Skip to main content
SpringerLink
Log in

Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results

  • Regular Article
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

One-dimensional multi-component Fermi or Bose systems with strong zero-range interactions can be described in terms of local exchange coefficients and mapping the problem into a spin model is thus possible. For arbitrary external confining potentials the local exchanges are given by highly non-trivial geometric factors that depend solely on the geometry of the confinement through the single-particle eigenstates of the external potential. To obtain accurate effective Hamiltonians to describe such systems one needs to be able to compute these geometric factors with high precision which is difficult due to the computational complexity of the high-dimensional integrals involved. An approach using the local density approximation would therefore be a most welcome approximation due to its simplicity. Here we assess the accuracy of the local density approximation by going beyond the simple harmonic oscillator that has been the focus of previous studies and consider some double-wells of current experimental interest. We find that the local density approximation works quite well as long as the potentials resemble harmonic wells but break down for larger barriers. In order to explore the consequences of applying the local density approximation in a concrete setup we consider quantum state transfer in the effective spin models that one obtains. Here we find that even minute deviations in the local exchange coefficients between the exact and the local density approximation can induce large deviations in the fidelity of state transfer for four, five, and six particles.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

    Article  ADS  Google Scholar 

  2. M. Lewenstein et al., Adv. Phys. 56, 243 (2007)

    Article  ADS  Google Scholar 

  3. T. Esslinger, Ann. Rev. Condens. Matter Phys. 1, 129 (2010)

    Article  ADS  Google Scholar 

  4. Y. Nishida, D.T. Son, Phys. Rev. D 76, 086004 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  5. D.T. Son, Phys. Rev. D 78, 046003 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  6. K. Balasubramanian, J. McGreevy, Phys. Rev. Lett. 101, 061601 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  7. J. Maldacena, D. Martelli, Y. Tachikawa, J. High Energy Phys. 10, 072 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  8. A. Adams, K. Balasubramanian, J. McGreevy, J. High Energy Phys. 11, 059 (2008)

    ADS  Google Scholar 

  9. N.T. Zinner, A.S. Jensen, J. Phys. G: Nucl. Part. Phys. 40, 053101 (2013)

    Article  ADS  Google Scholar 

  10. M.A. Baranov, M. Dalmonte, G. Pupillo, P. Zoller, Chem. Rev. 112, 5012 (2012)

    Article  Google Scholar 

  11. H. Moritz, T. Stöferle, M. Köhl, T. Esslinger, Phys. Rev. Lett. 91, 250402 (2003)

    Article  ADS  Google Scholar 

  12. T. Stöferle, H. Moritz, C. Schori, M. Köhl, T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004)

    Article  ADS  Google Scholar 

  13. T. Kinoshita, T. Wenger, D.S. Weiss, Science 305, 1125 (2004)

    Article  ADS  Google Scholar 

  14. B. Paredes et al., Nature 429, 277 (2004)

    Article  ADS  Google Scholar 

  15. T. Kinoshita, T. Wenger, D.S. Weiss, Nature 440, 900 (2005)

    Article  ADS  Google Scholar 

  16. E. Haller et al., Science 325, 1224 (2009)

    Article  ADS  Google Scholar 

  17. E. Haller et al., Nature 466, 597 (2010)

    Article  ADS  Google Scholar 

  18. M. Olshanii, Phys. Rev. Lett. 81, 938 (1998)

    Article  ADS  Google Scholar 

  19. L.W. Tonks, Phys. Rev. 50, 955 (1936)

    Article  ADS  Google Scholar 

  20. M.D. Girardeau, J. Math. Phys. 1, 516 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. G. Pagano et al., Nat. Phys. 10, 198 (2014)

