Symmetry-Protected Scattering in Non-Hermitian Linear Systems
L. Jin* and Z. Song
School of Physics, Nankai University, Tianjin 300071, China
Abstract ：Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal ($C$ and $K$) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity ($Q$ symmetry) or the inversion ($P$) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symmetries, the symmetric features on the transmission and reflection interchange. The odd-parity symmetries including the particle-hole symmetry, chiral symmetry, and sublattice symmetry cannot ensure the scattering to be symmetric. These guiding principles are valid for both Hermitian and non-Hermitian linear systems. Our findings provide fundamental insights into symmetry and scattering ranging from condensed matter physics to quantum physics and optics.
收稿日期: 2020-12-01
出版日期: 2021-01-04
[1] Beenakker C W J 1997 Rev. Mod. Phys. 69 731 [2] Potton R J 2004 Rep. Prog. Phys. 67 717 [3] Sounas D L and Alù A 2017 Nat. Photon. 11 774 [4] Yu Z and Fan S 2009 Nat. Photon. 3 91 [5] Fan L, Wang J, Varghese L T, Shen H, Niu B, Xuan Y, Weiner A M and Qi M 2012 Science 335 447 [6] Chang L, Jiang X, Hua S, Yang C, Wen J, Jiang L, Li G, Wang G and Xiao M 2014 Nat. Photon. 8 524 [7] Li X Q, Zhang X Z, Zhang G and Song Z 2015 Phys. Rev. A 91 032101 [8] Jin L and Song Z 2018 Phys. Rev. Lett. 121 073901 [9] Du L, Zhang Y and Wu J H 2020 Sci. Rep. 10 1113 [10] Bi L, Hu J, Jiang P, Kim D H, Dionne G F, Kimerling L C and Ross C A 2011 Nat. Photon. 5 758 [11] Ramezani H, Lin Z, Kalish S, Kottos T, Kovanis V and Vitebskiy I 2012 Opt. Express 20 26200 [12] El-Ganainy R, Kumar P and Levy M 2013 Opt. Lett. 38 61 [13] El-Ganainy R, Eisfeld A, Levy M and Christodoulides D N 2013 Appl. Phys. Lett. 103 161105 [14] Ramezani H, Kalish S, Vitebskiy I and Kottos T 2014 Phys. Rev. Lett. 112 043904 [15] Lepri S and Casati G 2011 Phys. Rev. Lett. 106 164101 [16] Bender N, Factor S, Bodyfelt J D, Ramezani H, Christodoulides D N, Ellis F M and Kottos T 2013 Phys. Rev. Lett. 110 234101 [17] Shi Y, Yu Z and Fan S 2015 Nat. Photon. 9 388 [18] Huang R, Miranowicz A, Liao J Q, Nori F and Jing H 2018 Phys. Rev. Lett. 121 153601 [19] Sounas D L and Alù A 2017 Phys. Rev. Lett. 118 154302 [20] Fang K, Yu Z and Fan S 2012 Phys. Rev. Lett. 108 153901 [21] Fang K, Yu Z and Fan S 2012 Nat. Photon. 6 782 [22] Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M and Szameit A 2013 Nature 496 196 [23] Tzuang L D, Feng K, Nussenzveig P, Fan S and Lipson M 2014 Nat. Photon. 8 701 [24] Li E, Eggleton B J, Fang K and Fan S 2014 Nat. Commun. 5 3225 [25] Longhi S 2014 Opt. Lett. 39 5892 [26] Mukherjee S, Di Liberto M, Öhberg P, Thomson R R and Goldman N 2018 Phys. Rev. Lett. 121 075502 [27] Cooper N R, Dalibard J and Spielman I B 2019 Rev. Mod. Phys. 91 015005 [28] Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M, Schuster D, Simon J, Zilberberg O and Carusotto I 2019 Rev. Mod. Phys. 91 015006 [29] Lumer Y, Bandres M A, Heinrich M, Maczewsky L J, Herzig-Sheinfux H, Szameit A and Segev M 2019 Nat. Photon. 13 339 [30] Kremer M, Petrides I, Meyer E, Heinrich M, Zilberberg O and Szameit A 2020 Nat. Commun. 11 907 [31] Dutt A, Lin Q, Yuan L, Minkov M, Xiao M and Fan S 2020 Science 367 59 [32] Krasnok A, Baranov D, Li H N, Miri M A, Monticone F and Alù A 2019 Adv. Opt. Photon. 