Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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2. The <strong>Inhomogeneous</strong> <strong>Bogoliubov</strong> Hamiltonian<br />
at the external potential and (ii) scattering processes between atoms. The<br />
above expansion can be nicely written in terms <strong>of</strong> Feynman diagrams, see<br />
box 2.1.<br />
The first-order correction reads<br />
Φ (1)<br />
k<br />
−Vk<br />
=<br />
2µ + ɛ0 Φ0 =: S(k)VkΦ0 , (2.22)<br />
k<br />
where the linear response function S(k) contains the kinetic energy <strong>of</strong> a<br />
free particle ɛ 0 k = �2 k 2 /(2m) and the chemical potential µ. Atoms from<br />
the originally homogeneous condensate are scattered once by the external<br />
potential (first-order diagram in box 2.1). The condensate wave function<br />
shows an imprint <strong>of</strong> the external potential, similar to the one described by<br />
the Thomas-Fermi formula (2.20), but variations on length scales shorter<br />
than the healing length ξ are suppressed. This smoothing is due to the cost<br />
that was neglected in the Thomas-Fermi formula.<br />
<strong>of</strong> the kinetic energy ɛ0 k<br />
In figure 2.2, the density depression <strong>of</strong> a rather narrow impurity potential<br />
is shown. The density dip <strong>of</strong> the first-order result (2.22) is broader and<br />
shallower than that predicted by the Thomas-Fermi formula. The realspace<br />
representation is given as the convolution <strong>of</strong> the bare potential with<br />
a smoothing kernel given by the d-dimensional inverse Fourier transform <strong>of</strong><br />
[2µ + ɛ0 k ]−1 .<br />
The second order contains double scattering processes and reads<br />
� �<br />
�<br />
Φ (1)<br />
q . (2.23)<br />
Φ (2)<br />
k<br />
1<br />
= S(k)<br />
L d<br />
2<br />
q<br />
Vk−q + 3gΦ0Φ (1)<br />
k−q<br />
The processes contained in this formula are double scattering at the external<br />
potential and interaction <strong>of</strong> two single-scattered particles, see box 2.1. In<br />
figure 2.2 it is demonstrated, that the second order leads to a very satisfying<br />
agreement with the exact solution <strong>of</strong> the Gross-Pitaevskii equation, even for<br />
a rather strong potential.<br />
Rephrasing in terms <strong>of</strong> density and smoothed potential<br />
Motivated by the Thomas-Fermi formula (2.20), one can cast the imprint <strong>of</strong><br />
the potential into a smoothed potential ˜ V (r), which is expanded in orders<br />
<strong>of</strong> the small parameter<br />
28<br />
n0(r) = 1<br />
g<br />
�<br />
µ − ˜ V (1) (r) − ˜ V (2) �<br />
(r) + . . .<br />
. (2.24)