From Bloch model to the rate equations
Open Access
- 1 January 2004
- journal article
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete & Continuous Dynamical Systems
- Vol. 11 (1) , 1-26
- https://doi.org/10.3934/dcds.2004.11.1
Abstract
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 justiflcation 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 difierential equations. Diophantine estimates play a key role in the analysis. 1. Introduction. In this article, we address an asymptotic model as " ! 0 for the scaled Bloch equations "2@t‰(t;n;m) = ¡(i!(n;m) + ∞(n;m))‰(t;n;m) + i" X k •Keywords
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