Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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