PhD thesis in English

3. Rotating ideal BEC3.2 Finite number of energy eigenvalues and semiclassicalcorrectionsIn the previous section we have described a numerical approach that is capableof providing a large number of accurate energy eigenvalues for a general quantumsystem. For instance, we are able to calculate typically 10 4 energy eigenvalues forthe considered BEC potential (3.4). In this section we discuss in more detail howthe finiteness of numerically available energy eigenstates affects the calculation ofthermodynamic properties of Bose-Einstein condensates.As outlined at the beginning of this Chapter, the information on single-particleeigenvalues is sufficient for calculating the condensation temperature according toEq. (3.9). Below the condensation temperature, the ground-state occupancy followsfrom solving Eq. (3.8). In practical calculations, however, one is inevitably forcedto restrict the sum over j in the cumulant expansion (3.8) to some finite cutoff J,resulting in the following approximation for the number of thermal atomsN − N 0 ≈J∑ ∞∑e −jβ(En−E0) . (3.11)j=1 n=1Thus, the ground-state occupancy N 0 depends not only on the particle number Nand the temperature T, but also on the cumulant cutoff J. In particular, whenwe solve Eq. (3.11) for the (inverse) condensation temperature β c , obtained fromthe condition N 0 = 0, we will get the solution in the form β c (J), with an explicitdependence on J. The exact condensation temperature β c is only obtained in thelimit J → ∞.Fig. 3.3 illustrates the J-dependence resulting from Eq. (3.11) for both a nonrotatingand a critically rotating condensate. As expected, the sum saturates forhigh values of J to some finite number N − N 0 . By tuning the temperature in sucha way that the sum saturates at the desired value of the total number of atoms Nin the system, which corresponds to N 0 = 0, one is, in principle, able to extract thevalue of the condensation temperature T c .Although the results in Fig. 3.3 suggest that this approach can be appliedstraightforwardly, a closer look at the results for numerically calculated values ofN − N 0 reveals several problems that have to be addressed. At first we have toinvestigate how the results depend on the number of energy eigenstates used in the66