Letters in Mathematical Physics

, Volume 107, Issue 7, pp 1215–1233 | Cite as

Completely positive dynamical semigroups and quantum resonance theory

  • Martin KönenbergEmail author
  • Marco Merkli


Starting from a microscopic system–environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system–environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory.


Open quantum system Quantum resonances Dynamical semigroup Complete positivity 

Mathematics Subject Classification

82C10 81S22 



This work has been supported by an NSERC Discovery Grant and an NSERC Discovery Grant Accelerator. We thank an anonymous referee for useful comments and for inciting us to explain the derivation of (2.11), resulting in Sect. 2.1.


  1. 1.
    Alicki, R., Lendi, K.: Quantum dynamical semigroups and applications, Lecture notes on physics, vol. 717. Springer Verlag, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Araki, H., Woods, E.J.: Representation of the canonical commutation relations describing a nonrelativistic infinite free bose gas. J. Math. Phys. 4, 637–662 (1963)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Benatti, F., Floreanini, R.: Open quantum dynamics: complete positivity and entanglement. Int. J. Mod. Phys. B 19, 3063 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bach, V., Fröhlich, J., Sigal, I.M.: Return to equilibrium. J. Math. Phys. 41(6), 3985–4060 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Breuer, H.-P., Petruccione, F.: The theory of open quantum systems. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  6. 6.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, vol. 1 and 2. Springer Verlag, New York (1979)CrossRefzbMATHGoogle Scholar
  7. 7.
    Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davies, E.B.: Markovian master equations, II. Math. Annalen 219, 147–158 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Derezinski, J., Jaksic, V., Pillet, C.-A.: Perturbation theory of \(W^*\)-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15(5), 447–489 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dümke, R., Spohn, H.: The proper form of the generator in the weak coupling limit. Z. Phys. B 34, 419–422 (1979)ADSCrossRefGoogle Scholar
  11. 11.
    Fröhlich, J., Merkli, M.: Another return of “return to equilibrium”. Commun. Math. Phys. 251(2), 235–262 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gardiner, C.W., Zoller, P.: Quantum noise. Springer series in synergetics, 3rd edn. Springer, Berlin (2004)Google Scholar
  13. 13.
    Jaksic, V., Pillet, C.-A.: From resonances to master equations. Ann. Inst. H. Poincaré Phys. Théor. 67(4), 425–445 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jaksic, V., Pillet, C.-A.: On a model for quantum friction. II. Fermi’s golden rule and dynamics at positive temperature. Commun. Math. Phys. 176(3), 619–644 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Könenberg, M., Merkli, M.: On the irreversible dynamics emerging from quantum resonances. J. Math. Phys. 57, 033302 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Merkli, M.: Entanglement evolution via quantum resonances. J. Math. Phys. 52(9), 092201 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Merkli, M., Berman, G.P., Sayre, R.: Electron transfer reactions: generalized Spin–Boson approach. J. Math. Chem. 51(3), 890–913 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Merkli, M., Berman, G.P., Song, H.: Multiscale dynamics of open three-level quantum systems with two quasi-degenerate levels. J. Phys. A Math. Theor. 48, 275304 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Merkli, M., Berman, G.P., Sayre, R.T., Gnanakaran, S., Könenberg, M., Nesterov, A.I., Song, H.: Dynamics of a chlorophyll dimer in collective and local thermal environments. J. Math. Chem. 54(4), 866–917 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Merkli, M., Sigal, I.M., Berman, G.P.: Resonance theory of decoherence and thermalization. Ann. Phys. 323, 373–412 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Merkli, M., Sigal, I.M., Berman, G.P.: Decoherence and thermalization. Phys. Rev. Lett. 98(13), 130401–130405 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Merkli, M., Sigal, I.M., Berman, G.P.: Dynamics of collective decoherence and thermalization. Ann. Phys. 323(12), 3091–3112 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mukamel, S.: Principles of nonlinear spectroscopy. Oxford series in optical and imaging sciences. Oxford University Press, Oxford (1995)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada
  2. 2.Fachbereich MathematikUniversität StuttgartStuttgartGermany

Personalised recommendations