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Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices

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Abstract

Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities. Moiré photonic and optical lattices — two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones — are a new emerging investigative tool for studying nonlinear localized waves of diverse types. Herein, a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction (linear nonlocality) and moiré optical lattices is investigated. Specifically, the flat-band feature is well preserved in shallow moiré optical lattices which, interact with the defocusing nonlinearity of the media, can support fundamental gap solitons, bound states composed of several fundamental solitons, and topological states (gap vortices) with vortex charge s = 1 and 2, all populated inside the finite gaps of the linear Bloch-wave spectrum. Employing the linear-stability analysis and direct perturbed simulations, the stability and instability properties of all the localized gap modes are surveyed, highlighting a wide stability region within the first gap and a limited one (to the central part) for the third gap. The findings enable insightful studies of highly localized gap modes in linear nonlocality (fractional) physical systems with shallow moiré patterns that exhibit extremely flat bands.

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References

  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003

    Google Scholar 

  2. O. Morsch and M. Oberthaler, Dynamics of Bose–Einstein condensates in optical lattices, Rev. Mod. Phys. 78(1), 179 (2006)

    Article  ADS  Google Scholar 

  3. Y. V. Kartashov, B. A. Malomed, and L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys. 83(1), 247 (2011)

    Article  ADS  Google Scholar 

  4. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides, Phys. Rep. 518(1–2), 1 (2012)

    Article  ADS  Google Scholar 

  5. V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88(3), 035002 (2016)

    Article  ADS  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Bragg grating solitons, Phys. Rev. Lett. 76(10), 1627 (1996)

    Article  ADS  Google Scholar 

  8. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Gap solitons in waveguide arrays, Phys. Rev. Lett. 92(9), 093904 (2004)

    Article  ADS  Google Scholar 

  9. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett. 98(10), 103901 (2007)

    Article  ADS  Google Scholar 

  10. B. Eiermann, T. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, Bright Bose–Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett. 92(23), 230401 (2004)

    Article  ADS  Google Scholar 

  11. Th. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, Nonlinear self-trapping of matter waves in periodic potentials, Phys. Rev. Lett. 94(2), 020403 (2005)

    Article  ADS  Google Scholar 

  12. F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays, Phys. Rev. Lett. 106(9), 093901 (2011)

    Article  ADS  Google Scholar 

  13. C. Bersch, G. Onishchukov, and U. Peschel, Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices, Phys. Rev. Lett. 109(9), 093903 (2012)

    Article  ADS  Google Scholar 

  14. L. Zeng and J. Zeng, Gap-type dark localized modes in a Bose–Einstein condensate with optical lattices, Adv. Photonics 1(4), 046004 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  15. J. Shi and J. Zeng, Self-trapped spatially localized states in combined linear-nonlinear periodic potentials, Front. Phys. 15(1), 12602 (2020)

    Article  ADS  Google Scholar 

  16. J. Li and J. Zeng, Dark matter-wave gap solitons in dense ultra-cold atoms trapped by a one-dimensional optical lattice, Phys. Rev. A 103(1), 013320 (2021)

    Article  ADS  Google Scholar 

  17. J. Chen and J. Zeng, Dark matter-wave gap solitons of Bose–Einstein condensates trapped in optical lattices with competing cubic–quintic nonlinearities, Chaos Solitons Fractals 150, 111149 (2021)

    Article  MathSciNet  Google Scholar 

  18. Z. Chen and J. Zeng, Localized gap modes of coherently trapped atoms in an optical lattice, Opt. Express 29(3), 3011 (2021)

    Article  ADS  Google Scholar 

  19. Z. Chen and J. Zeng, Two-dimensional optical gap solitons and vortices in a coherent atomic ensemble loaded on optical lattices, Commun. Nonlinear Sci. Numer. Simul. 102, 105911 (2021)

    Article  MathSciNet  Google Scholar 

  20. Z. Chen and J. Zeng, Nonlinear localized modes in one-dimensional nanoscale dark-state optical lattices, Nanophotonics 11(15), 3465 (2022)

