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Quantum droplets in two-dimensional optical lattices

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Abstract

We study the stability of zero-vorticity and vortex lattice quantum droplets (LQDs), which are described by a two-dimensional (2D) Gross–Pitaevskii (GP) equation with a periodic potential and Lee–Huang–Yang (LHY) term. The LQDs are divided in two types: onsite-centered and offsite-centered LQDs, the centers of which are located at the minimum and the maximum of the potential, respectively. The stability areas of these two types of LQDs with different number of sites for zero-vorticity and vorticity with S = 1 are given. We found that the μ–N relationship of the stable LQDs with a fixed number of sites can violate the Vakhitov–Kolokolov (VK) criterion, which is a necessary stability condition for nonlinear modes with an attractive interaction. Moreover, the μ–N relationship shows that two types of vortex LQDs with the same number of sites are degenerated, while the zero-vorticity LQDs are not degenerated. It is worth mentioning that the offsite-centered LQDs with zero-vorticity and vortex LQDs with S = 1 are heterogeneous.

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References

  1. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Moscow: Nauka publishers, 1974

    Google Scholar 

  2. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Observation of two-dimensional spatial solitary waves in a quadratic medium, Phys. Rev. Lett. 74(25), 5036 (1995)

    Article  ADS  Google Scholar 

  3. X. Liu, K. Beckwitt, and F. Wise, Two-dimensional optical spatiotemporal solitons in quadratic media, Phys. Rev. E 62(1), 1328 (2000)

    Article  ADS  Google Scholar 

  4. D. Mihalache, D. Mazilu, L. C. Crasovan, B. A. Malomed, and F. Lederer, Three-dimensional spinning solitons in the cubic–quintic nonlinear medium, Phys. Rev. E 61(6), 7142 (2000)

    Article  ADS  Google Scholar 

  5. S. Konar, M. Mishra, and S. Jana, Nonlinear evolution of cosh-Gaussian laser beams and generation of flat top spatial solitons in cubic–quintic nonlinear media, Phys. Lett. A 362(5–6), 505 (2007)

    Article  ADS  Google Scholar 

  6. E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett. 110(1), 013901 (2013)

    Article  ADS  Google Scholar 

  7. Y. Y. Wang, L. Chen, C. Q. Dai, J. Zheng, and Y. Fan, Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity, Nonlinear Dyn. 90(2), 1269 (2017)

    Article  MathSciNet  Google Scholar 

  8. C. Q. Dai, R. P. Chen, Y. Y. Wang, and Y. Fan, Dynamics of light bullets in inhomogeneous cubic–quanticseptimal nonlinear media with PT-symmetric potentials, Nonlinear Dyn. 87(3), 1675 (2017)

    Article  Google Scholar 

  9. Y. Chen, L. Zheng, and F. Xu, Spatiotemporal vector and scalar solitons of the coupled nonlinear Schringer equation with spatially modulated cubic–quantic-septimal nonlinearities, Nonlinear Dyn. 93(4), 2379 (2018)

    Article  Google Scholar 

  10. J. Li, Y. Zhu, J. Han, W. Qin, C. Dai, and S. Wang, Scalar and vector multipole and vortex solitons in the spatially modulated cubic–quintic nonlinear media, Nonlinear Dyn. 91(2), 757 (2018)

    Article  Google Scholar 

  11. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Steady-state spatial screening solitons in photorefractive materials with external applied field, Phys. Rev. Lett. 73(24), 3211 (1994)

    Article  ADS  Google Scholar 

  12. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Nonlocal spatial soliton interactions in nematic liquid crystals, Opt. Lett. 27(16), 1460 (2002)

    Article  ADS  Google Scholar 

  13. P. Pedri and L. Santos, Two-dimensional bright solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 95(20), 200404 (2005)

    Article  ADS  Google Scholar 

  14. I. Tikhonenkov, B. A. Malomed, and A. Vardi, Anisotropic solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 100(9), 090406 (2008)

    Article  ADS  Google Scholar 

  15. J. Huang, X. Jiang, H. Chen, Z. Fan, W. Pang, and Y. Li, Quadrupolar matter-wave soliton in two-dimensional free space, Front. Phys. 10(4), 100507 (2015)

