We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is asymptotically solution to a linear Boltzmann equation, in which transition rates are appropriate time averages of the potential. This study provides a mathematical justification of the approximation of Bloch equations by rate equations, as described in e.g. [Lou91]. The techniques used stem from manipulations on the density matrix and the averaging theory for ordinary differential equations. Diophantine estimates play a key role in the analysis.
Publié le : 2004-07-05
Classification:
linear Boltzmann equation,
rate equations,
Bloch model,
averaging,
Diophantine estimates.,
density matrix,
Diophantine estimates,
34C27;81V80,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
@article{hal-00319994,
author = {Bid\'egaray-Fesquet, Brigitte and Castella, Fran\c cois and Degond, Pierre},
title = {From the Bloch model to the rate equations},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00319994}
}
Bidégaray-Fesquet, Brigitte; Castella, François; Degond, Pierre. From the Bloch model to the rate equations. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00319994/