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Logarithmic decays of unstable states

  • Filippo GiraldiEmail author
Regular Article

Abstract

It is known that the survival amplitude of unstable quantum states deviates from exponential relaxations and exhibits decays that depend on the integral and analytic properties of the energy distribution density. In the same scenario, model independent dominant logarithmic decays t −1−α0log t of the survival amplitude are induced over long times by special conditions on the energy distribution density. While the instantaneous decay rate exhibits the dominant long time relaxation 1 /t, the instantaneous energy tends to the minimum value of the energy spectrum with the dominant logarithmic decay 1/(tlog 2 t) over long times. Similar logarithmic relaxations have already been found in the dynamics of short range potential systems with even dimensional space or in the Weisskopf-Wigner model of spontaneous emission from a two-level atom. Here, logarithmic decays are obtained as a pure model independent quantum effect in general unstable states.

Keywords

Atomic Physics 

References

  1. 1.
    L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Prog. Phys. 41, 587 (1978)ADSCrossRefGoogle Scholar
  2. 2.
    S. Krylov, V.A. Fock, Zh. Eksp. Teor. Fiz. 17, 93 (1947)Google Scholar
  3. 3.
    A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Publ. Inc., 1999) Google Scholar
  4. 4.
    L.A. Khalfin, Zh. Eksp. Teor. Fiz. 33, 1371 (1957) Google Scholar
  5. 5.
    L.A. Khalfin, Sov. Phys. JETP 6, 1053 (1958)ADSzbMATHGoogle Scholar
  6. 6.
    R.E.A.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, New York, 1934)Google Scholar
  7. 7.
    C. Rothe, S.I. Hintschich, A.P. Monkman, Phys. Rev. Lett. 96, 163601 (2006) ADSCrossRefGoogle Scholar
  8. 8.
    A. Peres, Ann. Phys. 129, 33 (1980)ADSCrossRefGoogle Scholar
  9. 9.
    T. Jittoh, S. Matsumoto, J. Sato, Y. Sato, K. Takeda, Phys. Rev. A 71, 012109 (2005) ADSCrossRefGoogle Scholar
  10. 10.
    D.S. Onley, A. Kumar, Am. J. Phys. 60, 432 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    K. Urbanowski, Eur. Phys. J. D 54, 25 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    F.W.J. Olver, Asymoptotic and Special Functions (Academic Press, New York, 1974)Google Scholar
  13. 13.
    A. Erdelyi, Asymptotic Expansions (Dover Publ. Inc., New York, 1956)Google Scholar
  14. 14.
    E.T. Copson, Asymptotic Expansions (Cambridge University Press, 1965)Google Scholar
  15. 15.
    K. Urbanowski, Cent. Eur. J. Phys. 7, 696 (2009)CrossRefGoogle Scholar
  16. 16.
    K. Urbanowski, Eur. Phys. J. C 58, 151 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    P.J. Aston, Europhys. Lett. 97 52001 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    P.J. Aston, Europhys. Lett. 101 42002 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    C. Roth, S.I. Hintschich, A.P. Monkman, Phys. Rev. Lett. 71, 163601 (2006) ADSCrossRefGoogle Scholar
  20. 20.
    P. Facchi, S. Pascazio, Physica A 271, 133 (1999) ADSCrossRefGoogle Scholar
  21. 21.
    E. Torrontegui, J.G. Muga, J. Martorell, D.W.L. Sprung, Phys. Rev. A 80, 012703 (2009) ADSCrossRefGoogle Scholar
  22. 22.
    M. Nowakowski, N.G. Kelkar, AIP Conf. Proc. 1030, 250 (2008) ADSCrossRefGoogle Scholar
  23. 23.
    I. Joichi, Sh. Matsumoto, M. Yoshimura, Phys. Rev. D 58, 045004 (1998) ADSCrossRefGoogle Scholar
  24. 24.
    H. Nakazato, M. Namiki, S. Pascazio, Int. J. Mod. Phys. B 10, 247 (1996)ADSCrossRefGoogle Scholar
  25. 25.
    S.R. Wilkinson et al., Nature 387, 575 (1997) ADSCrossRefGoogle Scholar
  26. 26.
    P.T. Greenland, Nature 355, 298 (1988) ADSCrossRefGoogle Scholar
  27. 27.
    F. Giacosa, Found. Phys. 42, 1262 (2012) ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    L.M. Krauss, J. Dent, Phys. Rev. Lett. 100, 171301 (2008) ADSCrossRefMathSciNetGoogle Scholar
  29. 29.
    K. Urbanowski, Phys. Rev. Lett. 107, 209001 (2011) ADSCrossRefGoogle Scholar
  30. 30.
    N.G. Kelkar, M. Nowakowski, K.P. Khemchandani, Phys. Rev. C 70, 024601 (2004) ADSCrossRefGoogle Scholar
  31. 31.
    E.B. Norman, S.B. Gazes, S.G. Crane, D.A. Bennett, Phys. Rev. Lett. 60, 2246 (1988) ADSCrossRefGoogle Scholar
