PhD thesis in English

3. Rotating ideal BEC4.0·10 53.5·10 53.0·10 52.5·10 520 40 60 80 100 120N - N 02.0·10 51.5·10 51.0·10 50.5·10 50η=1.0η=0.8η=0.6η=0.4η=0.0T [nK]Figure 3.6: Number of thermally excited atoms N −N 0 as a function of the temperatureT for different values of the rotation frequency and the quartic anharmonicityκ = κ BEC . The discretization parameters are given in Table 3.3, and the results arecalculated by taking into account semiclassical corrections. The dashed line correspondsto the number of atoms N = 3 · 10 5 in the experiment [12]. For comparison,the full lines depict the semiclassical results from Ref. [65].the exact expression (3.9) we obtain∞∑j=J+1 n=1∞∑e −jβc(En−E0) ≈ ∆β cJ∑j=1 n=1∞∑j(E n − E 0 ) e −jβc(En−E0) . (3.15)The term j(E n −E 0 ) within the sum can be obtained by setting N 0 = 0 and applyingthe partial derivative ∂/∂β c to Eq. (3.11):[ ]∂N ∂ ∑ ∞−∆β c ≈ 1 − ∆β c∂β c ∂β cj=J+1 n=1∞∑e −jβc(En−E0) . (3.16)Note that the derivative of the particle number N with respect to β c is not equal tozero, since N is here effectively defined by the sum (3.9). Therefore, we have instead∂N∂β c= −∞∑j=1 n=1∞∑j (E n − E 0 ) e −jβc(En−E0) . (3.17)Clearly, the right-hand side is a negative quantity that does not depend on J. How-71