International Journal of Theoretical Physics

, Volume 51, Issue 11, pp 3536–3550 | Cite as

Analysis Technique for Exceptional Points in Open Quantum Systems and QPT Analogy for the Appearance of Irreversibility

  • Savannah Garmon
  • Ingrid Rotter
  • Naomichi Hatano
  • Dvira Segal


We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. We also report the precise number of EPs occurring in an OQS with a continuum described by a quadratic dispersion curve. In particular, the number of EPs occurring in a bare discrete Hamiltonian of dimension n D is given by n D(n D−1); if this discrete Hamiltonian is then coupled to continuum (or continua) to form an OQS, the interaction with the continuum generally produces an enlarged discrete solution space that includes a greater number of EPs, specifically \(2^{n_{\mathrm{C}}} (n_{\mathrm{C}} + n_{\mathrm{D}} ) [ 2^{n_{\mathrm{C}}} (n_{\mathrm{C}} + n_{\mathrm{D}} ) - 1] \), in which n C is the number of (non-degenerate) continua to which the discrete sector is attached. Finally, we offer a heuristic quantum phase transition analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.


Exceptional points Open quantum systems Resonant state Time irreversibility Exponential decay Phase transition 



We would like to thank E.C.G. Sudarshan, G. Ordonez, J. Pixley and S. Kirchner for stimulating discussion on the QPT analogy and T. Petrosky for helpful comments on an earlier draft. The research of S.G. was supported by CQIQC, the Sloan research fellowship of D.S. and MPI-PKS. The research of N.H. was supported by Grant-in-Aid for Scientific Research No. 17340115 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.


