Vector valley Hall edge solitons in the photonic lattice with type-II Dirac cones

Abstract

Topological edge solitons represent a significant research topic in the nonlinear topological photonics. They maintain their profiles during propagation, due to the joint action of lattice potential and nonlinearity, and at the same time are immune to defects or disorders, thanks to the topological protection. In the past few years topological edge solitons were reported in systems composed of helical waveguide arrays, in which the time-reversal symmetry is effectively broken. Very recently, topological valley Hall edge solitons have been demonstrated in straight waveguide arrays with the time-reversal symmetry preserved. However, these were scalar solitary structures. Here, for the first time, we report vector valley Hall edge solitons in straight waveguide arrays arranged according to the photonic lattice with innate type-II Dirac cones, which is different from the traditional photonic lattices with type-I Dirac cones such as honeycomb lattice. This comes about because the valley Hall edge state can possess both negative and positive dispersions, which allows the mixing of two different edge states into a vector soliton. Our results not only provide a novel avenue for manipulating topological edge states in the nonlinear regime, but also enlighten relevant research based on the lattices with type-II Dirac cones.

Graphical abstract

Keywords

valley Hall effect / topological edge soliton / photonic topological insulator

References

[1] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[2] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[3] R. Süsstrunk and S. D. Huber, Observation of phononic helical edge states in a mechanical topological insulator, Scienc. 349(6243), 47 (2015)

[4] S. D. Huber, Topological mechanics, Nat. Phys. 12(7), 621 (2016)

[5] C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, Acoustic topological insulator and robust one-way sound transport, Nat. Phys. 12(12), 1124 (2016)

[6] Y. G. Peng, C. Z. Qin, D. G. Zhao, Y. X. Shen, X. Y. Xu, M. Bao, H. Jia, and X. F. Zhu, Experimental demonstration of anomalous Floquet topological insulator for sound, Nat. Commun. 7(1), 13368 (2016)

[7] A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, Polariton Z topological insulator, Phys. Rev. Lett. 114(11), 116401 (2015)

[8] S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, Exciton–polariton topological insulator, Naturee 562(7728), 552 (2018)

[9] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultracold fermions, Nature 515(7526), 237 (2014)

[10] N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Direct imaging of topological edge states in cold-atom systems, Proc. Natl. Acad. Sci. USA 110(17), 6736 (2013)

[11] F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100(1), 013904 (2008)

[12] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461(7265), 772 (2009)

[13] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Robust optical delay lines with topological protection, Nat. Phys. 7(11), 907 (2011)

[14] A. B. Khanikaev, S. Hossein Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Photonic topological insulators, Nat. Mater. 12(3), 233 (2013)

[15] N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nat. Phys. 7(6), 490 (2011)

[16] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature 496(7444), 196 (2013)

[17] L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, Observation of photonic anomalous Floquet topological insulators, Nat. Commun. 8(1), 13756 (2017)

[18] S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. Öhberg, N. Goldman, and R. R. Thomson, Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice, Nat. Commun. 8(1), 13918 (2017)

[19] L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics, Nat. Photonics 8(11), 821 (2014)

[20] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91(1), 015006 (2019)

[21] M. Kim, Z. Jacob, and J. Rho, Recent advances in 2D, 3D and higher-order topological photonics, Light Sci. Appl. 9(1), 130 (2020)

[22] M. Segev and M. A. Bandres, Topological photonics: Where do we go from here? Nanophoton. 10(1), 425 (2020)

[23] H. F. Wang, B. Y. Xie, P. Zhan, M. H. Lu, and Y. F. Chen, Research progress of topological photonics, Acta Phy. Sin. 68(22), 224206 (2019) (in Chinese)

[24] H. Wang, S. K. Gupta, B. Xie, and M. Lu, Topological photonic crystals: A review, Front. Optoelectron. 13(1), 50 (2020)

[25] H. Liu, B. Xie, H. Cheng, J. Tian, and S. Chen, Topological photonic states in artificial microstructures, Chin. Opt. Lett. 19(5), 052602 (2021)

[26] H. Liu, H. N. Wang, B. Y. Xie, H. Cheng, J. G. Tian, and S. Q. Chen, Progress of two-dimensional photonic topological insulators, Chin. Opt. 14(4), 935 (2021)

[27] D. J. Bisharat, R. J. Davis, Y. Zhou, P. R. Bandaru, and D. F. Sievenpiper, Photonic topological insulators: A beginner’s introduction, IEEE Antennas Propag. Mag. 63(3), 112 (2021)

[28] D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7(2), 021306 (2020)

[29] Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, Active topological photonics, Nanophoton. 9(3), 547 (2020)

