- 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

Confined few-particle systems beyond mean-field theory adopting Gaussian-type orbitals and Morse interparticle interaction

Matee ur Rehman1,*, Paul Winter1,†, Fabio Revuelta2, and Alejandro Saenz1

  • *Contact author: rehmanmatee@physik.hu-berlin.de
  • †Present address: Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany.
PDF

Phys. Rev. A 112, 063320 – Published 22 December, 2025

DOI: https://doi.org/10.1103/pf98-jls2

Abstract

Recent advancements in optical tweezers enable the trapping of arbitrary numbers of neutral atoms and molecules, and even arrays of tweezers with variable geometry can be realized. These fascinating breakthroughs require novel full-dimensional beyond mean-field treatments for systems with more than two confined particles spread over traps that are arranged arbitrarily in space. In this work, the suitability of a quantum-chemistry inspired approach adopting Cartesian Gaussians as basis functions is investigated. For this purpose, the six-dimensional integrals associated with a realistic atom-atom interaction described by a Morse model potential were implemented. The performance, correctness, and efficiency of the implementation is assessed by comparing full configuration-interaction calculations (exact diagonalizations) for two atoms in an isotropic harmonic trap with quasiexact reference results.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (91)

  1. B. M. Spar, E. Guardado-Sanchez, S. Chi, Z. Z. Yan, and W. S. Bakr, Phys. Rev. Lett. 128, 223202 (2022).
  2. L. Pause, T. Preuschoff, D. Schäffner, M. Schlosser, and G. Birkl, Phys. Rev. Res. 5, L032009 (2023).
  3. D. Bloch, B. Hofer, S. R. Cohen, A. Browaeys, and I. Ferrier-Barbut, Phys. Rev. Lett. 131, 203401 (2023).
  4. H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler et al., Phys. Rev. Lett. 123, 170503 (2019).
  5. A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine, J. Ye, and A. M. Kaufman, Nature (London) 588, 408 (2020).
  6. A. M. Kaufman and K.-K. Ni, Nat. Phys. 17, 1324 (2021).
  7. Y. Yu, K. Wang, J. D. Hood, L. R. B. Picard, J. T. Zhang, W. B. Cairncross, J. M. Hutson, R. Gonzalez-Ferez, T. Rosenband, and K.-K. Ni, Phys. Rev. X 11, 031061 (2021).
  8. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
  9. M. Aymar, J. Phys. B 11, 1413 (1978).
  10. C. Ti, Y. Shen, M.-T. Ho Thanh, Q. Wen, and Y. Liu, Sci. Rep. 10, 20099 (2020).
  11. D. K. Ruttley, A. Guttridge, S. Spence, R. C. Bird, C. R. Le Sueur, J. M. Hutson, and S. L. Cornish, Phys. Rev. Lett. 130, 223401 (2023).
  12. D. K. Ruttley, A. Guttridge, T. R. Hepworth, and S. L. Cornish, PRX Quantum 5, 020333 (2024).
  13. P.-I. Schneider, S. Grishkevich, and A. Saenz, Phys. Rev. A 80, 013404 (2009).
  14. D. Capecchi, C. Cantillano, M. J. Mark, F. Meinert, A. Schindewolf, M. Landini, A. Saenz, F. Revuelta, and H.-C. Nägerl, Phys. Rev. Lett. 131, 213002 (2023).
  15. G. Bougas, S. I. Mistakidis, P. Giannakeas, and P. Schmelcher, Phys. Rev. A 106, 043323 (2022).
  16. G. Bougas, S. I. Mistakidis, and P. Schmelcher, Phys. Rev. A 100, 053602 (2019).
  17. G. Bougas, S. Mistakidis, P. Giannakeas, and P. Schmelcher, New J. Phys. 23, 093022 (2021).
  18. S. Sala, P.-I. Schneider, and A. Saenz, Phys. Rev. Lett. 109, 073201 (2012).
