Singular limit in a nonlinear quantum field

  • Gilbert ReinischEmail author
Regular Article


The self-consistent mean-fied Schrödinger-Poisson model of quantum-dot helium is described in the mixed-state approximation of a two-level nonlinear quantum system (G. Reinisch, M. Gazeau, Eur. Phys. J. Plus 131, 220 (2016)). We investigate the strong nonlinear limit of vanishing harmonic confinement. We find that the two corresponding trajectories in the appropriate phase-space bifurcate from the Thomas-Fermi fixed point to a final singular statistical equilibrium state that is defined by equal population of the two energy levels as a result of eigenstate overlap \( \sim 1/\sqrt{2}\) between their respective nonlinear eigenstates. Further increase of nonlinearity, i.e. of eigenstate overlap, yields population inversion and therefore instability. At equilibrium, a black-body residual photon field is set up when considering the single-photon lowest-order QED description of the Coulomb interaction between the two electrons. We point out the possible interest of such an equilibrium state in the search of the Cooper-pair “pairing glue” of bound electron pairs in the degenerate Fermi gas of high-temperature superconductors, as well as its link with a recently renewed debate about the physical existence of QED mediating virtual photons (G. Jaeger, Entropy 21, 141 (2019)).


  1. 1.
    W. Kohn, L. Sham, Phys. Rev. 140, A1133 (1965)ADSCrossRefGoogle Scholar
  2. 2.
    C. Cohen-Tannoudji, J. Dupont-Roc, G. Gryndberg, Photons et Atomes: Introduction à l’électrodynamique quantique (EDP Sciences, Editions du CNRS, Paris, 1987)Google Scholar
  3. 3.
    M. Ehrhardt, A. Zisowsky, J. Comput. Appl. Math. 187, 1 (2006)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    G. Reinisch, V. Gudmundsson, Physica D 241, 902 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    R. Bullough, F. Calogero, P. Caudrey, A. Degasperis, L. Faddeev, H. Gibbs, R. Hirota, G. Lamb, A. Luther, D. McLaughlin, Solitons (Springer, 1980)Google Scholar
  6. 6.
    G. Reinisch, J. Pacheco, P. Valiron, Phys. Rev. A 63, 042505 (2001)ADSCrossRefGoogle Scholar
  7. 7.
    G. Reinisch, Phys. Rev. A 70, 033613 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    G. Reinisch, V. Gudmundsson, Eur. Phys. J. B 84, 699 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    G. Reinisch, V. Gudmundsson, A. Manolescu, Phys. Lett. A 378, 1566 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    G. Reinisch, M. Gazeau, Eur. Phys. J. Plus 131, 220 (2016)CrossRefGoogle Scholar
  11. 11.
    J. Zaanen, A modern, but way too short history of the theory of superconductivity at a high temperature, in 100 Years of Superconductivity (CRC Press, 2011) Chapt. 2.4, arXiv:1012.5461v2Google Scholar
  12. 12.
    G. Jaeger, Entropy 21, 141 (2019)ADSCrossRefGoogle Scholar
  13. 13.
    D. Pfannkuche, V. Gudmundsson, P. Maksym, Phys. Rev. B 47, 2244 (1993)ADSCrossRefGoogle Scholar
  14. 14.
    W. Kohn, Phys. Rev. 123, 1242 (1961)ADSCrossRefGoogle Scholar
  15. 15.
    S.M. Reimann, M. Manninen, Rev. Mod. Phys. 74, 1283 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover, 1996)Google Scholar
  17. 17.
    H. Eleuch, I. Rotter, Eur. Phys. J. D 69, 229 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    W.H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill Book Company, 1964)Google Scholar
  19. 19.
    J.A. Wheeler, The “Past” and the “Delayed-Choice” Double-slit Experiment, in Mathematical Foundations of Quantum Theory, edited by A.R. Marlow (Academic Press, 1978) pp. 9--47Google Scholar
  20. 20.
    I.D. Lawrie, A Unified Grand Tour of Theoretical Physics (Adam Hilger, 1990)Google Scholar
  21. 21.
    F. Mandl, Introduction to Quantum Field Theory (Interscience, 1959)Google Scholar
  22. 22.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon Press, London, 1958)Google Scholar
  23. 23.
    C. Cohen-Tannoudji, J. Dupont-Roc, G. Gryndberg, Processus d’interaction entre photons et atomes (EDP Sciences, Editions du CNRS, Paris, 1988)Google Scholar
  24. 24.
    V. Berestetski, E. Lifchitz, L. Pitayevski, Quantum electrodynamics (Mir, Moskow, 1989)Google Scholar
  25. 25.
    R.P. Feynman, QED: The Strange Theory of Light and Matter (Princeton, 1986) Chapt. 4Google Scholar
  26. 26.
    D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations: Introduction for Scientists and Engineers (Oxford University Press, 2007)Google Scholar
  27. 27.
    K. Briggs, PhD Thesis, University of Melbourne (1997)Google Scholar
  28. 28.
    L.N. Cooper, Phys. Rev. 104, 1189 (1956)ADSCrossRefGoogle Scholar
  29. 29.
    J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    M.R. Schafroth, Phys. Rev. 100, 463 (1955)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    A.J. Leggett, Nat. Phys. 2, 134 (2006)CrossRefGoogle Scholar
  32. 32.
    P.W. Anderson, Phys. Rev. 115, 2 (1959)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    P.W. Anderson, Science 235, 1196 (1987)ADSCrossRefGoogle Scholar
  34. 34.
    P. Monthoux, A. Balatsky, D. Pines, Phys. Rev. B 46, 14803 (1992)ADSCrossRefGoogle Scholar
  35. 35.
    S. Chakravarthy, A. Sudbö, P.W. Anderson, S. Strong, Science 261, 337 (1993)ADSCrossRefGoogle Scholar
  36. 36.
    G. Reinisch, Phys. Rev. A 56, 3409 (1997)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université de la Côte d’Azur - Observatoire de la Côte d’AzurNice CedexFrance
  2. 2.Science InstituteUniversity of Iceland, Dunhaga 3ReykjavikIceland

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