    Article  ADS  Google Scholar 

  16. F. Maucher, N. Henkel, M. Saffman, W. Królikowski, S. Skupin, and T. Pohl, Rydberg-induced solitons: Threedimensional self-trapping of matter waves, Phys. Rev. Lett. 106(17), 170401 (2011)

    Article  ADS  Google Scholar 

  17. Y. Xu, Y. Zhang, and C. Zhang, Bright solitons in a two-dimensional spin–orbit-coupled dipolar Bose–Einstein condensate, Phys. Rev. A 92(1), 013633 (2015)

    Article  ADS  Google Scholar 

  18. X. Jiang, Z. Fan, Z. Chen, W. Pang, Y. Li, and B. A. Malomed, Two-dimensional solitons in dipolar Bose–Einstein condensates with spin–orbit coupling, Phys. Rev. A 93(2), 023633 (2016)

    Article  ADS  Google Scholar 

  19. Y. Li, Y. Liu, Z. Fan, W. Pang, S. Fu, and B. A. Malomed, Two-dimensional dipolar gap solitons in free space with spin–orbit coupling, Phys. Rev. A 95(6), 063613 (2017)

    Article  ADS  Google Scholar 

  20. X. Chen, Z. Deng, X. Xu, S. Li, Z. Fan, Z. Chen, B. Liu, and Y. Li, Nonlinear modes in spatially confined spin–orbit-coupled Bose–Einstein condensates with repulsive nonlinearity, Nonlinear Dyn. 101(1), 569 (2020)

    Article  Google Scholar 

  21. Z. Ye, Y. Chen, Y. Zheng, X. Chen, and B. Liu, Symmetry breaking of a matter-wave soliton in a double-well potential formed by spatially confined spin–orbit coupling, Chaos Solitons Fractals 130, 109418 (2020)

    Article  MathSciNet  Google Scholar 

  22. B. Liu, R. Zhong, Z. Chen, X. Qin, H. Zhong, Y. Li, and B. A. Malomed, Holding and transferring matter-wave solitons against gravity by spin–orbit-coupling tweezers, New J. Phys. 22(4), 043004 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  23. B. Liao, S. Li, C. Huang, Z. Luo, W. Pang, H. Tan, B. A. Malomed, and Y. Li, Anisotropic semi-vortices in dipolar spinor condensates controlled by Zeeman splitting, Phys. Rev. A 96(4), 043613 (2017)

    Article  ADS  Google Scholar 

  24. B. Liao, Y. Ye, J. Zhuang, C. Huang, H. Deng, W. Pang, B. Liu, and Y. Li, Anisotropic solitary semivortices in dipolar spinor condensates controlled by the two-dimensional anisotropic spin–orbit coupling, Chaos Solitons Fractals 116, 424 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. S. Liu, B. Liao, J. Kong, P. Chen, J. Lü, Y. Li, C. Huang, and Y. Li, Anisotropic semi vortices in spinor dipolar Bose–Einstein condensates induced by mixture of Rashba–Dresselhaus coupling, J. Phys. Soc. Jpn. 87(9), 094005 (2018)

    Article  ADS  Google Scholar 

  26. W. Pang, H. Deng, B. Liu, J. Xu, and Y. Li, Twodimensional vortex solitons in spin–orbit-coupled dipolar Bose–Einstein condensates, Appl. Sci. (Basel) 8(10), 1771 (2018)

    Article  Google Scholar 

  27. X. Cui, Spin–orbit-coupling-induced quantum droplet in ultracold Bose–Fermi mixtures, Phys. Rev. A 98(2), 023630 (2018)

    Article  ADS  Google Scholar 

  28. M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature 539(7628), 259 (2016)

    Article  ADS  Google Scholar 

  29. L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose–Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6(4), 041039 (2016)

    Google Scholar 

  30. C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Quantum liquid droplets in a mixture of Bose–Einstein condensates, Science 359(6373), 301 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  31. D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose–Bose mixture, Phys. Rev. Lett. 115(15), 155302 (2015)

    Article  ADS  Google Scholar 

  32. D. S. Petrov and G. E. Astrakharchik, Ultradilute lowdimensional liquids, Phys. Rev. Lett. 117(10), 100401 (2016)

    Article  ADS  Google Scholar 

  33. P. Zin, M. Pylak, T. Wasak, M. Gajda, and Z. Idziaszek, Quantum Bose–Bose droplets at a dimensional crossover, Phys. Rev. A 98, 051603(R) (2018)

