Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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