Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

Fixed particle number
3.4. Deriving physical quantities from the self-energy
Let us have a look at the density profile (2.24) of the condensate in presence
of a disorder potential V (r). In the disorder average the first-order
contribution (2.25) vanishes because V = 0. In contrast, the second-
order contribution (2.25) survives the disorder average. With VqVk−q =
(2π) d δ(k)V 2
0 σ d Cd(qσ) (3.11) and Cd(qσ) ≥ 0 (3.12), the shift of the mean
density is found to be positive:
n(r) − n∞
n∞
= − ˜ V (2) (r)
µ
= σdV 2
0
µ 2
� d d q
(2π) dCd(qσ)
q2ξ2 (2 + q2ξ2 > 0 (3.56)
) 2
At constant chemical potential, the disorder attracts more particles from the
grand canonical reservoir into the system. The integral (3.56) is significant
for σ ≈ ξ, because then, the fraction q 2 ξ 2 /(2 + q 2 ξ 2 ) 2 and the disorder
correlation function overlap efficiently.
The counter-intuitive increase of the particle density can be understood
more directly [14] from the spatial average of the stationary Gross-
Pitaevskii equation (2.19). When the potential is switched on, the right
hand side of equation (2.19) does not change, because V (r) has zero
mean. By integration by parts one sees that the kinetic energy term
− �2
� d d r
2m Vol (∇�n0(r)) 2 /n0(r) has a negative average, which has to be com-
pensated by a positive correction of the averaged interaction term, i.e. an
increase of the average density.
For an experiment it appears very unnatural that the particle number
should change while ramping up the disorder. Once the condensate is created,
the atom number and the average condensate density should stay constant
when the disorder potential is ramped up. For the rest of this work,
we switch to the canonical ensemble with fixed atom number. The increase
of the average density due to the disorder is compensated by an appropriate
shift of the chemical potential ∆µ. Consider the disorder average of (2.24)
with ˜ V (1) = 0 and n∞ = µ/g replaced by (µ + ∆µ)/g
�
µ + ∆µ
1 −
g
˜ V (2) �
(r) !
=
µ
µ
. (3.57)
g
Consequently, the chemical potential is shifted by
∆µ
µ = ˜ V (2) (r)
µ
(3.56) σ
= − dV 2
0
µ 2
� d d q
(2π) dC(qσ)
q2ξ2 (2 + q2ξ2 < 0, (3.58)
) 2
up to higher orders in V 2
0 /µ 2 . The correction to ReΣ is of higher order,
but the shift ∆µ corrects also the clean dispersion relation ɛk by ∆µ ∂ɛk
∂µ .
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