Gustavo Petro – Bogotá Humana (2012-2016)

In Sec. 5.3 several cases were presented, which demonstrated that the results of QiwiB calculations depend on the number of single-particle functions M. We found that a simple two-mode model is not sufficient anymore to model the system properly. This is especially true for small particle numbers as can be seen in Fig. 5.5. More single-particle functions lead to a better description of the full many-body system. Therefore, effects found in the two-mode model, that are very sensitive to changes of the initial parameters, may change or even disappear for calculations withM >2.

To give an example, we present calculations forN = 80andM = 2in Fig. 5.11, right at the edge of the full reflection regime of theT RLwindow atV0= 4.82. The top panel in Fig. 5.11 shows the time evolution of the natural orbital populations. The initial total state in the QiwiB calculations was very similar to an exact soliton solution obtained from the GP approach, that is ρ1/N = 0.99 at t = 0. After the scattering event, once the soliton is positioned at a distance of several soliton widths from the well, ρ1 is still highly populated with ∼ 93%. However, during the impact with the well, ρ1 drops to ∼ 56.4%. The magnitude of this drop is about the same as in calculations, which are not shown here, forV0= 4.83. There, we find that where after the collision the initial soliton splits into a reflected (ρ1≈0.52) and a trapped part (ρ1 ≈0.48). ForV0 = 4.82similar results are found from the density plot in the bottom panel of Fig. 5.11, which reveals that during and shortly after the impact with the well, the initial soliton starts to split into

Figure 5.11: Density (left) and natural orbital populations (right) for N = 80

and M = 2 at V0 = 4.82 (top graphs) and V0 = 4.5 (bottom graphs). For the former, the initial soliton fragments into a reflected and a trapped part. After

t ≈ 50the trapped soliton starts to escape from the well and moves towards the previously reflected part with a higher centre of mass velocity. Att≈100the two solitons begin to merge and partially regenerate the initial state of one soliton. For

V = 4.5 the interaction time of the incoming soliton with the well is negligible, and therefore reflects completely without fragmenting.

n=1 n=2 n=3 n=4

M=2 55% 45% - -

M=3 60% 32% 8% -

M=4 59% 29% 11% 1%

Table 5.1: Population numbers for the nth natural orbital for N = 80 particles andM = 2,3,4 modes, and the potential depth V0 ≈4.82att≈60. For M = 2 a threshold of ≈ 45% for the population of the second highest occupied natural orbital can be identified, above which the trapped soliton remains bound to the well until the end of our simulations. This threshold is significantly reduced for

M = 3andM = 4, and it remains unclear if the final state of two reflected solitons in Fig. 5.11 is still existent forM → ∞.

two solitons, a reflected one and a trapped one.

To our knowledge, there is no clear explanation for the initial break-up during the impact on the well, but it may be related to the length of the time period during which it interacts with the well. For V0 = 4.5 the soliton density barely enters the well and is reflected very quickly and we find ρ1 > 0.975 at all times (see bottom panel of Fig. 5.11). This is in stark contrast to V0= 4.82, where the soliton almost completely covers the well for a finite period of time. This could lead to dynamical processes that allow for a coupling between several Fock states and therefore an increased population of natural orbitals other than the highest occupied one.

However, Fig. 5.11 also shows that a short amount of time after the scattering process, the trapped soliton escapes the well travelling into the same direction as the previously reflected soliton. The velocities of both solitons clearly differ, and therefore it is possible for the previously trapped soliton to catch up with the other one. Hence, they collide at t ≈ 120 and the natural orbital populations, which are shown in the top panel of Fig. 5.11, change to form only one macroscopically populated natural orbital, i.e. only one coherent soliton.

ForM = 2 modes we can identify a threshold on the population of the second natural orbital for the temporal trapping. Once the second natural orbital is occu- pied by more than∼45% of the particles, the initial soliton gets partially trapped

by the well for the remainder of our simulations, which usually run untilt= 200. Increasing the number of modes M now alters this effect. For M = 3and M = 4

the threshold on the second natural orbital population for the temporal trapping is significantly reduced, and even though the differences between the results for

M = 3andM = 4are much smaller compared to those forM = 2(see Table 5.1), it is still unclear whether the results are converged in this parameter regime and the splitting into two reflected solitons, as shown in Fig. 5.11, is physical.

This example, together with the results from Sec. 5.3, show that calculations for M = 2 can be misleading and a detailed study of the convergence properties is needed.