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Low Density Limit and the Quantum Langevin equation for the Heat Bath

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Date
2009
Link to item file
http://hal.archives-ouvertes.fr/hal-00265934/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Repeated quantum interactions; quantum stochastique differentiel equation (or quantum Langevin equation); low density limit; Poisson processes
Journal issue
Open Systems & Information Dynamics
Volume
16
Number
4
Publication date
2009
Publisher
World Scientific
DOI
http://dx.doi.org/10.1142/S1230161209000268
URI
https://basepub.dauphine.fr/handle/123456789/3670
Collections
  • CEREMADE : Publications
Metadata
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Author
Dhahri, Ameur
Type
Article accepté pour publication ou publié
Abstract (EN)
We consider a repeated quantum interaction model describing a small system $\Hh_S$ in interaction with each one of the identical copies of the chain $\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another during the same short time intervals $[0,h]$. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order $h$. After giving the GNS representation, we establish the relation between the time scale $h$ and the classical low density limit. We introduce a chemical potential $\mu$ related to the time $h$ as follows: $h^2=e^{\beta\mu}$. We further prove that the solution of the associated discrete evolution equation converges strongly, when $h$ tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.

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