Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
We investigate the robustness of the stable points by varying both δ and
ω slightly (by 10 −4 ) and determining the maximum Lyapunov exponent
λ = maxk λk from a numerical integration of (6.25). The inverse Lyapunov
exponent is a measure for the robustness and is mapped to the radius in
the graphical representation in the parameter space figure 6.12. In the
dissymmetry of the points + cos(F t) and − cos(F t), we recover the result
from above. The stable points are arranged on a regular pattern, with the
most stable points arranged on lines. With increasing denominator ν2 the
robustness drops very rapidly.
6.7. Conclusions (Part II)
We have treated the problem of Bloch oscillations with a time-dependent
interaction in the framework of the one-dimensional tight-binding model,
i.e. for a deep lattice potential with a strong transverse confinement. For
smooth wave packets, the bounded-time argument allows identifying a class
of interactions g(t) that lead to periodic dynamics.
Beyond the mere existence of these periodic solutions, we have set up
two complementary methods for the quantitative description of stability
and decay of the Bloch oscillating wave packet. The collective-coordinates
approach is valid as long as the wave packet is essentially conserved. This
approach is capable of describing on the one hand the centroid and the
breathing dynamics in the periodic cases, and on the other hand the beginning
of the decay in the unstable cases, for example at constant interaction.
The other approach, the linear stability analysis of the infinite wave
packet, is suitable for the quantitative description of the decay of wide wave
packets. More precisely, the relevant excitations have to be well decoupled
in k-space from the original wave packet around k = 0. Perturbations like
the off-phase perturbation in subsection 6.6.4 are well described. Other perturbations,
like a constant interaction, are missed, because the width of the
wave packet is not properly included in the ansatz of an infinitely wide wave
packet. Together, the two approaches provide a rather complete picture of
the wave-packet dynamics.
The most remarkable physical results of this part are firstly the existence
of long-living Bloch oscillations despite non-zero interaction if the interaction
g(t) respects the time-reversal symmetry (6.16). Secondly, a modulation
of the interaction that enhances the breathing of the wave packet can make
the Bloch oscillation more robust with respect to perturbations.
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