Skip to main content
SpringerLink
Log in

Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions

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

Abstract

We derive 35 five-parametric classes of the quantum time-dependent two-state models solvable in terms of the general Heun functions. Each of the classes is defined by a pair of generating functions the first of which is referred to as the amplitude- and the second one as the detuning-modulation function. The classes suggest numerous families of specific field configurations with different physical properties generated by appropriate choices of the transformation of the independent variable, real or complex. There are many families of models with constant detuning or constant amplitude, numerous classes of chirped pulses of controllable amplitude and/or detuning, families of models with double or multiple (periodic) crossings, periodic amplitude modulation field configurations, etc. The detuning modulation function is the same for all the derived classes. This function involves four arbitrary parameters, that is, two more than the previously known hypergeometric classes. These parameters in general are complex and should be chosen so that the resultant detuning is real for the applied (arbitrary) complex-valued transformation of the independent variable. The generalization of the detuning modulation function to the four-parametric case is the most notable extension since many useful properties of the two-state models described by the Heun equation are due to namely the additional parameters involved in this function. Many of the derived amplitude modulation functions present different generalizations of the known hypergeometric models. In several cases the generalization is achieved by multiplying the amplitude modulation function of the corresponding prototype hypergeometric class by an extra factor including an additional parameter. Finally, many classes suggest amplitude modulation functions having forms not discussed before. We present several families of constant-detuning field configurations generated by a real transformation of the independent variable. The members of these families are symmetric or asymmetric two-peak finite-area pulses with controllable distance between the peaks and controllable amplitude of each of the peaks. We show that the edge shapes, the distance between the peaks as well as the amplitude of the peaks are controlled almost independently, by different parameters. We identify the parameters controlling each of the mentioned features and discuss other basic properties of pulse shapes. We show that the pulse edges may become step-wise functions and determine the positions of the limiting vertical-wall edges. We show that the pulse width is controlled by only two of the involved parameters. For some values of these parameters the pulse width diverges and for some other values the pulses become infinitely narrow. We show that the effect of the two mentioned parameters is almost similar, that is, both parameters are able to independently produce pulses of almost the same shape and width. We determine the conditions for generation of pulses of almost indistinguishable shape and width, and present several such examples. Finally, we present a constant-amplitude periodic level-crossing model and several families of constant-detuning field configurations generated by complex transformations of the independent variable.

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.

Similar content being viewed by others

References

  1. L.D. Landau, Phys. Z. Sowjetunion 2, 46 (1932)

    Google Scholar 

  2. C. Zener, Proc. R. Soc. London A 137, 696 (1932)

    Article  ADS  Google Scholar 

  3. E. Majorana, Nuovo Cimento 9, 43 (1932)

    Article  Google Scholar 

  4. E.C.G. Stückelberg, Helv. Phys. Acta 5, 369 (1932)

    Google Scholar 

  5. M.S. Child, Molecular Collision Theory (Academic Press, London, 1974)

  6. E.E. Nikitin, S.Ya. Umanski, Theory of Slow Atomic Collisions (Springer-Verlag, Berlin, 1984)

  7. H. Nakamura, Nonadiabatic Transition: Concepts, Basic Theories and Applications (World Scientific, Singapore, 2012)

  8. B.W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990)

  9. B.W. Shore, Manipulating Quantum Structures Using Laser Pulses (Cambridge University Press, New York, 2011)

  10. H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping (Springer-Verlag, New York, 1999)

  11. A.P. Kazantsev, G.I. Surdutovich, V.P. Yakovlev, Mechanical Action of Light on Atoms (World Scientific, Singapore, 1990)

