4. Two-dimensional solitons 63
4.5. Collisions of solitons
4.5.4. Collisions including finite angular momentum
the rearranging of the atomic dipoles is the reason for the binding mechanism in the creation of the soliton.
The newly created soliton can be seen as a bound state that influences the scattering process – a mechanism comparable to the scenario known from scattering theory and Feshbach resonances. There, the bound state would correspond to a resonance. However, this analogy should not be taken too far as we face a highly nonlinear scattering process here and a decent amount of particles are scattered out of the soliton during the collision. Furthermore, the reduction to only two scattering channels is obviously an oversimplification. A complete description of soliton collisions in the sense of scattering theory is a highly nontrivial task for the reasons given above and in the beginning of this section and is beyond the scope of this thesis.
4.5. Collisions of solitons
(a) (b) (c)
Figure 4.11.: Absorption images of two colliding solitons with nonzero angular mo-mentum obtained by simulations on a grid (left columns) and the variational approach (right columns). All absorption images have been normalized to the maximum value.
The absolute value of the initial momentum of each soliton is k = 10 and the field of view is 1.4×1.4(135×135µm). The figures (a), (b) and (c) show calculations for a difference of φ= 0, φ =π/2 and φ =π in phase, where the left column is the result of the grid calculations and the column on the right hand-side presents the results of the variational ansatz. For all three variational calculations six GWPs (three for each soliton) were used. The variational calculation is able to reproduce the transient ring-like structure during the collision for a difference of φ=π in phase and yields the correct results for the configuration at the end of all three calculations.
in the case with zero angular momentum, leading to condensates with smaller extension at t= 0.049 than those in figure 4.5a.
For phase differenceφ=π/2 the absorption images in figure4.11b show a similar behavior as in the case with vanishing angular momentum, resulting once again in
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Ekin
t var. φ = 0
var. φ = π/2 var. φ = π num. φ = 0 num. φ = π/2 num. φ = π
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Figure 4.12.: Kinetic energy as a function of time for the collisions with angular momentum shown in figure 4.11. The kinetic energy increases while the condensates merge. After the split up, the condensates have a lower kinetic energy than before, which indicates a transfer of kinetic to internal energy, thus resulting in excitations of the condensates. The inset shows the kinetic energy obtained by the variational calculations for large time scales. The oscillatory behavior indicates the excitation of the solitons. Note the larger kinetic energy obtained by the grid calculations shortly after the split up at t ≈ 0.035. This is due to the finite grid size, which manifests itself in (non-physical) oscillatory modulations of the wave function’s amplitude at large times.
an asymmetric situation, where after the collision the condensates do not have the same amplitudes anymore. But for a finite impact factor one may actually speak of a merged condensate at t = 0.031. Finally the collision with a difference of φ=π (figure 4.11c) shows the condensates effectively repelling each other, but in this case the finite impact factor leads to a transient ring-like structure due to the node of the wave function at x = 0. The extension of the condensates after the collision is much larger compared to the case with no difference in phase, which means that the amount of transferred kinetic energy in internal energy is larger than in the former case. The reason for this behavior is the reduced destructive interference due to the finite impact factor.
A soliton configuration including angular momentum is suited best to show how the transfer of kinetic energy affects the spatial distribution of the condensate. In
4.5. Collisions of solitons
figure 4.12we show the kinetic energy as a function of time for the collisions with angular momentum. We compare the curves in figure 4.12 with the absorption images in figure 4.11 and see that a larger transfer of kinetic energy implies a larger size of the condensate at t = 0.049. The slightly smaller transfer, i.e. the higher kinetic energy after the collision, observed at the end of the full-numerical calculations (this leads to a larger extend of the solitons after the collision, cf.
figure4.11) originates from finite grid sizes and thus has no physical meaning [34].
Variational calculations show an oscillation of the kinetic energy for large time scales, which corresponds to the excitation of the solitons.
The amount of transferred kinetic energy is lower than in the collision with vanishing impact factor, leading to condensates with smaller extension at t = 0.049 than their corresponding condensates in the simulation presented above.
Regardless of a finite impact factor, both simulations for φ = π/2 (figures 4.5b and 4.11b) show an asymmetric behavior in the absorption images of the left and
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∆σ
t
∆x 3 GWP
∆z 3 GWP
∆x 1 GWP
∆z 1 GWP
Figure 4.13.: Variance ∆σ of a single soliton (right soliton in the absorption images in the upper panel; note that this is the left soliton in the starting configuration) as a function of time. The bold lines show the variance of the soliton represented by 3 GWPs, the normal lines show the variance of the dominant GWPg0 after the collision.
right outgoing solitary waves. But for finite angular momentum one may actually speak of a merged condensate at t= 0.031.
In figure4.13the variance ∆σ =hσ2i−hσi2 withσ =x, zis plotted as a function of time. The variance has been calculated for the three GWPs representing the solitons on the left-hand side in the starting configuration and for the GWP which has the largest amplitude after the collision process. This dominant GWPg0shows an oscillatory behavior while the other GWPs with much smaller amplitudes can be interpreted as particles leaving the soliton. This effect can hardly be seen in the absorption images in the upper panel of figure 4.13. However, the absorption images prove the existence of a soliton even on this large time scale. However, we expect this to be difficult to observe in an actual experiment due to the very large time scale.