Since the first experimental achievement of Bose Einstein condensates (BEC) in 1995 and the award of the Nobel Prize for Physics in 2001, the properties of these gaseous quantum fluids have been the focus of international interest in condensed matter physics. This monograph is dedicated to the mathematical modeling of some specific experiments which display vortices and to a rigorous analysis of features emerging experimentally. In contrast to a classical fluid, a quantum fluid such as a Bose Einstein condensate can rotate only through the nucleation of quantized vortices beyond some critical velocity. There are two interesting regimes: one close to the critical velocity, where there is only one vortex that has a very special shape; and another one at high rotation values, for which a dense lattice is observed. One of the key features related to superfluidity is the existence of these vortices. We address this issue mathematically and derive information on their shape, number and location. In the dilute limit of the experiments, the condensate is well described by a mean field theory and a macroscopic wave function solving the so-called Gross Pitaevskii equation. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. We prove existence of solutions that have properties consistent with the experimental observations. Open problems related to recent experiments are presented.
|Series:||Progress in Nonlinear Differential Equations and Their Applications , #67|
|Product dimensions:||6.10(w) x 9.25(h) x 0.36(d)|
Table of Contents
The Physical Experiment and Their Mathematical Modeling.- The Mathematical Setting: A Survey of the Main Theorems.- Two-Dimensional Model for otating Condensate.- Other Trapping Potentials.- High-Velocity and Quantam Hall Regime.- Three-Dimensional Rotating Condensate.- Superfluid Flow Around an Obstacle.- Further Open Problems.