PhD thesis in English

3. Rotating ideal BECT c [nK]12011010090807060504040353025201510κ BEC , N=1·10 4SC0 0.2 0.4 0.6 0.8 1ηκ BEC , N=3·10 50 0.2 0.4 0.6 0.8 1SCFigure 3.8: The condensation temperature as a function of the rotation frequencyfor the condensate of N = 3 · 10 5 and N = 1 · 10 4 atoms of 87 Rb, with the quarticanharmonicity of the trap κ = κ BEC . The discretization parameters are given inTable 3.3. The full lines correspond to the semiclassical approximation for T c fromRef. [65].obtained from this approach in the next section, we will compare our numericallyexact results with the semiclassical approximation in more detail, and identify theparameter ranges where a full numerical treatment becomes necessary.3.3.2 Ground-state occupancyThe ground-state occupancy is the next important global property of Bose-Einsteincondensates we will look into. Below the condensation temperature a non-trivialfraction of atoms is in the ground state, thus yielding a macroscopic value of theoccupancy ratio N 0 /N.Using the same approach as above, we can calculate the ground-state occupancyfrom Eq. (3.8). After determining the ground-state energy E 0 from an exact diagonalizationof the evolution operator, we obtain the occupancy asN 0N = 1 − 1 N∞∑ [ejβE 0Z 1 (jβ) − 1 ] . (3.22)j=1In order to calculate N 0 /N, we need the full single-particle energy spectrum. For74