  19. E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Reichsöllner, V. Melezhik, P. Schmelcher, and H.-C. Nägerl, Phys. Rev. Lett. 104, 153203 (2010).
  20. M. Valiente and K. Mølmer, Phys. Rev. A 84, 053628 (2011).
  21. V. S. Melezhik and P. Schmelcher, Phys. Rev. A 84, 042712 (2011).
  22. S. Sala, G. Zürn, T. Lompe, A. N. Wenz, S. Murmann, F. Serwane, S. Jochim, and A. Saenz, Phys. Rev. Lett. 110, 203202 (2013).
  23. S. Sala and A. Saenz, Phys. Rev. A 94, 022713 (2016).
  24. V. N. Efimov, Sov. J. Nucl. Phys. 12, 589 (1971) [Yad. Fiz. 12, 1080 (1970)].
  25. V. N. Efimov, Phys. Lett. B 33, 563 (1970).
  26. B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. 83, 1751 (1999).
  27. F. Werner and Y. Castin, Phys. Rev. Lett. 97, 150401 (2006).
  28. F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. D'Incao, H.-C. Nägerl, and R. Grimm, Phys. Rev. Lett. 102, 140401 (2009).
  29. P. Naidon and S. Endo, Rep. Prog. Phys. 80, 056001 (2017).
  30. M. D. Higgins and C. H. Greene, Phys. Rev. A 106, 023304 (2022).
  31. G. Bougas, S. I. Mistakidis, P. Schmelcher, C. H. Greene, and P. Giannakeas, Phys. Rev. Res. 5, 043134 (2023).
  32. J. R. Armstrong, N. T. Zinner, D. V. Fedorov, and A. S. Jensen, J. Phys. B 44, 055303 (2011).
  33. X. Y. Yin, D. Blume, P. R. Johnson, and E. Tiesinga, Phys. Rev. A 90, 043631 (2014).
  34. P. Jeszenszki, A. Y. Cherny, and J. Brand, Phys. Rev. A 97, 042708 (2018).
  35. P. Jeszenszki, A. Alavi, and J. Brand, Phys. Rev. A 99, 033608 (2019).
  36. S. F. Boys, Proc. R. Soc. Lond. A 200, 542 (1950).
  37. D. Rakshit and D. Blume, Phys. Rev. A 86, 062513 (2012).
  38. G. M. Barca and P.-F. Loos, J. Chem. Phys. 147, 024103 (2017).
  39. B. D. Esry and C. H. Greene, Phys. Rev. A 60, 1451 (1999).
  40. F. Brauneis, H.-W. Hammer, S. M. Reimann, and A. G. Volosniev, Phys. Rev. A 111, 013303 (2025).
  41. In fact, already within the mean-field description, the standard δ pseudopotential employing the trap-free scattering length does not lead to an accurate description. Instead, an energy-dependent scattering length [91] is usually adopted that is, however, only available after a full treatment of the correct interparticle interaction together with the confining trap potential.
  42. S. Sala, J. Förster, and A. Saenz, Phys. Rev. A 95, 011403(R) (2017).
  43. J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  44. M. Srednicki, Phys. Rev. E 50, 888 (1994).
  45. N. Shiraishi and K. Matsumoto, Nat. Commun. 12, 5084 (2021).
  46. S. Moudgalya, N. Regnault, and B. A. Bernevig, Phys. Rev. B 98, 235156 (2018).
  47. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Nat. Phys. 14, 745 (2018).
  48. M. Serbyn, D. A. Abanin, and Z. Papić, Nat. Phys. 17, 675 (2021).
  49. F. Revuelta, E. G. Vergini, R. M. Benito, and F. Borondo, Phys. Rev. E 85, 026214 (2012).
  50. R. J. Bartlett and M. Musiał, Rev. Mod. Phys. 79, 291 (2007).
  51. L. Cao, S. Krönke, O. Vendrell, and P. Schmelcher, J. Chem. Phys. 139, 134103 (2013).
  52. K. M. Daily and D. Blume, Phys. Rev. A 81, 053615 (2010).
  53. J. Rotureau, I. Stetcu, B. R. Barrett, M. C. Birse, and U. van Kolck, Phys. Rev. A 82, 032711 (2010).
  54. C. H. Greene, P. Giannakeas, and J. Pérez-Ríos, Rev. Mod. Phys. 89, 035006 (2017).
  55. E. Hiyama and M. Kamimura, Front. Phys. 13, 132106 (2018).
  56. D. Blume, Rep. Prog. Phys. 75, 046401 (2012).
  57. J. von Stecher and C. H. Greene, Phys. Rev. A 80, 022504 (2009).
