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Many interesting phenomena, such as quantization of Landau levels and quantum Hall effect, can occur in an electronic system under a strong magnetic field. However, photons do not carry charge, and they do not have many properties induced by external magnetic fields, either. Recently, the pseudomagnetic field, an artificial synthetic gauge field, has attracted intense research interest in classical wave systems, in which the propagation of the wave can be manipulated like in a real magnetic field. The photonic crystal is an optical structure composed of periodic material distributions and provides a good platform for studying the control of electromagnetic waves. In this work, we construct a uniform pseudomagnetic field by introducing uniaxial linear gradient deformation of metallic rods in a two-dimensional photonic crystal. The strong pseudomagnetic field leads to the quantization of photonic Landau levels in photonic crystal. The sublattice polarization of n = 0 Landau level is also demonstrated in our simulations. Unlike the real magnetic field, the pseudomagnetic fields of photonic crystal is opposite in two inequivalent energy valleys, and the time-reversal symmetry of the system is not broken. Our designed gradient photonic crystals support the transport of edge state in the gap between n = 0 and n = ±1 Landau levels. The edge state can propagate unidirectionally when it is excited by a chiral source. When a gaussian beam impinges on the photonic crystal, the propagating paths of two splitting beams can be controlled, which gives rise to the bend of two beams. Two photonic crystals with opposite pseudomagnetic fields are assembled together, and the interesting phenomenon of “snake-state” can be obtained. Our proposal opens the way for designing information processing devices by manipulating electromagnetic waves.
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Keywords:
- photonic crystal /
- pseudomagnetic field /
- Landau level /
- edge state
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图 1 (a) 二维三角晶格光子晶体原胞示意图; (b)狄拉克点的偏移量与(q – p)/p呈线性关系; (c)—(e) 分别为当q = 5, 6和7 mm时光子晶体的能带图
Figure 1. (a) Schematic of a unit cell of a two-dimensional triangular photonic crystal; (b) linear relation between the shift of Dirac point and (p – q)/p; (c)–(e) band diagram of photonic crystal for q = 5, 6 and 7 mm, respectively.
图 2 朗道能级量子化 (a)梯度光子晶体结构示意图; (b)光子晶体的本征模式电场分布图; (c), (d) 在y方向上有37层金属柱的光子晶体的投影能带; (e), (f) 分别为y方向有19层和91层金属柱的光子晶体的投影能带
Figure 2. Landau quantization: (a) Schematic of a gradient photonic crystal; (b) electric field distribution of eigenmode of photonic crystal; (c), (d) projected band of a photonic crystal with 37 layers along y direction; (e), (f) projected band of photonic crystals with 19 and 91 layers, respectively.
图 3 n = 0朗道能级的子晶格极化 (a), (b) 激发源分别放在梯度光子晶体A和B子晶格位置时电场积分频谱; (c), (d)激发源在A子晶格时的电场分布; (e), (f) 激发源在B子晶格时的电场分布
Figure 3. Sublattice polarization of n = 0 Landau level: (a), (b) Integration of electric field when the excitation source is placed at A and B sublattices of a gradient photonic crystal; (c), (d) the electric field distribution for the excitation source at A sublattice; (e), (f) electric field distribution for the excitation source at B sublattice.
图 4 (a)—(d) 梯度光子晶体中下边界处的边界态传输情况; (e), (f) 上边界处边界态在不同频率时的传输情况; (g), (h) 手性激发源激发单向传输的边界态
Figure 4. (a)–(d) Transport of edge state on the bottom side of a gradient photonic crystal; (e), (f) edge state on the upper side at different frequencies; (g), (h) unidirectional transport of edge state excited by a chiral source.
图 5 (a) 光子晶体(金属圆柱)中的光束分裂; (b) 梯度光子晶体中的赝磁场导致分裂的两束波呈向内弯曲现象; (c)减小光子晶体中的赝磁场, 波束弯曲效应减弱; (d) 反转光子晶体的梯度变化, 两束波都向外弯折
Figure 5. (a) Beam splitting in a photonic crystal composed of circular metallic rods; (b) bend of two beams in a gradient photonic crystal caused by pseudomagnetic field; (c) weakness of bending wave beam in decreasing pseudomagnetic field; (d) bending wave beams in inverted gradient photonic crystal.
图 6 (a)具有相反赝磁场的光子晶体结构示意图; (b), (c)具有过渡区的光子晶体投影能带; (d) 频率为8.84 GHz时“蛇态”的电场分布
Figure 6. (a) Schematic diagram of two photonic crystal structures with opposite pseudomagnetic fields; (b), (c) projected bands of gradient photonic crystals with transition regions; (d) electric field distribution of snake state at 8.84 GHz.
