PhD thesis in English

3. Rotating ideal BECof the rotation frequency Ω once the total particle number N is large enough andthe trap anharmonicity κ is small enough. The latter condition implies that theunderlying potential (3.4) has a small curvature around its minimum, and hencethe corresponding density of energy levels is sufficiently high. However, in thiscontext the question arises how accurate the semiclassical approximation is, forwhich system parameters it is not anymore sufficient for a precise description ofBEC phenomena, as well as when it finally breaks down, requiring a full quantummechanicaltreatment of the system.In order to analyze the problem more quantitatively, it is mandatory to determinethe single-particle energy eigenvalues and eigenfunctions fully quantum mechanically.In this Chapter we show how the exact diagonalization of a time-evolutionoperator, presented in Chapter 2, is applied for studying both global and local propertiesof fast-rotating Bose-Einstein condensates. To this end we proceed as follows:first we calculate a large number of energy eigenvalues and eigenfunctions for the anharmonicpotential (3.4). Afterwards, we discuss how a finite number of numericallyavailable energy eigenvalues affects the results and how they can be improved byintroducing systematic semiclassical corrections. On the basis of this precise numericalsingle-particle information, we study global properties of a rotating condensate.Finally, we calculate local properties of the condensate, such as density profiles andTOF absorption pictures.To begin with, we rewrite Eq. (1.3) for the total number of particles in a moreconvenient form in terms of the single-particle partition function Z 1 (β), defined asZ 1 (β) =∞∑e −βEn . (3.5)n=0To do this, we single out the contribution of the ground state and use the Taylor’sexpansion 1/(1−x) = ∑ ∞n=0 xn (valid for |x| < 1), to derive the following expression:N =∞∑n=01e β(En−µ) − 1 = B 0(µ, T) +∞∑n=1 j=1∞∑e −jβ(En−µ) . (3.6)60