    Article  Google Scholar 

  22. F. Serwane et al., Science 332, 336 (2011)

    Article  ADS  Google Scholar 

  23. G. Zürn et al., Phys. Rev. Lett. 108, 075303 (2012)

    Article  ADS  Google Scholar 

  24. G. Zürn et al., Phys. Rev. Lett. 111, 175302 (2013)

    Article  ADS  Google Scholar 

  25. A. Wenz et al., Science 342, 457 (2013)

    Article  ADS  Google Scholar 

  26. S. Murmann et al., Phys. Rev. Lett. 114, 080402 (2015)

    Article  ADS  Google Scholar 

  27. S. Murmann et al., Phys. Rev. Lett. 115, 215301 (2015)

    Article  ADS  Google Scholar 

  28. S. Zöllner, H.-D. Meyer, P. Schmelcher, Phys. Rev. A 74, 063611 (2006)

    Article  ADS  Google Scholar 

  29. S. Zöllner, H.-D. Meyer, P. Schmelcher, Phys. Rev. A 75, 043608 (2007)

    Article  ADS  Google Scholar 

  30. S. Zöllner, H.-D. Meyer, P. Schmelcher, Phys. Rev. Lett. 100, 040401 (2008)

    Article  Google Scholar 

  31. F. Deuretzbacher, K. Bongs, K. Sengstock, D. Pfannkuche, Phys. Rev. A 75, 013614 (2007)

    Article  ADS  Google Scholar 

  32. E. Tempfli, S. Zöllner, P. Schmelcher, New J. Phys. 11, 073015 (2009)

    Article  ADS  Google Scholar 

  33. M.D. Girardeau, Phys. Rev. A 83, 011601(R) (2011)

    Article  ADS  Google Scholar 

  34. I. Brouzos, P. Schmelcher, Phys. Rev. Lett. 108, 045301 (2012)

    Article  ADS  Google Scholar 

  35. I. Brouzos, A. Förster, Phys. Rev. A 89, 053632 (2014)

    Article  ADS  Google Scholar 

  36. B. Wilson, A. Förster, C.C.N. Kuhn, I. Roditi, D. Rubeni, Phys. Lett. A 378, 1065 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  37. N.T. Zinner et al., Europhys. Lett. 107, 60003 (2014)

    Article  ADS  Google Scholar 

  38. M.A. Garcia-March et al., Phys. Rev. A 92, 033621 (2015)

    Article  ADS  Google Scholar 

  39. A.S. Dehkharghani et al., Sci. Rep. 5, 10675 (2015)

    Article  ADS  Google Scholar 

  40. P. Massignan, J. Levinsen, M.M. Parish, Phys. Rev. Lett. 115, 247202 (2015)

    Article  ADS  Google Scholar 

  41. L. Guan, S. Chen, Y. Wang, Z.-Q. Ma, Phys. Rev. Lett. 102, 160402 (2009)

    Article  ADS  Google Scholar 

  42. C.N. Yang, Chin. Phys. Lett. 26, 120504 (2009)

    Article  ADS  Google Scholar 

  43. X.-W. Guan, Z.-Q. Ma, Phys. Rev. A 85, 033632 (2012)

    Article  ADS  Google Scholar 

  44. D. Rubeni, A. Förster, I. Roditi, Phys. Rev. A 86, 043619 (2012)

    Article  ADS  Google Scholar 

  45. I. Brouzos, P. Schmelcher, Phys. Rev. A 87, 023605 (2013)

    Article  ADS  Google Scholar 

  46. P.O. Bugnion, G.J. Conduit, Phys. Rev. A 87, 060502(R) (2013)

    Article  ADS  Google Scholar 

  47. S.E. Gharashi, D. Blume, Phys. Rev. Lett. 111, 045302 (2013)

    Article  ADS  Google Scholar 

  48. T. Sowiński, T. Graß, O. Dutta, M. Lewenstein, Phys. Rev. A 88, 033607 (2013)

    Article  ADS  Google Scholar 

  49. S.E. Gharashi, X.Y. Yin, D. Blume, Phys. Rev. A 89, 023603 (2014)

    Article  ADS  Google Scholar 

  50. A.G. Volosniev et al., Few-Body Syst. 55, 839 (2014)

    Article  ADS  Google Scholar 

  51. X. Cui, T.-L. Ho, Phys. Rev. A 89, 023611 (2014)

    Article  ADS  Google Scholar 

  52. N.J.S. Loft et al., Eur. Phys. J. D 69, 65 (2015)

    Article  ADS  Google Scholar 

  53. R. Lundmark, C. Forssén, J. Rotureau, Phys. Rev. A 91, 041601(R) (2015)

    Article  ADS  Google Scholar 

  54. S.E. Gharashi, X.Y. Yin, D. Blume, Phys. Rev. A 91, 013620 (2015)

    Article  ADS  Google Scholar 

  55. F. Nur Ünal, B. Hetényi, M.Ö. Oktel, Phys. Rev. A 91, 053625 (2015)

    Article  ADS  Google Scholar 

  56. A.G. Volosniev, H.-W. Hammer, N.T. Zinner, e-print arXiv:1507.00186 (2015)

  57. M.D. Girardeau, Phys. Rev. A 82, 011607(R) (2010)

    Article  ADS  Google Scholar 

  58. L. Guan, S. Chen, Phys. Rev. Lett. 105, 175301 (2010)

    Article  ADS  Google Scholar 

  59. G.E. Astrakharchik, I. Brouzos, Phys. Rev. A 88, 021602(R) (2013)

    Article  ADS  Google Scholar 

  60. A.G. Volosniev et al., Nat. Commun. 5, 5300 (2014)

    Article  ADS  Google Scholar 

  61. E.J. Lindgren et al., New J. Phys. 16, 063003 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  62. F. Deuretzbacher et al., Phys. Rev. A 90, 013611 (2014)

    Article  ADS  Google Scholar 

  63. A.G. Volosniev et al., Phys. Rev. A 91, 023620 (2015)

    Article  ADS  Google Scholar 

  64. L. Yang, L. Guan, H. Pu, Phys. Rev. A 91, 043634 (2015)

    Article  ADS  Google Scholar 

  65. J. Levinsen, P. Massignan, G.M. Bruun, M.M. Parish, Sci. Adv. 1, e1500197 (2015)

    Article  ADS  Google Scholar 

  66. T. Sowiński, M. Gajda, K. Rza¸żewski, Europhys. Lett. 109, 26005 (2015)

    Article  ADS  Google Scholar 

  67. T. Grining et al., Phys. Rev. A 92, 061601 (2015)

    Article  ADS  Google Scholar 

  68. M.D. Girardeau, M. Olshanii, Phys. Rev. A 70, 023608 (2004)

    Article  ADS  Google Scholar 

  69. M.D. Girardeau, A. Minguzzi, Phys. Rev. Lett. 99, 230402 (2007)

    Article  ADS  Google Scholar 

  70. S. Zöllner, H.-D. Meyer, P Schmelcher, Phys. Rev. A 78, 013629 (2008)

    Article  ADS  Google Scholar 

  71. X.-W. Guan, M.T. Batchelor, J.-Y. Lee, Phys. Rev. A 78, 023621 (2008)

    Article  ADS  Google Scholar 

  72. F. Deuretzbacher et al., Phys. Rev. Lett. 100, 160405 (2008)

    Article  ADS  Google Scholar 

  73. B. Fang, P. Vignolo, M. Gattobigio, C. Miniatura, A. Minguzzi, Phys. Rev. A 84, 023626 (2011)

    Article  ADS  Google Scholar 

  74. N.L. Harshman, Phys. Rev. A 86, 052122 (2012)

    Article  ADS  Google Scholar 

  75. M.A. Garcia-March, Th. Busch, Phys. Rev. A 87, 063633 (2013)

    Article  ADS  Google Scholar 

  76. M.A. Garcia-March et al., Phys. Rev. A 88, 063604 (2013)

    Article  ADS  Google Scholar 

  77. N.L. Harshman, Phys. Rev. A 89, 033633 (2014)

    Article  ADS  Google Scholar 

  78. S. Campbell, M.A. Garcia-March, T. Fogarty, Th. Busch, Phys. Rev. A 90, 013617 (2014)

    Article  ADS  Google Scholar 

  79. M.A. Garcia-March et al., New J. Phys. 16, 103004 (2014)

    Article  ADS  Google Scholar 

  80. P. D’Amico, M. Rontani, J. Phys. B 47, 065303 (2014)

    Article  ADS  Google Scholar 

  81. P. D’Amico, M. Rontani, Phys. Rev. A 91, 043610 (2015)

    Article  ADS  Google Scholar 

  82. N.P. Mehta, Phys. Rev. A 89, 052706 (2014)

    Article  ADS  Google Scholar 

  83. M.A. Garcia-March et al., Phys. Rev. A 90, 063605 (2014)

    Article  ADS  Google Scholar 

  84. R.E. Barfknecht, I. Brouzos, A. Förster, Phys. Rev. A 91, 043640 (2015)

    Article  ADS  Google Scholar 

  85. A.G. Volosniev, H.-W. Hammer, N.T. Zinner, Phys. Rev. A 92, 023623 (2015)

    Article  ADS  Google Scholar 

  86. A.S. Dehkharghani, A.G. Volosniev, N.T. Zinner, Phys. Rev. A 92, 031601(R) (2015)

    Article  ADS  Google Scholar 

  87. N.L. Harshman, Few-body Syst. 57, 45 (2016)

    Article  Google Scholar 

  88. D. Pȩcak, M. Gajda, T. Sowiński, New J. Phys. 18, 013030 (2016)

    Article  Google Scholar 

  89. T. Graß, Phys. Rev. A 92, 023634 (2015)

    Article  ADS  Google Scholar 

  90. A.S. Dehkharghani, A.G. Volosniev, N.T. Zinner, e-print arXiv:1511.01702 (2015)

  91. D.S. Petrov, G.V. Shlyapnikov, J.T.M. Walraven, Phys. Rev. Lett. 85, 3745 (2001)

    Article  ADS  Google Scholar 

  92. A. Recati, P.O. Fedichev, W. Zwerger, P. Zoller, Phys. Rev. Lett. 90, 020401 (2003)

    Article  ADS  Google Scholar 

  93. G.E. Astrakharchik, D. Blume, S. Giorgini, L.P. Pitaevskii, Phys. Rev. Lett. 93, 050402 (2004)