[30] M. Parto, Y. G. N. Liu, B. Bahari, M. Khajavikhan, and D. N. Christodoulides, Non-Hermitian and topological photonics: Optics at an exceptional point, Nanophoton. 10(1), 403 (2020)

[31] B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, Nonreciprocal lasing in topological cavities of arbitrary geometries, Science 358(6363), 636 (2017)

[32] G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Topological insulator laser: Theory, Science 359(6381), eaar4003 (2018)

[33] M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Topological insulator laser: Experiments, Science 359(6381), eaar4005 (2018)

[34] Y. V. Kartashov and D. V. Skryabin, Two-dimensional topological polariton laser, Phys. Rev. Lett. 122(8), 083902 (2019)

[35] S. K. Ivanov, Y. Q. Zhang, Y. V. Kartashov, and D. V. Skryabin, Floquet topological insulator laser, APL Photonics 4(12), 126101 (2019)

[36] Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong, and Q. J. Wang, Electrically pumped topological laser with valley edge modes, Nature 578(7794), 246 (2020)

[37] H. Zhong, Y. D. Li, D. H. Song, Y. V. Kartashov, Y. Q. Zhang, Y. P. Zhang, and Z. Chen, Topological valley Hall edge state lasing, Laser Photonics Rev. 14(7), 2000001 (2020)

[38] Y. Gong, S. Wong, A. J. Bennett, D. L. Huffaker, and S. S. Oh, Topological insulator laser using valley-Hall photonic crystals, ACS Photonics 7(8), 2089 (2020)

[39] D. Smirnova, A. Tripathi, S. Kruk, M. S. Hwang, H. R. Kim, H. G. Park, and Y. Kivshar, Room-temperature lasing from nanophotonic topological cavities, Light Sci. Appl. 9(1), 127 (2020)

[40] W. Noh, H. Nasari, H. M. Kim, Q. Le-Van, Z. Jia, C. H. Huang, and B. Kanté, Experimental demonstration of single-mode topological valley-Hall lasing at telecommunication wavelength controlled by the degree of asymmetry, Opt. Lett. 45(15), 4108 (2020)

[41] L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, Nonlinearity-induced photonic topological insulator, Science 370(6517), 701 (2020)

[42] Y. V. Kartashov and D. V. Skryabin, Bistable topological insulator with exciton-polaritons, Phys. Rev. Lett. 119(25), 253904 (2017)

[43] W. Zhang, X. Chen, Y. V. Kartashov, D. V. Skryabin, and F. Ye, Finite-dimensional bistable topological insulators: From small to large, Laser Photonics Rev. 13(11), 1900198 (2019)

[44] Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Self-localized states in photonic topological insulators, Phys. Rev. Lett. 111(24), 243905 (2013)

[45] S. Mukherjee and M. C. Rechtsman, Observation of Floquet solitons in a topological bandgap, Science 368(6493), 856 (2020)

[46] M. J. Ablowitz, C. W. Curtis, and Y. P. Ma, Linear and nonlinear traveling edge waves in optical honeycomb lattices, Phys. Rev. A 90(2), 023813 (2014)

[47] M. J. Ablowitz and J. T. Cole, Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons, Phys. Rev. A 96(4), 043868 (2017)

[48] M. J. Ablowitz and Y. P. Ma, Strong transmission and reflection of edge modes in bounded photonic graphene, Opt. Lett. 40(20), 4635 (2015)

[49] D. Leykam and Y. D. Chong, Edge solitons in nonlinear photonic topological insulators, Phys. Rev. Lett. 117(14), 143901 (2016)

[50] S. K. Ivanov, Y. V. Kartashov, L. J. Maczewsky, A. Szameit, and V. V. Konotop, Edge solitons in Lieb topological Floquet insulator, Opt. Lett. 45(6), 1459 (2020)

[51] S. K. Ivanov, Y. V. Kartashov, L. J. Maczewsky, A. Szameit, and V. V. Konotop, Bragg solitons in topological Floquet insulators, Opt. Lett. 45(8), 2271 (2020)

[52] S. K. Ivanov, Y. V. Kartashov, A. Szameit, L. Torner, and V. V. Konotop, Vector topological edge solitons in Floquet insulators, ACS Photonics 7(3), 735 (2020)

[53] S. K. Ivanov, Y. V. Kartashov, M. Heinrich, A. Szameit, L. Torner, and V. V. Konotop, Topological dipole Floquet solitons, Phys. Rev. A 103(5), 053507 (2021)

[54] S. K. Ivanov, Y. V. Kartashov, and V. V. Konotop, Four-wave mixing Floquet topological solitons, Opt. Lett. 46(19), 4710 (2021)

