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Markovian properties of the spin-boson model

  • Ameur Dhahri
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)

Abstract

We systematically compare the Hamiltonian and Markovian approaches of quantum open system theory, in the case of the spin-boson model. We first give a complete proof of the weak coupling limit and we compute the Lindblad generator of this model. We study properties of the associated quantum master equation such as decoherence, detailed quantum balance and return to equilibrium at inverse temperature 0 < β ≤ ∞. We further study the associated quantum Langevin equation, its associated interaction Hamiltonian. We finally give a quantum repeated interaction model describing the spin‐boson system where the associated Markovian properties are satisfied without any assumption.

Keywords

Markovian Property Quantum Open System Weak Coupling Limit Quantum Master Equation Quantum Dynamical Semigroup 
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.

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References

  1. [AK00].
    L. Accardi, S. Kozyrev: Quantum interacting particle systems. Volterra International School (2000).Google Scholar
  2. [AFL90].
    L. Accardi, A. Frigerio, Y.G. Lu: Weak coupling limit as a quantum functional central limit theorem.Com. Math. Phys. 131, 537–570 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [ALV02].
    L. Accardi, Y.G. Lu, I. Volovich: Quantum theory and its stochastic limit. Springer-Verlag Berlin (2002).zbMATHGoogle Scholar
  4. [AL87].
    R. Alicki, K. Lendi: Quantum dynamical semigroups and applications. Lecture Notes in physics, 286. Springer-Verlag Berlin (1987).zbMATHCrossRefGoogle Scholar
  5. [AJ07].
    S. Attal, A. Joye: The Langevin Equation for a Quantum Heat Bath.J. Func. Analysis, 247, p. 253–288 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  6. [AP06].
    S. Attal, Y. Pautrat: From Repeated to Continuous Quantum Interactions.Annales Institut Henri Poincaré, (Physique Théorique) 7, p. 59–104 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [B06].
    A. Barchielli: Continual Measurements in Quantum Mechanics. Quantum Open systems. Vol III: Recent developments. Springer Verlag, Lecture Notes in Mathematics, 1882 (2006).Google Scholar
  8. [BR96].
    O. Bratteli, D.W. Robinson: Operator algebras and Quantum Statistical Mechanics II, Volume 2. Springer-Verlag New York Berlin Heidelberg London Paris Tokyo, second edition (1996).Google Scholar
  9. [C04].
    R. Carbone: Optimal Log-Sobolev Inequality and Hypercontractivity for positive semigroups on \(M_2(\mathbb{C})\),Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 7, No. 3 317–335 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [D74].
    E.B. Davies: Markovian Master equations.Comm. Math. Phys. 39, 91–110 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [D76a].
    E.B. Davies: Markovian Master Equations II.Math. Ann. 219, 147–158 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  12. [D80].
    E.B. Davies: One-Parameter Semigroups. Academic Press London New York Toronto Sydney San Francisco (1980).zbMATHGoogle Scholar
  13. [D76b].
    E.B. Davies: Quantum Theory of Open Systems. Academic Press, New York and London (1976).zbMATHGoogle Scholar
  14. [DJ03].
    J. Derezinski, V. Jaksic: Return to Equilibrium for Pauli-Fierz Systems.Annales Institut Henri Poincaré 4, 739–793 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [DJP03].
    J. Derezinski, V. Jaksic, C.A. Pillet: Perturbation theory of W*-dynamics, KMS-states and Liouvillean,Rev. Math. Phys. 15, 447–489 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [DF06].
    J. Derezinski, R. Fruboes: Fermi Golden Rule and Open Quantum Systems, Quantum Open systems. Vol III: Recent developments. Springer Verlag, Lecture Notes in Mathematics, 1882 (2006).Google Scholar
  17. [F06].
    F. Fagnola: Quantum Stochastic Differential Equations and Dilation of Completely Positive Semigroups. Quantum Open systems. Vol II: The Markovian approach. Springer Verlag, Lecture Notes in Mathematics, 1881 (2006).Google Scholar
  18. [F99].
    F. Fagnola: Quantum Markovian Semigroups and Quantum Flows. Proyecciones, Journal of Math. 18, n.3 1–144 (1999).MathSciNetGoogle Scholar
  19. [F93].
    F. Fagnola: Characterization of Isometric and Unitary Weakly Differentiable Cocycles in Fock space. Quantum Probability and Related Topics VIII 143 (1993). Google Scholar
  20. [FR06].
    F. Fagnola, R. Rebolledo: Nets of the Qualitative behaviour of Quantum Markov Semigroups. Quantum Open systems. Vol III: Recent developments. Springer Verlag, Lecture Notes in Mathematics, 1882 (2006).Google Scholar
  21. [FR98].
    F. Fagnola, R. Rebolledo: The Approach to equilibrium of a class of quantum dynamical semigroups.Inf. Q. Prob. and Rel. Topics, 1(4), 1–12 (1998).MathSciNetGoogle Scholar
  22. [HP84].
    R.L Hudson, K.R. Parthasarathy: Quantum Ito's formula and stochastic evolutions,Comm. Math. Phys. 93, no 3, pp.301–323 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [G01].
    M. Gregoratti: The Hamiltonian Operator Associated with Some quantum Stochastic EvolutionsCom. Math. Phys. 222, 181–200 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [JP96a].
    V. Jaksic, C.A. Pillet: On a model for quantum friction II: Fermi's golden rule and dynamics at positive temperature.Comm. Math. Phys. 178, 627 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  25. [JP96b].
    V. Jaksic, C.A. Pillet: On a model for quantum friction III: Ergodic properties of the spin-boson system.Comm. Math. Phys. 178, 627 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  26. [M95].
    P. A. Meyer: Quantum Probability for Probabilists. Second edition. Lect Not. Math. 1538, Berlin: Springer-Verlag (1995).zbMATHGoogle Scholar
  27. [P92].
    K. R. Parthasarathy: An Introduction to Quantum Stochastic Calculus. Birkhäuser Verlag: Basel. Boston. Berlin (1992).zbMATHCrossRefGoogle Scholar
  28. [R06].
    R. Rebolledo: Complete Positivity and Open Quantum Systems. Quantum Open systems. Vol II: The Markovian approach. Springer Verlag, Lecture Notes in Mathematics, 1881 (2006). Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.CeremadeUMR CNRS 7534Université Paris Dauphine Place de Lattre de TassignyFrance

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