  1. 1.
    Heiss, W.D.: Phases of wave functions and level repulsion. Eur. Phys. J. D 7, 1 (1999) ADSCrossRefGoogle Scholar
  2. 2.
    Dembowski, C., Gräf, H.-D., Harney, H.L., Heine, A., Heiss, W.D., Rehfeld, H., Richter, A.: Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett. 86, 787 (2001) ADSCrossRefGoogle Scholar
  3. 3.
    Dembowski, C., Dietz, B., Gräf, H.-D., Harney, H.L., Heine, A., Heiss, W.D., Richter, A.: Observation of a chiral state in a microwave cavity. Phys. Rev. Lett. 90, 034101 (2003) ADSCrossRefGoogle Scholar
  4. 4.
    Lee, S.-B., Yang, J., Moon, S., Lee, S.-Y., Shim, J.-B., Kim, S.W., Lee, J.-H., An, K.: Observation of an exceptional point in a chaotic optical microcavity. Phys. Rev. Lett. 103, 134101 (2009) ADSCrossRefGoogle Scholar
  5. 5.
    Rotter, I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A, Math. Theor. 42, 153001 (2009) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Rubinstein, J., Sternberg, P., Ma, Q.: Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. Phys. Rev. Lett. 99, 167003 (2007) ADSCrossRefGoogle Scholar
  7. 7.
    Moiseyev, N., Friedland, S.: Association of resonance states with the incomplete spectrum of finite complex-scaled Hamiltonian matrices. Phys. Rev. A 22, 618 (1980) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Lefebvre, R., Atabek, O., Šindelka, M., Moiseyev, N.: Resonance coalescence in molecular photodissociation. Phys. Rev. Lett. 103, 123003 (2009) ADSCrossRefGoogle Scholar
  9. 9.
    Cartarius, H., Main, J., Wunner, G.: Exceptional points in atomic spectra. Phys. Rev. Lett. 99, 173003 (2007) ADSCrossRefGoogle Scholar
  10. 10.
    Hernández, E., Jáuegui, A., Mondragón, A.: Non-Hermitian degeneracy of two unbound states. J. Phys. A, Math. Gen. 39, 10087 (2006) ADSMATHCrossRefGoogle Scholar
  11. 11.
    Zirnbauer, M.R., Verbaarschot, J.J.M., Weidenmüller, H.A.: Destruction of order in nuclear spectra by a residual GOE interaction. Nucl. Phys. A 411, 161 (1983) ADSCrossRefGoogle Scholar
  12. 12.
    Cejnar, P., Heinze, S., Macek, M.: Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions. Phys. Rev. Lett. 99, 100601 (2007) ADSCrossRefGoogle Scholar
  13. 13.
    Kato, T.: Perturbation Theory for Linear Operators, pp. 62–66. Springer, Berlin (1980) MATHGoogle Scholar
  14. 14.
    Petrosky, T., Ting, C.-O., Garmon, S.: Strongly coupled matter field and nonanalytic decay rate of dipole molecules in a waveguide. Phys. Rev. Lett. 94, 043601 (2005) ADSCrossRefGoogle Scholar
  15. 15.
    Longhi, S.: Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model. Phys. Rev. B 80, 165125 (2009) ADSCrossRefGoogle Scholar
  16. 16.
    Garmon, S., Nakamura, H., Hatano, N., Petrosky, T.: Two-channel quantum wire with an adatom impurity: role of the van Hove singularity in the quasibound state in continuum, decay rate amplification, and the Fano effect. Phys. Rev. B 80, 115318 (2009) ADSCrossRefGoogle Scholar
  17. 17.
    Bustos-Marún, R.A., Coronado, E.A., Pastawski, H.M.: Buffering plasmons in nanoparticle waveguides at the virtual-localized transition. Phys. Rev. B 82, 035434 (2010) ADSCrossRefGoogle Scholar
  18. 18.
    Bulgakov, E.N., Rotter, I., Sadreev, A.F.: Phase rigidity and avoided level crossings in the complex energy plane. Phys. Rev. E 74, 056204 (2006) ADSCrossRefGoogle Scholar
  19. 19.
    Jung, C., Müller, M., Rotter, I.: Phase transitions in open quantum systems. Phys. Rev. E 60, 114 (1999) ADSCrossRefGoogle Scholar
  20. 20.
    Longhi, S.: Nonexponential decay via tunneling in tight-binding lattices and the optical Zeno effect. Phys. Rev. Lett. 97, 110402 (2006) ADSCrossRefGoogle Scholar
  21. 21.
    Longhi, S.: Bound states in the continuum in a single-level Fano-Anderson model. Eur. Phys. J. B 57, 45 (2007) ADSCrossRefGoogle Scholar
  22. 22.
    Tanaka, S., Garmon, S., Ordonez, G., Petrosky, T.: Electron trapping in a one-dimensional semiconductor quantum wire with multiple impurities. Phys. Rev. B 76, 153308 (2007) ADSCrossRefGoogle Scholar
  23. 23.
    Hatano, N., Sasada, K., Nakamura, H., Petrosky, T.: Some properties of the resonant state in quantum mechanics and its computation. Prog. Theor. Phys. 119, 187 (2008) ADSMATHCrossRefGoogle Scholar
  24. 24.
    Sasada, K., Hatano, N., Ordonez, G.: Resonant spectrum analysis of the conductance of an open quantum system and three types of Fano parameter. J. Phys. Soc. Jpn. 80, 104707 (2011) ADSCrossRefGoogle Scholar
  25. 25.
    Hatano, N.: Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states. Fortschr. Phys. (2012). doi: 10.1002/prop.201200064 Google Scholar
  26. 26.
    Jarlebring, E., Kvaal, S., Michiels, W.: Computing all pairs (λ,μ) such that λ is a double eigenvalue of A+μB. KU Leuven, TW559 (2010) Google Scholar
  27. 27.
    Lefebvre, R., Moiseyev, N.: Localization of exceptional points with Padé approximants. J. Phys. B, At. Mol. Opt. Phys. 43, 095401 (2010) ADSCrossRefGoogle Scholar
  28. 28.
    Uzdin, R., Lefebvre, R.: Finding and pinpointing exceptional points of an open quantum system. J. Phys. B, At. Mol. Opt. Phys. 43, 235004 (2010) ADSCrossRefGoogle Scholar
  29. 29.
    Garmon, S., Petrosky, T., Nikulina, Y., Segal, D.: Amplification of non-Markovian decay due to bound state absorption into continuum. Fortschr. Physik (2012). arXiv:1204.6141
  30. 30.
    Fuchs, B.A., Levin, V.I.: Functions of a Complex Variable, vol. II. Pergamon Press, New York (1961) MATHGoogle Scholar
  31. 31.
    Economou, E.N.: Green’s Functions in Quantum Physics, 2nd edn. Springer, Berlin (1983) Google Scholar
  32. 32.
    Chiu, C.B., Misra, B., Sudarshan, E.C.G.: The evolution of unstable states and a resolution of Zeno’s paradox. Phys. Rev. D 16, 520 (1977) MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Fonda, L., Ghirardi, G.C., Rimini, A.: Decay theory of unstable quantum systems. Rep. Prog. Phys. 41, 587 (1978) ADSCrossRefGoogle Scholar
  34. 34.
    Martorell, J., Muga, J.G., Spring, D.W.L.: Quantum Post-Exponential Decay. In: Lect. Notes Phys., vol. 789, p. 239. Springer, Berlin (2009) Google Scholar
  35. 35.
    Dente, A.D., Bustos-Marún, R.A., Pastawski, H.M.: Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule. Phys. Rev. A 78, 062116 (2008) ADSCrossRefGoogle Scholar
  36. 36.
    Mostafazadeh, A.: Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009) ADSCrossRefGoogle Scholar
  37. 37.
    Knopp, K.: Theory of Functions, Pt. 2, pp. 126–129. Dover, New York Google Scholar
  38. 38.
    Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (2001) Google Scholar
  39. 39.
    Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Experimental study of active LRC circuits with \(\mathcal {PT}\)-symmetries. Phys. Rev. A 84, 040101(R) (2011) ADSCrossRefGoogle Scholar
  40. 40.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \(\mathcal {PT}\)-symmetry. Phys. Rev. Lett. 80, 5243 (1998) MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Savannah Garmon
    • 1
  • Ingrid Rotter
    • 2
  • Naomichi Hatano
    • 3
  • Dvira Segal
    • 1
  1. 1.Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum ControlUniversity of TorontoTorontoCanada
  2. 2.Max-Planck-Institut für Physik Komplexer SystemeDresdenGermany
  3. 3.Institute of Industrial ScienceUniversity of TokyoTokyoJapan

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