PhD thesis in English

3. Rotating ideal BEC3.0·10 5 0 20 40 60 80 100 120 1402.5·10 5With SC corr.Without SC corr.2.0·10 5N - N 01.5·10 51.0·10 50.5·10 53.00·10 52.96·10 53.04·10 5 0 20 40 60 80 100 120 1400E max / − hω ⊥Figure 3.5: Number of thermally excited atoms N − N 0 calculated as a function ofE max with and without semiclassical corrections, calculated with a large cumulantcutoff J = 10 4 to eliminate the J-dependence. The results correspond to a criticallyrotating condensate with the same parameters as in Fig. 3.3. The horizontal linecorresponds to N = 301834 which represents the exact value at T c = 63.30 nK.where the superscript denotes that only the x−y part of the potential is considered.When this semiclassical correction is taken into account, the numerical resultsshow almost no dependence on E max , as can be seen from Fig. 3.5. Here we haveused an excessively large value of the cumulant cutoff J = 10 4 in order to completelyeliminate any J dependence. From the inset in this graph we also see that E maxmust be chosen in accordance with the value estimated in the previous section forthe maximal reliable energy eigenvalue obtained by numerical diagonalization. Ifwe use a value E max larger than this, we will be underestimating the higher part ofthe energy spectra, and obtain incorrect results. For a critically rotating condensatewith the anharmonicity κ = κ BEC the estimated value of E max from Table 3.3 isaround 90 ω, which agrees with the results from the inset of Fig. 3.5. If we use thisvalue for E max and calculate properties of the condensate using numerically obtainedeigenstates below E max with semiclassical corrections according to Eq. (3.14), we willobtain the exact results with very high accuracy.69