Decays of Unstable Quantum Systems

  • Charis AnastopoulosEmail author


This paper is a pedagogical yet critical introduction to the quantum description of unstable systems, mostly at the level of a graduate quantum mechanics course. Quantum decays appear in many different fields of physics, and their description beyond the exponential approximation is the source of technical and conceptual challenges. In this article, we present both general methods that can be adapted to a large class of problems, and specific elementary models to describe phenomena like photo-emission, beta emission and tunneling-induced decays. We pay particular attention to the emergence of exponential decay; we analyze the approximations that justify it, and we present criteria for its breakdown. We also present a detailed model for non-exponential decays due to resonance, and an elementary model describing decays in terms of particle-detection probabilities. We argue that the traditional methods for treating decays face significant problems outside the regime of exponential decay, and that the exploration of novel regimes of current interest requires new tools.


Quantum decays Non-exponential decays Lee model Tunneling Particle detection 



Research was supported by Grant No. E611 from the Research Committee of the University of Patras via the ”K. Karatheodoris” program.


  1. 1.
    Pauli, W.: The principles of quantum mechanics. In: Flugge, S. (ed.) Encyclopedia of Physics, vol. 5/1. Springer, Berlin (1958)Google Scholar
  2. 2.
    Muga, J.C., Mayato, R.S., Equisquiza, I.L.: Time in quantum mechanics, vol. 1. Springer, Berlin (2008)Google Scholar
  3. 3.
    Muga, J.G., Ruschhaupt, A., Del Campo, A.: Time in quantum mechanics, vol. 2. Springer, Berlin (2010)zbMATHGoogle Scholar
  4. 4.
    Muga, J.C., Leavens, J.R.: Arrival time in quantum mechanics. Phys. Rep. 338, 353 (2000)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hauge, E.H., Støvneng, J.A.: Tunneling, Times: A critical review. Rev. Mod. Phys. 61, 917 (1989)ADSCrossRefGoogle Scholar
  6. 6.
    Olkhovsky, V.S., Recami, E.: Recent developments in the time analysis of tunnelling processes. Phys. Rep. 214, 339 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    Landauer, R., Martin, T.: Barrier interaction time in tunneling. Rev. Mod. Phys. 66, 17 (1994)ADSCrossRefGoogle Scholar
  8. 8.
    Fonda, L., Ghirardi, G.C., Rimini, A.: Decay theory of unstable quantum systems. Rep. Prog. Phys. 41, 587 (1978)ADSCrossRefGoogle Scholar
  9. 9.
    Peres, A.: Non-exponential decay law. Ann. Phys. 129, 33 (1980)ADSCrossRefGoogle Scholar
  10. 10.
    Gorin, T., Prosen, T., Seligman, T., Znidaric, N.: Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    Lee, T.D.: Some special examples in renormalizable field theory. Phys. Rev. 95, 1329 (1954)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Rosenstein, B., Horwitz, L.P.: Probability current versus charge current of a relativistic particle. J. Phys. A: Math. Gen. 18, 2115 (1985)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Landau, L., Lifschitz, E.: Quantum mechanics: Non-relativistic theory 2 (ed.) . Pergamon Press, Oxford (1965)Google Scholar
  14. 14.
    Perelemov, A.M., Popov, V.S., Terent’ev, M.V.: Ionization of atoms in an alternating electric field. Soviet Phys. JETP 23, 924 (1966)ADSGoogle Scholar
  15. 15.
    Bracken, A.J., Melloy, G.F.: Probability backflow and a new dimensionless quantum number. J. Phys. A27, 2197 (1994)ADSMathSciNetzbMATHGoogle Scholar
  16. 16.
    Winter, R.G.: Evolution of a quasi-stationary state. Phys. Rev. 123, 1503 (1961)ADSCrossRefGoogle Scholar
  17. 17.
    Ekstein, H., Siegert, A.J.F.: On a reinterpretation of decay experiments. Ann. Phys. (N.Y.) 68, 509 (1971)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Heisenberg, W.: Quantum theory and its interpretations, reprinted at quantum theory and measurement,. In: Wheeler, J.A., Zurek, W.H. (eds.) . Princeton University Press, Princeton (1983)Google Scholar
  19. 19.
    Fonda, L., Ghirardi, G.C., Omero, C., Rimini, A., Weber, T.: Quantum theory of sequential decay processes. Phys. Rev. D18, 4757 (1978)ADSGoogle Scholar
  20. 20.
    Anastopoulos, C., Savvidou, N.: Time-of-arrival probabilities for general particle detectors,. Phys. Rev. A86, 012111 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    Anastopoulos, C., Savvidou, N.: Time-of-arrival correlations. Phys. Rev. A95, 032105 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    Anastopoulos, C., Savvidou, N.: Time of arrival and localization of relativistic particles. arXiv:1807-06533
  23. 23.
