PhD thesis in English

3. Rotating ideal BECTable 3.3: Maximal reliable numerically calculated energy eigenvalue E max of thexy-part of the BEC potential (3.4) for different values of η = Ω/ω, estimated fromcomparing the numerically obtained density of states ρ(E) with the semiclassicalapproximation. The numerical diagonalization was done using level p = 21 effectiveaction. The spacing ∆ was always chosen so that L/∆ = 100, and the propagationtime was t = 0.2 for κ = κ BEC and t = 0.05 for κ = 10 3 κ BEC . The total number ofreliable energy eigenstates is in all cases of the order of 10 4 .κ = κ BEC κ = 10 3 κ BECη E max /ω L E max /ω L0.0 140 14.2 190 3.900.2 140 14.4 190 3.900.4 140 15.0 180 3.910.6 140 16.3 180 3.920.8 130 18.6 180 3.941.0 90 22.3 170 3.961.04 90 23.2 170 3.96BEC experiments, but we consider it for the sake of completeness. When the temperatureis sufficiently high, so that effects of higher energy eigenstates cannot beneglected, the inverse temperature β becomes a small parameter. Thus, it becomespossible to calculate numerically the single-particle partition function as a sum ofdiagonal amplitudes, i.e.Z 1 (β) = Tr e −βĤ ≈ ∑ jA(j∆,j∆; β)∆ d , (3.10)where ∆ represents the spatial spacing, as before, the values of j are defined byj ∈ [−L/∆, L/∆] d , with the spatial cutoff L chosen in such a way as to ensurethe localization of the evolution matrix within the interval [−L, L] d , and transitionamplitudes for small β can be calculated directly using the effective action approach[10, 44].65