Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Fixed particle number<br />
3.4. Deriving physical quantities from the self-energy<br />
Let us have a look at the density pr<strong>of</strong>ile (2.24) <strong>of</strong> the condensate in presence<br />
<strong>of</strong> a disorder potential V (r). In the disorder average the first-order<br />
contribution (2.25) vanishes because V = 0. In contrast, the second-<br />
order contribution (2.25) survives the disorder average. With VqVk−q =<br />
(2π) d δ(k)V 2<br />
0 σ d Cd(qσ) (3.11) and Cd(qσ) ≥ 0 (3.12), the shift <strong>of</strong> the mean<br />
density is found to be positive:<br />
n(r) − n∞<br />
n∞<br />
= − ˜ V (2) (r)<br />
µ<br />
= σdV 2<br />
0<br />
µ 2<br />
� d d q<br />
(2π) dCd(qσ)<br />
q2ξ2 (2 + q2ξ2 > 0 (3.56)<br />
) 2<br />
At constant chemical potential, the disorder attracts more particles from the<br />
grand canonical reservoir into the system. The integral (3.56) is significant<br />
for σ ≈ ξ, because then, the fraction q 2 ξ 2 /(2 + q 2 ξ 2 ) 2 and the disorder<br />
correlation function overlap efficiently.<br />
The counter-intuitive increase <strong>of</strong> the particle density can be understood<br />
more directly [14] from the spatial average <strong>of</strong> the stationary Gross-<br />
Pitaevskii equation (2.19). When the potential is switched on, the right<br />
hand side <strong>of</strong> equation (2.19) does not change, because V (r) has zero<br />
mean. By integration by parts one sees that the kinetic energy term<br />
− �2<br />
� d d r<br />
2m Vol (∇�n0(r)) 2 /n0(r) has a negative average, which has to be com-<br />
pensated by a positive correction <strong>of</strong> the averaged interaction term, i.e. an<br />
increase <strong>of</strong> the average density.<br />
For an experiment it appears very unnatural that the particle number<br />
should change while ramping up the disorder. Once the condensate is created,<br />
the atom number and the average condensate density should stay constant<br />
when the disorder potential is ramped up. For the rest <strong>of</strong> this work,<br />
we switch to the canonical ensemble with fixed atom number. The increase<br />
<strong>of</strong> the average density due to the disorder is compensated by an appropriate<br />
shift <strong>of</strong> the chemical potential ∆µ. Consider the disorder average <strong>of</strong> (2.24)<br />
with ˜ V (1) = 0 and n∞ = µ/g replaced by (µ + ∆µ)/g<br />
�<br />
µ + ∆µ<br />
1 −<br />
g<br />
˜ V (2) �<br />
(r) !<br />
=<br />
µ<br />
µ<br />
. (3.57)<br />
g<br />
Consequently, the chemical potential is shifted by<br />
∆µ<br />
µ = ˜ V (2) (r)<br />
µ<br />
(3.56) σ<br />
= − dV 2<br />
0<br />
µ 2<br />
� d d q<br />
(2π) dC(qσ)<br />
q2ξ2 (2 + q2ξ2 < 0, (3.58)<br />
) 2<br />
up to higher orders in V 2<br />
0 /µ 2 . The correction to ReΣ is <strong>of</strong> higher order,<br />
but the shift ∆µ corrects also the clean dispersion relation ɛk by ∆µ ∂ɛk<br />
∂µ .<br />
77