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Lie algebraic treatment of the quadratic invariants for a quantum system

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Abstract

We consider the problem of the time-dependent degenerate parametric amplifier. We obtain the quadratic invariant and use it to derive the wave function via its su(1, 1) algebraic basis and a unitary transformation to the time-dependent Schrödinger equation for the parametric amplifier. We obtain the real and the complex invariants, which we use to solve the time-dependent Cauchy problem. Using different integrability conditions, we find the most general solution, which we analyze extensively, providing details of the calculations.

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References

  1. M. Hillery, D. Yu., and J. Bergou, Phys. Rev. A, 49, 1288–1295 (1994).

    Article  ADS  Google Scholar 

  2. N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, and H. J. Kimble, Phys. Rev. Lett., 75, 3426–3429 (1995).

    Article  ADS  Google Scholar 

  3. C. C. Gerry, J. Opt. Soc. Amer. B, 8, 685–690 (1991).

    Article  ADS  Google Scholar 

  4. C. C. Gerry, Phys. Rev. A, 35, 2146–2149 (1987); C. C. Gerry and E. E. Hach, Phys. Lett. A, 174, 185–189 (1993).

    Article  ADS  Google Scholar 

  5. I. Jex and V. Bužek, J. Modern Opt., 40, 771–783 (1993).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. M. Zahler and Y. Ben Aryeh, Phys. Rev. A, 43, 6368–6378 (1991).

    Article  ADS  Google Scholar 

  7. M. S. Abdalla, A. A. El-Orany Faisal, and J. Perina, Nuovo Cimento Soc. Ital. Fis. B, 116, 137–154 (2001).

    ADS  Google Scholar 

  8. V. Bužek and G. Drobný, Phys. Rev. A, 65, 053410 (2002).

  9. A. A. El-Orany Faisel, S. S. Hassan, and M. S. Abdalla, J. Optics B, 5, 396–404 (2003).

    ADS  Google Scholar 

  10. C. W. Gardiner and C. M. Savage, Opt. Commun., 50, 173–178 (1984).

    Article  ADS  Google Scholar 

  11. A. Karpati, P. Adam, J. Janszky, M. Bertolotti, and C. Sibilia, J. Optics B, 2, 133–139 (2000).

    ADS  MathSciNet  Google Scholar 

  12. X.-G. Wang, S.-H. Pan, and G.-Z. Yang, European J. Phys., 10, 415–422 (2000).

    ADS  Google Scholar 

  13. M. S. Abdalla and M. A. Bashir, J. Egyptian Math. Soc., 5, 171–182 (1997).

    MATH  MathSciNet  Google Scholar 

  14. H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D, 35, 1831–1834 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  15. H.-Y. Fan and J. VanderLinde, Phys. Rev. A, 39, 2987–2993 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  16. C. F. Lo, Phys. Rev. A, 42, 6752–6755 (1990).

    Article  ADS  Google Scholar 

  17. M. Xin and W. Rhodes, Phys. Rev. A, 41, 4625–4631 (1990).

    Article  ADS  Google Scholar 

  18. M. S. Abdalla and M. A. Bashir, Quantum Semiclass. Opt., 10, 415–423 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  19. M. A. Al-Gwaiz, M. S. Abdalla, and S. Deshmukh, J. Phys. A, 27, 1275–1282 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. M. S. Abdalla, Nuovo Cimento Soc. Ital. Fis. B, 112, 1549–1554 (1997).

    MathSciNet  Google Scholar 

  21. M. A. Bashir and M. S. Abdalla, Phys. Lett. A, 204, 21–25 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. E. Y. C. Lu, Phys. Rev. A, 8, 1053–1061 (1973); Nuovo Cimento Lett., 3, 585–589 (1972).

    Article  ADS  Google Scholar 

  23. M. S. Abdalla, S. S. Hassan, and A.-S. F. Obada, Phys. Rev. A, 34, 4869–4874 (1986); M. S. Abdalla and M. M. Nassar, Ann. Phys., 324, 637–669 (2009).

