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Journal of Statistical Physics

, Volume 126, Issue 6, pp 1241–1283 | Cite as

Weak Coupling and Continuous Limits for Repeated Quantum Interactions

  • Stéphane Attal
  • Alain Joye
Article

Abstract

We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical sub-systems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ, the reference system is coupled to one new element of the chain only, by means of an interaction of strength λ. We consider three asymptotic regimes of the parameters λ and τ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales T and whose generator can be computed: the weak coupling limit regime λ → 0, τ = 1, the regime τ → 0, λ2τ → 0 and the critical case λ2τ = 1, τ → 0. The first two regimes are perturbative in nature and the effective generators they determine is such that a non-trivial invariant sub-algebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator.

Keywords

Markovian approximation repeated quantum interactions 

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References

  1. 1.
    1. R. Alicki and M. Fannes, Quantum Dynamical Systems, Oxford University Press, (2001).Google Scholar
  2. 2.
    2. S. Attal and M. Lindsay (Eds), Quantum Probability Communictions XI and XII, World Scientific, (2003).Google Scholar
  3. 3.
    3. S. Attal and Y. Pautrat, From repeated to continuous quantum interactionsInteractions, Annales Henri Poincar , 7:59–104 (2006).Google Scholar
  4. 4.
    4. A. Buchleitner and K. Hornberger (Eds), Coherent Evolution in Noisy Environments, Springer Lecture Notes in Physics 611, (2002).Google Scholar
  5. 5.
    5. O. Brattelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Texts and Monographs in Physics, Springer, New York, Heidelberg, Berlin, (1981).Google Scholar
  6. 6.
    6. E. B. Davies, Markovian Master Equations, Comm. Math. Phys. 39:91–110 (1974).MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    7. E. B. Davies, One-Parameter Semigroups, Academic Press, (1980).Google Scholar
  8. 8.
    8. E. B. Davies, Quantum Theory of Open Systems, Academic Press, (1976).Google Scholar
  9. 9.
    9. J. Derezinski and V. Jaksic, On the Nature of Fermi Golden Rule of Open Quantum Systems, J. Stat. Phys. 116: 411–423 (2004).CrossRefMathSciNetGoogle Scholar
  10. 10.
    10. E. B. Davies and H. Spohn, Open Quantum Systems with Time-Dependent Hamiltonians and their Linear Response J. Stat. Phys. 19:511–523 (1978).CrossRefMathSciNetGoogle Scholar
  11. 11.
    11. T. Kato, Perturbation Theory for Linear Operators, Springer, (1980).Google Scholar
  12. 12.
    12. J. Lebowitz and H. Spohn, Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs, Adv. Chem. Phys. 39:109–142 (1978).CrossRefGoogle Scholar
  13. 13.
    13. J. M. Lindsay and H. Maassen, Stochastic Calculus for Quantum Brownian Motion of a non-minimal variance. In Mark Kac Seminar of probability in Physics, Syllabus (pp. 1987–1992). CWI Syllabus 32, Amsterdam (1992).Google Scholar
  14. 14.
    14. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, (1983).Google Scholar
  15. 15.
    15. T. Wellens, A. Buchleitner, B. Kuemmerer, and H. Maassen, Quantum state preparation via asymptotic completeness, Phys. Rev. Lett. 85:3361 (2000).CrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Institut Girard DesarguesUniversité de Lyon 1Villeurbanne CedexFrance
  2. 2.Institut FourierUniversité de Grenoble 1St.-Martin d’Hères CedexFrance

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