Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

1
0.01
0.0001
1e-06
1e-08
1e-10
k = 0
k = 0.15
k = 0.21
k = 0.44
k = 0.61
125 150 175 200 225
t/TB
6.6. Dynamical instabilities
Figure 6.10.: Growth of the most unstable mode. Momentum density |Ψk| 2 for selected
k-modes in the unstable case g(t) = g1 sin(F t). The original wave function is centered
around k = 0. The solid line marks the growth rate λmax of the most unstable mode
as predicted by equation (6.32). The growth of this mode precedes the damping of the
centroid motion that sets in at t ≈ 200TB. Numerical parameters: σ0 = 100, g1n0 = 0.01,
F = 0.2.
6.6.4. Robustness with respect to small perturbations
An important application for the linear stability analysis is the question
“How sensitive are the long-living Bloch oscillations of (6.16) to small perturbations?”
Robustness of rigid and breathing wave packets
As shown in figure 6.6(c), the breathing wave packet with g(t) = g0 cos(F t)
is long living, in accordance with the time-reversal condition (6.12). But its
interaction parameter g(t) always has the same sign as the mass m, thus,
the stability criterion for solitons (6.11) is never fulfilled. The wave packet
would decay if one stopped the Bloch oscillation by switching off the force
F . It survives only because of the time-reversal argument of section 6.3.
How robust with respect to external perturbations can such a wave packet
be? Should not a soliton that is stable at all times be more robust? Even
cold-atom experiments suffer from slight imperfections, such as residual
uncertainties in the magnetic field controlling the interaction term g(t).
For instance, in the Innsbruck experiment [40], the magnetic field is controlled
up to 1 mG. The slope of 61a0/G at the zero of the Feshbach resonance
turns this into an uncertainty ∆a = 0.06a0 in the scattering length,
a0 ≈ 5.3 10 −11 m being the Bohr radius. This is converted to the uncertainty
of the dimensionless tight-binding interaction ∆g ≈ 0.4. Note that
this uncertainty ∆g is larger than the interaction gr = −4/ξ0 needed to
create a rigid soliton of only moderate width ξ0 � 10. From this point of
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