Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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4. Disorder—Results and Limiting Cases<br />
MN<br />
0<br />
0.1<br />
10 20<br />
kσ<br />
30 40 50<br />
0.08<br />
0.06<br />
0.04<br />
0.02<br />
0<br />
�0.02<br />
1 2 3 4 5<br />
σ/ξ<br />
Figure 4.12.: Relative disorder correction MN = (ɛk − ɛk)µ/V 2<br />
0 <strong>of</strong> the dispersion relation<br />
at kξ = 10, V0 = 0.03µ in one dimension. Solid line: Full correction (3.60) for finite<br />
kξ = 10. Dashed line: Limit (4.40) M (0)<br />
N (σ/ξ) for kξ → ∞. Dots: Numerical integration<br />
in a system <strong>of</strong> size L = 200σ, averaged over 50 realizations <strong>of</strong> disorder [Each realization<br />
<strong>of</strong> disorder was normalized to mean value zero and rms value |V0|]. The red circle marks<br />
kσ = 0.5, cf. figure 4.14<br />
(4.38) is positive although the chemical potential has been lowered in order<br />
to keep the particle number constant. At fixed µ, the correction would be<br />
double, Λµ = 2ΛN.<br />
For the 1D speckle potential, the integral (4.38) can be solved analytically<br />
M (0)<br />
N<br />
= z<br />
2<br />
�<br />
arctan<br />
� 1<br />
z<br />
�<br />
− z log<br />
�<br />
1 + 1<br />
z2 ��<br />
, z = σ<br />
√ . (4.40)<br />
2ξ<br />
In figure 4.12, this limit is compared to the full correction at a finite value<br />
kξ = 10.<br />
Numerical test: <strong>Bogoliubov</strong> quasiparticles at kξ = 10. Analogously to<br />
the procedure in section 4.3, we determine numerically the correction to the<br />
dispersion relation in the particle regime kξ = 10, as function <strong>of</strong> the correlation<br />
length σ. In figure 4.12, the numerical results are shown together with<br />
the full Born prediction (3.60) and the limit (4.40), showing good agreement.<br />
In contrast to the low-energy excitations, where the 1D correction<br />
[(4.15) and (4.30)] is always negative, it is here found to be positive. Only<br />
for σ/ξ � kξ, which is outside <strong>of</strong> the plot range, negative values are found.<br />
The correction ΛN = MNµ/ɛk is much weaker than in the hydrodynamic<br />
104