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.
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References
1. R. Alicki and M. Fannes, Quantum Dynamical Systems, Oxford University Press, (2001).
2. S. Attal and M. Lindsay (Eds), Quantum Probability Communictions XI and XII, World Scientific, (2003).
3. S. Attal and Y. Pautrat, From repeated to continuous quantum interactionsInteractions, Annales Henri Poincar , 7:59–104 (2006).
4. A. Buchleitner and K. Hornberger (Eds), Coherent Evolution in Noisy Environments, Springer Lecture Notes in Physics 611, (2002).
5. O. Brattelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Texts and Monographs in Physics, Springer, New York, Heidelberg, Berlin, (1981).
6. E. B. Davies, Markovian Master Equations, Comm. Math. Phys. 39:91–110 (1974).
7. E. B. Davies, One-Parameter Semigroups, Academic Press, (1980).
8. E. B. Davies, Quantum Theory of Open Systems, Academic Press, (1976).
9. J. Derezinski and V. Jaksic, On the Nature of Fermi Golden Rule of Open Quantum Systems, J. Stat. Phys. 116: 411–423 (2004).
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).
11. T. Kato, Perturbation Theory for Linear Operators, Springer, (1980).
12. J. Lebowitz and H. Spohn, Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs, Adv. Chem. Phys. 39:109–142 (1978).
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).
14. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, (1983).
15. T. Wellens, A. Buchleitner, B. Kuemmerer, and H. Maassen, Quantum state preparation via asymptotic completeness, Phys. Rev. Lett. 85:3361 (2000).
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Attal, S., Joye, A. Weak Coupling and Continuous Limits for Repeated Quantum Interactions. J Stat Phys 126, 1241–1283 (2007). https://doi.org/10.1007/s10955-006-9085-z
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DOI: https://doi.org/10.1007/s10955-006-9085-z