 2019 arXiv: 1904.09466 v1
[60]	Hadad Y, Khanikaev A B, Alù A 2016 Phys. Rev. B 93 155112 Google Scholar
[61]	Mingaleev S F, Kivshar Y S 2001 Phys. Rev. Lett. 86 5474 Google Scholar
[62]	Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W 2018 arXiv: 1812.08326
[63]	Hu H, Huang B, Zhao E, Liu W V 2019 arXiv: 1905.03727 v1
[64]	Ezawa M 2018 Phys. Rev. B 98 201402 Google Scholar
[65]	Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S, Bernevig B A, Neupert T 2018 Sci. Adv. 4 eaat0346 Google Scholar
[66]	Ezawa M 2018 Phys. Rev. Lett. 120 026801 Google Scholar
[67]	Khalaf E 2018 Phys. Rev. B 97 205136 Google Scholar
[68]	van Miert G, Ortix C 2018 Phys. Rev. B 98 081110 Google Scholar
[69]	Călugăru D, Juričić V, Roy B 2019 Phys. Rev. B 99 041301 Google Scholar
[70]	Kunst F K, van Miert G, Bergholtz E J 2018 Phys. Rev. B 97 241405 Google Scholar
[71]	Ezawa M 2018 Phys. Rev. B 97 155305 Google Scholar
[72]	Ezawa M 2018 Phys. Rev. B 98 045125 Google Scholar
[73]	Yasutomo O, Feng L, Ryota K, Katsuyuki W, Katsunori W, Yasuhiko A, Satoshi I 2018 arXiv: 1812.10171
[74]	Peterson C W, Benalcazar W A, Hughes T L, Bahl G 2018 Nature 555 346 Google Scholar
[75]	Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147 Google Scholar
[76]	Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903 Google Scholar
[77]	Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698 Google Scholar
[78]	Zak J 1989 Phys. Rev. Lett. 62 2747 Google Scholar
[79]	Malkova N, Hromada I, Wang X, Bryant G, Chen Z 2009 Opt. Lett. 34 1633 Google Scholar
[80]	Poli C, Bellec M, Kuhl U, Mortessagne F, Schomerus H 2015 Nat. Commun. 6 6710 Google Scholar
[81]	Keil R, Zeuner J M, Dreisow F, Heinrich M, Tunnermann A, Nolte S, Szameit A 2013 Nat. Commun. 4 1368
[82]	Schomerus H 2013 Opt. Lett. 38 1912 Google Scholar
[83]	Meier E J, An F A, Gadway B 2016 Nat. Commun. 7 13986 Google Scholar
[84]	Ling C W, Xiao M, Chan C T, Yu S F, Fung K H 2015 Opt. Express 23 2021 Google Scholar
[85]	Slobozhanyuk A P, Poddubny A N, Miroshnichenko A E, Belov P A, Kivshar Y S 2015 Phys. Rev. Lett. 114 123901 Google Scholar
[86]	Parto M, Wittek S, Hodaei H, Harari G, Bandres M A, Ren J, Rechtsman M C, Segev M, Christodoulides D N, Khajavikhan M 2018 Phys. Rev. Lett. 120 113901 Google Scholar
[87]	Zhao H, Miao P, Teimourpour M H, Malzard S, El-Ganainy R, Schomerus H, Feng L 2018 Nat. Commun. 9 981 Google Scholar
[88]	St-Jean P, Goblot V, Galopin E, Lemaître A, Ozawa T, Le Gratiet L, Sagnes I, Bloch J, Amo A 2017 Nat. Photonics 11 651 Google Scholar
[89]	Kitagawa T, Broome M A, Fedrizzi A, Rudner M S, Berg E, Kassal I, Aspuru-Guzik A, Demler E, White A G 2012 Nat. Commun. 3 882 Google Scholar
[90]	Tarasinski B, Asbóth J K, Dahlhaus J P 2014 Phys. Rev. A 89 042327 Google Scholar
[91]	Barkhofen S, Nitsche T, Elster F, Lorz L, Gábris A, Jex I, Silberhorn C 2017 Phys. Rev. A 96 033846 Google Scholar
[92]	Cardano F, D'Errico A, Dauphin A, Maffei M, Piccirillo B, de Lisio C, De Filippis G, Cataudella V, Santamato E, Marrucci L, Lewenstein M, Massignan P 2017 Nat. Commun. 8 15516 Google Scholar
[93]	Yannopapas V 2011 Phys. Rev. B 84 195126 Google Scholar
[94]	Minkov M, Savona V 2016 Optica 3 200 Google Scholar
[95]	Liu K, Shen L, He S 2012 Opt. Lett. 37 4110 Google Scholar
[96]	Umucalılar R O, Carusotto I 2011 Phys. Rev. A 84 043804 Google Scholar
[97]	He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. U S A 113 4924 Google Scholar
[98]	Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photonics 7 1001 Google Scholar
[99]	Pasek M, Chong Y D 2014 Phys. Rev. B 89 075113 Google Scholar
[100]	Lu L, Fu L, Joannopoulos J D, Soljačić M 2013 Nat. Photonics 7 294 Google Scholar
