- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Second-order density-dependent geometric vector potentials: Modifying chiral soliton and ring-soliton dynamics in Bose-Einstein condensates with nonlocal interactions

Christopher Gaafele1,*, Conrad Bertrand Tabi1,†, and Timoléon Crépín Kofané1,2,‡

  • *Contact author: christopher.gaafele@studentmail.biust.ac.bw
  • †Contact author: tabic@biust.ac.bw
  • ‡Contact author: kofanet@biust.ac.bw
PDF

Phys. Rev. E 113, 054211 – Published 11 May, 2026

DOI: https://doi.org/10.1103/jncz-jm9q

Abstract

We consider a cigar-shaped Bose-Einstein condensate (BEC) of two-level atoms subjected to laser-induced Rabi coupling. By expanding the geometric vector potential to second order in the density-dependent fields and developing a quasi-one-dimensional (1D) reduction of the full three-dimensional problem via wave-function factorization, we derive an extended chiral Gross-Pitaevskii (GP) equation. We also derive traveling-wave solutions of the extended chiral GP equation using a projection (orthogonality) method, obtaining both bright and dark chiral solitons whose amplitudes and widths depend explicitly on the propagation direction. The chiral character of these solutions is confirmed through analytical analysis and numerical simulations. A BEC confined in a ring-shaped trap is also analyzed, leading to chiral soliton ring geometries. We show that a subtle interplay among density-dependent gauge potentials, confinement-induced interaction corrections, and higher-order current nonlinearities can give rise to alternative topological states, including chiral solitons and chiral ring solitons, in quasi-1D ultracold Bose gases. These results illustrate how higher-order corrections enrich BEC dynamics and provide means to control spatiotemporal structures in density-dependent condensates.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (58)

