during the collision and was in excellent agreement with numerically exact grid calculations. We varied the relative phase difference of the two solitons and showed that the amount of transferred kinetic energy depends on the phase difference.
Furthermore, we confirmed in long-time calculations that the solitons survive the scattering in a certain range of the initial speed. However, it turns out that below a critical initial momentum the solitons glue together and finally merge to one soliton which is stabilized by the emission of particles which leave the condensate.
If the solitons possess zero or very small initial speed, the repulsion of the dipole-dipole interaction cannot be overcome and the dipole-dipole barrier leads to a fully elastic scattering of the solitons.
Self-organization and pattern formation in dipolar BECs have been the physical effects we aimed for in chapter5. To gain insight into the mechanisms of such con-densates we investigated a model system for larger periodic potential, viz. a BEC in a triple-well potential. It turned out that the structure of the stationary states and their influence on the dynamics appears to be quite complex. The main tasks here were the investigation of phase transitions and of metastable states. These states are present even in regions of the parameter space, where no stable ground state exists and the collapse of the condensate wave function is expected. We showed that the metastable states play a crucial role in determining the shape of the ground state wave function and are able to dynamically stabilize the condensate in the absence of a stable ground state. We performed a detailed analysis for two different values of the dipole strength including real-time evolutions of metastable states and dynamical variations of the scattering length and were thereby able to clarify the character of the phase transitions. Furthermore, we were able to show the signatures of metastable states in our dynamical simulations.
An open system with interesting mathematical properties has been addressed in chapter6. PT-symmetric systems can possess a real eigenvalue spectrum in linear quantum mechanics. However, in the nonlinear GPE the PT symmetry can be broken as the wave function enters the Hamiltonian and thus determines its sym-metry properties additionally to the external potential. We investigated a dipolar BEC in aPT-symmetric double-well potential in the attractive and repulsive con-figuration. We revealed the existence of an additional PT-broken state, compared to a system without dipolar interaction. We tuned both the scattering length and the strength of the imaginary potential to show the arrangement of states and their connections. It turned out that there is a small interval in the scattering length, where two degenerate PT-broken states emerge in a pitchfork bifurcation. Out-side this interval PT-symmetric and PT-broken states exist separately, although all stationary states disappear for a scattering length small enough. Furthermore, we investigated the stability of all states and discussed the role of the PT-broken states in this context. We performed real-time calculations and showed that a
7.1. Outlook
dynamical stabilization mechanism, as already found in chapter 5, is present here as well. We confirmed the picture gained from the analysis of the repulsive con-figuration by calculations in the attractive concon-figuration. These supported the result that the qualitative picture is characteristic for a BEC in a PT-symmetric double-well potential, independent of the orientation of the dipoles.
7.1. Outlook
It is founded in the nature of science that interesting results are interrelated with new questions arising. In our investigations of soliton collisions we found signatures of effects which can be described in terms of scattering theory. The development of such a scattering theory for this system is an interesting task. The results obtained for dipolar BECs in triple-well potentials and the findings for the PT -symmetric double-well potential suggest the investigation of systems with more coupled wells which leads to the description of dipolar BECs in optical lattices in the mean-field limit. On this way an apparent next step could be the investigation of a PT-symmetric triple-well potential. As a PT-symmetric system can be seen as a subsystem of a larger Hermitian system it would be interesting to investigate the role of PT-broken and metastable states in the generation of novel quantum phases like a supersolid. Moreover, it would be interesting to investigate for both of these systems the dependency on the dipole strength, what might also lead to the tangent bifurcation of thePT-broken states coincide with the pitchfork bifurcation in the PT-symmetric potential. For the PT-symmetric potential the development of an adequate matrix model states another interesting step in the understanding of the mechanisms in dipolar BECs.
Apart from the systems investigated here, the method developed in this thesis is applicable to various systems as a large number of external potentials can be implemented easily. For example, the description of a dipolar BEC in an external toroidal trap resulting in a self-induced Josephson junction [120] could be generated by Gaussian potentials which are already implemented.