  32. 32.
    K.O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    M. Gadella, G. Pronko, arXiv:1106.5782v1 (2011) and references thereinGoogle Scholar
  34. 34.
    M. Miyamoto, J. Math. Phys. 47, 082103 (2006) ADSCrossRefMathSciNetGoogle Scholar
  35. 35.
    M. Miyamoto, Open Syst. Inf. Dyn. 13, 291 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    A. Jensen, T. Kato, Duke Math. J. 46, 583 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    V.F. Weisskopf, E.P. Wigner, Z. Phys. 63, 54 (1930)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    E. Merzbacher, Quantum Mechanics, 2nd edn. (Wiley, New York, 1970)Google Scholar
  39. 39.
    W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973)Google Scholar
  40. 40.
    L. Davidovich, Ph. D. thesis, University of Rochester, 1976Google Scholar
  41. 41.
    P.L. Knight, P.W. Milonni, Phys. Lett. A 56, 275 (1976)ADSCrossRefGoogle Scholar
  42. 42.
    K. Wodkiewicz, J.H. Eberly, Ann. Phys. 101, 574 (1976) ADSCrossRefGoogle Scholar
  43. 43.
    J. Seke, W.N. Herfort, Phys. Rev. A 38, 833 (1988) and references thereinADSCrossRefGoogle Scholar
  44. 44.
    M. Murata, J. Funct. Anal. 49, 10 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    M.L. Goldberger, Collision Theory (Willey, New York, 1964)Google Scholar
  46. 46.
    J. Mostowski, K. Wodkiewicz, Bul. Acad. Polon. Sci. 21, 1027 (1973) Google Scholar
  47. 47.
    D.G. Arbo, M.A. Castagnino, F.H. Gaioli, S. Iguri, Physica A227, 469 (2000) ADSCrossRefMathSciNetGoogle Scholar
  48. 48.
    M. Nowakowski, N.G. Kelkar, AIP Conf. Proc. 1030, 250 (2008) ADSCrossRefGoogle Scholar
  49. 49.
    K. Urbanowski, J. Piskorski, J. Nucl. Part. Phys. 2, 71 (2012)CrossRefGoogle Scholar
  50. 50.
    K.M. Sluis, E.A. Gislason, Phys. Rev. A 43, 4581 (1991) ADSCrossRefGoogle Scholar
  51. 51.
    R.E. Parrot, J. Lawrence, Europhys. Lett. 57, 632 (2002)ADSCrossRefGoogle Scholar
  52. 52.
    J. Lawrence, J. Opt. B 4, S446 (2002)ADSCrossRefGoogle Scholar
  53. 53.
    R. Santra, J.M. Shainline, C.H. Greene, Phys. Rev. A 71, 032703 (2005) ADSCrossRefGoogle Scholar
  54. 54.
    N.M. Temme, Special Functions An Introduction to the Classical Functions of Mathematical Physics (John Wiley & Sons, 1996)Google Scholar
  55. 55.
    A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, in Integrals and Series (Gordon and Breach Science Publishers, Amsterdam, 1986), Vols. 1–5Google Scholar
  56. 56.
    E.C. Titchmarsh, The Theory of functions, 2nd edn. (Oxford University Press, 1997)Google Scholar
  57. 57.
    E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd edn. (New York, Chelsea Pub. Co., 1986) Google Scholar
  58. 58.
    A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, in Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vols. 1, 2Google Scholar
  59. 59.
    F. Mainardi, Fractional Calcolus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010) Google Scholar
  60. 60.
    K. Urbanowski, Phys. Rev. A 50, 2847 (1994) ADSCrossRefGoogle Scholar
  61. 61.
    K. Urbanowski, Open Syst. Inf. Dyn. 20, 1340008 (2013) CrossRefMathSciNetGoogle Scholar
  62. 62.
    K. Urbanowski, J. Skorek, Int. J. Mod. Phys. 8, 4355 (1993)ADSCrossRefGoogle Scholar
  63. 63.
    L.P. Horwitz, J.P. Marchand, Helv. Phys. Acta 42, 801 (1969)MathSciNetGoogle Scholar
  64. 64.
    D.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1941)Google Scholar
  65. 65.
    R. Wong, Asymptotic Approximations of Integrals (Academic Press, Boston, 1989)Google Scholar
  66. 66.
    R. Wong, Y.-Q. Zhao, Proc. R. Soc. London A 458, 625 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  67. 67.
    F. Giraldi, F. Petruccione, J. Phys. A 46, 015304 (2013) ADSCrossRefMathSciNetGoogle Scholar
  68. 68.
    F. Giraldi, F. Petruccione, J. Phys. A 45, 285306 (2012) CrossRefMathSciNetGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Quantum Research Group, School of Chemistry and PhysicsUniversity of KwaZulu-Natal and National Institute for Theoretical PhysicsDurbanSouth Africa
  2. 2.Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM), c/o Istituto Nazionale di Alta Matematica Francesco Severi Citta’ UniversitariaRomaItaly

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