Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

7. Quantum simulations of evaporatively cooled Bose condensatesFigure 7.8: Single-trajectory calculation of the atom density (a) in Fourier space n(k) and (b) incoordinate space n(x) for a three-dimensional Bose condensate undergoing an ‘RF-scalpel’ evaporation.Time has been normalised by t 0 =0.79ms, momentumbyk 0 =1.32×10 6 m −1 and positionby x 0 =0.76µm.Unlike the previous case, the effect of the scalpel cutting into the spatial distribution canbe seen, to produce a final distribution which occupies a small volume. Also unlike theprevious case, there are no large peaks in the final k-distribution, but rather the finaldistribution is small and smooth 3 . One reason for this small peak momentum densityis that since the final x-distribution is now much narrower, the k-distribution must becorrespondingly larger if the distribution is transform limited. The waveforms must atleast obey Heisenberg’s uncertainty principle for the x and k variables:∆x∆k ≥ 1 2 . (7.24)Equality corresponds to transform limited, or minimum uncertainty, distributions. In thesimulation shown, ∆x∆k ≃ 0.53, which is very close to being transform limited. As thislimit is approached, the occupied volume in k-space can no longer be compressed, leadingto a high loss rate through the higher k-modes which are now damped.Thus the simulation has resulted in (almost) the lowest energy configuration for thegiven spatial confinement. What is not occurring is a large occupation of this state, since3 That the final ‘intensity’ a 1a ∗ 2 is negative shows that the system is not in a coherent state, but rathera superposition of coherent states.164