Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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ω<br />
3<br />
ωB<br />
5<br />
4<br />
2<br />
1<br />
− cos(ωt) sin(ωt) cos(ωt) − sin(ωt)<br />
6.6. Dynamical instabilities<br />
− cos(ωt)<br />
Figure 6.12.: Stability map showing the positions <strong>of</strong> the stable cases in the ω-δ-plane,<br />
according to (6.16). The size <strong>of</strong> the ellipses represents the robustness <strong>of</strong> the periodic<br />
cases against detuning in δ, and ω, respectively. At a given detuning <strong>of</strong> 10 −4 the lifetime<br />
<strong>of</strong> the Bloch excitation is determined by mink 1/λk, which is mapped to the radii. The<br />
largest radii correspond to a lifetime <strong>of</strong> 5TB or more, the smallest to 1TB or less. On<br />
the ω-axis, all rational numbers ν1/ν2 with ν2 < 12 have been taken into account.<br />
Parameters: F = 0.2, gn0 = 1.<br />
smaller than those <strong>of</strong> the rigid soliton (g0 = gr = −0.06). The Lyapunov exponents<br />
provide a rather faithful portrait <strong>of</strong> the k-space evolution obtained<br />
by the numerics, plotted in the lower panels <strong>of</strong> figure 6.11(a) stroboscopically,<br />
i.e., at integer multiples <strong>of</strong> TB. Notably, excitations grow exclusively<br />
in the intervals with the largest Lyapunov exponents. The predicted growth<br />
rate <strong>of</strong> the most unstable mode (indicated by the vertical line) agrees very<br />
well with the numerical data (inset in the upper panels <strong>of</strong> figure 6.11(a)).<br />
The growth <strong>of</strong> the Fourier components is directly reflected in the k-space<br />
broadening <strong>of</strong> the wave packet (figure 6.11(b)). The k-space broadening is an<br />
experimentally accessible quantity that signals the decay <strong>of</strong> the wave packet<br />
and the destruction <strong>of</strong> Bloch oscillations in real space. The rigid soliton<br />
shows greater resilience than the strongly antibreathing wave packet, but<br />
the breathing wave packet survives even longer.<br />
Robustness map<br />
Similarly to the above study, we also investigate the robustness <strong>of</strong> other<br />
stable modulations g(t) from (6.16), namely monochromatic modulations<br />
g(t) = g cos(ωt + δ). In the ω-δ plane, there are only stable points, rather<br />
than regions <strong>of</strong> stability. How sensitive is the Bloch oscillation to slight<br />
experimental imperfections in frequency and phase?<br />
135