    Article  ADS  Google Scholar 

  34. Y. Li, Z. Luo, Y. Liu, Z. Chen, C. Huang, S. Fu, H. Tan, and B. A. Malomed, Two-dimensional solitons and quantum droplets supported by competing self- and crossinteractions in spin–orbit-coupled condensates, New J. Phys. 19(11), 113043 (2017)

    Article  ADS  Google Scholar 

  35. T. Ilg, J. Kumlin, L. Santos, D. S. Petrov, and H. P. Büchler, Dimensional crossover for the beyond-mean-field correction in Bose gases, Phys. Rev. A 98(5), 051604 (2018)

    Article  ADS  Google Scholar 

  36. Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Two-dimensional vortex quantum droplets, Phys. Rev. A 98(6), 063602 (2018)

    Article  ADS  Google Scholar 

  37. X. Zhang, X. Xu, Y. Zheng, Z. Chen, B. Liu, Ch. Huang, B. A. Malomed, and Y. Li, Semidiscrete quantum droplets and vortices, Phys. Rev. Lett. 123(13), 133901 (2019)

    Article  ADS  Google Scholar 

  38. Z. Lin, X. Xu, Z. Chen, Z. Yan, Z. Mai, and B. Liu, Twodimensional vortex quantum droplets get thick, Commun. Nonlinear Sci. Numer. Simul. 93, 105536 (2020)

    Article  MATH  Google Scholar 

  39. B. Liu, H. Zhang, R. Zhong, X. Zhang, X. Qin, X. Huang, Y. Li, and B. A. Malomed, Symmetry breaking of quantum droplets in a dual-core trap, Phys. Rev. A 99(5), 053602 (2019)

    Article  ADS  Google Scholar 

  40. G. E. Astrakharchik and B. A. Malomed, Dynamics of one-dimensional quantum droplets, Phys. Rev. A 98(1), 013631 (2018)

    Article  ADS  Google Scholar 

  41. Zh. Zhou, X. Yu, Y. Zou, and H. Zhong, Dynamics of quantum droplets in a one-dimensional optical lattice, Commun. Nonlinear Sci. Numer. Simul. 78, 104881 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zh. Zhou, B. Zhu, H. Wang, and H. Zhong, Stability and collisions of quantum droplets in PT-symmetric dualcore couplers, Commun. Nonlinear Sci. Numer. Simul. 91, 105424 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. F. Wächtler and L. Santos, Quantum filaments in dipolar Bose–Einstein condensates, Phys. Rev. A 93, 061603(R) (2016)

    Article  ADS  Google Scholar 

  44. F. Wächtler and L. Santos, Ground-state properties and elementary excitations of quantum droplets in dipolar Bose–Einstein condensates, Phys. Rev. A 94(4), 043618 (2016)

    Article  ADS  Google Scholar 

  45. D. Baillie, R. M. Wilson, R. N. Bisset, and P. B. Blakie, Self-bound dipolar droplet: A localized matter wave in free space, Phys. Rev. A 94, 021602(R) (2016)

    Article  ADS  Google Scholar 

  46. D. Edler, C. Mishra, F. Wächtler, R. Nath, S. Sinha, and L. Santos, Quantum fluctuations in quasi-onedimensional dipolar Bose–Einstein condensates, Phys. Rev. Lett. 119(5), 050403 (2017)

    Article  ADS  Google Scholar 

  47. I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of quantum droplets in a strongly dipolar Bose gas, Phys. Rev. Lett. 116(21), 215301 (2016)

    Article  ADS  Google Scholar 

  48. I. Ferrier-Barbut, M. Wenze, F. Böttcher, T. Langen, M. Isoard, S. Stringari, and T. Pfau, Scissors mode of dipolar quantum droplets of dysprosium atoms, Phys. Rev. Lett. 120(16), 160402 (2018)

    Article  ADS  Google Scholar 

  49. A. Cidrim, F. E. A. dos Santos, E. A. L. Henn, and T. Macrí, Vortices in self-bound dipolar droplets, Phys. Rev. A 98(2), 023618 (2018)

    Article  ADS  Google Scholar 

  50. R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A 94(3), 033619 (2016)

    Article  ADS  Google Scholar 

  51. Y. Sekino and Y. Nishida, Quantum droplet of onedimensional bosons with a three-body attraction, Phys. Rev. A 97, 011602(R) (2018)