  1. M. H. Anderson, J. R. Ensher, M. R. Mathews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269, 198 (1995).
  2. K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75, 3969 (1995).
  3. C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75, 1687 (1995).
  4. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Dark solitons in Bose-Einstein condensates, Phys. Rev. Lett. 83, 5198 (1999).
  5. B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, Watching dark solitons decay into vortex rings in a Bose-Einstein condensate, Phys. Rev. Lett. 86, 2926 (2001).
  6. J. Denschlag, J. E. Simsarian, D., L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, Generating solitons by phase engineering of a Bose-Einstein condensate, Science 287, 97 (2000).
  7. K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Formation and propagation of matter-wave soliton trains, Nature ( London) 417, 150 (2002).
  8. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Casting, and C. Salomon, Formation of a matter-wave bright soliton, Science 296, 1290 (2002).
  9. G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions, Phys. Rev. A 67, 063610 (2003).
  10. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett. 83, 2498 (1999).
  11. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett. 84, 806 (2000).
  12. S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Görlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, Observation of vortex phase singularities in Bose-Einstein condensates, Phys. Rev. Lett. 87, 080402 (2001).
  13. J. R. Abo-Shaer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science 292, 476 (2001).
  14. P. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell, Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates, Phys. Rev. Lett. 89, 100403 (2002).
  15. E. Wamba, A. Mohamadou, and T. C. Kofane, Modulational instability of a trapped Bose-Einstein condensate with two- and three-body interactions, Phys. Rev. E 77, 046216 (2008).
  16. E. Wamba, A. Mohamadou, and T. C. Kofane, A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap, J. Phys. B 41, 225403 (2008).
  17. B. P. Anderson and M. A. Kasevich, Macroscopic quantum interference from atomic tunnel arrays, Science 282, 1686 (1998).
  18. S. Liu, H. Xiong, Z. Xu, and G. Huang, Interference patterns of Bose-condensed gases in a two-dimensional optical lattice, J. Phys. B 36, 2083 (2003).
  19. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature (London) 415, 39 (2002).
  20. D. van Oosten, P. van der Straten, and H. T. C. Stoof, Quantum phases in an optical lattice, Phys. Rev. A 63, 053601 (2001).
  21. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40, 546 (1989).
  22. J. K. Pachos and P. L. Knight, Quantum computation with a one-dimensional optical lattice, Phys. Rev. Lett. 91, 107902 (2003).
  23. M. Aidelsburger, M. Atala, S. Nascimbéne, S. Trotzky, Y.-A. Chen, and I. Bloch, Experimental realization of strong effective magnetic fields in an optical lattice, Phys. Rev. Lett. 107, 255301 (2011).
  24. N. Goldman, J. Beugnon, and F. Gerbier, Identifying topological edge states in 2D optical lattices using light scattering, Eur. Phys. J. Spec. Top. 217, 135 (2013).
  25. Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, Synthetic magnetic fields for ultracold neutral atoms, Nature (London) 462, 628 (2009).
  26. Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose-Einstein condensate in a uniform light-induced vector potential, Phys. Rev. Lett. 102, 130401 (2009).
  27. J. Struck, C. Ölschläger, R. L. Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, Quantum simulation of frustrated classical magnetism in triangular optical lattices, Science 333, 996 (2011).
  28. J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011).
  29. Q. Du, Quantized vortices in BEC and superconductors, Proc. Appl. Math. Mech. 7, 1023901 (2007).
  30. N. Hatano, R. Shirasaki, and H. Nakamura, Non-Abelian gauge field theory of the spin-orbit interaction and a perfect spin filter, Phys. Rev. A 75, 032107 (2007).
  31. S. Pradhan, Hofstadter butterfly in the FalicovKimball model on some finite 2D lattices, J. Phys. Condens. Matter. 28, 505502 (2016).
  32. M. J. Edmonds, M. Valiente, G. Juzeliūnas, L. Santos, and P. Öhberg, Simulating an interacting gauge theory with ultracold Bose gases, Phys. Rev. Lett. 110, 085301 (2013).
  33. R. J. Dingwall and P. Öhberg, Stability of matter-wave solitons in a density-dependent gauge theory, Phys. Rev. A 99, 023609 (2019).
  34. P. Öhberg and E. M. Wright, Quantum time crystals and interacting gauge theories in atomic Bose-Einstein condensates, Phys. Rev. Lett. 123, 250402 (2019).
  35. U. Aglietti, L. Griguolo, R. Jackiw, S.-Y. Pi, and D. Seminara, Anyons and chiral solitons on a line, Phys. Rev. Lett. 77, 4406 (1996).
  36. E. Harikumar, C. N. Kumar, and M. Sivakumar, Chiral solitons in a current coupled Schrödinger equation with self-interaction, Phys. Rev. D 58, 107703 (1998).
  37. A. Syrwid, A. Kosior, and K. Sacha, Comment on “quantum time crystals and interacting gauge theories in atomic Bose-Einstein condensates”, Phys. Rev. Lett. 124, 178901 (2020).
  38. A. Syrwid, A. Kosior, and K. Sacha, Lack of a genuine time crystal in a chiral soliton model, Phys. Rev. Res. 2, 032038(R) (2020).
  39. P. Öhberg and E. M. Wright, Öhberg and Wright reply:, Phys. Rev. Lett. 124, 178902 (2020).
  40. R. Gao, X. Qiao, Y-E. Ma, Y. Jian, A-X. Zhang, and J-K. Xue, Chiral matter-wave soliton in a Bose-Einstein condensate under density-dependent gauge potential, Phys. Lett. A 446, 128283 (2022).
  41. R. Gao, X. Qiao, Y-E. Ma, Y. Jian, A-X. Zhang, and J-K. Xue, Interaction of chiral solitons in a Bose-Einstein condensate under density-dependent gauge potential, Europhys. Lett. 141, 55003 (2023).
  42. I. A. Bhat, S. Sivaprakasam, and B. A. Malomed, Modulational instability and soliton generation in chiral Bose-Einstein condensates with zero-energy nonlinearity, Phys. Rev. E 103, 032206 (2021).
  43. C. B. Tabi, C. Gaafele, E. Wamba, and T. C. Kofané, Nonlinear chiral modes in Bose-Einstein condensates with zero-energy and higher-order residual nonlinearities, Phys. Lett. A 559, 130902 (2025).
  44. H. Fu, Y. Wang, and B. Gao, Beyond the Fermi pseudopotential: A modified Gross-Pitaevskii equation, Phys. Rev. A 67, 053612 (2003).
  45. A. Collin, P. Massignan, C. J. Pethick. Energy-dependent effective interactions for dilute many-body systems, Phys. Rev. A 75, 013615 (2007).
  46. M. J. Edmonds, Few- to many-vortex states of density-angular-momentum–coupled Bose-Einstein condensates, Phys. Rev. A 104, 043310 (2021).
  47. R. Boral, S. K. Sarkar, M. J. Edmonds, P. Muruganandam, and P. K. Mishra, Excitation spectrum of vortex-lattice modes in a rotating condensate with a density-dependent gauge potential, Phys. Rev. A 112, 043308 (2025).
  48. M. Edmonds and M. Nitta, Vortex patterns of atomic Bose-Einstein condensates in a density-dependent gauge potential, Phys. Rev. A 102, 011303(R) (2020).
  49. I. A. Bhat and B. Dey, Hidden vortices and Feynman rule in Bose-Einstein condensates with density-dependent gauge potential, Phys. Rev. E 110, 024208 (2024).
  50. I. A. Bhat, T. Mithun, and B. Dey, Vortex nucleation in rotating Bose-Einstein condensates with density-dependent gauge potential, Phys. Rev. E 107, 044210 (2023).
  51. C. S. Chisholm, A. Frölian, E. Neri, R. Ramos, L. Tarruell, and A. Celi, Realizing a 1D topological gauge theory in an optically dressed BEC, Nature (London) 608, 293 (2022).
  52. C. S. Chisholm, A. Frölian, E. Neri, R. Ramos, and A. Celi, Encoding a one-dimensional topological gauge theory in a Raman-coupled Bose-Einstein condensate, Phys. Rev. Res. 4, 043088 (2022).
  53. Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys. 61, 763 (1989).
  54. Y. Kodama and M. J. Ablowitz, Perturbations of solitons and solitary waves, Stud. Appl. Math. 64, 225 (1981).
  55. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  56. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge University Press, Cambridge, England, 2008).
  57. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity, apt_slash_aptnewblock (Oxford University Press, Oxford, 2016).
  58. M. J. Edmonds, M. Valiente, and P. Öhberg, Elementary excitations of chiral Bose-Einstein condensates, Europhys. Lett. 110, 36004 (2015).

Outline

Information

Phys. Rev. E 113, 054211– Published 11 May, 2026
  • Received 30 November 2025
  • Accepted 20 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation