Abstract
We establish the existence of two weak coupling regime effective dynamics for an open quantum system of repeated interactions (vanishing strength and individual interaction duration, respectively). This generalizes known results (Attal and Joye in J. Stat. Phys. 126:1241–1283, 2007) in that the von Neumann algebras describing the system and the chain element may not be of finite type. Then (but now assuming that the small system is of finite type), we prove that both effective dynamics capture the long-term behavior of the system: existence of a unique asymptotic state for them implies the same property for the respective exact dynamics—provided that the perturbation parameter is sufficiently small. The zero-th order term in a power series expansion in the perturbation parameter of such an asymptotic state is given by the asymptotic state of the effective dynamics. We conclude by working out the case in which the small system and the chain element are spins.
Similar content being viewed by others
References
Accardi, L., Lu, Y., Volovich, I.: Quantum Theory and Its Stochastic Limit. Springer, Berlin (2002)
Attal, S.: Quantum noises. In: Attal, S., Joye, A., Pillet, C.A. (eds.) Open Quantum Systems II. Lecture Notes in Mathematics, vol. 1881. Springer, Berlin (2006)
Attal, S., Joye, A.: Weak coupling and continuous limits for repeated quantum interactions. J. Stat. Phys. 126, 1241–1283 (2007)
Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Ann. Henri Poincare 7(1), 59–104 (2006)
Barchielli, A.: Continuous measurements in quantum mechanics and quantum stochastic calculus. In: Attal, S., Joye, A., Pillet, C.A. (eds.) Open Quantum Systems III. Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006)
Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1987)
Bruneau, L., Joye, A., Merkli, M.: Infinite products of random matrices and repeated interaction dynamics. Prepublication
Bruneau, L., Joye, A., Merkli, M.: Random repeated interaction quantum systems. Commun. Math. Phys. (2008, to appear). 0710.5908
Bruneau, L., Joye, A., Merkli, M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 247, 310–344 (2006)
Davies, E.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974)
Davies, E.: One-Parameter Semigroups. Academic Press, San Diego (1980)
Derezinski, J., Früboes, R.: Fermi golden rule and open quantum systems. In: Attal, S., Joye, A., Pillet, C.A. (eds.) Open Quantum Systems III. Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)
Lebowitz, J., Spohn, H.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109–142 (1978)
Orszag, M.: Quantum Optics. Springer, Berlin (1997)
Wellens, T., Buchleitner, A., Kümmerer, B., Maassen, H.: Quantum state preparation via asymptotic completeness. Phys. Rev. Lett. 85(16), 3361–3364 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Mariana Huerta.
This work was partially funded by Nucleus Millennium Information and Randomness P04-069-F.
Rights and permissions
About this article
Cite this article
Vargas, R. Repeated Interaction Quantum Systems: Van Hove Limits and Asymptotic States. J Stat Phys 133, 491–511 (2008). https://doi.org/10.1007/s10955-008-9605-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9605-0