PhD thesis in English

3. Rotating ideal BECsition from the position ⃗r to the position ⃗r for the imaginary time t = jβ.While both definitions (1.12) and (3.23) are mathematically equivalent when oneis able to exactly calculate infinitely many energy eigenstates and required transitionamplitudes for an arbitrary propagation time, the first definition is more suitable forlow temperatures, when the number of relevant energy eigenstates is moderate, andthe second one is suitable for high temperatures, when the imaginary propagationtime β is small, and the short-time expansion can be successfully applied.3.4.2 Time-of-flight graphs for BECsIn typical BEC experiments, a trapping potential is switched off and the gas isallowed to expand freely during a short flight time t which is of the order of severaltens of milliseconds. Afterwards an absorption picture is taken which maps thedensity profile to the plane perpendicular to the laser beam. For the ideal Bosecondensate, the density profile after time t is given byn(⃗r, t) = N 0 |ψ 0 (⃗r, t)| 2 + ∑ n≥1B n (E 0 , T)|ψ n (⃗r, t)| 2 , (3.24)where the density profile has to be integrated along the imaging axis, and the eigenstatesψ n (⃗r, t) are propagated according to the free Hamiltonian, containing only thekinetic term, since the trapping potential is switched off. If the energy eigenstatesare available exactly, either analytically or numerically, their propagation in timecan be calculated by performing two consecutive Fourier transformations:ψ n (⃗r, t) =∫ d ⃗ k d ⃗ R(2π) 3 ei[⃗ k·(⃗r− ⃗ R)−ω ⃗k t] ψ n ( ⃗ R) , (3.25)where the term e −iω ⃗ k t accounts for a free-particle propagation in ⃗ k-space, ω ⃗k = 2 k 2 /(2M). In practical applications, when the energy eigenstates are calculatedby a numerical diagonalization of space-discretized transition amplitudes, the naturalway to calculate the above free-particle time evolution is to use Fast FourierTransform (FFT) numerical libraries.For high temperatures we can use a mathematically equivalent definition of thedensity profile which is derived again from using the cumulant expansion of occu-79