    Anastopoulos, C.: Time-of-arrival probabilities and quantum measurements. III. Decay of unstable states. J. Math. Phys. 49, 022103 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Glauber, R.J.: The quantum theory of optical coherence. Phys. Rev. 130, 2529 (1963)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Anastopoulos, C., Shresta, S., Hu, B.L.: Non-markovian entanglement dynamics of two qubits interacting with a common electromagnetic field. Q. Inf. Proc. 8, 549 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    del Campo, A., Molina-Vilaplana, J., Sonner, J.: Scrambling the spectral form factor: Unitarity constraints and exact results. Phys. Rev. D95, 126008 (2017)ADSMathSciNetGoogle Scholar
  28. 28.
    Chenu, A., Egusquiza, I.L., Molina-Vilaplana, J., del Campo, A.: Quantum work statistics, loschmidt echo and information scrambling. Sci. Rep. 8, 12634 (2018)ADSCrossRefGoogle Scholar
  29. 29.
    Landsmann, A.S., Keller, U.: Attosecond science and the tunnelling time problem. Phy. Rep. 547, 1 (2015)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Litvinov, Y.A., et al.: Observation of non-exponential orbital electron capture decays of hydrogen-like 140,P r and 142 P m ions. Phys. Lett. B664, 162 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Kienle, P., et al.: High-resolution measurement of the time-modulated orbital electron capture and of the decay of hydrogen-like 142 P m 60+ ions. Phys. Lett. B 726, 638 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    Bohm, D., Pines, D.: A collective description of electron interactions: III. Coulomb interactions in a degenerate electron gas. Phys. Rev. 92, 609 (1953)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gell-Mann, M., Brueckner, K.A.: Correlation energy of an electron gas at high density. Phys. Rev. 106, 364 (1957)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Bloch, C.: Une formulation unifie de la thorie des ractions nuclaires. Nucl. Phys. 4, 53 (1957)CrossRefGoogle Scholar
  35. 35.
    Namiki, M.: One-particle motions in many-particle systems and the optical model in nuclear reactions. Prog. Theor. Phys. 23, 629 (1960)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Nakazato, H., Namiki, M., Pascazio, S.: Temporal behavior of quantum mechanical systems. Int. J. Mod. Phys. B10, 247 (1996)ADSCrossRefGoogle Scholar
  37. 37.
    Weisskopf, W., Wigner, E.P.: Berechnung der Natrlichen Linienbreite auf Grund der Diracschen Lichttheorie. Zeit. Phys. 63, 54 (1930)ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Barnett, S.M., Radmore, P.M.: Methods in theoretical quantum optics. Clarendon Press, Oxford (1997). Appendix 6zbMATHGoogle Scholar
  39. 39.
    Van Hove, L.: Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21, 517 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Misra, B., Sudarshan, E.C.G.: The Zeno’s paradox in quantum theory. J. Math. Phys. 18, 756 (1977)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Facchi, P., Pascazio, S.: Quantum Zeno dynamics: Mathematical and physical aspects. J. Phys. A: Math. Theor. 493001, 41 (2008)zbMATHGoogle Scholar
  42. 42.
    Mandelstam, L., Tamm, I.: The uncertainty relation between energy and time in non-relativistic quantum mechanics. J. Phys (USSR) 9, 249 (1945)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Bhattacharyya, K.: Quantum decay and the Mandelstam-Tamm energy inequality. J. Phys. A Math. Gen. 16, 2993 (1983)ADSCrossRefGoogle Scholar
  44. 44.
    Dodonov, V.V., Dodonov, A.V.: Energy–time and frequency–time uncertainty relations: Exact inequalities. Phys. Scr. 90, 074049 (2015)ADSCrossRefGoogle Scholar
  45. 45.
    Wilkinson, S.R., Bharucha, C.F., Fischer, M.C., Madison, K.W., Morrow, P.R., Niu, Q., Sundaram, B., Raizen, Mark G.: Experimental evidence for non-exponential decay in quantum tunnelling. Nature 387, 575 (1997)ADSCrossRefGoogle Scholar
  46. 46.
    Fischer, M.C., Gutirrez-Medina, B., Raizen, M.G.: Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys. Rev. Lett. 87, 040402 (2001)ADSCrossRefGoogle Scholar
  47. 47.
    Margolus, N., Levitin, L.B.: The maximum speed of dynamical evolution. Physica D120, 188 (1998)ADSGoogle Scholar
  48. 48.