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[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
Google Scholar
[2] Thouless D J, Kohmoto M, Nightingale M P, Den Nijs M 1982 Phys. Rev. Lett. 49 405
Google Scholar
[3] Guinea F, Katsnelson M I, Geim A K 2010 Nat. Phys. 6 30
Google Scholar
[4] Levy N, Burke S A, Meaker K L, Panlasigui M, Zettl A, Guinea F, Castro Neto A H, Crommie M F 2010 Science 329 544
Google Scholar
[5] Salerno G, Ozawa T, Price H M, Carusotto I 2015 2D Mater. 2 034015
Google Scholar
[6] Salerno G, Ozawa T, Price H M, Carusotto I 2017 Phys. Rev. B 95 245418
Google Scholar
[7] Lantagne-Hurtubise É, Zhang X X, Franz M 2020 Phys. Rev. B 101 085423
Google Scholar
[8] Wu B L, Wei Q, Zhang Z Q, Jiang H 2021 Chin. Phys. B 30 030504
Google Scholar
[9] Brendel C, Peano V, Painter O J, Marquardt F 2017 Proc. Natl. Acad. Sci. 114 3390
[10] Abbaszadeh H, Souslov A, Paulose J, Schomerus H 2017 Phys. Rev. Lett. 119 195502
Google Scholar
[11] Yang Z J, Gao F, Yang Y H, Zhang B L 2017 Phys. Rev. Lett. 118 194301
Google Scholar
[12] Wen X H, Qiu C Y, Qi Y J, Ye L P, Ke M Z, Zhang F, Liu Z Y 2019 Nat. Phys. 15 352
Google Scholar
[13] Luo J C, Feng L Y, Huang H B, Chen J J 2019 Phys. Lett. A 383 125974
Google Scholar
[14] Rechtsman M C, Zeuner J M, Tünnermann A, Nolte S 2013 Nat. Photonics 7 153
Google Scholar
[15] Schomerus H, Halpern N Y 2013 Phys. Rev. Lett. 110 013903
Google Scholar
[16] Jamadi O, Rozas E, Salerno G, Milicevic M, Ozawa T, Sagnes I, Lemaitre A, Gratiet L L, Harouri A, Carusotto I, Bloch J, Amo A 2020 Light Sci. Appl. 9 144
Google Scholar
[17] Bellec M, Poli C, Kuhl U, Mortessagne F, Schomerus H 2020 Light Sci. Appl. 9 146
Google Scholar
[18] Mann C R, Horsley S A R, Mariani E 2020 Nat. Photonics 14 669
Google Scholar
[19] Guglielmon J, Rechtsman M C, Weinstein M I 2021 Phys. Rev. A 103 013505
Google Scholar
[20] Huang Z T, Hong K B, Lee R K, Pilozzi L, Conti C, Wu S J, Lu T C 2021 arXiv: 211010050 [physics. optics]
[21] Jamotte M, Goldman N, Di Liberto M 2022 Commun. Phys. 5 30
Google Scholar
[22] Deng F S, Sun Y, Wang X, Xue R, Li Y, Jiang H T, Shi Y L, Chang K, Chen H 2014 Opt. Express 22 23605
Google Scholar
[23] Deng F S, Li Y M, Sun Y, Wang X, Guo Z W, Shi Y L, Jiang H T, Chang K, Chen H 2015 Opt. Lett. 40 3380
Google Scholar
[24] Oroszlány L, Rakyta P, Kormányos A, Lambert C J, Cserti J 2008 Phys. Rev. B 77 081403
Google Scholar
[25] Ghosh T K, De Martino A, Häusler W, Dell L, Egger R 2008 Phys. Rev. B 77 081404
Google Scholar
[26] Williams J R, Marcus C M 2011 Phys. Rev. Lett. 107 046602
Google Scholar
[27] Taychatanapat T, Tan J Y, Yeo Y, Watanabe K, Taniguchi T, Ozyilmaz B 2015 Nat. Commun 6 6093
Google Scholar
[28] Rickhaus P, Makk P, Liu M H, Tovari E, Weiss M, Maurand R, Richter K, Schonenberger C 2015 Nat. Commun. 6 6470
Google Scholar
[29] Cohnitz L, Häusler W, Zazunov A, Egger R 2015 Phys. Rev. B 92 085422
Google Scholar
[30] Ren Y N, Zhuang Y C, Sun Q F, He L 2022 Phys. Rev. Lett. 129 076802
Google Scholar
[31] Yan M, Deng W Y, Huang X Q, Wu Y, Yang Y, Lu J Y, Liu Z Y 2021 Phys. Rev. Lett. 127 136401
Google Scholar
[32] Zuo C Y, Qi J J, Lu T L, Bao Z Q, Li Y 2022 Phys. Rev. B 105 195420
Google Scholar
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