[55] S. K. Ivanov, Y. V. Kartashov, and V. V. Konotop, Floquet defect solitons, Opt. Lett. 46(21), 5364 (2021)

[56] Y. V. Kartashov and D. V. Skryabin, Modulational instability and solitary waves in polariton topological insulators, Optica 3(11), 1228 (2016)

[57] D. R. Gulevich, D. Yudin, D. V. Skryabin, I. V. Iorsh, and I. A. Shelykh, Exploring nonlinear topological states of matter with exciton-polaritons: Edge solitons in kagome lattice, Sci. Rep. 7(1), 1780 (2017)

[58] C. Li, F. Ye, X. Chen, Y. V. Kartashov, A. Ferrando, L. Torner, and D. V. Skryabin, Lieb polariton topological insulators, Phys. Rev. B 97(8), 081103 (2018)

[59] Y. Q. Zhang, Y. V. Kartashov, and A. Ferrando, Interface states in polariton topological insulators, Phys. Rev. A 99(5), 053836 (2019)

[60] D. A. Smirnova, L. A. Smirnov, D. Leykam, and Y. S. Kivshar, Topological edge states and gap solitons in the nonlinear Dirac model, Laser Photonics Rev. 13(12), 1900223 (2019)

[61] W. Zhang, X. Chen, Y. V. Kartashov, V. V. Konotop, and F. Ye, Coupling of edge states and topological Bragg solitons, Phys. Rev. Lett. 123(25), 254103 (2019)

[62] H. Zhong, S. Xia, Y. Zhang, Y. Li, D. Song, C. Liu, and Z. Chen, Nonlinear topological valley Hall edge states arising from type-II Dirac cones, Adv. Photonics 3(05), 056001 (2021)

[63] Q. Tang, B. Ren, V. O. Kompanets, Y. V. Kartashov, Y. Li, and Y. Zhang, Valley Hall edge solitons in a photonic graphene, Opt. Express 29(24), 39755 (2021)

[64] B. Ren, H. Wang, V. O. Kompanets, Y. V. Kartashov, Y. Li, and Y. Zhang, Dark topological valley Hall edge solitons, Nanophoton. 10(13), 3559 (2021)

[65] Z. Y. Zhang, R. Wang, Y. Q. Zhang, Y. V. Kartashov, F. Li, H. Zhong, H. Guan, K. Gao, F. L. Li, Y. P. Zhang, and M. Xiao, Observation of edge solitons in photonic graphene, Nat. Commun. 11(1), 1902 (2020)

[66] J. W. Liu, F. L. Shi, X. T. He, G. J. Tang, W. J. Chen, X. D. Chen, and J. W. Dong, Valley photonic crystals, Adv. Phys. X 6(1), 1905546 (2021)

[67] K. C. Jin, H. Zhong, Y. D. Li, F. W. Ye, Y. P. Zhang, F. L. Li, C. L. Liu, and Y. Q. Zhang, Parametric type-II Dirac photonic lattices, Adv. Quantum Technol. 3(7), 2000015 (2020)

[68] K. T. Wang, F. Xu, B. Wang, Y. Yu, and Y. Wei, Transport features of topological corner states in honeycomb lattice with multihollow structure, Front. Phys. (Beijing.) 17(4), 43501 (2021)

[69] S. Li, Z. M. Yu, Y. Yao, and S. A. Yang, Type-II topological metals, Front. Phys. (Beijing.) 15(4), 43201 (2020)

[70] S. Stützer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, Photonic topological Anderson insulators, Nature 560(7719), 461 (2018)

[71] M. S. Kirsch, Y. Zhang, M. Kremer, L. J. Maczewsky, S. K. Ivanov, Y. V. Kartashov, L. Torner, D. Bauer, A. Szameit, and M. Heinrich, Nonlinear second-order photonic topological insulators, Nat. Phys. 17(9), 995 (2021)

[72] D. Tan, Z. Wang, B. Xu, and J. Qiu, Photonic circuits written by femtosecond laser in glass: Improved fabrication and recent progress in photonic devices, Adv. Photonics 3(02), 024002 (2021)

[73] Y. V. Kartashov, G. E. Astrakharchik, B. A. Malomed, and L. Torner, Frontiers in multidimensional self-trapping of nonlinear fields and matter, Nat. Rev. Phys. 1(3), 185 (2019)

[74] B. A. Malomed and D. Mihalache, Nonlinear waves in optical and matter-wave media: A topical survey of recent theoretical and experimental results, Rom. J. Phys. 64, 106 (2019)

[75] D. Mihalache, Localized structures in optical and matter-wave media: A selection of recent studies, Rom. Rep. Phys. 73, 403 (2021)

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