In the present application the scope of theoretically investigating the formation, relaxation, decay and re-relaxation of a few number of quantized vortices in a gas of Bose-Einstein condensed atoms is being proposed. Bose-Einstein condensates in dilute atomic gases have been playing a central role in modern fundamental and applied physics. Various experiments and novel ideas the past two decades have paved the way for synthesing quantum degenerate matter with, until recently, unknown properties and an unprecedented amount of control. At the same time, atomic gases at (almost) zero absolute temperature have proven to be ideal laboratories for testing theories of physics. Such an example is the search and control of quantum turbulence in ultracold atomic gases.Given the above, I discuss and propose numerical experiments that involve few quantum vortices that carry single or multiple charge. Based on prior research, systems of few (up to six) interacting vortices moving in two-dimensions can exhibit chaotic motion. The interplay between regular and chaotic motion, as well as probable onset of turbulence is to be scrutinized. Furthermore, the formation and relaxation of vortices in two and three spatial dimensions, by a moving object (thread) in a quantum gas is to be examined in systems consisting of few particles. In such cases fragmentation and loss of coherence is expected to take place. Moreover, I want to examine the fate of 'big vortices', that is, quantum vortices that carry topological charge larger than one. For the investigation of the above scenarios I use a novel many-body multi-configurational method, that is herein reviewed and discussed, and makes it possible to predict new phenomena and states of the gas. This method goes beyond the traditional mean-field theory and, hence, fragmented and correlated states are well described. Preliminary results show, so far, that fragmentation quickly develops in the above configurations, both at the stage of vortex formation and the decay processes. Last, the little-explored topic of thermalization is discussed and viewed as a possible emergence of temperature at a closed, initially at zero-temperature quantum system. Aside projects that involve the development of complementary numerical codes are also discussed.
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