  12. P. Meystre, Atom Optics (Springer Verlag, New York, 2001)

  13. H. Nakamura, Int. Rev. Phys. Chem. 10, 123 (1991)

    Article  Google Scholar 

  14. H. Nakamura, Ann. Rev. Phys. Chem. 48, 299 (1997)

    Article  ADS  Google Scholar 

  15. C. Zhu, Y. Teranishi, H. Nakamura, Adv. Chem. Phys. 117, 127 (2001)

    Google Scholar 

  16. in Electron Transfer in Inorganic, Organic, and Biological Systems, edited by J. Bolton, N. Mataga, G. Mclendon (American Chemical Society, Washington D.C., 1991), Vol. 228

  17. D. DeVault, Quantum Mechanical Tunnelling in Biological Systems (Cambridge University Press, Cambridge, 1984)

  18. D.E. Shaw et al., Science 330, 341 (2010)

    Article  ADS  Google Scholar 

  19. W.H. Zurek, U. Dorner, P. Zoller, Phys. Rev. Lett. 95, 105701 (2005)

    Article  ADS  Google Scholar 

  20. B. Damski, Phys. Rev. Lett. 95, 035701 (2005)

    Article  ADS  Google Scholar 

  21. R. Barankov, A. Polkovnikov, Phys. Rev. Lett. 101, 076801 (2008)

    Article  ADS  Google Scholar 

  22. J. Dziarmaga, Adv. Phys. 59, 1063 (2010)

    Article  ADS  Google Scholar 

  23. F. Gaitan, Phys. Rev. A 68, 052314 (2003)

    Article  ADS  Google Scholar 

  24. D.M. Berns, W.D. Oliver, S.O. Valenzuela, A.V. Shytov, K.K. Berggren, L.S. Levitov, T.P. Orlando, Phys. Rev. Lett. 97, 150502 (2006)

    Article  ADS  Google Scholar 

  25. K. Smith-Mannschott, M. Chuchem, M. Hiller, T. Kottos, D. Cohen, Phys. Rev. Lett. 102, 230401 (2009)

    Article  ADS  Google Scholar 

  26. A.M. Ishkhanyan, Eur. Phys. Lett. 90, 30007 (2010)

    Article  ADS  Google Scholar 

  27. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Muller, E. Courtade, M. Anderlini, E. Arimondo, Phys. Rev. Lett. 91, 230406 (2003)

