Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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6. Bloch Oscillations and Time-Dependent Interactions<br />
Figure 6.8: The initial contraction <strong>of</strong><br />
the wave packet shown in (6.6(e))<br />
(red dots) can be described by collective<br />
coordinates (blue). The<br />
green line shows the approximation<br />
(6.22). F = 0.2, gs = 1,<br />
σ0 = 10.<br />
√ w<br />
10<br />
9<br />
8<br />
7<br />
6<br />
1 2 3 4<br />
t/TB<br />
5 6 7 8<br />
assume w = σ 2 0 as constant. Using (6.20c), we compute the change <strong>of</strong> the<br />
momentum b due to the interaction g(t)<br />
∆b(t) = b(t) − b(t)|g=0 = I<br />
2<br />
� t<br />
3<br />
w− 2<br />
0 dt<br />
0<br />
′ g(t ′ ). (6.21)<br />
Inserting this into the equation <strong>of</strong> motion (6.20d) for the width degree <strong>of</strong><br />
freedom w, we understand that the modulation with double frequency from<br />
figure 6.6(f) has no average effect on the width.<br />
Sine modulation<br />
The modulation g(t) = gs sin(F t), considered in figure 6.6(e), however, leads<br />
to an average growth<br />
w(t) ≈ w0 − 2 Igs<br />
F √ t. (6.22)<br />
w0<br />
This prediction is shown in figure 6.8, together with the full solution <strong>of</strong><br />
(6.20) and the width extracted from the integration <strong>of</strong> the tight-binding<br />
equation <strong>of</strong> motion. A sine like modulation <strong>of</strong> the interaction contracts or<br />
broadens the wave packet, before destroying it.<br />
Offset and cosine modulation<br />
Let us consider an interaction parameter with a finite time average and a<br />
cosine modulation g(t) = g0 + gc cos(F t). In the case gc = 0, this covers<br />
the case <strong>of</strong> constant interaction shown in figure 6.6(b). The modulation<br />
is compatible with the time reversal, but the <strong>of</strong>fset is expected to destroy<br />
the wave packet. We are interested in the centroid motion (6.20b). We<br />
expect a damping <strong>of</strong> the oscillation due to the momentum broadening term<br />
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