    Article  ADS  Google Scholar 

  24. H. R. Lewis and W. B. Riesenfeld, J. Math. Phys., 10, 1458–1473 (1969).

    Article  ADS  MathSciNet  Google Scholar 

  25. H. R. Lewis Jr., Phys. Rev. Lett., 18, 510–512 (1967); J. Math. Phys., 9, 1976–1986 (1968).

    Article  ADS  Google Scholar 

  26. V. P. Ermakov, Appl. Anal. Discrete Math., 2, 123–145 (2008).

    Article  MathSciNet  Google Scholar 

  27. I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, Phys. Rev. D, 2, 1371–1385 (1970).

    Article  ADS  Google Scholar 

  28. I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, J. Math. Phys., 14, 576–582 (1973).

    Article  ADS  Google Scholar 

  29. V. V. Dodonov and V. I. Man’ko, Phys. Rev. A, 20, 550–560 (1979).

    Article  ADS  Google Scholar 

  30. M. S. Abdalla and P. G. L. Leach, J. Phys. A, 36, 12205–12221 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. M. S. Abdalla and P. G. L. Leach, J. Phys. A, 38, 881–893 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. S. Steinberg, J. Differential Equations, 26, 404–434 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  33. G. Dattoli, M. Richetta, G. Schettini, and A. Torre, J. Math. Phys., 31, 2856–2863 (1990).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. G. Dattoli, J. C. Gallado, and A. Torre, Riv. Nuovo Cimento, 11, 1–79 (1988).

    Google Scholar 

  35. A. O. Barut and L. Girardello, Comm. Math. Phys., 21, 41–55 (1971).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. A. O. Barut, “Unified algebraic construction of representations of compact and non-compact Lie algebras and Lie groups,” in: Mathematical Methods of Theoretical Physics (Lect. Theoret. Phys., Vol. 9A, W. E. Brittin, A. O. Barut, and M. Guenin, eds.), Gordon and Breach, New York (1967), pp. 125–171.

    Google Scholar 

  37. V. de Alfaro, S. Fubini, and G. Furlan, Nuovo Cimento Soc. Ital. Fis. A, 34, 569–612 (1984).

    Article  ADS  Google Scholar 

  38. K. Andriopoulos and P. G. L. Leach, J. Phys. A, 38, 4365–4374 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. K. Andriopoulos and P. G. L. Leach, J. Nonlinear Math. Phys. (Suppl. 1), 12, 32–42 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  40. P. G. L. Leach and K. Andriopoulos, Phys. Scripta, 77, 015002 (2008).

    Google Scholar 

  41. R. L. Lemmer and P. G. L. Leach, Arab. J. Math. Sci., 5, 1–17 (1999).

    MATH  MathSciNet  Google Scholar 

  42. M. Znojil and P. G. L. Leach, J. Math. Phys., 33, 2785–2794 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. P. G. L. Leach, J. Math. Phys., 18, 1608–1611 (1977).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. J.-R. Burgan, “Sur les groupes de transformation en physique mathématique: Application aux fluides de l’éspace des phases et à la mécanique quantique,” Dissertation, Univ. d’Orléans, Orléans (1978).

    Google Scholar 

  45. J.-R. Burgan, M. R. Feix, E. Fijalkow, J. Gutierrez, and A. Munier, “Utilisation des groupes de transformation pour la résolution des équations aux dérivées partielles,” in: Applied Inverse Problems (Lect. Notes Phys., Vol. 85, P. C. Sabatier, ed.), Springer, Berlin (1978), pp. 248–273.

    Chapter  Google Scholar 

  46. E. Pinney, Proc. Amer. Math. Soc., 1, 681 (1950).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia

    M. Sebawe Abdalla

  2. School of Mathematical Sciences, University of KwaZulu-Natal, Durban, Republic of South Africa

    P. G. L. Leach

Authors
  1. M. Sebawe Abdalla
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  2. P. G. L. Leach
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Correspondence to M. Sebawe Abdalla.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 142–161, April, 2009.

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Abdalla, M.S., Leach, P.G.L. Lie algebraic treatment of the quadratic invariants for a quantum system. Theor Math Phys 159, 535–550 (2009). https://doi.org/10.1007/s11232-009-0043-1

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