A perturbation algorithm for the pointers of Franke–Gorini–Kossakowski–Lindblad–Sudarshan equation


This paper is devoted to the study of behavior of open quantum systems consistently based on the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation which covers evolution in situations when decoherence can be distinguished. We focus on the quantum measurement operation which is determined by final stationary states of an open system—so called pointers. We find pointers by applying the FGKLS equation to asymptotically constant density matrix. In seeking pointers, we have been able to propose a perturbative scheme of calculation, if we take the interaction components with an environment to be weak. Thus, the Lindblad operators can be used in some way as expansion parameters for perturbation theory. The scheme we propose is different for the cases of non-degenerate and degenerate Hamiltonian. We illustrate our scheme by particular examples of quantum harmonic oscillator with spin in external magnetic field. The efficiency of the perturbation algorithm is demonstrated by its comparison with the exact solution.

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  1. 1.

    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2003)

    Google Scholar 

  2. 2.

    M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, Berlin, 2007)

    Google Scholar 

  3. 3.

    U. Weiss, Quantum Dissipative Systems, 3rd edn. (World Scientific, Singapore, 2007)

    Google Scholar 

  4. 4.

    D. Calvania, A. Cuccoli, N.I. Gidopoulos, P. Verrucchi, PNAS 110, 6748 (2013)

    ADS  Article  Google Scholar 

  5. 5.

    C. Foti, T. Heinosaari, S. Maniscalco, P. Verrucchi, Quantum 3, 179 (2019)

    Article  Google Scholar 

  6. 6.

    G. Lindblad, Commun. Math. Phys. 48, 119 (1976)

    ADS  Article  Google Scholar 

  7. 7.

    G. Lindblad, Rep. Math. Phys. 10, 393 (1976)

    ADS  Article  Google Scholar 

  8. 8.

    V.A. Franke, Theor. Math. Phys. 27, 406 (1976)

    Article  Google Scholar 

  9. 9.

    V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)

    ADS  Article  Google Scholar 

  10. 10.

    A.A. Andrianov, J. Taron, R. Tarrach, Phys. Lett. B 507, 200 (2001)

    ADS  Article  Google Scholar 

  11. 11.

    F. Benatti, R. Floreanini, Nucl. Phys. B 488, 335 (1997)

    ADS  Article  Google Scholar 

  12. 12.

    F. Benatti, R. Floreanini, Nucl. Phys. B 511, 550 (1998)

    ADS  Article  Google Scholar 

  13. 13.

    J.P. Blaizot, M.A. Escobedo, JHEP 1806, 034 (2018)

    ADS  Article  Google Scholar 

  14. 14.

    N. Armesto, F. Dominguez, A. Kovner, M. Lublinsky, V.V. Skokov, JHEP 05, 025 (2019)

    ADS  Article  Google Scholar 

  15. 15.

    R. Gambini, R.A. Porto, J. Pullin, Gen. Relativ. Gravit. 39, 1143 (2007)

    ADS  Article  Google Scholar 

  16. 16.

    L.D. Landau, Z. Physik 45, 430 (1927). (see Collected Papers of L. D. Landau, Pergamon Press, 1965, p. 8)

    ADS  Article  Google Scholar 

  17. 17.

    J. von Neumann, Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 245 (1927)

    MATH  Google Scholar 

  18. 18.

    W.H. Zurek, Phys. Rev. D 24, 1516 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    W.H. Zurek, Phys. Rev. D 26, 1862 (1982)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    S. Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, Cambridge, 2015)

    Google Scholar 

  21. 21.

    S. Weinberg, Phys. Rev. A 90, 042102 (2014)

    ADS  Article  Google Scholar 

  22. 22.

    F. Benatti, A. Nagy, H. Narnhofer, J. Phys. A 44, 155303 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    B.M. Villegas-Martínez, F. Soto-Eguibar, H.M. Moya-Cessa, Adv. Math. Phys. 2016, 9265039 (2016)

    Article  Google Scholar 

  24. 24.

    A.C.Y. Li, F. Petruccione, J. Koch, Sci. Rep. 4, 4887 (2014)

    ADS  Article  Google Scholar 

  25. 25.

    E.A. Gòmez, J.D. Castaño-Yepesa, S.P. Thirumuruganandhamc, Results Phys. 10, 353 (2018)

    ADS  Article  Google Scholar 

  26. 26.

    I. Bengtsson, K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  27. 27.

    T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin, 1966)

    Google Scholar 

  28. 28.

    W.D. Heiss, J. Phys. A 45, 444016 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    A.A. Andrianov, M.V. Ioffe, O.O. Novikov, J. Phys. A 52, 425301 (2019)

    ADS  MathSciNet  Article  Google Scholar 

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The research was supported by RFBR Grant No. 18-02-00264-a. The work of A.A.A. was funded by the Grant FPA2016-76005-C2-1-P and Grant 2017SGR0929 (Generalitat de Catalunya).

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Correspondence to M. V. Ioffe.

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Andrianov, A.A., Ioffe, M.V., Izotova, E.A. et al. A perturbation algorithm for the pointers of Franke–Gorini–Kossakowski–Lindblad–Sudarshan equation. Eur. Phys. J. Plus 135, 531 (2020). https://doi.org/10.1140/epjp/s13360-020-00540-3

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