    Article  ADS  Google Scholar 

  52. C. Staudinger, F. Mazzanti, and R. E. Zillich, Self-bound Bose mixtures, Phys. Rev. A 98(2), 023633 (2018)

    Article  ADS  Google Scholar 

  53. V. Cikojevic, K. Dželalija, P. Stipanovic, and L. V. Markic, Ultradilute quantum liquid drops, Phys. Rev. B 97, 140502(R) (2018)

    Article  ADS  Google Scholar 

  54. P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi, and L. Tarruell, Bright soliton to quantum droplet transition in a mixture of Bose–Einstein condensates, Phys. Rev. Lett. 120(13), 135301 (2018)

    Article  ADS  Google Scholar 

  55. G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wolswijk, F. Minardi, M. Modugno, G. Modugno, M. Inguscio, and M. Fattori, Self-bound quantum droplets in atomic mixtures, Phys. Rev. Lett. 120(23), 235301 (2018)

    Article  ADS  Google Scholar 

  56. A. Cappellaro, T. Macrì, and L. Salasnich, Collective modes across the soliton–droplet crossover in binary Bose mixtures, Phys. Rev. A 97(5), 053623 (2018)

    Article  ADS  Google Scholar 

  57. A. Pricoupenko and D. S. Petrov, Dimer–dimer zero crossing and dilute dimerized liquid in a one-dimensional mixture, Phys. Rev. A 97(6), 063616 (2018)

    Article  ADS  Google Scholar 

  58. A. Cappellaro, T. Macrí, G. F. Bertacco, and L. Salasnich, Equation of state and self-bound droplet in Rabicoupled Bose mixtures, Sci. Rep. 7(1), 13358 (2017)

    Article  ADS  Google Scholar 

  59. N. Westerberg, K. E. Wilson, C. W. Duncan, D. Faccio, E. M. Wright, P. Öhberg, and M. Valiente, Self-bound droplets of light with orbital angular momentum, Phys. Rev. A 98(5), 053835 (2018)

    Article  ADS  Google Scholar 

  60. E. Shamriz, Zh. Chen, B. A. Malomed, and H. Sakaguchi, Singular mean-field states: A brief review of recent results, Condens. Matter 5, 20 (2020)

    Article  Google Scholar 

  61. Z. Luo, W. Pang, B. Liu, Y. Li, and A. B. Malomed, A new form of liquid matter: Quantum droplets, Front. Phys. (2021) (submitted), arXiv: 2009.01061

    Google Scholar 

  62. T. D. Lee, K. S. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev. 106(6), 1135 (1957)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  64. H. Zhang, F. Chen, C. Yu, L. Sun, and D. Xu, Tunable ground-state solitons in spin–orbit coupling Bose–Einstein condensates in the presence of optical lattices, Chin. Phys. B 26(8), 080304 (2017)

    Article  ADS  Google Scholar 

  65. R. Campbell, and G. L. Oppo, Stationary and traveling solitons via local dissipation in Bose–Einstein condensates in ring optical lattices, Phys. Rev. A 94(4), 043626 (2016)

    Article  ADS  Google Scholar 

  66. X. Zhu, H. Li, Z. Shi, Y. Xiang, and Y. He, Gap solitons in spin–orbit-coupled Bose–Einstein condensates in mixed linear–nonlinear optical lattices, J. Phys. At. Mol. Opt. Phys. 50(15), 155004 (2017)

    Article  ADS  Google Scholar 

  67. F. Li, F. Zong, and Y. Wang, Vortical solitons of threedimensional Bose–Einstein condensates under both a bichromatic optical lattice and anharmonic potentials, Chin. Phys. Lett. 30(6), 060306 (2013)

    Article  ADS  Google Scholar 

  68. Sh. Chen, Q. Guo, S. Xu, M. R. Belic, Y. Zhao, D. Zhao, and J. He, Vortex solitons in Bose–Einstein condensates with spin–orbit coupling and Gaussian optical lattices, Appl. Math. Lett. 92, 15 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  69. Z. He, Z. Zhang, Sh. Zhu, and W. Liu, Oscillation and fission behavior of bright–bright solitons in two-species Bose–Einstein condensates trapped in an optical potential, Acta Physica Sinica 63, 190502 (2014)

