Fundamental modes in waveguide pipe twisted by saturated double-well potential

Abstract

We study fundamental modes trapped in a rotating ring with a saturated nonlinear double-well potential. This model, which is based on the nonlinear Schrödinger equation, can be constructed in a twisted waveguide pipe in terms of light propagation, or in a Bose–Einstein condensate (BEC) loaded into a toroidal trap under a combination of a rotating-out-of-phase linear potential and nonlinear pseudopotential induced by means of a rotating optical field and the Feshbach resonance. Three types of fundamental modes are identified in this model, one symmetric and the other two asymmetric. The shape and stability of the modes and the transitions between different modes are investigated in the first rotational Brillouin zone. A similar model used a Kerr medium to build its nonlinear potential, but we replace it with a saturated nonlinear medium. The model exhibits not only symmetry breaking, but also symmetry recovery. A specific type of unstable asymmetric mode is also found, and the evolution of the unstable asymmetric mode features Josephson oscillation between two linear wells. By considering the model as a configuration of a BEC system, the ground state mode is identified among these three types, which characterize a specific distribution of the BEC atoms around the trap.

Keywords

Twisted double-well waveguide / saturated nonlinear potential / symmetry breaking / symmetry recovery

References

[1] Q. Gong and X. Hu, Ultrafast photonic crystal optical switching, Front. Phys. 1, 171 (2006)

[2] Y. Liu, F. Qin, F. Zhou, Q. Meng, D. Zhang, and Z. Li, Ultrafast optical switching in Kerr nonlinear photonic crystals, Front. Phys. 5, 244 (2010)

[3] M. Shen, B. Li, L. Ge, W. Chen, and D. Wu, Stability of vortex solitons under competing local and nonlocal cubic nonlinearities, Opt. Commun. 338, 27 (2015)

[4] Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, Modulation instability in metamaterials with saturable nonlinearity, J. Opt. Soc. Am. B 28(4), 908 (2011)

[5] T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, Travelling solitary waves in the discrete Schrodinger equation with saturable nonlinearity: Existence, stability and dynamics, Physica D 237(4), 551 (2008)

[6] F. Setzpfandt, A. A. Sukhorukov, and T. Pertsch, Discrete quadratic solitons with competing secondharmonic components, Phys. Rev. A 84(5), 053843 (2011)

[7] V. Lutsky and B. A. Malomed, One- and twodimensional solitons supported by singular modulation of quadratic nonlinearity, Phys. Rev. A 91(2), 023815 (2015)

[8] M. Shen, Y. Y. Lin, C. C. Jeng, and R. K. Lee, Vortex pairs in nonlocal nonlinear media, J. Opt. 14(6), 065204 (2012)

[9] M. Shen, J. S. Gao, and L. J. Ge, Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media, Sci. Rep. 5, 9814 (2015)

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

[11] Y. S. Kivshar, Nonlinear Tamm states and surface effects in periodic photonic structures, Laser Phys. Lett. 5(10), 703 (2008)

[12] Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, Soliton topology versus discrete symmetry in optical lattices, Phys. Rev. Lett. 95(12), 123902 (2005)

[13] Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, Defect-mediated discrete solitons in optically induced photorefractive lattices, Phys. Rev. A 80(4), 043824 (2009)

[14] W. M. Liu, B. Wu, and Q. Niu, Nonlinear effects in interference of Bose–Einstein condensates, Phys. Rev. Lett. 84(11), 2294 (2000)

[15] Z. X. Liang, Z. D. Zhang, and W. M. Liu, Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential, Phys. Rev. Lett. 94(5), 050402 (2005)

[16] A. C. Ji, W. M. Liu, J. L. Song, and F. Zhou, Dynamical creation of fractionalized vortices and vortex lattices, Phys. Rev. Lett. 101(1), 010402 (2008)

[17] E. A. Ostrovskaya and Y. S. Kivshar, Matter-wave gap solitons in atomic band-gap structures, Phys. Rev. Lett. 90(16), 160407 (2003)

[18] N. K. Efremidis and D. N. Christodoulides, Lattice solitons in Bose–Einstein condensates, Phys. Rev. A 67(6), 063608 (2003)

[19] H. Sakaguchi and B. A. Malomed, Matter-wave solitons in nonlinear optical lattices, Phys. Rev. E 72(4), 046610 (2005)

[20] B. B. Baizakov, B. A. Malomed, and M. Salerno, Matterwave solitons in radially periodic potentials, Phys. Rev. E 74(6), 066615 (2006)

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

[22] H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010)

[23] Y. Li, B. A. Malomed, M. Feng, and J. Zhou, Double symmetry breaking of solitons in one-dimensional virtual photonic crystals, Phys. Rev. A 83(5), 053832 (2011)

[24] Y. Li, B. A. Malomed, J. Wu, W. Pang, S. Wang, and J. Zhou, Quasicompactons in inverted nonlinear photonic crystals, Phys. Rev. A 84(4), 043839 (2011)

