Skip to main content
SpringerLink
Log in

Analytical results for periodically-driven two-level models in relation to Heun functions

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

We introduce three different types of periodically-driven multiparametric two-level models whose analytical solutions are given in terms of Heun functions. These results are applied to obtain exact analytical results for certain types of periodic potentials and asymmetric double-well potentials. In particular, it is shown that under special parameter conditions, an experimentally realised periodic potential supports the exact in-gap solutions. In the asymmetric double-well potentials, some exact results of the bound-state wave functions and associated energies are found in explicit form.

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

Similar content being viewed by others

References

  1. H Ciftci, R L Hall, N Saad and E Dogu, J. Phys. A: Math. Theor. 43, 415206 (2010)

    Article  Google Scholar 

  2. Y-Z Zhang, J. Phys. A: Math. Theor. 45, 065206 (2012)

    Article  ADS  Google Scholar 

  3. M Hortaçsu, arXiv:1101.0471

  4. M L Glasser and E Montaldi, Phys. Rev. E 48, R2339 (1993)

    Article  ADS  Google Scholar 

  5. P Dorey, J Suzuki and R Tateo, J. Phys. A: Math. Gen. 37, 2047 (2004)

    Article  ADS  Google Scholar 

  6. A O Gogolin, Phys. Rev. Lett. 84, 1760 (2000)

    Article  ADS  Google Scholar 

  7. A Turbiner, Phys. Rev. A 50, 5335 (1994)

    Article  ADS  Google Scholar 

  8. J Karwowski, J. Phys.: Conf. Ser. 104, 012033 (2008)

    Google Scholar 

  9. P Loos and P M W Gill, Phys. Rev. Lett. 103, 123008 (2009)

    Article  ADS  Google Scholar 

  10. P Loos and P M W Gill, Phys. Rev. Lett. 108, 083002 (2012)

    Article  ADS  Google Scholar 

  11. S Dorosz, T Platini and D Karevski, Phys. Rev. E 77, 051120 (2008)

    Article  ADS  Google Scholar 

  12. A V Turbiner, Commun. Math. Phys. 118, 467 (1988)

    Article  ADS  Google Scholar 

  13. A G Ushveridze, Quasi-exactly solvable models in quantum mechanics (Institute of Physics Publishing, Bristol, 1994)

    MATH  Google Scholar 

  14. B Chen, Y Wu and Q Xie, J. Phys. A: Math. Theor. 46, 035301 (2013)

    Article  ADS  Google Scholar 

  15. Q Xie, J. Phys. A: Math. Theor. 44, 285302 (2011)

    Article  Google Scholar 

  16. Q Xie, J. Phys. A: Math. Theor. 45, 175302 (2012)

    Article  ADS  Google Scholar 

  17. Q Xie, L Yan, L Wang and J Fu, Pramana – J. Phys. 86, 965 (2016)

    Article  ADS  Google Scholar 

  18. I I Rabi, Phys. Rev. 49, 324 (1936); 51, 652 (1937)

  19. H Zhong, Q Xie, M T Batchelor and C Lee, J. Phys. A 46, 415302 (2013)

    Article  MathSciNet  Google Scholar 

  20. H Zhong, Q Xie, X Guan, M T Batchelor, K Gao and C Lee, J. Phys. A 47, 045301 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  21. M Grifoni and P Hänggi, Phys. Rep. 304, 229 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  22. S-I Chua and D A Telnov, Phys. Rep. 390, 1 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  23. Q Xie and W Hai, Phys. Rev. A 82, 032117 (2010)

    Article  ADS  Google Scholar 

  24. P K Jha and Y V Rostovtsev, Phys. Rev. A 81, 033827 (2010)

    Article  ADS  Google Scholar 

  25. P K Jha and Y V Rostovtsev, Phys. Rev. A 82, 015801 (2010)

    Article  ADS  Google Scholar 

  26. T A Shahverdyan, T A Ishkhanyan, A E Grigoryan and A M Ishkhanyan, J. Contemp. Phys. (Armenian Ac. Sci.) 50, 211 (2015)

