Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.2. Interacting BEC and Gross-Pitaevskii mean-field
Box 2.1: Feynman diagrams of the condensate function (2.21)
The constituents of the diagrams are
• particles from the k = 0 mode |
• response function S(k) = from (2.22)
• potential scattering Vq =
• particle-particle scattering g =
Drawing Feynman diagrams. Starting from Φ (0) = | , diagrams
of order n are constructed from diagrams of order n ′ < n by
• attaching a potential scattering, e.g.
| · · = |
• by combination of several diagrams, e.g.
| · | · | · · = 3 |
The combinatorial factor three comes from permutations.
The first diagrams read
Φ = |
����
Φ (0)
+ |
� �� �
Φ (1)
+ |
�
+ 3 |
��
Φ
| �
(2)
|
|
|
+ . . .
Computing the diagrams. Each potential contributes to the momentum.
At the vertices, the momentum is conserved, so the outgoing
momentum (open end) is the sum of all momentum transfers by the
external potentials.
Φ (2b)
q
= |
q ′
|
q − q ′
q − q ′
q ′ | q
= 1
L d
2
�
Vq ′S(q′ ) Vq−q ′S(|q − q′ |) g S(q)
Finally, all free momenta are summed over.
The diagrams presented in this box are equivalent to the real-space
diagrams in [98], when taken in the case of a real ground-state wave
function.
q ′
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