PhD thesis in English

ently explored to illustrate the versatility of possibilities. Once that the propertiesof a weakly interacting quantum gas were understood and explored to some extent,more complex and interesting phases of matter came to the focus of experimentalresearch. Progress in the last decade has led to the realization of optical latticeswhich practically allow simulations of condensed matter and other systems withinultracold atoms framework, in the sense of Feynman’s quantum simulator [8]. Phasediagrams of strongly interacting quantum gases with long-range dipolar interactionare currently explored. Beside the equilibrium properties, non-equilibrium and dynamicalphenomena in these systems are also major research topic. Most recentadvances include the realization of non-Abelian gauge fields, as well as bosonic systemswith the spin-orbit type of coupling. Beside bosonic atoms, experiments arenow performed with ultracold fermionic atoms and even with ultracold molecules(emergent field of ultracold chemistry). Even a photonic BEC has been producedrecently [17]. Of course, the exciting experimental developments are led and closelyfollowed by theoretical studies. Some of those will be presented in this thesis.1.2 Few basic facts on Bose-Einstein condensationTo set up the stage and to sharpen our intuition on the facets of cold bosonicgases, first we review a textbook knowledge on the Bose-Einstein condensation of anoninteracting gas. Initial ideas about this type of phase transition were based onthe considerations of a homogenous gas of particles. However, as will be explainedlater in this Chapter, all experimental realizations include an external confining trap.Most widely used is a harmonic trap and hence here we consider thermodynamicproperties of harmonically trapped noninteracting bosons [18, 19].1.2.1 Noninteracting bosonic gas in the harmonic trapThe average occupation of a single-particle state of the energy E n at temperature Tin the gas of noninteracting particles is given by the Bose-Einstein distribution asB n (µ, T) =1e β(En−µ) − 1 , (1.2)where β = 1/k B T is the inverse temperature and µ stands for the chemical potential.Formally speaking, we use the grand canonical ensemble description and assume3