Variational approaches to dipolar Bose-Einstein condensates

Abstract

In this work several aspects of dipolar Bose-Einstein condensates are investigated. These condensates have attracted much interest since their experimental realization and offer the chance to explore the novel, fascinating, and sometimes unexpected physics originating from the dipole-dipole interaction. For the theoretical description the time-dependent variational principle with an ansatz of coupled Gaussian wave packets is applied to the analysis of this form of matter. In order to describe physical systems, where the shape of the condensate wave function cannot be approximated adequately by origin-centered Gaussian functions, translational and rotational degrees of freedom of the particular wave packets are included. This provides not only a full-fledged alternative method to full-numerical grid calculations that are demanding in computation time, used for the determination of ground states and investigation of the real-time dynamics, but also allows for the calculation of excited states and the examination of stability properties and offers the possibility for a deeper physical understanding of dipolar condensates. The variational method is applied to three different physical systems. In the first one anisotropic quasi-two-dimensional solitons are investigated. Ground states and small excitations are presented and the full flexibility of the approach is utilized in the simulation of the collision of two solitons. Thereby simulations of collisions are performed, where initially the solitons are in the repelling side-by-side configuration and move towards each other with a specific momentum. It is shown that the results are in excellent agreement with full-numerical grid calculations. Collisions with various initial parameters such as the initial momentum and impact factor are performed, allowing to classify the qualitative interactions of two solitons and illuminating the effects of the solitons' two-dimensional nature on the collision process. In the second system dipolar Bose-Einstein condensates in a triple-well potential are investigated. This constitutes a well-suited model system for periodic optical potentials with important contributions of the non-local and anisotropic dipole-dipole interaction, which show a variety of effects such as self-organization and formation of patterns. Here, a macroscopic sample of dipolar bosons in the mean-field limit is addressed. The analysis goes beyond the calculation of ground states and clarifies the role of excited and metastable states in such systems. In particular, the formation of phases is found to originate from the interplay of several states with distinct stability properties. As some of the phases are formed by metastable states, special attention is paid to the characteristics of phase transitions in real time and the dynamical stabilization of the condensate. The dynamical simulations that support these results, constitute a way for an experimental observation of these findings. The third physical application is an open quantum system. Here, an external potential models gain and loss of particles and features parity with simultaneous time-reversal symmetry. For these dipolar condensates real eigenvalues of the nonlinear Hamiltonian are found, preserving the PT-symmetry of the external potential, as well as states that break this symmetry. A detailed stability analysis of these, in general complex states, is performed. The comparison with a similar system featuring a non-dipolar condensate demonstrates the role of the dipolar interaction for novel effects. Furthermore, the real-time dynamics is investigated, which reveals, among other effects, long-living oscillations of the two wells' population.