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

60 Groundstate of a BEC in an optical lattice
is related to the fact that in a 1D system one has µ ∝ Nl. In contrast, for a 3D system with
radial trapping one obtains µ ∝ √ Nl from Eq.(5.23).
As mentioned above, the average density profile nl(r⊥) varies slowly as a function of the
site index l. We obtain a smooth macroscopic density profile nM(r⊥,z) by replacing the
discrete index l by the continous variable z = ld This is convenient in devising a hydrodynamic
formalism (see section 9): The particular profile (5.24) becomes
nM(r⊥,z)= 1
µ − µgn=0 −
˜g
m
2 ω2 zz 2 − m
2 ω2 ⊥r 2
⊥ , (5.32)
with µ given by Eq.(5.29). The extension of the condensate along z is given by
2(µ − µgn=0)
Z =
mω2 . (5.33)
z
Obviously, the profile (5.32) mimics a TF-profile of a condensate without lattice. This confirms
the statement made in section 5.2 that if µopt =˜gn + µgn=0 the system can be described as
if there was no lattice as long as g is replaced by ˜g.
Before concluding this section, we would like to emphasize that it is important to do the
average (5.21) before applying the LDA. The strong modulation of the wavefunction generated
by the lattice contributes a large kinetic energy which would be, mistakenly, discarded within
a LDA. The situation is different in a system made up of only very few wells (produced for
example by raising a few barriers in a TF-condensate). In this case, the condensate in each
well might be well described by the TF-approximation (see for example [57]). In the LDA
proposed in this section, the large kinetic energy caused by the lattice, is contained in the
chemical potential µopt(nl(r⊥)).
The effects described in this section, in particular the dependence of µ (5.29) and R0 (5.28)
on s, have been investigated by [71]. In this experiment, the chemical potential and the radius
of the groundstate in the combined potential of harmonic trap and one-dimensional optical
lattice is determined from the measurement of the radial size of the cloud after a time of
free flight. The results for the chemical potential are depicted in Fig.5.10 together with the
theoretical prediction (5.29).
The generalization of the results presented in this section to two-dimensional cubic lattices is
straightforward. For a two-dimensional lattice in the x, y-directions, the smoothed macroscopic
density profile is given by
nM(r⊥,z)= 1
µ − µgn=0 −
˜g
m
2 ω2 zz 2 − m
2 ω2 ⊥r 2
⊥ , (5.34)
with the effective coupling constant
and
˜g = gd 2
d/2
|ϕgn=0(x, y)|
−d/2
4 dxdy , (5.35)
µ = ¯h¯ω
2
a
15Ntot
aho
2/5 ˜g
+ µgn=0 , (5.36)
g