- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Analytical study of a finite-range impurity in a one-dimensional Bose gas

T. Alper Yoğurt* and Matthew T. Eiles†

  • *Contact author: ayogurt@pks.mpg.de
  • †Contact author: meiles@pks.mpg.de
PDF

Phys. Rev. A 112, 053306 – Published 7 November, 2025

DOI: https://doi.org/10.1103/gqd6-t881

Abstract

One-dimensional Bose gases present an interesting setting to study the physics of Bose polarons, as density fluctuations play an enhanced role due to reduced dimensionality. Theoretical descriptions of this system have predominantly relied on contact pseudopotentials to model the impurity-bath interaction, leading to unphysical results in the strongly coupled limit. In this work, we analytically solve the Gross-Pitaevskii equation, using a square-well potential instead of a zero-range potential, for the ground-state wave function of a static impurity. We compute perturbative corrections arising from infinitesimally slow impurity motion. The polaron energy and effective mass remain finite in the strongly coupled regime, in contrast to the divergent behavior obtained using a contact potential. In this limit, we characterize the polaron properties in terms of the dimensionless ratio w̃≡w/ξ between the interaction range w of the impurity-bath potential and the coherence length ξ of the Bose gas. The effective mass exhibits a 1/w̃ scaling. The energy of the attractive polaron scales as −1/w̃3, whereas the repulsive polaron features subleading corrections to the dark soliton energy at the order w̃3.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (57)

