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

56 Groundstate of a BEC in an optical lattice
V/ER
6
5
4
3
2
1
0
−25 −20 −15 −10 −5 0 5 10 15 20 25
Figure 5.7: The z-dependence of the combined potential of optical lattice and harmonic trap
(5.17) with a lattice depth of s =5and ¯hωz =0.02ER.
z/d
and the variation on the scale Z is slow. Situation (5.20) is typical of current experiments. It
implies that many sites of the lattice are occupied and that the site occupation numbers vary
slowly as a function of the site index. For example, in the experiment [70] atoms are loaded
into ∼ 200 sites.
Local Density Approximation
Given (5.20), one can generalize the local density approximation (LDA) to describe harmonically
trapped condensates in a lattice. This procedure avoids the solution of the full problem
(5.18).
Let us consider the average density at site l
nl(r⊥) = 1
ld+d/2
n(r⊥,z) dz , (5.21)
d ld−d/2
where n(r⊥,z)=|Ψ(r⊥,z)| 2 is the density obtained by solving the GP-equation (5.18). With
l replacing the continous variable z, expression (5.21) defines an average density profile of the
condensate in the trap. It is a smooth function of r⊥ and varies slowly as a function of the
index l since many sites are occupied as a consequence of condition (5.20). Basically, the idea
is now to apply a LDA to the average profile nl(r⊥).