    Article  Google Scholar 

  21. J. Li, Y. Zhang, and J. Zeng, Matter-wave gap solitons and vortices in three-dimensional parity–time-symmetric optical lattices, iScience 25(4), 104026 (2022)

    Article  ADS  Google Scholar 

  22. J. Li, Y. Zhang, and J. Zeng, 3D nonlinear localized gap modes in Bose–Einstein condensates trapped by optical lattices and space-periodic nonlinear potentials, Adv. Photon. Res. 3(7), 2100288 (2022)

    Article  Google Scholar 

  23. J. Qin and L. Zhou, Supersolid gap soliton in a Bose–Einstein condensate and optical ring cavity coupling system, Phys. Rev. E 105(5), 054214 (2022)

    Article  ADS  Google Scholar 

  24. J. Yang and Y. Zhang, Spin–orbit-coupled spinor gap solitons in Bose–Einstein condensates, Phys. Rev. A 107(2), 023316 (2023)

    Article  ADS  Google Scholar 

  25. Z. Chen, Z. Wu, and J. Zeng, Light gap bullets in defocusing media with optical lattices, Chaos Solitons Fractals 174, 113785 (2023)

    Article  MathSciNet  Google Scholar 

  26. C. Huang and L. Dong, Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice, Opt. Lett. 41(24), 5636 (2016)

    Article  ADS  Google Scholar 

  27. C. Huang, C. Li, H. Deng, and L. Dong, Gap Solitons in fractional dimensions with a quasi-periodic lattice, Ann. Phys. 531(9), 1900056 (2019)

    Article  MathSciNet  Google Scholar 

  28. J. Xie, X. Zhu, and Y. He, Vector solitons in nonlinear fractional Schrödinger equations with parity–time-symmetric optical lattices, Nonlinear Dyn. 97(2), 1287 (2019)

    Article  Google Scholar 

  29. L. Zeng and J. Zeng, One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential, Nonlinear Dyn. 98, 985 (2019)

    Article  Google Scholar 

  30. L. Zeng and J. Zeng, Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and non-linearities, Commun. Phys. 3(1), 26 (2020)

    Article  MathSciNet  Google Scholar 

  31. X. Zhu, F. Yang, S. Cao, J. Xie, and Y. He, Multipole gap solitons in fractional Schrödinger equation with parity–time-symmetric optical lattices, Opt. Express 28(2), 1631 (2020)

    Article  ADS  Google Scholar 

  32. L. Zeng, M. R. Belić, D. Mihalache, J. Shi, J. Li, S. Li, X. Lu, Y. Cai, and J. Li, Families of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction, Nonlinear Dyn. 108(2), 1671 (2022)

    Article  Google Scholar 

  33. Y. Y. Bao, S. R. Li, Y. H. Liu, and T. F. Xu, Gap solitons and non-linear Bloch waves in fractional quantum coupler with periodic potential, Chaos Solitons Fractals 156, 111853 (2022)

    Article  Google Scholar 

  34. X. Liu, B. A. Malomed, and J. Zeng, Localized modes in nonlinear fractional systems with deep lattices, Adv. Theory Simul. 5(4), 2100482 (2022)

    Article  Google Scholar 

  35. L. Dong and C. Huang, Double-hump solitons in fractional dimensions with a PT-symmetric potential, Opt. Express 26(8), 10509 (2018)

    Article  ADS  Google Scholar 

  36. C. Huang and L. Dong, Beam propagation management in a fractional Schrödinger equation, Sci. Rep. 7(1), 5442 (2017)

    Article  ADS  Google Scholar 

  37. N. Laskin, Fractional Quantum Mechanics, World Scientific, 2018

  38. B. A. Malomed, Optical solitons and vortices in fractional media: A mini-review of recent results, Photonics 8(9), 353 (2021)

    Article  Google Scholar 

  39. S. Liu, Y. Zhang, B. A. Malomed, and E. Karimi, Experimental realizations of the fractional Schrödinger equation in the temporal domain, Nat. Commun. 14(1), 222 (2023)

