PhD thesis in English

3. Rotating ideal BEC3.3 Global properties of rotating BECsIn this section we will apply the presented approach to calculate different globalproperties of rotating BECs. First, we will calculate condensation temperature,and then, we will present phase diagrams defined in terms of condensate-fractiondependence on the temperature for different trap parameters. Additionally, wewill compare numerical results with semiclassical values and identify when the fullnumerical treatment becomes necessary.3.3.1 Condensation temperatureIf we take into account semiclassical corrections, as explained in the previous section,we can calculate, for instance, the condensation temperature of the condensatefor different rotation frequencies. This implies that we have to find the temperaturefor which the number of thermal atoms saturates precisely at the total number ofatoms N. In practice, this works the other way around: for a given condensationtemperature T c we numerically calculate the particle number in the system usingEq. (3.11), which gives the number of atoms in the system required for a condensationtemperature to be equal to T c . This procedure is implemented in Fig. 3.6 forseveral values of the rotation frequency Ω in units of η = Ω/ω. For example, forT c = 63.14 nK we see that the corresponding number of particles is N = 3 · 10 5 ,which coincides with the value for a critically rotating condensate in the experimentof Dalibard and collaborators [12].In principle, such a procedure is only applicable for low-accuracy calculations ofthe critical temperature, since otherwise one has to use very large values of the cutoffJ which would practically slow-down numerical calculations. If one is interested inmore precise results, a suitable J-dependence must be properly taken into account.In order to be able to efficiently extract the correct value of β c , we will derive ananalytical estimate of the asymptotic error ∆β c = β c − β c (J), which is introducedby the presence of the cutoff J. Note that always ∆β c > 0, since β c (J) < β c has tocompensate the missing terms in the sum (3.11).If we insert β c = β c (J) + ∆β c into Eq. (3.11), the error ∆β c can considered tobe small for sufficiently large value of the cutoff J. By comparing Eq. (3.11) with70