    Article  Google Scholar 

  70. Z. Li and Q. Li, Dark soliton interaction of spinor Bose–Einstein condensates in an optical lattice, Ann. Phys. 322(8), 1961 (2007)

    Article  ADS  MATH  Google Scholar 

  71. Ch. Song, J. Li, and F. Zong, Dynamic stability and manipulation of bright matter-wave solitons by optical lattices in Bose–Einstein condensates, Chin. Phys. B 21(2), 020306 (2012)

    Article  ADS  Google Scholar 

  72. Z. D. Li, P. B. He, L. Li, J. Q. Liang, and W. M. Liu, Magnetic soliton and soliton collisions of spinor Bose–Einstein condensates in an optical lattice, Phys. Rev. A 71(5), 053611 (2005)

    Article  ADS  Google Scholar 

  73. A. Muñoz Mateo, V. Delgado, M. Guilleumas, R. Mayol, and J. Brand, Nonlinear waves of Bose–Einstein condensates in rotating ring-lattice potentials, Phys. Rev. A 99(2), 023630 (2019)

    Article  ADS  Google Scholar 

  74. X. Zhao, Y. Zhang, and W. Liu, Magnetic excitation of ultra-cold atoms trapped in optical lattice, Acta Physica Sinica 68, 043703 (2019)

    Article  Google Scholar 

  75. G. Verma, U. D. Rapol, and R. Nath, Generation of dark solitons and their instability dynamics in two-dimensional condensates, Phys. Rev. A 95(4), 043618 (2017)

    Article  ADS  Google Scholar 

  76. Z. Fan, J. Mai, Z. Chen, M. Xie, and Z. Luo, Matterwave soliton buffer realized by a tailored one-dimensional lattice, Mod. Phys. Lett. B 32(06), 1850070 (2018)

    Article  ADS  Google Scholar 

  77. H. Li, S. Xu, M. R. Belic, and J. Cheng, Threedimensional solitons in Bose–Einstein condensates with spin–orbit coupling and Bessel optical lattices, Phys. Rev. A 98(3), 033827 (2018)

    Article  ADS  Google Scholar 

  78. Z. Zhou, H. Zhong, B. Zhu, F. Xiao, K. Zhu, and J. Tan, Collision dynamics of dissipative matter-wave solitons in a perturbed optical lattice, Chin. Phys. Lett. 33(11), 110301 (2016)

    Article  ADS  Google Scholar 

  79. L. Dong, W. Qi, P. Peng, L. Wang, H. Zhou, and C. Huang, Multi-stable quantum droplets in optical lattice, Nonlinear Dynamics, 2020

    Google Scholar 

  80. A. Mock, Paritytime-symmetry breaking in two-dimensional photonic crystals: Square lattice, Phys. Rev. A 93(6), 063812 (2016)

    Article  ADS  Google Scholar 

  81. L. Salasnich and F. Toigo, Pair condensation in the BCS–BEC crossover of ultracold atoms loaded onto a twodimensional square lattice, Phys. Rev. A 86(2), 023619 (2012)

    Article  ADS  Google Scholar 

  82. R. Zaera, J. Vila, J. Fernandez-Saez, and M. Ruzzene, Propagation of solitons in a two-dimensional nonlinear square lattice, Int. J. Non-linear Mech. 106, 188 (2018)

    Article  ADS  Google Scholar 

  83. Zh. Niu, Y. Tai, J. Shi, and W. Zhang, Bose–Einstein condensates in an eightfold symmetric optical lattice, Chin. Phys. B 29(5), 056103 (2020)

    Article  ADS  Google Scholar 

  84. H. Chen, Y. Liu, Q. Zhang, Y. Shi, W. Pang, and Y. Li, Dipolar matter-wave solitons in two-dimensional anisotropic discrete lattices, Phys. Rev. A 93(5), 053608 (2016)

    Article  ADS  Google Scholar 

  85. Y. Gao and S. Chu, Optical induction of non-diffracting discrete photonic lattice, Superlattices Microstruct. 78, 163 (2015)

    Article  Google Scholar 

  86. K. Xie, A. D. Boardman, Q. Li, Z. Shi, H. Jiang, H. Xia, Z. Hu, J. Zhang, W. Zhang, Q. Mao, L. Hu, T. Yang, F. Wen, and E. Wang, Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices, Opt. Express 25(24), 30349 (2017)