11 892 [33] Jalas D, Petrov A, Eich M, Freude W, Fan S, Yu Z, Baets R, Popović M, Melloni A, Joannopoulos J D, Vanwolleghem M, Doerr C R and Renner H 2013 Nat. Photon. 7 579 [34] Fan S, Baets R, Petrov A, Yu Z, Joannopoulos J D, Freude W, Melloni A, Popović M, Vanwolleghem M, Jalas D, Eich M, Krause M, Renner H, Brinkmeyer E and Doerr C R 2012 Science 335 38 [35] Yin X and Zhang X 2013 Nat. Mater. 12 175 [36] Chong Y D, Ge L, Cao H and Stone A D 2010 Phys. Rev. Lett. 105 053901 [37] Wan W, Chong Y, Ge L, Noh H, Stone A D and Cao H 2011 Science 331 889 [38] Baranov D G, Krasnok A, Shegai T, Alù A and Chong Y 2017 Nat. Rev. Mater. 2 17064 [39] Baranov D G, Krasnok A and Alù A 2017 Optica 4 1457 [40] Longhi S 2018 Opt. Lett. 43 2122 [41] Trainiti G, Ra'di Y, Ruzzene M and Alù A 2019 Sci. Adv. 5 eaaw3255 [42] Zhong Q, Simonson L, Kottos T and El-Ganainy R 2020 Phys. Rev. Res. 2 013362 [43] Burke P C, Wiersig J and Haque M 2020 Phys. Rev. A 102 012212 [44] Cannata F, Dedonder J P and Ventura A 2007 Ann. Phys. (N.Y.) 322 397 [45] Jin L, Zhang X Z, Zhang G and Song Z 2016 Sci. Rep. 6 20976 [46] Muga J G, Palao J P, Navarro B and Egusquiza I L 2004 Phys. Rep. 395 357 [47] Longhi S 2010 Phys. Rev. A 82 031801(R) [48] Longhi S 2011 J. Phys. A 44 485302 [49] Chong Y D, Ge L and Stone A D 2011 Phys. Rev. Lett. 106 093902 [50] Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H and Christodoulides D N 2011 Phys. Rev. Lett. 106 213901 [51] Feng L, Ayache M, Huang J, Xu Y L, Lu M H, Chen Y F, Fainman Y and Scherer A 2011 Science 333 729 [52] Kalish S, Lin Z and Kottos T 2012 Phys. Rev. A 85 055802 [53] Ge L, Chong Y D and Stone A D 2012 Phys. Rev. A 85 023802 [54] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167 [55] Feng L, Xu Y L, Fegadolli W S, Lu M H, Oliveira J E B, Almeida V R, Chen Y F and Scherer A 2013 Nat. Mater. 12 108 [56] Castaldi G, Savoia S, Galdi V, Alù A and Engheta N 2013 Phys. Rev. Lett. 110 173901 [57] Schomerus H 2013 Philos. Trans. R. Soc. A 371 20120194 [58] Ahmed Z 2013 Phys. Lett. A 377 957 [59] Ambichl P, Makris K G, Ge L, Chong Y, Stone A D and Rotter S 2013 Phys. Rev. X 3 041030 [60] Savoia S, Castaldi G and Galdi V 2014 Phys. Rev. B 89 085105 [61] Mostafazadeh A 2014 J. Phys. A 47 505303 [62] Wu J H, Artoni M and Rocca G C L 2014 Phys. Rev. Lett. 113 123004 [63] Ramezani H, Li H K, Wang Y and Zhang X 2014 Phys. Rev. Lett. 113 263905 [64] Fleury R, Sounas D and Alù A 2015 Nat. Commun. 6 5905 [65] Gear J, Liu F, Chu S T, Rotter S and Li J 2015 Phys. Rev. A 91 033825 [66] Huang Y, Veronis G and Min C 2015 Opt. Express 23 029882 [67] Miri M A, Eftekhar M A, Facao M, Abouraddy A F, Bakry A, Razvi M A N, Alshahrie A, Alù A and Christodoulides D N 2016 J. Opt. 18 075104 [68] Zhao H, Fegadolli W S, Yu J, Zhang Z, Ge L, Scherer A and Feng L 2016 Phys. Rev. Lett. 117 193901 [69] Wu J H, Artoni M and Rocca G C L 2016 Sci. Rep. 6 35356 [70] Ge L and Feng L 2016 Phys. Rev. A 94 043836 [71] Wong Z J, Xu Y, Kim J, Brien K O, Wang Y, Feng L and Zhang X 2016 Nat. Photon. 10 796 [72] Christensen J, Willatzen M, Velasco V R and Lu M H 2016 Phys. Rev. Lett. 116 207601 [73] Ramezani H, Wang Y, Yablonovitch E and Zhang X 2016 IEEE J. Sel. Top. Quantum Electron. 