Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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6. Bloch Oscillations and Time-Dependent Interactions<br />
Table 6.1.: Collective-coordinates parameters for Gaussian and soliton-shaped wave packets<br />
shape Ã(u) K I<br />
1 u2<br />
− −<br />
Gaussian (2π) 4 e 4<br />
soliton<br />
� π<br />
4 √ 3<br />
�<br />
cosh<br />
� π u<br />
2 √ 3<br />
�� −1<br />
1<br />
4<br />
1<br />
1<br />
4 4 √ π<br />
� �<br />
π 2 1<br />
3 4 √ �<br />
π<br />
π 3<br />
with the free variables center-<strong>of</strong>-mass position x(t) = 〈n〉 = �<br />
2<br />
n n|Ψn(t)|<br />
and variance w(t) = � (n − x(t)) 2� and their conjugate momenta p(t) and<br />
b(t), respectively. The momentum p is included in equation (6.17), according<br />
to its generating role −i∂pΨn = nΨn. Similarly, b is included in the wave<br />
function as<br />
1 −<br />
A(z, w, b) = w 4e ibz2Ã(z/ √ w) . (6.18)<br />
The even, real function Ã(u) is normalized to one with standard deviation<br />
one, � du| Ã(u)|2 = 1 = � du u2 | Ã(u)|2 .<br />
The collective-coordinates ansatz (6.17) is inserted into the Hamiltonian<br />
(6.6). Taylor-expanding the discrete gradient to second order, the effective<br />
Hamiltonian is found as<br />
�<br />
Hcc = F x − 2 cos p 1 − K + 4b2w2 �<br />
+ I<br />
2w<br />
g(t)<br />
√ , (6.19)<br />
w<br />
with the kinetic integral K = � du| Ã′ (u)| 2 �<br />
and the interaction integral I =<br />
1<br />
2 du|Ã(u)| 4 . In table 6.1, these parameters are given for a Gaussian and<br />
for a soliton-shaped wave packet.<br />
By construction, the collective coordinates obey the canonical equations<br />
<strong>of</strong> motion<br />
˙p = − ∂Hcc<br />
∂x<br />
˙x = ∂Hcc<br />
∂p<br />
˙b = − ∂Hcc<br />
∂w = K − 4w2b2 w2 ˙w = ∂Hcc<br />
∂b<br />
= −F, (6.20a)<br />
�<br />
= 2 sin p 1 − K + 4b2w2 �<br />
= sin p<br />
2w<br />
� 2 − (∆k) 2� , (6.20b)<br />
cos p + 1<br />
3<br />
I g(t)w− 2,<br />
2<br />
(6.20c)<br />
= 8wb cos p. (6.20d)<br />
Within the scope <strong>of</strong> collective coordinates, the momentum variance <strong>of</strong> the<br />
wave packet is given as (∆k) 2 = K/w +4b 2 w. It appears in the Hamiltonian<br />
126<br />
� 3<br />
2