    Article  ADS  Google Scholar 

  27. A M Ishkhanyan, T A Shahverdyan and T A Ishkhanyan, Eur. Phys. J. D 69, 10 (2015)

    Article  ADS  Google Scholar 

  28. A M Ishkhanyan and A E Grigoryan, J. Phys. A: Math. Theor. 47, 465205 (2014)

    Article  ADS  Google Scholar 

  29. A M Ishkhanyan and A E Grigoryan, arXiv:1402.1330

  30. A Bambini and M Lindberg, Phys. Rev. A 30 794 (1984)

    Article  ADS  Google Scholar 

  31. F T Hioe, Phys. Rev. A 30, 2100 (1984)

    Article  ADS  Google Scholar 

  32. F T Hioe and C E Carroll, Phys. Rev. A 32, 1541 (1985)

    Article  ADS  Google Scholar 

  33. C E Carroll and F T Hioe, Phys. Rev. A 41, 2835 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  34. A Ronveaux (Ed.), Heun’s differential equations (Oxford University Press, Oxford, 1995)

    MATH  Google Scholar 

  35. S Y Slavyanov and W Lay, Special functions: A unified theory based on singularities (Oxford University Press, Oxford, 2000)

    MATH  Google Scholar 

  36. J W F Olver , W D Lozier, F R Boisvert and W C Clark (Eds), NIST Handbook of mathematical functions (Cambridge University Press, New York, 2010)

    MATH  Google Scholar 

  37. J M Gomez Llorente and J Plata, Phys. Rev. A 45, R6958 (1992)

    Article  ADS  Google Scholar 

  38. P Kràl, Rev. Mod. Phys. 79, 53 (2007)

    Article  ADS  Google Scholar 

  39. W Hai, K Hai and Q Chen, Phys. Rev. A 87, 023403 (2013)

    Article  ADS  Google Scholar 

  40. W Lay and S Y Slavyanov, J. Phys. A: Math. Gen. 31, 4249 (1998)

    Article  ADS  Google Scholar 

  41. W Lay, Theor. Mater. Phys. 101, 1413 (1994)

    Article  Google Scholar 

  42. N Malkova, I Hromada, X Wang, G Bryant and Z Chen, Opt. Lett. 34, 1633 (2009)

    Article  ADS  Google Scholar 

  43. N Malkova, I Hromada, X Wang, G Bryant and Z Chen, Phys. Rev. A 80, 043806 (2009)

    Article  ADS  Google Scholar 

  44. A M Ishkhanyan, Ann. Phys. (N.Y.) 388, 456 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  45. W Kohn, Phys. Rev. 15, 809 (1959)

    Article  ADS  Google Scholar 

  46. Q Xie and C Lee, Phys. Rev. A 85, 063802 (2012)

    Article  ADS  Google Scholar 

  47. A Erdélyi, Duke Math. J. 9, 48 (1942)

    Article  MathSciNet  Google Scholar 

  48. T A Ishkhanyan, T A Shahverdyan and A M Ishkhanyan, arXiv:1403.7863

  49. A M Manukyan, T A Ishkhanyan, M V Hakobyan and A M Ishkhanyan, Int. J. Diff. Equ. Appl. 13, 231 (2014)

    Google Scholar 

  50. T A Shahverdyan, V M Redkov and A M Ishkhanyan, Nonlinear Phenom. Complex Syst. 19, 395 (2016)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11565011.

Author information

Authors and Affiliations

  1. College of Physics and Electronic Engineering, Hainan Normal University, Haikou , 571158, China

    Qiongtao Xie

Authors
  1. Qiongtao Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiongtao Xie.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xie, Q. Analytical results for periodically-driven two-level models in relation to Heun functions. Pramana - J Phys 91, 19 (2018). https://doi.org/10.1007/s12043-018-1596-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-018-1596-z

Keywords

PACS Nos

Search

Navigation