Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

From Bloch model to the rate equations

Pages: 1 - 26, Volume 11, Issue 1, July 2004      doi:10.3934/dcds.2004.11.1

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B. Bidégaray-Fesquet - LMC - IMAG, UMR 5523 (CNRS-UJF-INPG), B.P. 53, 38041 Grenoble Cedex 9, France (email)
F. Castella - IRMAR, UMR 6625 (CNRS-UR1), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France (email)
Pierre Degond - Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France (email)

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 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.

Keywords:  Density matrix, Bloch model, rate equations, linear Boltzmann equation, averaging, Diophantine estimates.
Mathematics Subject Classification:  Primary: 34E99; Secondary: 81V80.

Received: January 2003;      Revised: December 2003;      Available Online: April 2004.