Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
independently of the particular initial state. The internal dynamics of
these solutions has not been characterized yet.
• For the largest part of the parameter space [g(t) not fulfilling (6.16),
for example g(t) ∝ sin(F t), g(t) = g0], no periodicity is predicted.
The wave packet is expected to decay, but we do not know how and
how fast.
Let us start the investigations of the open question by directly integrating
the discrete Gross-Pitaevskii equation (6.5), using the forth-order Runge-
Kutta method [104]. As initial state, we choose the Gaussian wave packet,
the ground state of the harmonic trap. For experimental applications, this
appears more generic than the specific soliton shape. We then compute the
time evolution with different functions g(t). The resulting real-space plots
are shown in figure 6.6. The linear Bloch oscillation shown in figure 6.6(a)
is indeed long living. At constant interaction g(t) = g0 (figure 6.6(b)) the
wave packet decays.
Whenever the time-reversal condition (6.16) is fulfilled, the wave packet
is long-living, (c) and (d). When equation (6.16) is not fulfilled [figure 6.6
(e) and (f)], we recognize two decay mechanisms:
• a smooth contraction (or spreading) of the wave packet, figure 6.6(e)
• the growth of perturbations on a short scale, figure 6.6(e) and figure
6.6(f)
We investigate these decay mechanisms and the internal dynamics in more
detail by means of a collective-coordinates ansatz and a linear stability analysis
in the following sections.
6.5. Collective coordinates
We employ a collective-coordinates ansatz [139] in order to describe the
most important degrees of freedom of a Bloch oscillating wave packet. In
this approach, we assume that the wave packet is essentially conserved and
describe the dynamics of its center of mass (first moment) and its width
(second moment).
Similarly to the smooth-envelope ansatz (6.7), the discrete wave function
is cast into the form
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Ψn(t) = e ip(t)n A(n − x(t), w(t), b(t))e iφ(t) , (6.17)