[101]	Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622 Google Scholar
[102]	Yang B, Guo Q, Tremain B, Liu R, Barr L E, Yan Q, Gao W, Liu H, Xiang Y, Chen J 2018 Science 359 1013 Google Scholar
[103]	Dubcek T, Kennedy C J, Lu L, Ketterle W, Soljacic M, Buljan H 2015 Phys. Rev. Lett. 114 225301 Google Scholar
[104]	Roy S, Kolodrubetz M, Goldman N, Grushin A G 2018 2D Mater. 5 024001
[105]	Gao W, Yang B, Lawrence M, Fang F, Beri B, Zhang S 2016 Nat. Commun. 7 12435 Google Scholar
[106]	Xiao M, Lin Q, Fan S 2016 Phys. Rev. Lett. 117 057401 Google Scholar
[107]	Lin Q, Xiao M, Yuan L, Fan S 2016 Nat. Commun. 7 13731 Google Scholar
[108]	Kawakami T, Hu X 2017 Phys. Rev. B 96 235307 Google Scholar
[109]	Yan Q, Liu R, Yan Z, Liu B, Chen H, Wang Z, Lu L 2018 Nat. Phys. 14 461 Google Scholar
[110]	Yan Z, Bi R, Shen H, Lu L, Zhang S C, Wang Z 2017 Phys. Rev. B 96 041103 Google Scholar
[111]	Bi R, Yan Z, Lu L, Wang Z 2017 Phys. Rev. B 96 201305 Google Scholar
[112]	Lu L, Gao H, Wang Z 2018 Nat. Commun. 9 5384 Google Scholar
[113]	Slobozhanyuk A P, Khanikaev A B, Filonov D S, Smirnova D A, Miroshnichenko A E, Kivshar Y S 2016 Sci. Rep. 6 22270 Google Scholar
[114]	Slobozhanyuk A, Mousavi S H, Ni X, Smirnova D, Kivshar Y S, Khanikaev A B 2016 Nat. Photonics 11 130
[115]	Yang Y, Gao Z, Xue H, Zhang L, He M, Yang Z, Singh R, Chong Y, Zhang B, Chen H 2019 Nature 565 622 Google Scholar
[116]	Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljačić M 2016 Nat. Phys. 12 337 Google Scholar
[117]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[118]	Ochiai T 2017 Phys. Rev. A 96 043842 Google Scholar
[119]	Xu Y, Wang S T, Duan L M 2016 Phys. Rev. Lett. 118 045701
[120]	Zhen B, Hsu C W, Igarashi Y, Lu L, Kaminer I, Pick A, Chua S L, Joannopoulos J D, Soljacic M 2015 Nature 525 354 Google Scholar
[121]	Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C, Szameit A 2017 Nat. Mater. 16 433 Google Scholar
[122]	Zhao H, Qiao X, Wu T, Midya B, Longhi S, Feng L 2019 Science 365 1163 Google Scholar
[123]	Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079
[124]	Martinez Alvarez V M, Barrios Vargas J E, Berdakin M, Foa Torres L E F 2018 Eur. Phys. J. Spec. Top. 227 1295 Google Scholar
[125]	Cerjan A, Raman A, Fan S 2016 Phys. Rev. Lett. 116 203902 Google Scholar
[126]	Pan M, Zhao H, Miao P, Longhi S, Feng L 2018 Nat. Commun. 9 1308 Google Scholar
[127]	Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803 Google Scholar
[128]	Kleinman D A 1962 Phys. Rev. 126 1977 Google Scholar
[129]	Adler E 1964 Phys. Rev. 134 A728 Google Scholar
[130]	Lumer Y, Plotnik Y, Rechtsman M C, Segev M 2013 Phys. Rev. Lett. 111 263901 Google Scholar
[131]	Jia N, Schine N, Georgakopoulos A, Ryou A, Clark L W, Sommer A, Simon J 2018 Nat. Phys. 14 550 Google Scholar
[132]	Roushan P, Neill C, Megrant A, Chen Y, Babbush R, Barends R, Campbell B, Chen Z, Chiaro B, Dunsworth A, Fowler A, Jeffrey E, Kelly J, Lucero E, Mutus J, O’Malley P J J, Neeley M, Quintana C, Sank D, Vainsencher A, Wenner J, White T, Kapit E, Neven H, Martinis J 2016 Nat. Phys. 13 146
[133]	Reinhard A, Volz T, Winger M, Badolato A, Hennessy K J, Hu E L, Imamoğlu A 2011 Nat. Photonics 6 93
[134]	Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Dan B, Preiss P M, Grusdt F, Kaufman A M, Greiner M 2017 Nature 546 519 Google Scholar
[135]	Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W 2019 Phys. Rev. Lett. 122 233902 Google Scholar
[136]	Ota Y, Liu F, Katsumi R, Watanabe K, Wakabayashi K, Arakawa Y, Iwamoto S 2019 Optica 6 786 Google Scholar

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