    Article  ADS  Google Scholar 

  87. M. Metcalf, G. Chern, M. D. Ventra, and C. Chien, Matter-wave propagation in optical lattices: Geometrical and flat-band effects, J. Phys. At. Mol. Opt. Phys. 49(7), 075301 (2016)

    Article  ADS  Google Scholar 

  88. D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belic, and M. Xiao, Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice, Ann. Phys. (Berlin) 529(9), 1700149 (2017)

    Article  ADS  Google Scholar 

  89. Q. E. Hoq, P. G. Kevrekidis, and A. R. Bishop, Discrete solitons and vortices in anisotropic hexagonal and honeycomb lattices, J. Opt. 18(2), 024008 (2016)

    Article  ADS  Google Scholar 

  90. L. H. Haddad, C. M. Weaver, and L. D. Carr, The nonlinear Dirac equation in Bose–Einstein condensates (I): Relativistic solitons in armchair nanoribbon optical lattices, New J. Phys. 17(6), 063033 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. V. E. Vekslerchik, Solitons of a vector model on the honeycomb lattice, J. Phys. A Math. Theor. 49(45), 455202 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  92. R. Zhong, N. Huang, H. Li, H. He, J. Lü, C. Huang, and Z. P. Chen, Matter-wave solitons supported by quadrupole quadrupole interactions and anisotropic discrete lattices, Int. J. Mod. Phys. B 32(09), 1850107 (2018)

    Article  ADS  MATH  Google Scholar 

  93. Q. Wang and Z. Deng, Multi-pole solitons in nonlocal nonlinear media with fourth-order diffraction, Results in Physics 17, 103056 (2020)

    Article  Google Scholar 

  94. H. Wang, X. Ren, J. Huang, and Y. Weng, Evolution of vortex and quadrupole solitons in the complex potentials with saturable nonlinearity, J. Opt. 20(12), 125504 (2018)

    Article  ADS  Google Scholar 

  95. G. Chen, Y. Liu, and H. Wang, Mixed-mode solitons in quadrupolar BECs with spin–orbit coupling, Commun. Nonlinear Sci. Numer. Simul. 48, 318 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  96. Y. V. Kartashov and D. A. Zezyulin, Stable multiring and rotating solitons in two-dimensional spin–orbit-coupled, Bose–Einstein condensates with a radially periodic potential, Phys. Rev. Lett. 122(12), 123201 (2019)

    Article  ADS  Google Scholar 

  97. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, New York: Cambridge University Press, 2002

    Google Scholar 

  98. L. M. Chiofalo, S. Succi, and P. M. Tosi, Ground state of trapped interacting Bose–Einstein condensates by an explicit imaginary time algorithm, Phys. Rev. E 62(5), 7438 (2000)

    Article  ADS  Google Scholar 

  99. J. Yang and T. I. Lakoba, Accelerated imaginary-time evolution methods for the computation of solitary waves, Stud. Appl. Math. 120, 265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  100. I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, Finite-band solitons in the Kronig–Penney model with the cubic–quintic nonlinearity, Phys. Rev. E 71, 016613 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  101. R. Driben, B. A. Malomed, A. Gubeskys, and J. Zyss, Cubic–quintic solitons in the checkerboard potential, Phys. Rev. E 76, 066604 (2007)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11905032 and 11874112, the Key Research Projects of General Colleges in Guangdong Province through Grant No. 2019KZDXM001, the Foundation for Distinguished Young Talents in Higher Education of Guangdong through Grant No. 2018KQNCX279, and the Special Funds for the Cultivation of Guangdong College Students Scientific and Technological Innovation (No. xsjj202005zra01).

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  1. School of Physics and Optoelectronic Engineering, Foshan University, Foshan, 528000, China

    Yi-Yin Zheng, Shan-Tong Chen, Zhi-Peng Huang, Shi-Xuan Dai, Bin Liu, Yong-Yao Li & Shu-Rong Wang

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  1. Yi-Yin Zheng
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Zheng, YY., Chen, ST., Huang, ZP. et al. Quantum droplets in two-dimensional optical lattices. Front. Phys. 16, 22501 (2021). https://doi.org/10.1007/s11467-020-1011-3

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  • DOI: https://doi.org/10.1007/s11467-020-1011-3

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