    Article  ADS  Google Scholar 

  40. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556(7699), 43 (2018)

    Article  ADS  Google Scholar 

  41. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556(7699), 80 (2018)

    Article  ADS  Google Scholar 

  42. S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, and E. Kaxiras, Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle, Phys. Rev. B 95(7), 075420 (2017)

    Article  ADS  Google Scholar 

  43. C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, Localization–delocalization wavepacket transition in Pythagorean aperiodic potentials, Sci. Rep. 6(1), 32546 (2016)

    Article  ADS  Google Scholar 

  44. P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, Localization and delocalization of light in photonic moiré lattices, Nature 577(7788), 42 (2020)

    Article  ADS  Google Scholar 

  45. Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, Optical soliton formation controlled by angle twisting in photonic moiré lattices, Nat. Photonics 14(11), 663 (2020)

    Article  ADS  Google Scholar 

  46. X. R. Mao, Z. K. Shao, H. Y. Luan, S. L. Wang, and R. M. Ma, Magic-angle lasers in nanostructured moiré superlattice, Nat. Nanotechnol. 16(10), 1099 (2021)

    Article  ADS  Google Scholar 

  47. Y. V. Kartashov, F. Ye, V. V. Konotop, and L. Torner, Multi-frequency solitons in commensurate-incommensurate photonic moiré lattices, Phys. Rev. Lett. 127(16), 163902 (2021)

    Article  ADS  Google Scholar 

  48. Y. V. Kartashov, Light bullets in moiré lattices, Opt. Lett. 47(17), 4528 (2022)

    Article  ADS  Google Scholar 

  49. S. K. Ivanov, V. V. Konotop, Y. V. Kartashov, and L. Torner, Vortex solitons in moiré optical lattices, Opt. Lett. 48(14), 3797 (2023)

    Article  ADS  Google Scholar 

  50. A. A. Arkhipova, Y. V. Kartashov, S. K. Ivanov, S. A. Zhuravitskii, N. N. Skryabin, I. V. Dyakonov, A. A. Kalinkin, S. P. Kulik, V. O. Kompanets, S. V. Chekalin, F. Ye, V. V. Konotop, L. Torner, and V. N. Zadkov, Observation of linear and nonlinear light localization at the edges of moiré arrays, Phys. Rev. Lett. 130(8), 083801 (2023)

    Article  ADS  Google Scholar 

  51. S. S. Sunku, G. X. Ni, B. Y. Jiang, H. Yoo, A. Sternbach, A. S. McLeod, T. Stauber, L. Xiong, T. Taniguchi, K. Watanabe, P. Kim, M. M. Fogler, and D. N. Basov, Photonic crystals for nano-light in moiré graphene superlattices, Science 362(6419), 1153 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  52. W. J. M. Kort-Kamp, F. J. Culchac, R. B. Capaz, and F. A. Pinheiro, Photonic spin Hall effect in bilayer graphene moiré superlattices, Phys. Rev. B 98(19), 195431 (2018)

    Article  ADS  Google Scholar 

  53. G. Hu, Q. Ou, G. Si, Y. Wu, J. Wu, Z. Dai, A. Krasnok, Y. Mazor, Q. Zhang, Q. Bao, C. W. Qiu, and A. Alù, Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers, Nature 582(7811), 209 (2020)

    Article  ADS  Google Scholar 

  54. M. Chen, X. Lin, T. H. Dinh, Z. Zheng, J. Shen, Q. Ma, H. Chen, P. Jarillo-Herrero, and S. Dai, Configurable phonon polaritons in twisted α-MoO3, Nat. Mater. 19(12), 1307 (2020)

    Article  ADS  Google Scholar 

  55. A. González-Tudela and J. I. Cirac, Cold atoms in twisted-bilayer optical potentials, Phys. Rev. A 100(5), 053604 (2019)