  1. J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, F. Minardi, A. Kantian, and T. Giamarchi, Quantum dynamics of impurities in a one-dimensional Bose gas, Phys. Rev. A 85, 023623 (2012).
  2. N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M. Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, and J. J. Arlt, Observation of attractive and repulsive polarons in a Bose-Einstein condensate, Phys. Rev. Lett. 117, 055302 (2016).
  3. M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A. Cornell, and D. S. Jin, Bose polarons in the strongly interacting regime, Phys. Rev. Lett. 117, 055301 (2016).
  4. F. Grusdt, N. Mostaan, E. Demler, and L. A. P. Ardila, Impurities and polarons in bosonic quantum gases: A review on recent progress, Rep. Prog. Phys. 88, 066401 (2025).
  5. P. Massignan, R. Schmidt, G. E. Astrakharchik, A. İmamoglu, M. Zwierlein, J. J. Arlt, and G. M. Bruun, Polarons in atomic gases and two-dimensional semiconductors, arXiv:2501.09618.
  6. H. Fröhlich, Electrons in lattice fields, Adv. Phys. 3, 325 (1954).
  7. T. D. Lee, F. E. Low, and D. Pines, The motion of slow electrons in a polar crystal, Phys. Rev. 90, 297 (1953).
  8. J. T. Devreese, Fröhlich polarons: Lecture course including detailed theoretical derivations – 10th edition, arXiv:1611.06122.
  9. W. Li and S. Das Sarma, Variational study of polarons in Bose-Einstein condensates, Phys. Rev. A 90, 013618 (2014).
  10. A. Shashi, F. Grusdt, D. A. Abanin, and E. Demler, Radio-frequency spectroscopy of polarons in ultracold Bose gases, Phys. Rev. A 89, 053617 (2014).
  11. F. Grusdt and E. Demler, New theoretical approaches to Bose polarons, in Quantum Matter at Ultralow Temperatures (IOS, Amsterdam, 2016), pp. 325–411.
  12. Y. E. Shchadilova, R. Schmidt, F. Grusdt, and E. Demler, Quantum dynamics of ultracold Bose polarons, Phys. Rev. Lett. 117, 113002 (2016).
  13. F. Grusdt, G. E. Astrakharchik, and E. Demler, Bose polarons in ultracold atoms in one dimension: Beyond the Fröhlich paradigm, New J. Phys. 19, 103035 (2017).
  14. B. Kain and H. Y. Ling, Analytical study of static beyond-Fröhlich Bose polarons in one dimension, Phys. Rev. A 98, 033610 (2018).
  15. S. Van Loon, W. Casteels, and J. Tempere, Ground-state properties of interacting Bose polarons, Phys. Rev. A 98, 063631 (2018).
  16. F. Scazza, M. Zaccanti, P. Massignan, M. M. Parish, and J. Levinsen, Repulsive Fermi and Bose polarons in quantum gases, Atoms 10, 55 (2022).
  17. J. Jager, R. Barnett, M. Will, and M. Fleischhauer, Strong-coupling Bose polarons in one dimension: Condensate deformation and modified Bogoliubov phonons, Phys. Rev. Res. 2, 033142 (2020).
  18. P. Massignan, N. Yegovtsev, and V. Gurarie, Universal aspects of a strongly interacting impurity in a dilute Bose condensate, Phys. Rev. Lett. 126, 123403 (2021).
  19. N. Yegovtsev, P. Massignan, and V. Gurarie, Strongly interacting impurities in a dilute Bose condensate, Phys. Rev. A 106, 033305 (2022).
  20. N. Yegovtsev and V. Gurarie, Effective mass and interaction energy of heavy Bose polarons at unitarity, Phys. Rev. A 108, L051301 (2023).
  21. R. Alhyder, V. Colussi, M. Čufar, J. Brand, A. Recati, and G. M. Bruun, Lattice Bose polarons at strong coupling and quantum criticality, SciPost Phys. 19, 002 (2025).
  22. T. Hartweg, T. Gupta, and G. Pupillo, The Bose-Hubbard polaron from weak to strong coupling, arXiv:2508.00486.
  23. T. Giamarchi, Quantum Physics in One Dimension, International Series of Monographs on Physics Vol. 121 (Clarendon, Oxford, 2003).
  24. M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys. 83, 1405 (2011).
  25. F. M. Cucchietti and E. Timmermans, Strong-coupling polarons in dilute gas Bose-Einstein condensates, Phys. Rev. Lett. 96, 210401 (2006).
  26. M. Bruderer, W. Bao, and D. Jaksch, Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling, Europhys. Lett. 82, 30004 (2008).
  27. K. Sacha and E. Timmermans, Self-localized impurities embedded in a one-dimensional Bose-Einstein condensate and their quantum fluctuations, Phys. Rev. A 73, 063604 (2006).
  28. D. Breu, E. Vidal Marcos, M. Will, and M. Fleischhauer, Impurities in a trapped one-dimensional Bose gas of arbitrary interaction strength: Localization-delocalization transition and absence of self-localization, Phys. Rev. A 111, 023326 (2025).
  29. A. G. Volosniev and H.-W. Hammer, Analytical approach to the Bose-polaron problem in one dimension, Phys. Rev. A 96, 031601(R) (2017).
  30. M. Will, G. E. Astrakharchik, and M. Fleischhauer, Polaron interactions and bipolarons in one-dimensional Bose gases in the strong coupling regime, Phys. Rev. Lett. 127, 103401 (2021).
  31. A. Petković and Z. Ristivojevic, Mediated interaction between polarons in a one-dimensional Bose gas, Phys. Rev. A 105, L021303 (2022).
  32. L. Parisi and S. Giorgini, Quantum Monte Carlo study of the Bose-polaron problem in a one-dimensional gas with contact interactions, Phys. Rev. A 95, 023619 (2017).
  33. R. Schmidt and T. Enss, Self-stabilized Bose polarons, SciPost Phys. 13, 054 (2022).
  34. W. Casteels, J. Tempere, and J. Devreese, Polaronic properties of an impurity in a Bose-Einstein condensate in reduced dimensions, Phys. Rev. A 86, 043614 (2012).
  35. A. S. Dehkharghani, A. G. Volosniev, and N. T. Zinner, Quantum impurity in a one-dimensional trapped Bose gas, Phys. Rev. A 92, 031601(R) (2015).
  36. J. Akram and A. Pelster, Numerical study of localized impurity in a Bose-Einstein condensate, Phys. Rev. A 93, 033610 (2016).
  37. V. Pastukhov, Impurity states in the one-dimensional Bose gas, Phys. Rev. A 96, 043625 (2017).
  38. S. I. Mistakidis, A. G. Volosniev, and P. Schmelcher, Induced correlations between impurities in a one-dimensional quenched Bose gas, Phys. Rev. Res. 2, 023154 (2020).
  39. Z. Ristivojevic, Exact result for the polaron mass in a one-dimensional Bose gas, Phys. Rev. A 104, 052218 (2021).
  40. G. Panochko and V. Pastukhov, Mean-field construction for spectrum of one-dimensional Bose polaron, Ann. Phys. 409, 167933 (2019).
  41. F. K. Abdullaev and R. Galimzyanov, Bosonic impurity in a one-dimensional quantum droplet in the Bose–Bose mixture, J. Phys. B: At., Mol. Opt. Phys. 53, 165301 (2020).
  42. P. Leboeuf and N. Pavloff, Bose-Einstein beams: Coherent propagation through a guide, Phys. Rev. A 64, 033602 (2001).
  43. S. Majumdar and A. Petković, Relaxation dynamics of a mobile impurity injected in a one-dimensional Bose gas, arXiv:2507.10402.
  44. E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130, 1605 (1963).
  45. E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum, Phys. Rev. 130, 1616 (1963).
  46. V. E. Barlette, M. M. Leite, and S. K. Adhikari, Quantum scattering in one dimension, Eur. J. Phys. 21, 435 (2000).
  47. A. Farrell and B. P. van Zyl, S-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions, Can. J. Phys. 88, 817 (2010).
  48. Z. Z. Yan, Y. Ni, C. Robens, and M. W. Zwierlein, Bose polarons near quantum criticality, Science 368, 190 (2020).
  49. J. Etrych, G. Martirosyan, A. Cao, C. J. Ho, Z. Hadzibabic, and C. Eigen, Universal quantum dynamics of Bose polarons, Phys. Rev. X 15, 021070 (2025).
  50. Note that one also needs to subtract the interaction energy between the background particles of the uniform density n0 and the particles repelled (attracted) from (into) the vicinity of the impurity.
  51. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Vol. 55 (U.S. Government printing office, Washington, DC, 1948).
  52. D. F. Lawden, Elliptic Functions and Applications, Applied Mathematical Sciences Vol. 80 (Springer, Berlin, 2013).
  53. The function cd(x,ν) interpolates between the cos(x) in the limit ν→0 and a constant in the limit ν→1. The ground-state solution is given by the larger elliptic parameter ν, corresponding to a nodeless wave function. The solution with the smaller ν corresponds to an excited-state solution, with at least one node in the range of the impurity.
  54. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, Cambridge, England, 2008).
  55. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity, International Series of Monographs on Physics Vol. 164 (Oxford University, Oxford, 2016).
  56. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment, Vol. 45 (Springer, Berlin, 2008).
  57. M. Drescher, M. Salmhofer, and T. Enss, Theory of a resonantly interacting impurity in a Bose-Einstein condensate, Phys. Rev. Res. 2, 032011(R) (2020).

Outline

Information

Phys. Rev. A 112, 053306– Published 7 November, 2025
  • Received 15 August 2025
  • Accepted 20 October 2025
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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation