Skip to main content
SpringerLink
Log in

Periodic Ultranarrow Rods as 1D Subwavelength Optical Lattices

  • Published:
Journal of Low Temperature Physics Aims and scope Submit manuscript

Abstract

We report on ground-state properties of a one-dimensional, weakly interacting Bose gas constrained by an infinite multi-rod periodic structure at zero temperature. We solve the stationary Gross–Pitaevskii equation (GPE) to obtain the Bloch wave functions from which we give a semi-analytical solution for the density profile, as well as for the phase of the wave function in terms of the Jacobi elliptic functions, and the incomplete elliptic integrals of the first, second and third kind. Then, we determine numerically the energy of the ground state, the chemical potential and the compressibility of the condensate and show their dependence on the potential height, as well as on the interaction between the bosons. We show the appearance of loops in the energy band spectrum of the system for strong enough interactions, which appear at the edges of the first Brillouin zone for odd bands and at the center for even bands. We apply our model to predict the energy band structure of the Bose gas in an optical lattice with subwavelength spatial structure. To discuss the density range of the validity of the GPE predictions, we calculate the ground-state energies of the free Bose gas using the GPE, which we compare with the Lieb–Liniger exact energies.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269(5221), 198 (1995)

    Article  ADS  Google Scholar 

  2. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. Van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75(22), 3969 (1995)

    Article  ADS  Google Scholar 

  3. D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81(15), 3108 (1998)

    Article  ADS  Google Scholar 

  4. I. Bloch, Nat. Phys. 1(1), 23 (2005)

    Article  Google Scholar 

  5. C. Gross, I. Bloch, Science 357(6355), 995 (2017)

    Article  ADS  Google Scholar 

  6. M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Nature 415(6867), 39 (2002)

    Article  ADS  Google Scholar 

  7. E. Haller, R. Hart, M.J. Mark, J.G. Danzl, L. Reichsöllner, M. Gustavsson, M. Dalmonte, G. Pupillo, H.C. Nägerl, Nature 466(7306), 597 (2010)

    Article  ADS  Google Scholar 

  8. M. Ła̧cki, M.A. Baranov, H. Pichler, P. Zoller, Phys. Rev. Lett. 117(23), 233001 (2016)

    Article  ADS  Google Scholar 

  9. Y. Wang, S. Subhankar, P. Bienias, M. Ła̧cki, T.C. Tsui, M.A. Baranov, A.V. Gorshkov, P. Zoller, J.V. Porto, S.L. Rolston, Phys. Rev. Lett. 120(8), 083601 (2018)

    Article  ADS  Google Scholar 

  10. E. Merzbacher, Quantum Mechanics, 2nd edn. (Wiley, New York, 1969)

    MATH  Google Scholar 

  11. A.K. Fedorov, V.I. Yudson, G.V. Shlyapnikov, Phys. Rev. A 95(4), 043615 (2017)

    Article  ADS  Google Scholar 

  12. B.T. Seaman, L.D. Carr, M.J. Holland, Phys. Rev. A 71(3), 033609 (2005)

    Article  ADS  Google Scholar 

  13. S. Theodorakis, E. Leontidis, J. Phys. A. Math. Gen. 30(13), 4835 (1997)

    Article  ADS  Google Scholar 

  14. M. Machholm, C.J. Pethick, H. Smith, Phys. Rev. A At. Mol. Opt. Phys. 67(5), 15 (2003)

    Article  Google Scholar 

  15. W. Li, A. Smerzi, Phys. Rev. E 70(1), 016605 (2004)

    Article  ADS  Google Scholar 

  16. B.T. Seaman, L.D. Carr, M.J. Holland, Phys. Rev. A 71(3), 033622 (2005)

    Article  ADS  Google Scholar 

  17. X. Dong, B. Wu, Laser Phys. 17(2), 190 (2007)

    Article  ADS  Google Scholar 

  18. I. Danshita, S. Tsuchiya, Phys. Rev. A 75(3), 033612 (2007)

    Article  ADS  Google Scholar 

  19. A. Negretti, R. Gerritsma, Z. Idziaszek, F. Schmidt-Kaler, T. Calarco, Phys. Rev. B 90(15), 155426 (2014)

    Article  ADS  Google Scholar 

  20. R.de L. Kronig, W.G. Penney, Proc. R. Soc. A Math. Phys. Eng. Sci. 130(814), 499 (1931)

    Article  ADS  Google Scholar 

  21. E.P. Gross, Nuovo Cim. 20(3), 454 (1961)

    Article  ADS  Google Scholar 

  22. L.P. Pitaevskii, Sov. Phys. JETP 13(2), 451 (1961)

    Google Scholar 

  23. S. Subhankar, P. Bienias, P. Titum, T.C. Tsui, Y. Wang, A.V. Gorshkov, S.L. Rolston, J.V. Porto, arXiv:1906.07646v1 (2019)

  24. E.H. Lieb, W. Liniger, Phys. Rev. 130(4), 1605 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  25. L. Pitaevskii, S. Stringari, Bose–Einstein Condensation and Superfluidity, 2nd edn. (Oxford University Press, New York, 2016)

    Book  Google Scholar 

  26. M. Krämer, C. Menotti, L. Pitaevskii, S. Stringari, Eur. Phys. J. D 27(3), 247 (2003)

    Article  ADS  Google Scholar 

  27. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn. (Dover Publications Inc., New York, 1964)

    MATH  Google Scholar 

  28. O.A. Rodríguez, M.A. Solís, J. Low Temp. Phys. 183(3–4), 144 (2016)

    Article  ADS  Google Scholar 

  29. J. Rogel-Salazar, Eur. J. Phys. 34(2), 247 (2013)

    Article  Google Scholar 

  30. E.J. Mueller, Phys. Rev. A 66(6), 063603 (2002)

    Article  ADS  Google Scholar 

  31. T. Tsuzuki, J. Low Temp. Phys. 4(4), 441 (1971)

    Article  ADS  Google Scholar 

  32. V. Dunjko, V. Lorent, M. Olshanii, Phys. Rev. Lett. 86(24), 5413 (2001)

    Article  ADS  Google Scholar 

  33. M.O. Mewes, M.R. Andrews, N.J. Van Druten, D.M. Kurn, D.S. Durfee, W. Ketterle, Phys. Rev. Lett. 77(3), 416 (1996)

    Article  ADS  Google Scholar 

  34. P. Krüger, S. Hofferberth, I.E. Mazets, I. Lesanovsky, J. Schmiedmayer, Phys. Rev. Lett. 105(26), 265302 (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge partial support from Grants PAPIIT-DGAPA-UNAM IN-107616 and IN-110319 and CONACyT 221030.

Author information

Authors and Affiliations

  1. Posgrado en Ciencias Físicas, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, Ciudad de México, México

    Omar Abel Rodríguez-López

  2. Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, Ciudad de México, México

    M. A. Solís

Authors
  1. Omar Abel Rodríguez-López
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Solís
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omar Abel Rodríguez-López.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rodríguez-López, O.A., Solís, M.A. Periodic Ultranarrow Rods as 1D Subwavelength Optical Lattices. J Low Temp Phys 198, 190–208 (2020). https://doi.org/10.1007/s10909-019-02276-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10909-019-02276-6

Keywords

Search

Navigation