Correlated dynamics of ultracold bosonic atoms in three dimensions. Facing the challenge with a multi-configurational approach

In general, the understanding of microscopic physical processes in many-body systems needs an investigation starting from first principles, especially taking the impact of dimensionality and correlations into account. However, designing numerical ab-initio algorithms, which can handle the dimensional crossover and correlations is a very challenging task and strongly limited with respect to computational power. Because of this, most of the available algorithms in the literature focus on either one of these challenges. In this thesis, nevertheless, we tackle these challenges by designing a highly optimized algorithm taking both all correlations and the dimensional crossover into account and show its efficient application for confined, interacting bosonic ensembles. In doing so, the exponential scaling of complexity w.r.t. the number of particles can be tackled by the ab-initio Multi-Configuration Time-Dependent Hartree methods for Bosons (MCTDHB) [PRL 77, 033613 (2008)]. However, further different challenges occur when trying to model three-dimensional interaction potentials and in order to determine the most feasible interaction potential, we discuss different implementations for zero- as well as finite-range interaction potentials and rate them with respect to (i) the possibility to resolve correlations, (ii) a numerically efficient implementation and acceptable runtimes for desktop computers and (iii) numerically created artefacts due to approximations made. We show that repulsive short-range interaction potentials given as a product with respect to the dimensions are most suited for an implementation. Despite being most suitable, the use of these interaction potentials is highly challenging because they introduce a small length scale and, hence, need large number of grid points in a numerical treatment. We deal with this challenge by developing an efficient algorithm with respect to the number of grid points based on the Multi-Layer MCTDHB (ML-MCTDHB) method for ultracold bosons. In doings so by using a particularly tailored wave function ansatz, we derive equations of motion by using the Dirac-Frenkel variational principle and implement them in a highly optimized way, e.g., using parallel-processing. With our new wave function ansatz, the total numbers of grid point scales linear with the dimensions and not exponential as in the MCTDHB method. The algorithm is validated by comparison with possible analytical results and with other numerical methods available in the literature. The beneficial scaling of our approach is used to study the impact of dimensionality on bosonic ensembles in different trap geometries. This is achieved by changing the trap aspect ratio, defined as the quotient between the transversal and longitudinal trap frequencies, leading to a crossover from a quasi 1D to an isotropic confinement. Especially, we are interested in the interplay between spatial and particle correlations and, thereby, we employ the following three systems. (i) Two bosons interact in a harmonic trap with various aspect ratios: This system serves as a prototype in order to determine suitable numerical and physical parameters and to study the time scale on which the interaction can induce significant spatial and particle correlations. Furthermore, we study the general convergence behaviour, which indicates an algebraic decay of the natural populations in three dimensions. (ii) A bosonic ensemble tunnels between two wells separated in the longitudinal direction, which are embedded in an elongated harmonic trap: We show that at least two transversal modes are needed in order to resolve the time-dependent density profile correctly and identify the influence of the dimensionality on the evolution of the population imbalance. In addition, we study the validly of different approximation of the manybody wave function, such as the mean-field approximation or the adiabatic separation of the spatial dimensions. (iii) A bosonic ensemble, initially displaced from the trap centre, scatters off a barrier, placed in the trap centre: By studying the mechanisms of coherence loss, interesting for matter-wave interferometers, we find that for nearly isotropic traps, loss of coherence occurs between the region close to the barrier and outer regions, due to spatial correlations while for quasi one-dimensional traps incoherences rise between the two density fragments of the left and right side of the barrier, due to particle correlations. Furthermore, we can show how spatial and particle correlations modify the decay of the centre of mass oscillation. All these effects are enhanced if the aspect ratio is integer valued.