    Levitin, L.B., Toffoli, T.: The fundamental limit on the rate of quantum dynamics: The unified bound is tight. Phys. Rev. Lett. 103, 160502 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    Taddei, M.M., Escher, B.M., Davidovich, L., de Matos Filho, R.L.: Quantum speed limit for physical processes. Phys. Rev. Lett. 110, 050402 (2013)ADSCrossRefGoogle Scholar
  50. 50.
    del Campo, A., Egusquiza, I.L., Plenio, M.B., Huelga, S.F.: Quantum speed limits in open system dynamics. Phys. Rev. Lett. 110, 050403 (2013)ADSCrossRefGoogle Scholar
  51. 51.
    Shanahan, B., Chenu, A., Margolus, N., del Campo, A.: Quantum speed limits across the quantum-to-classical transition. Phys. Rev. Lett. 120, 070401 (2018)ADSCrossRefGoogle Scholar
  52. 52.
    Okuyama, M., Ohzeki, M.: Quantum speed limit is not quantum. Phys. Rev. Lett. 120, 070402 (2018)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Deffner, S., Campbell, S.: Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control. J. Phys. A: Math. Theor. 50, 453001 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Goldberger, M.L., Watson, K.M.: Collision theory. John, Wiley (1967)zbMATHGoogle Scholar
  55. 55.
    Newton, R.G.: Scattering theory of waves and particles. Springer-Verlag, Berlin (1982)zbMATHCrossRefGoogle Scholar
  56. 56.
    Weinberg, S.: Lectures on quantum mechanics. Cambridge University Press, Cambridge (2015)zbMATHCrossRefGoogle Scholar
  57. 57.
    Hellund, E.J.: The decay of resonance radiation by spontaneous emission. Phys. Rev. 89, 919 (1953)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Namiki, M., Mugibayashi, M.: On the radiation damping and the decay of an excited state. Prog. Theor. Phys. 10, 474 (1953)ADSzbMATHCrossRefGoogle Scholar
  59. 59.
    Rothe, C., Hintschich, S.I., Monkman, A.P.: Violation of the exponential-decay law at long times. Phys. Rev. Lett. 96, 163601 (2006)ADSCrossRefGoogle Scholar
  60. 60.
    Scully, M.O., Zubairy, M.S.: Quantum optics. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  61. 61.
    Anastopoulos, C., Hu, B.L.: Two-level atom-field interaction: Exact master equations for non-markovian dynamics, decoherence, and relaxation. Phys. Rev. A62, 033821 (2000)ADSCrossRefGoogle Scholar
  62. 62.
    Fonda, L., Ghirardi, G.C., Rimini, A.: Interpretation of the normalizable state in the lee model with form factor. Phys. Rev. 133, B196 (1964)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    Alzetta, R., d’ Ambrogio, E.: Evolution of a resonant state. Nucl. Phys. 82, 683 (1966)CrossRefGoogle Scholar
  64. 64.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, 10th edn. Dover, New York (1972)zbMATHGoogle Scholar
  65. 65.
    Fermi, E.: Tentativo di una Teoria dei Raggi β. Ric. Sci. 2, 12 (1933)zbMATHGoogle Scholar
  66. 66.
    Wilson, F.L.: Fermi’s theory of beta decay. Am. J. Phys. 36, 1150 (1968)ADSCrossRefGoogle Scholar
  67. 67.
    Cottingham, W.N., Greenwood, D.A.: An introduction to nuclear physics. Cambridge University Press, Cambridge (2004)Google Scholar
  68. 68.
    Cummings, N.I., Hu, B.L.: Dynamics of atom-field entanglement: Towards strong-coupling and non-markovian regimes. Phys. Rev. A77, 053823 (2008)ADSCrossRefGoogle Scholar
  69. 69.
    Lewenstein, M., Zakrzewski, J., Mossberg, T.W., Mostowski, J.: Non-exponential spontaneous decay in cavities and waveguides. J. Phys. B: At. Mol. Opt. Phys. 21, L9 (1988)ADSCrossRefGoogle Scholar
  70. 70.
    Raczyfiski, A., Zaremba, J.: Threshold effects in photoionization and photodetachment. Phys. Rep. 235, 1 (1993)ADSCrossRefGoogle Scholar
  71. 71.
    Jittoh, T., Matsumoto, S., Sato, J., Sato, Y., Takeda, K.: Nonexponential decay of an unstable quantum system: Small-Q-value s-wave decay. Phys. Rev. A 71, 012109 (2005)ADSCrossRefGoogle Scholar
  72. 72.
    Dinu, V., Jensen, A., Nenciu, G.: Nonexponential decay laws in perturbation theory of near threshold eigenvalues. J. Math. Phys. 50, 013516 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Gamow, G.: Zur Quantentheorie des Atomkernes. Zeit. Phys. 51, 204 (1928)ADSzbMATHCrossRefGoogle Scholar
  74. 74.
