Modélisation mathématique et analyse numérique des modèles de type Bloch pour les boîtes quantiques

Keywords :

• Boîtes quantiques
• Analyse mathématique
• Analyse numérique
• Calcul scientifique

Quantum dots are nanostructures confined in the three space directions. Since many decades, numerous studies have been devoted to these structures for their interesting electronic and optical properties.
In this thesis, we model the electronic behaviour of quantums dots thanks to a type Bloch model derived il the Heisenberg formalism. The closure of equations leads to a non linear model stemming from Coulomb and electron--phonon interactions. We study the qualitative properties of the obtained Bloch models (trace, hermicity, positivitiveness) and the Cauchy problem for the semi-classical model coupling Bloch and Maxwell equations to describe laser--quantum dot interaction. We derive also formally rate equations from the non-linear Bloch equations. The discretizations of one-dimensionnal Maxwell--Bloch equations involve splitting methods for the Bloch equations, which enable the preservation of the qualitative properties of the continuous model. The validation of the model and the study of the relevancy of some simplification is performed thanks to self-induced transparency and coherence-transfert test cases.

Directors:

• Mr Brigitte Bidegaray-Fesquet (Chargé de Recherche - CNRS )

Raporteurs:

• Mr Didier Felbacq (Professeur - Université de Montpellier II )
• Mr Christophe Besse (Professeur - Univ. Paul Sabatier Toulouse 3 )

Examinators:

• Mr Stéphane Descombes (Professeur - Université de Nice Sophia Antipolis )
• Mr Clément Jourdana (Maître de Conférence - Université Joseph Fourier )
• Mr Stéphane Labbé (Professeur - Université Joseph Fourier )