Annales Henri Poincaré

, Volume 16, Issue 6, pp 1397–1427 | Cite as

Overlapping Resonances in Open Quantum Systems

  • Marco MerkliEmail author
  • Haifeng Song


An N-level quantum system is coupled to a bosonic heat reservoir at positive temperature. We analyze the system–reservoir dynamics in the following regime: the strength λ of the system–reservoir coupling is fixed and small, but larger than the spacing σ of system energy levels. For vanishing σ there is a manifold of invariant system–reservoir states and for σ > 0 the only invariant state is the joint equilibrium. The manifold is invariant for σ = 0 but becomes quasi-invariant for σ > 0. Namely, on a first time-scale of the order 1/λ2, initial states approach the manifold. Then, they converge to the joint equilibrium state on a much larger time-scale of the order λ2/σ 2. We give a detailed expansion of the system–reservoir evolution showing the above scenario.


Density Matrix Reduce Density Matrix Simple Eigenvalue Open Quantum System Density Matrix Element 
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.


  1. 1.
    Alicki R.: Master equations for a damped nonlinear oscillator and the validity of the Markovian approximation. Phys. Rev. A 40(7), 4077–4081 (1989)CrossRefADSGoogle Scholar
  2. 2.
    Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Springer Lecture Notes in Physics 717 (1987)Google Scholar
  3. 3.
    Attal, S., Joye, A., Pillet, C.-A.: Open Quantum Systems I. The Hamiltonian Approach. Lecture Notes in Mathematics 1880, Springer, New York (2006)Google Scholar
  4. 4.
    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)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bach V., Fröhlich J., Sigal I.M.: Return to Equilibrium. J. Math. Phys. 41, 3985–4060 (1998)CrossRefADSGoogle Scholar
  6. 6.
    Bach V., Fröhlich J., Sigal I.M.: Quantum Electrodynamics of Confined Nonrelativistic Particles. Adv. Math. 137, 299–395 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bratteli O., Robinson D.W.: Operator algebras and quantum statistical mechanics I and II. Springer, New York (1987)CrossRefGoogle Scholar
  8. 8.
    Breuer H.-P., Petruccione F.: The theory of open quantum systems. Oxford University Press, Oxford (2006)Google Scholar
  9. 9.
    Celardo G.L., Izrailev F.M., Zelevinsky V.G., Berman G.P.: Transition from isolated to overlapping resonances in the open system of interacting fermions. Phys. Lett. B659, 170–175 (2008)CrossRefADSGoogle Scholar
  10. 10.
    Davies E.B.: Markovian Master Equations. Comm. Math. Phys. 39, 91–110 (1974)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Davies E.B.: Markovian Master Equations, II. Math. Ann. 219, 147–158 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Davies E.B.: A model of atomic radiation. Annales I.H.P., section A 28(1), 91–110 (1978)zbMATHGoogle Scholar
  13. 13.
    Derezinski J., Jaksic V., Pillet C.-A.: Perturbation theory of W*-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15(5), 447–489 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Derezinski J., Jaksic V.: Return to equilibrium for Pauli-Fierz systems. Ann. Henri Poincaré 4(4), 739–793 (2003)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    De Roeck W., Kupiainen A.: ‘Return to Equilibrium’ for Weakly Coupled Quantum Systems: A Simple Polymer Expansion. Comm. Math. Phys. 305, 797–862 (2011)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    De Roeck W., Kupiainen A.: Approach to Ground State and Time-Independent Photon Bound for Massless Spin-Boson Models. Ann. Henri Poincaré 14(2), 253–311 (2013)CrossRefADSzbMATHMathSciNetGoogle Scholar
  17. 17.
    Fröhlich J., Merkli M.: Another Return of “Return to Equilibrium”. Comm. Math. Phys. 251, 235–262 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gardiner, C.W., Zoller, P.: Quantum Noise. Springer series in synergetics, third edition (2004)Google Scholar
  19. 19.
    Jaksic V., Pillet C.-A.: On a model for quantum friction II. Fermi’s golden rule and dynamics at positive temperature. Comm. Math. Phys. 176, 619–644 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. 20.
    Jaksic V., Pillet C.-A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Comm. Math. Phys. 226, 131–162 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I.-O.: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer Physics and Astronomy, second edition (2003)Google Scholar
  22. 22.
    Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)CrossRefzbMATHGoogle Scholar
  23. 23.
    Merkli M.: Positive Commutators in Non-Equilibrium Quantum Statistical Mechanics. Comm. Math. Phys. 223, 327–362 (2001)CrossRefADSzbMATHMathSciNetGoogle Scholar
  24. 24.
    Merkli, M.: Entanglement Evolution via Quantum Resonances. J. Math. Phys. 52, Issue 9 (2011). doi: 10.1063/1.36376282011
  25. 25.
    Merkli M., Mück M., Sigal I.M.: Theory of Non-Equilibrium Stationary States as a Theory of Resonances. Ann. H. Poincaré 8, 1539–1593 (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Merkli M., Mück M., Sigal I.M.: Instability of equilibrium states for coupled heat reservoirs at different temperatures. J. Funct. Anal. 243, 87–120 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Merkli M., Sigal I.M., Berman G.P.: Resonance theory of decoherence and thermalization. Ann. Phys. 323, 373–412 (2008)CrossRefADSzbMATHMathSciNetGoogle Scholar
  28. 28.
    Mozyrsky D., Privman V.: Adiabatic Decoherence. J. Stat. Phys. 91, 787–799 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Mukamel, S.: Principles of Nonlinear Spectroscopy. Oxford Series in Optical and Imaging Sciences, Oxford University Press, Oxford (1995)Google Scholar
  30. 30.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  31. 31.
    Palma M.G., Suominen K.-A., Ekert A.: Quantum computers and dissipation. Proc. Soc. Lond. A 452, 567–584 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    Reed M., Simon B.: Methods of modern mahtematical physics. Vol. IV, Analysis of Operators. Academic Press, USA (1978)Google Scholar
  33. 33.
    Schlosshauer M.: Decoherence and the quantum-to-classical transition. The Frontiers collection. Springer, New York (2007)Google Scholar
  34. 34.
    Sokolov V.V., Zelevinski V.G.: Dynamics and statistics of unstable quantum states. Nucl. Phys. A504, 562–588 (1989)CrossRefADSGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

Personalised recommendations