Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

The aim of this chapter is first to give an introduction to the derivation of the Gross-Pitaevskii Equations (GPEs) that arise in the modeling of Bose-Einstein Condensates (BECs). In particular, we describe some physical problems related to stationary states, dynamics, multi-components BECs and the possibility of handling stochastic effects into the equation. Next, we explain how to compute the stationary (and ground) states of the GPEs through the imaginary time method (also called Conjugate Normalized Gradient Flow) and finite difference or pseudo-spectral discretization techniques. Examples are provided by using GPELab which is a Matlab toolbox dedicated to the numerical solution of GPEs. Finally, we explain how to discretize correctly the time-dependent GPE so that the schemes are physically admissible. We again provide some examples by using GPELab. Furthermore, extensions of the discretization schemes to some classes of stochastic (in time) GPEs are described and analyzed. 1 Modeling: Bose, Einstein, Gross & Pitaevskii 1.1 From the theory to the realization of Bose-Einstein condensates The discovery of Bose-Einstein Condensates (BECs), from their theoretical prediction by Bose & Einstein in 1925 to their first experimental realization in 1995 by Cornell and Wiemann, results from extraordinary scientific achievements that led to Xavier ANTOINE Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F54506, France & Inria Nancy Grand-Est/IECL ALICE, France. e-mail: xavier.antoine@ univ-lorraine.fr Romain DUBOSCQ Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F54506, France. e-mail: [email protected]