[25] Y. Li, W. Pang, S. Fu, and B. A. Malomed, Twocomponent solitons with a spatially modulated linear coupling: Inverted photonic crystals and fused couplers, Phys. Rev. A 85(5), 053821 (2012)

[26] G. Chen, S. Zhang, and M. Wu, Optical solitons in a trinal-chennel inverted nonlinear photonic crystal, J. Nonlinear Opt. Phys. Mater. 22(01), 1350012 (2013)

[27] W. Pang, H. Guo, G. Chen, and Z. Mai, Symmetry breaking bifurcation of two-component soliton modes in an inverted nonlinear random lattice, J. Phys. Soc. Jpn. 83(3), 034402 (2014)

[28] J. Deng, J. Liu, S. Tan, Z. Huang, and Y. Li, Propagation dynamic of a Gaussian in the inverted nonlinear photonic crystals, Optik (Stuttg.) 125(15), 4088 (2014)

[29] H. Guo, Z. Chen, J. Liu, and Y. Li, Fundamental modes in a waveguide pipe twisted by inverted nonlinear doublewell potential, Laser Phys. 24(4), 045403 (2014)

[30] A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica, Opt. Express 14(13), 6055 (2006)

[31] D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, Nonlinear refractive index of fs-laser-written waveguides in fused silica, Opt. Express 14(6), 2151 (2006)

[32] Y. Li, B. A. Malomed, M. Feng, and J. Zhou, Arrayed and checkerboard optical waveguides controlled by the electromagnetically induced transparency, Phys. Rev. A 82(6), 063813 (2010)

[33] J. Wu, M. Feng, W. Pang, S. Fu, and Y. Li, The transmission of quasi-discrete solitons in resonant waveguide arrays activated by the electromagnetically induced transparency, J. Nonlinear Opt. Phys. Mater. 20(02), 193 (2011)

[34] W. Pang, J. Wu, Z. Yuan, Y. Liu, and G. Chen, Lattice solitons in optical lattice controlled by electromagnetically induced transparency, J. Phys. Soc. Jpn. 80(11), 113401 (2011)

[35] H. Saito and M. Ueda, Emergence of Bloch bands in a rotating Bose–Einstein condensate, Phys. Rev. Lett. 93(22), 220402 (2004)

[36] S. Schwartz, M. Cozzini, C. Menotti, I. Carusotto, P. Bouyer, and S. Stringari, One-dimensional description of a Bose–Einstein condensate in a rotating closed-loop waveguide, New J. Phys. 8(8), 162 (2006)

[37] L. Wen, H. Xiong, and B. Wu, Hidden vortices in a Bose–Einstein condensate in a rotating double-well potential, Phys. Rev. A 82(5), 053627 (2010)

[38] Y. Li, W. Pang, and B. A. Malomed, Nonlinear modes and symmetry breaking in rotating double-well potentials, Phys. Rev. A 86(2), 023832 (2012)

[39] G. Chen, Z. Luo, J. Wu, and M. Wu, Switch between the types of the symmetry breaking bifurcation in optically induced photorefractive rotational double-well potential, J. Phys. Soc. Jpn. 82(3), 034401 (2013)

[40] Z. Luo, Y. Li, W. Pang, Y. Liu, and X. Wang, Double symmetry breaking of modes in dual-core rotating system, J. Phys. Soc. Jpn. 82(12), 124401 (2013)

[41] W. Pang, S. Fu, J. Wu, Y. Li, and Z. Mai, Nonlinear mode in rotating double-well potential with parity-time symmetry, Chin. Phys. B 23(10), 104214 (2014)

[42] J. Li, B. Liang, Y. Liu, P. Zhang, J. Zhou, S. O. Klimonsky, A. S. Slesarev, Y. D. Tretyakov, L. O’Faolain, and T. F. Krauss, Photonic crystal formed by the imaginary part of the refractive index, Adv. Mater. 22(24), 2676 (2010)

[43] M. Feng, Y. Liu, Y. Li, X. Xie, and J. Zhou, Light propagation in a resonantly absorbing waveguide array, Opt. Express 19(8), 7222 (2011)

[44] B. Liang, Y. Liu, J. Li, L. Song, Y. Li, J. Zhou, and K. S. Wong, Frabication of large-size photonic crystals by holographic lithography using a lens array, J. Micromech. Microeng. 22(3), 035013 (2012)

[45] Y. K. Liu, S. C. Wang, Y. Y. Li, L. Y. Song, X. S. Xie, M. N. Feng, Z. M. Xiao, S. Z. Deng, J. Y. Zhou, J. T. Li, K. Sing Wong, and T. F. Krauss, Effcient color routing with a dispersion-controlled waveguide array, Light Sci. Appl. 2(2), e52 (2013)