    Article  ADS  Google Scholar 

  56. T. Salamon, A. Celi, R. W. Chhajlany, I. Frérot, M. Lewenstein, L. Tarruell, and D. Rakshit, Simulating twistronics without a twist, Phys. Rev. Lett. 125(3), 030504 (2020)

    Article  ADS  Google Scholar 

  57. X. W. Luo and C. Zhang, Spin-twisted optical lattices: Tunable flat bands and Larkin–Ovchinnikov superfluids, Phys. Rev. Lett. 126(10), 103201 (2021)

    Article  ADS  Google Scholar 

  58. T. Ning, Y. Ren, Y. Huo, and Y. Cai, Efficient high harmonic generation in nonlinear photonic moiré superlattice, Front. Phys. 18(5), 52305 (2023)

    Article  ADS  Google Scholar 

  59. Z. Ma, W. J. Chen, Y. Chen, J. H. Gao, and X. C. Xie, Flat band localization due to self-localized orbital, Front. Phys. 18(6), 63302 (2023)

    Article  ADS  Google Scholar 

  60. Z. Chen, X. Liu, and J. Zeng, Electromagnetically induced moiré optical lattices in a coherent atomic gas, Front. Phys. 17(4), 42508 (2022)

    Article  ADS  Google Scholar 

  61. Z. Meng, L. Wang, W. Han, F. Liu, K. Wen, C. Gao, P. Wang, C. Chin, and J. Zhang, Atomic Bose–Einstein condensate in twisted-bilayer optical lattices, Nature 615(7951), 231 (2023)

    Article  ADS  Google Scholar 

  62. C. Huang, L. Dong, H. Deng, X. Zhang, and P. Gao, Fundamental and vortex gap solitons in quasiperiodic photonic lattices, Opt. Lett. 46(22), 5691 (2021)

    Article  ADS  Google Scholar 

  63. X. Liu and J. Zeng, Matter-wave gap solitons and vortices of dense Bose–Einstein condensates in moiré optical lattices, Chaos Solitons Fractals 174, 113869 (2023)

    Article  Google Scholar 

  64. X. Liu and J. Zeng, Gap solitons in parity–time symmetric moiré optical lattices, Photon. Res. 11(2), 196 (2023)

    Article  Google Scholar 

  65. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM: Philadelphia, 2010

    Book  Google Scholar 

  66. M. Cai and C. P. Li, On Riesz derivative, Fract. Calc. Appl. Anal. 22(2), 287 (2019)

    Article  MathSciNet  Google Scholar 

  67. S. Duo and Y. Zhang, Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications, Comput. Methods Appl. Mech. Eng. 355, 639 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  68. N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268(4–6), 298 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  69. N. Laskin, Fractional quantum mechanics, Phys. Rev. E (3), 3135 (2000)

  70. N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66(5), 056108 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  71. L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: From linear to nonlinear regimes, Opt. Express 24(13), 14406 (2016)

    Article  ADS  Google Scholar 

  72. L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, Modulational instability in fractional nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 48, 531 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  73. M. Vakhitov and A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16(7), 783 (1973)

    Article  ADS  Google Scholar 

  74. A. Ferrando, M. Zacarés, and M. A. García-March, Vorticity cutoff in nonlinear photonic crystals, Phys. Rev. Lett. 95(4), 043901 (2005)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) (No. 12074423), Young Scholar of Chinese Academy of Sciences in Western China (No. XAB2021YN18), and China Postdoctoral Science Foundation (No. 2023M733722).

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Authors and Affiliations

  1. Center for Attosecond Science and Technology, State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi’an, 710119, China

    Xiuye Liu & Jianhua Zeng

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Jianhua Zeng

  3. Collaborative lnnovation Center of Extreme Optics, Shanxi University, Taiyuan, 030006, China

    Jianhua Zeng

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  1. Xiuye Liu
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  2. Jianhua Zeng
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Correspondence to Jianhua Zeng.

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Liu, X., Zeng, J. Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices. Front. Phys. 19, 42201 (2024). https://doi.org/10.1007/s11467-023-1370-7

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