    Article  ADS  Google Scholar 

  28. M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, W. Ketterle, Phys. Rev. Lett. 78, 582 (1997)

    Article  ADS  Google Scholar 

  29. N.V. Vitanov, K.-A. Suominen, Phys. Rev. A 56, R4377 (1997)

    Article  ADS  Google Scholar 

  30. A. Ishkhanyan, Phys. Rev. A 81, 055601 (2010)

    Article  ADS  Google Scholar 

  31. I. Tikhonenkov, E. Pazy, Y.B. Band, M. Fleischhauer, A. Vardi, Phys. Rev. A 73, 043605 (2006)

    Article  ADS  Google Scholar 

  32. A.M. Ishkhanyan, B. Joulakian, K.-A. Suominen, Eur. Phys. J. D 48, 397 (2008)

    Article  ADS  Google Scholar 

  33. D. Sun, A. Abanov, V.L. Pokrovsky, Eur. Phys. Lett. 83, 16003 (2008)

    Article  ADS  Google Scholar 

  34. A. Ishkhanyan, B. Joulakian, K.-A. Suominen, J. Phys. B 42, 221002 (2009)

    Article  ADS  Google Scholar 

  35. F.R. Braakman, P. Barthelemy, C. Reichl, W. Wegscheider, L.M.K. Vandersypen, Nature Nanotechnology 8, 432 (2013)

    Article  ADS  Google Scholar 

  36. W. Wernsdorfer, R. Sessoli, Science 284, 133 (1999)

    Article  ADS  Google Scholar 

  37. A.F. Terzis, E. Paspalakis, J. Appl. Phys. 97, 023523 (2005)

    Article  ADS  Google Scholar 

  38. P. Földi, M.G. Benedict, F.M. Peeters, Phys. Rev. A 77, 013406 (2008)

    Article  ADS  Google Scholar 

  39. S.J. Parke, Phys. Rev. Lett. 57, 1275 (1986)

    Article  ADS  Google Scholar 

  40. W.C. Haxton, Phys. Rev. D 35, 2352 (1987)

    Article  ADS  Google Scholar 

  41. M. Blennow, A.Yu. Smirnov, Adv. High Energy Phys. 2013, 972485 (2013)

    Article  Google Scholar 

  42. E.E. Nikitin, Opt. Spectrosc. 6, 431 (1962)

    ADS  Google Scholar 

  43. E.E. Nikitin, Disc. Faraday Soc. 33, 14 (1962)

    Article  Google Scholar 

  44. E.E. Nikitin, Ann. Rev. Phys. Chem. 50, 1 (1999)

    Article  ADS  Google Scholar 

  45. N. Rosen, C. Zener. Phys. Rev. 40, 502 (1932)

    Article  ADS  MATH  Google Scholar 

  46. Yu.N. Demkov, M. Kunike, Vestn. Leningr. Univ. Fis. Khim. 16, 39 (1969)

    MathSciNet  Google Scholar 

  47. K.-A. Suominen, B.M. Garraway, Phys. Rev. A 45, 374 (1992)

    Article  ADS  Google Scholar 

  48. A. Bambini, P.R. Berman, Phys. Rev. A 23, 2496 (1981)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  50. F.T. Hioe, C.E. Carroll, J. Opt. Soc. Am. B 3, 497 (1985)

    Article  ADS  Google Scholar 

  51. C.E. Carroll, F.T. Hioe, J. Phys. A 19, 3579 (1986)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  52. A.M. Ishkhanyan, J. Phys. A 30, 1203 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. A.M. Ishkhanyan, Opt. Commun. 176, 155 (2000)

    Article  ADS  Google Scholar 

  54. A.M. Ishkhanyan, J. Contemp. Phys. (Armenian Ac. Sci.) 31, 10 (1996)

    Google Scholar 

  55. A.M. Ishkhanyan, J. Phys. A 33, 5539 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  56. C.E. Carroll, F.T. Hioe, J. Opt. Soc. Am. B 5, 1335 (1988)

    Article  ADS  Google Scholar 

  57. C.E. Carroll, F.T. Hioe, J. Phys. B 22, 2633 (1989)

    Article  ADS  Google Scholar 

  58. C.E. Carroll, F.T. Hioe, J. Phys. A 19, 1151 (1986)

    Article  ADS  Google Scholar 

  59. C.E. Carroll, F.T. Hioe, Phys. Rev. A 36, 724 (1987)

    Article  ADS  Google Scholar 

  60. C.E. Carroll, F.T. Hioe, Phys. Rev. A 42, 1522 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  61. A.M. Ishkhanyan, J. Phys. A 33, 5041 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  62. A.M. Ishkhanyan, K.-A. Suominen, Phys. Rev. A 65, 051403(R) (2002)

    Article  ADS  Google Scholar 

  63. T.A. Laine, S. Stenholm, Phys. Rev. A 53, 2501 (1996)

    Article  ADS  Google Scholar 

  64. N.V. Vitanov, S. Stenholm, Phys. Rev. A 55, 648 (1997)

    Article  ADS  Google Scholar 

  65. A.M. Ishkhanyan, Reports (Armenian Ac. Sci.) 102, 320 (2002)

    Google Scholar 

  66. A.M. Ishkhanyan, A.M. Manukyan, J. Contemp. Phys. (Armenian Ac. Sci.) 37, 1 (2002)

    Google Scholar 

  67. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, in Higher Transcendental Functions, (McGraw-Hill, New York, 1955), Vol. 3

