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Exceptional Points in a Non-Hermitian Extension of the Jaynes-Cummings Hamiltonian

  • Fabio BagarelloEmail author
  • Francesco Gargano
  • Margherita Lattuca
  • Roberto Passante
  • Lucia Rizzuto
  • Salvatore Spagnolo
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)

Abstract

We consider a generalization of the non-Hermitian \({\mathcal PT}\) symmetric Jaynes-Cummings Hamiltonian, recently introduced for studying optical phenomena with time-dependent physical parameters, that includes environment-induced decay. In particular, we investigate the interaction of a two-level fermionic system (such as a two-level atom) with a single bosonic field mode in a cavity. The states of the two-level system are allowed to decay because of the interaction with the environment, and this is included phenomenologically in our non-Hermitian Hamiltonian by introducing complex energies for the fermion system. We focus our attention on the occurrence of exceptional points in the spectrum of the Hamiltonian, clarifying its mathematical and physical meaning.

Keywords

Exceptional Point Photonic Crystal Slab Unitary Time Evolution Continuous Energy Spectrum Nonadiabatic Coupling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Financial support by the Julian Schwinger Foundation, MIUR, University of Palermo, GNFM is gratefully acknowledged.

References

  1. 1.
    W.C. Schieve, L.P. Horwitz, Quantum Statistical Mechanics (Cambridge Unibversity Press, Cambridge, 2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    G. Barton, Introduction to Advanced Field Theory (John Wiley & Sons, 1963)Google Scholar
  3. 3.
    A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Meth. Mod. Phys. 7, 1191 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    C.M. Bender, S. Boettcher, Real spectra of non-Hermitian Hamiltonian having \({\cal P}T\) symmetry. Phys. Rev. Lett. 80, 5243 (1998)Google Scholar
  5. 5.
    A. Mostafazadeh, Non-Hermitian Hamiltonians with a real spectrum and their physical applications. Pramana-J. Phys. 73, 269 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    C.M. Bender, M.V. Berry, A. Mandilara, Generalized \({\cal PT}\) symmetry and real spectra. J. Phys. A: Math. and Gen. 35, L467 (2002)Google Scholar
  7. 7.
    C.M. Bender, D.C. Brody, H.F. Jones, Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    C.M. Bender, Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Mostafazadeh, Pseudo-Hermiticity versus \({\cal P}T\)-symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205 (2002)Google Scholar
  10. 10.
    B. Zhen, C.W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J.D. Joannopoulos, M. Soljac̆ić, Spawning rings of exceptional points out of Dirac cones. Nature 525, 354 (2015)Google Scholar
  11. 11.
    W.D. Heiss, Exceptional points of non-Hermitian operators. J. Phys. A 37, 6 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin, 1966)CrossRefzbMATHGoogle Scholar
  13. 13.
    I. Rotter, J.P. Bird, A review of recent progress in the physics of open quantum systems: theory and experiment. arXiv:1507.08478, accepted for Report on Progress in Physics
  14. 14.
    E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89 (1963)CrossRefGoogle Scholar
  15. 15.
    F. Bagarello, M. Lattuca, R. Passante, L. Rizzuto, S. Spagnolo, Non-Hermitian Hamiltonian for a modulated Jaynes-Cummings model with \({\cal P}T\) symmetry. Phys. Rev. A 91, 042134 (2015)Google Scholar
  16. 16.
    F. Bagarello, Deformed canonical (anti-)commutation relations and non hermitian Hamiltonians, in Non-selfadjoint operators in quantum physics: Mathematical aspects, ed. by F. Bagarello, J.P. Gazeau, F. Szafraniec, M. Znojil (J. Wiley and Sons, 2015)Google Scholar
  17. 17.
    F. Bagarello, M. Lattuca, \({\cal D}\) pseudo bosons in quantum models. Phys. Lett. A 377, 3199 (2013)Google Scholar
  18. 18.
    I. Gilary, A.A. Mailybaev, N. Moiseyev, Time-asymmetric quantum-state-exchange mechanism. Phys. Rev. A 88, 010102(R) (2013)ADSCrossRefGoogle Scholar
  19. 19.
    G. Compagno, R. Passante, F. Persico, Atom-Field Interactions and Dressed Atoms (Cambridge University Press, Cambridge, 1995)CrossRefGoogle Scholar
  20. 20.
    F. Bagarello, F. Gargano, Model pseudofermionic systems: connections with exceptional points. Phys. Rev. A 89, 032113 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    F. Bagarello, F. Gargano, D. Volpe, \(\cal D\)-deformed harmonic oscillators. Int. J. Theor. Phys. 54(11), 4110 (2015)Google Scholar
  22. 22.
    W.D. Heiss, The physics of exceptional points. J. Phys. A: Math. Theor. 45, 444016 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I.E. Linington, B.M. Garraway, Control of atomic decay rates via manipulation of reservoir mode frequencies. J. Phys. B: At. Mol. Opt. Phys. 39, 3383 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    M. Müller, I. Rotter, Exceptional points in open quantum systems. J. Phys. A: Math. Gen. 41, 244018 (2008)Google Scholar
  25. 25.
    H. Eleuch, I. Rotter, Exceptional points in open and \({\cal P}T\) symmetric systems. Acta Polytech. 54, 106 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Fabio Bagarello
    • 1
    Email author
  • Francesco Gargano
    • 1
  • Margherita Lattuca
    • 2
  • Roberto Passante
    • 2
  • Lucia Rizzuto
    • 2
  • Salvatore Spagnolo
    • 2
  1. 1.Dipartimento di Energia, Informazione e Modelli MatematiciUniversità degli Studi di PalermoPalermoItaly
  2. 2.Dipartimento di Fisica e ChimicaUniversità degli Studi di PalermoPalermoItaly

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