Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

8.2 Static structure factor and sum rules 111
Static structure factor of the uniform system
In the uniform system, the sum (8.7) is exhausted by a single mode with the energy ¯hωuni(p)
(8.10). In this case the static structure factor obeys the Feynman relation
Suni(p) =
p2 2m , (8.22)
¯hωuni(p)
which can be derived using the f-sum rule (8.19) (see [1] chapter 7.6). For p → 0 the static
structure factor (8.22) behaves like
Suni(p) → |p|
2mcuni
, (8.23)
while the compressibility sum-rule (8.20) becomes
S(p, ω)
dω
¯hω =
p→0
Ntot
2mc2 ,
uni
(8.24)
where cuni = gn/m is the sound velocity of the uniform system. The suppression of Suni(p)
at small momenta is a direct consequence of phononic correlations between particles. For
large momenta, instead, the static structure factor (8.22) approaches unity (see dotted lines
in Figs.8.7 and 8.8).
Static structure factor of the system in a lattice
The structure factor (8.18) is obtained by summing up the excitation strengths Zj(p) (8.8) of
all bands
S(p) = 1
Zj(p) . (8.25)
Ntot
Results are presented in Figs.8.7 and 8.8 for gn =0, 0.02ER, 0.5ER at lattice depth s =5and
s =10respectively. For weak interactions (dashed lines) the static structure factor exhibits
characteristic oscillations, reflecting the contribution Z1(p) from the first band. This effect is
less pronounced for larger values of gn (solid lines) due to the suppression of Z1(p). In both
cases one observes a big difference with respect to the behaviour of S(p) in the uniform gas
(8.22) (dotted lines) and with respect to the non-interacting system (dash-dotted lines).
j