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

5.4 Effects of harmonic trapping 55
Figure 5.6: Density distribution after a time of free flight in the experiment [70]: Separation
of momentum components p =0(central peak) and p ± 2qB (lateral peaks). Figure taken
from [70].
5.4 Effects of harmonic trapping
The results obtained in the previous section can be used to describe the groundstate of a
condensate in the combined potential of optical lattice and harmonic trap
V = sER sin 2
πz
+
d
m
ω
2
2 zz 2 + ω 2 ⊥r 2
⊥ , (5.17)
where we have assumed radial symmetry of the harmonic trap ω⊥ = ωx = ωy in order to
simplify notation. The generalization of the results to anisotropic traps is immediate. For
convenience, we will denote the lattice site at the trap center by l = 0. The combined
potential (5.17) is depicted in Fig.5.7.
In the presence of the potential (5.17), the GPE for the groundstate takes the form
− ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+
d
m
ω
2
2 zz 2 + ω 2 ⊥r 2
⊥ + g |Ψ(r⊥,z)| 2
Ψ(r⊥,z)=µΨ(r⊥,z) .
(5.18)
The groundstate density profile
n(r⊥,z)=|Ψ(r⊥,z)| 2
(5.19)
varies rapidly on the length-scale d in the z-direction as discussed in the previous section. Yet,
due to the harmonic trap an additional length scale can come into play: If the axial size of the
condensate Z is much larger than the lattice period d, then the density profile (5.19) varies
on the scales d and Z. Provided the condensate is well described by the TF-approximation in
the absence of the lattice, then we have
d