[46] L. M. Chiofalo, S. Succi, and P. M. Tosi, Ground state of trapped interacting Bose–Einstein condensates by an explicit imaginarytime algorithm, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7438 (2000)

[47] J. Yang and T. I. Lakoba, Accelerated imaginarytime evolution methods for the computation of solitary waves, Stud. Appl. Math. 120(3), 265 (2008)

[48] L. Albuch and B. A. Malomed, Transitions between symmetric and asymmetric solitons in dual-core systems with cubicquintic nonlinearity, Math. Comput. Simul. 74(4-5), 312 (2007)

[49] G. Mazzarella and L. Salasnich, Spontaneous symmetry breaking and collapse in bosonic Josephson junctions, Phys. Rev. A 82(3), 033611 (2010)

[50] L. K. Lim, T. Troppenz, and C. M. Smith, Internal Josephson oscillations for distinct momenta Bose– Einstein condensates, Phys. Rev. A 84(5), 053609 (2011)

[51] J. Gillet, M. A. Garcia-March, T. Busch, and F. Sols, Tunneling, self-trapping, and manipulation of higher modes of a Bose–Einstein condensate in a double well, Phys. Rev. A 89, 023614 (2014)

[52] G. Szirmai, G. Mazzarella, and L. Salasnich, Tunneling dynamics of bosonic Josephson junctions assisted by a cavity field, Phys. Rev. A 91(2), 023601 (2015)

[53] J. Javanainen and R. Rajapakse, Bayesian inference to characterize Josephson oscillations in a double-well trap, Phys. Rev. A 92, 023613 (2015)

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

[55] R. Nath, P. Pedri, and L. Santos, Stability of dark solitons in three dimensional dipolar Bose–Einstein condensates, Phys. Rev. Lett. 101(21), 210402 (2008)

[56] I. Tikhonenkov, B. A. Malomed, and A. Vardi, Vortex solitons in dipolar Bose–Einstein condensates, Phys. Rev. A 78(4), 043614 (2008)

[57] Y. Li, J. Liu, W. Pang, and B. A. Malomed, Matterwave solitons supported by field-induced dipole-dipole repulsion with spatially modulated strength, Phys. Rev. A 88(5), 053630 (2013)

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

[59] S. K. Adhikari, Self-trapping of a dipolar Bose–Einstein condensate in a double well, Phys. Rev. A 89(4), 043609 (2014)

[60] 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)

[61] J. M. Junquera-Hernández, J. Sánchez-Marín, and D. Maynau, Molecular electric quadrupole moments calculated with matrix dressed SDCI, Chem. Phys. Lett. 359(3-4), 343 (2002)

[62] Y. Li, J. Liu, W. Pang, and B. A. Malomed, Lattice solitons with quadrupolar intersite interactions, Phys. Rev. A 88(6), 063635 (2013)

[63] 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)

[64] D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature 424(6950), 817 (2003)

[65] F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep. 463(1-3), 1 (2008)

[66] S. Flach and A. V. Gorbach, Discrete breathers Advances in theory and applications, Phys. Rep. 467(1-3), 1 (2008)

[67] X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, Discrete solitons and scattering of lattice wave in guiding arrays with a nonlinear PT-symmetric defect, Opt. Express 22(11), 13927 (2014)

[68] Z. Chen, J. Liu, S. Fu, Y. Li, and B. A. Malomed, Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers, Opt. Express 22(24), 29679 (2014)

[69] G. Chen, H. Huang, and M. Wu, Solitary vortices in two dimensional waveguide matrix, J. Nonlinear Opt. Phys. Mater. 24(01), 1550012 (2015)

[70] Z. Mai, S. Fu, J. Wu, and Y. Li, Discrete soliton in waveguide arrays with long-range linearly coupled effect, J. Phys. Soc. Jpn. 83(3), 034404 (2014)

[71] Z. Mai, W. Pang, J. Wu, and Y. Li, Symmetry breaking of discrete solitons in two-component waveguide arrays with long-range linearly coupled effect, J. Phys. Soc. Jpn. 84(1), 014401 (2015)

[72] J. Huang, H. Li, X. Zhang, and Y. Li, Transmission, reflection, scattering, and trapping of traveling discrete solitons by C and V point defects, Front. Phys. 10(2), 104201 (2015)

[73] Z. Chen, J. Huang, J. Chai, X. Zhang, Y. Li, and B. A. Malomed, Discrete solitons in self-defocusing systems with PT-symmetric defects, Phys. Rev. A 91(5), 053821 (2015)

[74] U. Al Khawaja, and A. A. Sukhorukov, Unidirectional ow of discrete solitons in waveguide arrays, Opt. Lett. 40(12), 2719 (2015)

[75] Z. Peng, H. Li, Z. Fan, X. Zhang, and Y. Li, All optical diode realized by one dimensional waveguide array, J. Nonlinear Opt. Phys. Mater. 24(02), 1550022 (2015)

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