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

5.4 Effects of harmonic trapping 57
Within this generalized LDA, the chemical potential at site l is given by
µl = µopt(nl(r⊥)) + m
2 (ω2 zl 2 d 2 + ω 2 ⊥r 2 ⊥) , (5.22)
where µopt(nl(r⊥)) is the chemical potential calculated at the average density nl(r⊥) in the
presence of the optical lattice potential only. The effect of axial trapping is accounted for
by the term mω2 zl2d2 /2 since the harmonic potential varies slowly on the scale d. Eq.(5.22)
fixes the radial density profile nl(r⊥) at the l-th site once the value of µl or, equivalently, the
number of atoms at well l
Rl
Nl =2πd r⊥dr⊥nl(r⊥) , (5.23)
0
is known. In Eq.(5.23), Rl is the radial size of the condensate at the l-th site, which is fixed
by the value of r⊥ where the density nl(r⊥) vanishes.
When equilibrium is established across the whole sample we have µl = µ for all l. Making
use of this fact and employing that
l Nl = Ntot, we can find the dependence of µ on the
total number of particles Ntot. This procedure also yields the single well occupation numbers
Nl and the number of sites occupied in the groundstate.
In the simple case in which the chemical potential without harmonic trap exhibits the linear
dependence on density µopt = µgn=0 +˜g(s)n (see Eq.(5.13)), one obtains for the radial density
profile
nl(r⊥) = 1
µ − µgn=0 −
˜g
m
2 ω2 zl 2 d 2 − m
2 ω2 ⊥r 2
⊥ . (5.24)
The well occupation numbers and transverse radii are given by
1 − l2
2 , (5.25)
where
Nl = N0
l2 m
Rl = R0 1 − l2
l2 1/2 m
lm =
2(µ − µgn=0)
mω 2 zd 2
, (5.26)
is the outer most occupied site and fixes the total number 2lm +1of occupied sites, and
R0 =
2(µ − µgn=0)
mω 2 ⊥
(5.27)
(5.28)
is the radial size at the trap center. To obtain an explicit expression for the chemical potential
µ and the occupation N0 of the central site, we apply the continuum approximation
l →
1/d dz to the normalization condition
l Nl = Ntot. This yields
µ = ¯h¯ω
2
a
15Ntot
aho
2/5 ˜g
+ µgn=0
g
(5.29)