Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1/F<br />
η<br />
0<br />
−1/F<br />
0<br />
a b<br />
m−1 η<br />
gr<br />
gb<br />
c d<br />
π 2π<br />
−p = F t<br />
6.4. Numerical examples<br />
0<br />
k − p<br />
Figure 6.5.: Left panel: Time evolution scheme <strong>of</strong> stable Bloch oscillations. The inverse<br />
mass m −1 , the interaction parameter <strong>of</strong> the rigid soliton (6.11) as well as that <strong>of</strong> a<br />
breathing soliton gb(t) = g0 cos(F t) are shown together with the bounded time η =<br />
sin(F t)/F as function <strong>of</strong> time or momentum −p = F t. Right panel: The k-space<br />
density [obtained by numerical integration <strong>of</strong> Eq. (6.5)] <strong>of</strong> a breathing wave packet is a<br />
function <strong>of</strong> η and thus strictly periodic in t: the points in time a and b as well as c and<br />
d show the same distribution, respectively.<br />
Again, the time reversal argument applies. Equation (6.15) is solved as<br />
function <strong>of</strong> the bounded time η ′ , which implies that the wave packet is<br />
conserved under modulations given as<br />
g(t) = sin(τ)P (cos(τ)) = �<br />
gn sin(nτ) = �<br />
n<br />
n<br />
�<br />
n � π<br />
gn sin F t − (2j + 1)�� .<br />
ν2 2<br />
(6.16)<br />
With a different but equivalent derivation, we have published this result in<br />
[129].<br />
6.4. Numerical examples<br />
So far, we have made the following predictions on the basis <strong>of</strong> the continuous<br />
nonlinear Schrödinger equation (6.8) with time-dependent mass and<br />
interaction:<br />
• The rigid soliton: With an initial state <strong>of</strong> soliton shape (6.10) and<br />
an interaction parameter modulated as g(t) = −|gr| cos(F t), the wave<br />
packet should completely conserve its shape.<br />
• Modulations g(t) that fulfill the simple time-reversal condition (6.12)<br />
or the generalized one (6.16) are predicted to lead to periodic solutions,<br />
123