PhD thesis in English

Chapter 6SummarySince the first experimental observation of Bose-Einstein condensation in thedilute vapors of alkali atoms in 1995, the field of ultracold atoms constantly expands.The experimental advances in the manipulation of quantum gases pave a way to thedevelopment of new technologies, and allow exploration of the quantum world withpreviously unseen precision and flexibility, uncovering at the same time new varietyof far reaching physical phenomena. In this thesis we have studied two particularphenomena that give new insights into the properties of cold quantum gases andhave been recently addressed experimentally.To begin with, in Chapter 1 we have introduced the aspects of the research in thefield of ultracold atoms, main concepts, and its position within the wide forefrontof modern physics. In Chapter 2 we have developed an efficient numerical methodfor calculation of the eigenspectrum and eigenvectors of a system in an arbitrarytrapping potential. The devised approach is based on the exact diagonalizationof the time-evolution operator and can be applied for numerical studies of generalfew-body systems. We have carefully explored different types of systematic errorsthat arise in the process of the spatial discretization of the time-evolution operator.We have shown analytically and numerically that the discretization error vanishesas the exponential of 1/∆ 2 , where ∆ represents the discretization spacing. Thus,the method highly outperforms in efficiency the approaches based on the real- spacediscretization of the Hamiltonian, which exhibit polynomial error in ∆. For thehighly accurate calculation of matrix elements of the evolution operator, necessaryfor the application of the method, we use the short-time expansion of transitionamplitudes in the propagation time to high-orders. The chief ingredients of this partof the procedure are higher-order effective actions, that were derived previously. Wehave demonstrated the advantages of this method by calculating highly accurateenergy spectra in a numerically efficient way for several one- and two-dimensionalmodels.Motivated by experimental studies of rotating ultra-cold quantum gases, in Chap-134