Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

ω
3
ωB
5
4
2
1
− cos(ωt) sin(ωt) cos(ωt) − sin(ωt)
6.6. Dynamical instabilities
− cos(ωt)
Figure 6.12.: Stability map showing the positions of the stable cases in the ω-δ-plane,
according to (6.16). The size of the ellipses represents the robustness of the periodic
cases against detuning in δ, and ω, respectively. At a given detuning of 10 −4 the lifetime
of the Bloch excitation is determined by mink 1/λk, which is mapped to the radii. The
largest radii correspond to a lifetime of 5TB or more, the smallest to 1TB or less. On
the ω-axis, all rational numbers ν1/ν2 with ν2 < 12 have been taken into account.
Parameters: F = 0.2, gn0 = 1.
smaller than those of the rigid soliton (g0 = gr = −0.06). The Lyapunov exponents
provide a rather faithful portrait of the k-space evolution obtained
by the numerics, plotted in the lower panels of figure 6.11(a) stroboscopically,
i.e., at integer multiples of TB. Notably, excitations grow exclusively
in the intervals with the largest Lyapunov exponents. The predicted growth
rate of the most unstable mode (indicated by the vertical line) agrees very
well with the numerical data (inset in the upper panels of figure 6.11(a)).
The growth of the Fourier components is directly reflected in the k-space
broadening of the wave packet (figure 6.11(b)). The k-space broadening is an
experimentally accessible quantity that signals the decay of the wave packet
and the destruction of Bloch oscillations in real space. The rigid soliton
shows greater resilience than the strongly antibreathing wave packet, but
the breathing wave packet survives even longer.
Robustness map
Similarly to the above study, we also investigate the robustness of other
stable modulations g(t) from (6.16), namely monochromatic modulations
g(t) = g cos(ωt + δ). In the ω-δ plane, there are only stable points, rather
than regions of stability. How sensitive is the Bloch oscillation to slight
experimental imperfections in frequency and phase?
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