  68. L.J. Slater, Generalized hypergeometric functions (Cambridge University Press, Cambridge, 1966)

  69. K. Heun, Math. Ann. 33, 161 (1889)

    Article  MATH  MathSciNet  Google Scholar 

  70. A. Ronveaux, Heun’s Differential Equations (Oxford University Press, London, 1995)

  71. S.Yu. Slavyanov, W. Lay, Special Functions (Oxford University Press, Oxford, 2000)

  72. in NIST Handbook of Mathematical Functions, edited by F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Cambridge University Press, New York, 2010), http://dlmf.nist.gov/31.12

  73. M. Hortacsu, in Proceedings of the 13th Regional Conference of Mathematical Physics, Antalya, Turkey, 2010, edited by U. Camci, I. Semiz (World Scientific, Singapore, 2013), pp. 23–39

  74. A.M. Ishkhanyan, A.E. Grigoryan, J. Phys. A 47, 465205 (2014)

    Article  ADS  Google Scholar 

  75. P.K. Jha, Yu.V. Rostovtsev, Phys. Rev. A 82, 015801 (2010)

    Article  ADS  Google Scholar 

  76. P.K. Jha, Yu.V. Rostovtsev, Phys. Rev. A 81, 033827 (2010)

    Article  ADS  Google Scholar 

  77. N. Svartholm, Math. Ann. 116, 413 (1939)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  79. A. Erdélyi, Q. J. Math. (Oxford) 15, 62 (1944)

    Article  MATH  Google Scholar 

  80. D. Schmidt, J. Reine Angew. Math. 309, 127 (1979)

    MATH  MathSciNet  Google Scholar 

  81. E.G. Kalnins, W. Miller, SIAM J. Math. Anal. 22, 1450 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  82. R.S. Sokhoyan, D.Yu. Melikdzanian, A.M. Ishkhanyan, J. Contemp. Physics (Armenian Ac. Sci.) 40, 1 (2005)

    Google Scholar 

  83. T.A. Ishkhanyan, T.A. Shahverdyan, A.M. Ishkhanyan, arXiv:1403.7863 (2014)

  84. T.A. Ishkhanyan, A.M. Ishkhanyan, AIP Adv. 4, 087132 (2014)

    Article  ADS  Google Scholar 

  85. E.S. Cheb-Terrab, J. Phys. A 37, 9923 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  86. A. Ishkhanyan, J. Phys. A 38, L491 (2005)

    Article  MathSciNet  Google Scholar 

  87. A. Ishkhanyan, J. Phys. A 34, L591 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  88. A. Ishkhanyan, K.-A. Suominen, J. Phys. A 36, L81 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  89. P.P. Fiziev, J. Phys. A 43, 035203 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  90. M.N. Hounkonnou, A. Ronveaux, Appl. Math. Comput. 209, 421 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  91. V.A. Shahnazaryan, T.A. Ishkhanyan, T.A. Shahverdyan, A.M. Ishkhanyan, Armenian J. Phys. 5, 146 (2012)

    Google Scholar 

  92. J.H. Lambert, Acta Helv. 3, 128 (1758)

  93. L. Euler, Acta Acad. Scient. Petropol. 2, 29 (1783)

  94. A.M. Ishkhanyan, Phys. Rev. A 61, 063611 (2000)

    Article  ADS  Google Scholar 

  95. A.M. Ishkhanyan, Laser Phys. 7, 1225 (1997)

    Google Scholar 

  96. A.M. Manukyan et al., IJDEA 13, 219 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Physical Research, NAS of Armenia, 0203, Ashtarak, Armenia

    Artur M. Ishkhanyan, Tigran A. Shahverdyan & Tigran A. Ishkhanyan

  2. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700, Russia

    Tigran A. Shahverdyan & Tigran A. Ishkhanyan

Authors
  1. Artur M. Ishkhanyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tigran A. Shahverdyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tigran A. Ishkhanyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artur M. Ishkhanyan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ishkhanyan, A.M., Shahverdyan, T.A. & Ishkhanyan, T.A. Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions. Eur. Phys. J. D 69, 10 (2015). https://doi.org/10.1140/epjd/e2014-50386-9

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/e2014-50386-9

Keywords

Search

Navigation