    Gurney, R.W., Condon, E.U.: Quantum mechanics and radioactive disintegration. Phys. Rev. 33, 127 (1929)ADSzbMATHCrossRefGoogle Scholar
  75. 75.
    García-Calderón, G., Mateos, J.L., Moshinsky, M.: Resonant spectra and the time evolution of the survival and nonescape probabilities. Phys. Rev. Lett. 74, 337 (1995)ADSCrossRefGoogle Scholar
  76. 76.
    Peshkin, M., Volya, A., Zelevinsky, V.: Non-exponential and oscillatory decays in quantum mechanics. Europhys. Lett. 107, 40001 (2014)ADSCrossRefGoogle Scholar
  77. 77.
    Bohm, D.: Quantum theory, p. 257. Prentice Hall, New York (1951)Google Scholar
  78. 78.
    Bohm, D., Wigner, E.P.: Lower limit for the energy derivative of the scattering phase shift. Phys. Rev. 98, 145 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Kudaka, S., Matsumoto, S.: Questions concerning the generalized hartman Eect. Phys. Lett. A375, 3259 (2011)ADSzbMATHCrossRefGoogle Scholar
  80. 80.
    Anastopoulos, C., Savvidou, N.: Quantum temporal probabilities in tunneling systems. Ann. Phys. 336, 281 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Muga, J.G., Delgado, F., del Campo, A., García-Calderón, G.: The role of initial state reconstruction in short and long time deviations from exponential decay. Phys. Rev. A73, 052112 (2006)ADSCrossRefGoogle Scholar
  82. 82.
    García-Calderón, G., Peierls, R.: Resonant states and their uses. Nucl. Phys. A265, 443 (1976)ADSCrossRefGoogle Scholar
  83. 83.
    del Campo, A., Garcia-Calderon, G., Muga, J.G.: Quantum transients. Phys. Rep. 476, 1 (2009)ADSCrossRefGoogle Scholar
  84. 84.
    Breuer, H.-P., Laine, E.-M., Piilo, J., Vacchini, B.: Colloquium: Non-Markovian dynamics in open quantum systems. Rev. Mod. Phys. 021002, 88 (2016)Google Scholar
  85. 85.
    Beau, M., Kiukas, J., Egusquiza, IL., del Campo, A.: Nonexponential quantum decay under environmental decoherence. Phys. Rev. Lett. 119, 130401 (2017)ADSCrossRefGoogle Scholar
  86. 86.
    del Campo, A., Delgado, F., García-Calderón, G., Muga, J.G.: Decay by tunneling of bosonic and fermionic tonks-girardeau gases. Phys. Rev. A74, 013605 (2006)ADSCrossRefGoogle Scholar
  87. 87.
    Taniguchi, T., Sawada, S.I.: Escape behavior of quantum two-particle systems with coulomb interactions. Phys. Rev. E83, 026208 (2011)ADSGoogle Scholar
  88. 88.
    García-Calderón, G., Mendoza-Luna, L.G.: Time evolution of decay of two identical quantum particles. Phys. Rev. A84, 032106 (2011)ADSCrossRefGoogle Scholar
  89. 89.
    del Campo, A.: Long-time behavior of many-particle quantum decay. Phys. Rev. A 84, 012113 (2011)ADSCrossRefGoogle Scholar
  90. 90.
    Marchewka, A., Granot, E.: Role of quantum statistics in multi-particle decay dynamics. Ann. Phys. 355, 348 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Pons, M., Sokolovski, D., del Campo, A.: Fidelity of fermionic-atom number states subjected to tunneling decay. Phys. Rev. A 85, 022107 (2012)ADSCrossRefGoogle Scholar
  92. 92.
    Hunn, S., Zimmermann, K., Hiller, M., Buchleitner, A.: Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach. Phys. Rev. A 87, 043626 (2013)ADSCrossRefGoogle Scholar
  93. 93.
    del Campo, A.: Exact quantum decay of an interacting many-particle system: the Calogero–Sutherland model. New. J. Phys. 18, 015014 (2016)ADSCrossRefGoogle Scholar
  94. 94.
    Khalfin, L.A.: Contribution to the decay theory of a quasi-stationary state, Sov. Phys.–JETP6, pp 1053 (1958)Google Scholar
  95. 95.
    Wiener, N., Paley, R.E.A.C.: Fourier transforms in the complex domain. Amer. Math. Soc. Theorem XII, 18 (1934)zbMATHGoogle Scholar
  96. 96.
    Leggett, A., Chakravarty, S., Dorsey, A., Fisher, M., Garg, A., Zwerger, W.: Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of PatrasPatrasGreece

Personalised recommendations