22 115 [74] Xiao S, Gear J, Rotter S and Li J 2016 New J. Phys. 18 085004 [75] Mostafazadeh A 2016 J. Phys. A 49 445302 [76] Gear J, Sun Y, Xiao S, Zhang L, Fitzgerald R, Rotter S, Chen H and Li J 2017 New J. Phys. 19 123041 [77] Wu J H, Artoni M and Rocca G C L 2017 Phys. Rev. A 95 053862 [78] Ge L and Feng L 2017 Phys. Rev. A 95 013813 [79] Kalozoumis P A, Morfonios C V, Kodaxis G, Diakonos F K and Schmelcher P 2017 Appl. Phys. Lett. 110 121106 [80] Zhang L L, Zhan G H, Li Z Z and Gong W J 2017 Phys. Rev. A 96 062133 [81] Jin L 2018 Phys. Rev. A 97 033840 [82] Luo J, Li J and Lai Y 2018 Phys. Rev. X 8 031035 [83] Gao F, Dong J R, Liu Y M 2018 Y. Zhang J. Opt. Soc. Am. B 35 2075 [84] Koutserimpas T T, Alù A and Fleury R 2018 Phys. Rev. A 97 013839 [85] Shen C, Li J, Peng X and Cummer S A 2018 Phys. Rev. Mater. 2 125203 [86] Wu N, Zhang C, Jin X R, Zhang Y Q and Lee Y P 2018 Opt. Express 26 3839 [87] Merkel A, Romero-García V, Groby J P, Li J and Christensen J 2018 Phys. Rev. B 98 201102(R) [88] Zhi Y, Yang X, Wu J, Du S, Cao P, Deng D and Liu C T 2018 Photon. Res. 6 579 [89] Wu H, Yang X, Tang Y, Tang X, Deng D, Liu H and Wei Z 2019 Ann. Phys. (Berlin) 531 1900120 [90] Simón M A, Buendía A, Kiely A, Mostafazadeh A and Muga J G 2019 Phys. Rev. A 99 052110 [91] Zhao Z, Guo C and Fan S 2019 Phys. Rev. A 99 033839 [92] Sweeney W R, Hsu C W, Rotter S and Stone A D 2019 Phys. Rev. Lett. 122 093901 [93] Sweeney W R, Hsu C W and Stone A D 2020 Phys. Rev. A 102 063511 [94] Novitsky A, Lyakhov D, Michels D, Pavlov A A, Shalin A S and Novitsky D V 2020 Phys. Rev. A 101 043834 [95] Zhu H, Yang X, Lin Z, Liu X, Yang X 2020 Opt. Commun. 459 124945 [96] Rotter S and Gigan S 2017 Rev. Mod. Phys. 89 015005 [97] Kawabata K, Higashikawa S, Gong Z, Ashida Y and Ueda M 2019 Nat. Commun. 10 297 [98] Kawabata K, Shiozaki K, Ueda M and Sato M 2019 Phys. Rev. X 9 041015 [99] Konotop V V, Yang J and Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 [100] Suchkov S V, Sukhorukov A A, Huang J, Dmitriev S V, Lee C and Kivshar Y S 2016 Laser & Photon. Rev. 10 177 [101] Feng L, El-Ganainy R and Ge L 2017 Nat. Photon. 11 752 [102] Longhi S 2017 Europhys. Lett. 120 64001 [103] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S and Christodoulides D N 2018 Nat. Phys. 14 11 [104] Miri M A and Alù A 2019 Science 363 eaar7709 [105] Özdemir S K, Rotter S, Nori F and Yang L 2019 Nat. Mater. 18 783 [106] Gupta S K, Zou Y, Zhu X Y, Lu M H, Zhang L J, Liu X P and Chen Y F 2019 Adv. Mater. 32 1903639 [107] Bernard D and LeClair A 2001 arXiv:cond-mat/0110649 [108] We emphasize two aspects in the definitions of symmetries that are distinct from the topological classification (Ref.[98]): (i) The signs $\epsilon _{\Bbbk }$ and $\epsilon _{q}$ in the $K$ and $Q$ symmetries, which are absent in the definitions of the $K$ and $Q$ symmetries for topological classification because $H_{\rm c}$ and $iH_{\rm c}$ have identical topological properties. However, the scattering properties of $H_{\rm c}$ and $iH_{\rm c}$ are not identical. The symmetries with opposite parities impose different scattering features. (ii) The even-parity $P$ symmetry. [109] The combination of two symmetries that belong to the same type gives the P symmetry. The combination of the P symmetry and any of the C, K, Q symmetries does not alter the C, K, Q symmetries. The combination of any two of the three symmetries C, K, Q produces the third symmetry. [110] Gou W, Chen T, Xie D, Xiao T, Deng T S, Gadway B, Yi W and Yan B 2020 Phys
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