    Article  ADS  Google Scholar 

  94. I.V. Tokatly, Phys. Rev. Lett. 93, 090405 (2004)

    Article  ADS  Google Scholar 

  95. G.E. Astrakharchik, Phys. Rev. A 72, 063620 (2005)

    Article  ADS  Google Scholar 

  96. G. E. Astrakharchik, J. Boronat, J. Casulleras, S. Giorgini, Phys. Rev. Lett. 95, 190407 (2005)

    Article  ADS  Google Scholar 

  97. G. Orso, Phys. Rev. Lett. 98, 070402 (2007)

    Article  ADS  Google Scholar 

  98. X.-J. Liu, H. Hu, P.D. Drummond, Phys. Rev. A 78, 023601 (2008)

    Article  ADS  Google Scholar 

  99. M. Colomé-Tatché, Phys. Rev. A 78, 033612 (2008)

    Article  ADS  Google Scholar 

  100. G. De Rosi, S. Stringari, Phys. Rev. A 92, 053617 (2015)

    Article  ADS  Google Scholar 

  101. M. Chenaeu et al., Nature 481, 484 (2012)

    Article  ADS  Google Scholar 

  102. T. Fukuhara et al., Nat. Phys. 9, 235 (2013)

    Article  MathSciNet  Google Scholar 

  103. T. Fukuhara et al., Nature 502, 76 (2013)

    Article  ADS  Google Scholar 

  104. S. Hild et al., Phys. Rev. Lett. 113, 147205 (2014)

    Article  ADS  Google Scholar 

  105. T. Fukuhara et al., Phys. Rev. Lett. 115, 035302 (2015)

    Article  ADS  Google Scholar 

  106. S. Trotzky et al., Science 319, 295 (2008)

    Article  ADS  Google Scholar 

  107. M. Ogata, H. Shiba, Phys. Rev. B 41, 2326 (1990)

    Article  ADS  Google Scholar 

  108. K.A. Matveev, Phys. Rev. B 70, 245319 (2004)

    Article  ADS  Google Scholar 

  109. X.-W. Guan, M.T. Batchelor, M. Takahashi, Phys. Rev. A 76, 043617 (2007)

    Article  ADS  Google Scholar 

  110. K.A. Matveev, A. Furusaki, Phys. Rev. Lett. 101, 170403 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  111. S. Bose, Phys. Rev. Lett. 91, 207901 (2003)

    Article  ADS  Google Scholar 

  112. S. Bose, Contemp. Phys. 48, 13 (2007)

    Article  ADS  Google Scholar 

  113. M. Christandl et al., Phys. Rev. A 71, 032312 (2005)

    Article  ADS  Google Scholar 

  114. M. Christandl et al., Phys. Rev. Lett. 92, 187902 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  115. G.M. Nikolopoulos et al., Europhys. Lett. 65, 297 (2004)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Aarhus University, 8000, Aarhus C, Denmark

    Oleksandr V. Marchukov, Emil H. Eriksen, Jonatan M. Midtgaard, Alex A.S. Kalaee, Dmitri V. Fedorov, Aksel S. Jensen & Nikolaj T. Zinner

Authors
  1. Oleksandr V. Marchukov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emil H. Eriksen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonatan M. Midtgaard
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alex A.S. Kalaee
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dmitri V. Fedorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Aksel S. Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Nikolaj T. Zinner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolaj T. Zinner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marchukov, O., Eriksen, E., Midtgaard, J. et al. Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results. Eur. Phys. J. D 70, 32 (2016). https://doi.org/10.1140/epjd/e2016-60489-x

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/e2016-60489-x

Keywords

Search

Navigation