Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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x<br />
xB<br />
1<br />
0.5<br />
0<br />
�0.5<br />
�1<br />
5 10 15 20<br />
t/TB<br />
25 30 35 40<br />
6.5. Collective coordinates<br />
Figure 6.9: The centroid motion <strong>of</strong> a<br />
Gaussian wave packet under the<br />
interaction parameter g(t) = g0 +<br />
gc cos(F t) shows a drift. The red<br />
dots are numerical results <strong>of</strong> (6.5),<br />
the blue line shows the collectivecoordinates<br />
prediction (6.20), and<br />
the green lines show the estimate<br />
for the envelope (6.23). Parameters:<br />
F = 0.2, g0 = 0.1, gc = 5,<br />
σ0 = 10.<br />
(∆k) 2 = K/w + 4b 2 w. Furthermore, if the product sin(p)(∆k) 2 has a nonzero<br />
time average, a drift <strong>of</strong> the centroid can occur.<br />
The perturbation g(t) enters the k-space width via b 2 . We make a separation<br />
<strong>of</strong> time scales and write ∆b 2 = B0(t)+Bc(t) cos(F t)+Bs(t) sin(F t)+. . . .<br />
In this ansatz, the Bx(t) vary on a much longer time scale than the Bloch<br />
period and their time derivatives are neglected. For equation (6.20b), only<br />
B0(t) = I2 g 2 0<br />
4w 3 0<br />
t 2 and Bs = I2 g0gs<br />
2F w 3 t are relevant and lead to<br />
x(t) ≈ 2<br />
F<br />
cos(F t)<br />
�<br />
1 − K + I2g 2 0t2 �<br />
/w0<br />
2w0<br />
+ 1<br />
2<br />
I2g0gc F w2 t<br />
0<br />
2<br />
(6.23)<br />
As expected, the <strong>of</strong>fset causes damping. More remarkably, the modulated<br />
interaction induces a drift <strong>of</strong> the oscillating wave packet. In figure 6.9, the<br />
estimate (6.23) is plotted together with the full solution <strong>of</strong> the collectivecoordinates<br />
equations <strong>of</strong> motion (6.20) and with the numerical results obtained<br />
from the integration <strong>of</strong> the tight-binding equation <strong>of</strong> motion (6.5).<br />
As expected, the agreement is initially very good, but deviations occur after<br />
about thirty Bloch periods, when the wave packet starts to decay and the<br />
collective-coordinates description breaks down.<br />
Conclusion<br />
The collective coordinates are useful for the description <strong>of</strong> the stable cases,<br />
where the wave packet is preserved, and for the description <strong>of</strong> the wavepacket<br />
decay in the very beginning. We have learned that the sine modulation<br />
directly contracts or broadens the wave packet and that a cosine<br />
modulation together with an <strong>of</strong>fset makes the wave packet drift. Later, the<br />
wave packet loses its smooth shape, and the collective-coordinates description<br />
is bound to fail. In particular, collective coordinates cannot grasp the<br />
short-scale perturbations that occur in figure 6.6(e) and figure 6.6(f).<br />
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