Ecole docorale PHAST - Physique et Astrophysuqye de Lyon Retour à l'université de Lyon

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Cours de l'ED-PHAST: 3 cours du Prof. Dionisios Margetis : "Bose-Einstein condensation beyond the mean field limit: Recent advances and challenges"





Monday, june 2th, 2014
Lecture 1: Bose-Einstein condensation of dilute atomic gases: An overview

Abstract: In this lecture, I will revisit fundamental physical concepts related intimately to the
Bose-Einstein condensation of dilute atomic gases at extremely low temperatures. The purpose of this lecture is to serve as an introduction 
to recent advances in this area (to be discussed in subsequent lectures). Starting from the pioneering prediction by Bose and Einstein
for non-interacting particles with integer spin (Bosons), I will proceed to discuss the important work
by Lee, Huang, and Yang for periodic systems of interacting Bosons. Particular emphasis will be placed on the concept of pair excitation and the role that this plays in the description of the phonon spectrum. Then, I will offer an overview of the significant works
by Gross, Pitaevskii and Wu for the non-periodic interacting atomic gas, particularly their formal, physically motivated
derivations of a nonlinear Schroedinger-type equation for the wave function of the macroscopic state, which is the mean field limit.

Tuesday june 3rd, 2014
Lecture 2: Bose-Einstein condensation beyond mean field: Many-body bound state
Abstract: In this lecture, I will discuss some possible applications of the pair excitation formalism, beyond the
mean field limit,  for trapped, dilute atomic gases of repulsively interacting bosons at zero temperature.
The goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic
Hamiltonian that accounts for certain scattering processes. In particular, for a scattering length with periodic microstructure, I will formally
describe via classical homogenization the lowest many-body bound state in slowly varying traps.

Wednesday, June 4th, 2014
Lecture 3: Mean field limit and corrections in Bose-Einstein condensation: Some rigorous results
Abstract: In this lecture, I will start with main, recent mathematical advances in the derivation of the mean field
limit for the time-dependent wave function of the macroscopic state of a dilute gas of Bosons. Particular emphasis will be placed
on the works by H.T. Yau and collaborators on the use of quantum kinetic hierarchies for the derivation of this limit. Then, I will
discuss how a correction to this mean field limit can be obtained rigorously by combining Wu's formalism for pair exitation in
non-translation invariant settings with recent works by Rodnianski and Schlein. Some open problems will be highlighted.


2 au 30 Juin 2014
Salle des Séminaires du Bât. Lippman - La Doua

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