Efficient numerical models for manybody physics of coldatoms

The goal of this project is to improve and develop numerical models for many-boson systems, beyond the mean-field approach of a single deterministic Gross-Pitaevskii equation in one dimension. These models aim to be more accurate than the mean-field approach but mainly aim to describe the dynamics of the quantum systems under typical experimental conditions with few hundreds to thousands of atoms with reasonable computation time. These models will be used to provide more precise simulations of dynamics of Bose-Einstein condensates (BEC) of ultra-cold atoms. Our goal is to provide simulations of the experiments that are carried out in the laboratory by the group of J. Schmiedmayer. Dynamic simulations represent a big challenge for efficient and accurate approximate mathematical models and their efficient implementation on appropriate computers. Here, we aim to combine restricted multiconfigurational Ansatz for many-body wave functions with restrictions also on the space dimensions. Restriction of the complexity of the wave function is a key point to allow numerical simulations of state-of-the-art cold-atoms experiments. Multiconfigurational methods are limited in their practical usefulness by the exponential growing of the configurational space and cannot be used for usual experimental conditions with hundreds to thousands of atoms. To avoid this limitation, we will consider restricted active spaces that constraint the number of configurations while large number of orbitals and/or atoms can be investigated. Restrictions on the space dimension will be based on the factorisation of the wave function in space. This is suggested by the strong anisotropy of the traps used in experiments. The tightly confined dimensions will be approximated by time-dependent parameterized analytical functions, while the loosely confined dimension will be discretized on a grid. Recent experiments are interested in dipolar-BEC, for which the interaction potential is singular. We will use a range separation into a singular short- and regular long-range potential to evaluate this complicated potential with high accuracy. The numerical efficiency of these models will be enhanced further by using algorithms compatible with graphic cards architecture to provide an efficient simulation toolbox. The developed methods will be directly applied to provide more precise simulations in collaboration with the experimentalists. The originality of this project relies on the restrictions on both the configurational space and the space dimension. This originates from discussions and the strong collaboration between the experimental physicists from Schmiedmayers laboratory and the applied mathematician and theoretical physicists, including the applicant, of Mausers group. Our goal is to provide a numerical framework that can efficiently provide trustable simulations of the on-going experiments.