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

5.4 Effects of harmonic trapping 59
that the density at the center of a TF-condensate without lattice is given by µ/g, where the
chemical potential is given by Eq.(5.29) with ˜g/g =1and µgn=0 =0. Hence, at the center
of the trap the ratio between the average density with lattice (5.24) and the density without
lattice is given by
nl=0(r⊥ =0;s)
n(r⊥ =0,z =0;s =0) =
−3/5 ˜g
. (5.31)
g
In Fig.5.9, we display this ratio as a function of lattice depth. Again, the effect is not very
dramatic, since the dependence of ˜g/g on s is weak. The estimate ˜g/g ∼ s1/4 valid in a deep
lattice (see chapter 6.2 below) implies that the average density at the trap center drops like
∼ s−3/20 .
The decrease of the average density at the trap center (5.31) is to be contrasted with an
increase of the non-averaged density (5.19) at r⊥ =0,z =0(peak density). This increase
arises due to the modulation of the density on the scale d which overcompensates the drop of
the average density (5.31). This effect will be evaluated quantitatively in chapter 6.2. It turns
out that the peak density grows like (˜g/g) 2/5 corresponding to an increase ∼ s1/10 .
n0(r⊥ =0;s)/n0(r⊥ =0;s=0)
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0 5 10 15 20 25 30
Figure 5.9: The average density at the central site l =0divided by its value at s =0as a
function of lattice depth (see Eq.(5.31)).
It is interesting to note that the integration (5.23) over the radial profile is crucial to obtain
the correct distribution (5.25). In a 1D system with the linear equation of state µopt =
µgn=0 +˜g(s)n, one would instead obtain the expression Nl = N0 1 − l2 /l2 m . This difference
s