  58. M. W. C. Sze, A. G. Sykes, D. Blume, and J. L. Bohn, Phys. Rev. A 97, 033608 (2018).
  59. M. Silkowski, M. Lesiuk, and R. Moszynski, J. Chem. Phys. 142, 124102 (2015).
  60. J. G. Balcerzak, M. Lesiuk, and R. Moszynski, Phys. Rev. A 96, 052510 (2017).
  61. S. Grishkevich and A. Saenz, Phys. Rev. A 80, 013403 (2009).
  62. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Courier Corporation, Garden City, NY, 1996).
  63. A. Grüneis, S. Hirata, Y.-y. Ohnishi, and S. Ten-No, J. Chem. Phys. 146, 080901 (2017).
  64. D. Casanova, WIREs Comput. Mol. Sci. 12, e1561 (2022).
  65. H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988).
  66. G. H. Booth, A. J. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009).
  67. N. M. Tubman, J. Lee, T. Y. Takeshita, M. Head-Gordon, and K. B. Whaley, J. Chem. Phys. 145, 044112 (2016).
  68. B. O. Roos, The Multiconfigurational (MC) Self-consistent Field (SCF) Theory (Springer, Berlin, 1992).
  69. P. Å. Malmqvist, A. Rendell, and B. O. Roos, J. Phys. Chem. 94, 5477 (1990).
  70. J. Olsen, Int. J. Quantum Chem. 111, 3267 (2011).
  71. B. O. Roos, P. R. Taylor, and P. E. Sigbahn, Chem. Phys. 48, 157 (1980).
  72. P.-I. Schneider, S. Grishkevich, and A. Saenz, Phys. Rev. A 87, 053413 (2013).
  73. G. Herzberg, Molecular Spectra and Molecular Structure. Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950), Vol. 1.
  74. P. Morse, Phys. Rev. 34, 57 (1929).
  75. R. A. Buckingham, Proc. R. Soc. Lond. A 168, 264 (1938).
  76. L. E. McMurchie and E. R. Davidson, J. Comput. Phys. 26, 218 (1978).
  77. Note that there are typos in the Eqs. (11) and (12) of Ref.  [60]. In both equations the term Eumn should be Evmn. Likewise, in Eq.  (12) the term Eu′m′n′ should be Ev′m′n′.
  78. C. C. M. Samson, W. Klopper, and T. Helgaker, Comput. Phys. Commun. 149, 1 (2002).
  79. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, New York, 1965).
  80. H. B. Schlegel and M. J. Frisch, Int. J. Quantum Chem. 54, 83 (1995).
  81. Here a factor of 2 was missing in Eq. (45) of Ref.  [59], which was subsequently corrected in Eq. (25) of Ref.  [60].
  82. S. Grishkevich, S. Sala, and A. Saenz, Phys. Rev. A 84, 062710 (2011).
  83. J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999).
  84. S. Sala, Ph.D. thesis, Humboldt-Universität zu Berlin, Germany, 2016.
  85. At these points the GTOs energies are lowered due to an avoided crossing with higher bound state(s) of the same symmetry, which are not shown in the graph. This lower energy accidentally leads to a better agreement with the quasiexact results.
  86. The variational principle only applies to the lowest state of each symmetry. For excited states it applies strictly only, if the state is orthogonal to all lower-lying exact states.
  87. B. Schulz, S. Sala, and A. Saenz, New J. Phys. 17, 065002 (2015).
  88. B. Schulz and A. Saenz, ChemPhysChem 17, 3747 (2016).
  89. T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 88, 173201 (2002).
  90. S. Grishkevich, P.-I. Schneider, Y. V. Vanne, and A. Saenz, Phys. Rev. A 81, 022719 (2010).
  91. D. Blume and C. H. Greene, Phys. Rev. A 65, 043613 (2002).

Outline

Information

Phys. Rev. A 112, 063320– Published 22 December, 2025
  • Received 15 September 2025
  • Accepted 25 November 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation