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

170 Condensate fraction
we obtain
∆Ntot
Ntot
Because smin =2/Nw ≪ 1, weexpand
∆Ntot
Ntot
= 1
1
ds
N smin
1 1
√
4
πs δκ sin(
= 1 1 2
N 4 π √ δκ arctanh
cos
= 1 1
N 4
= 1
N
and inserting smin =2/Nw we finally obtain
∆Ntot
Ntot
2 )
πsmin
2
. (12.69)
2
π √ δκ arctanh
1 − 1
8 π2s 2
min
1
2π √ δκ ln
4
. (12.70)
πsmin
= 1
N
1
2π √ δκ ln
2Nw
. (12.71)
π
Using the relation between δ and the effective mass (6.41) and the relation between sound
velocity, compressibility and effective mass (7.48) this result can also be written in the form
with
∆Ntot
Ntot
= ν ln
2Nw
, (12.72)
π
ν = m∗cd . (12.73)
2π¯hN
Hence, the quantum depletion can be made larger by increasing the lattice depth s (thereby
increasing m ∗ c), decreasing the number of particles per well N, or by increasing the number of
wells Nw. Yet, one can easily check that, unless N is of the order of unity or m ∗ is extremely
large, the value of ν always remains very small. To illustrate this point we plot in Fig.12.2 the
quantity Nν as a function of the lattice depth s as obtained for gn =0.5ER, showing that
for a small number of particles per site the depletion (12.72) is large.
Note that in a deep lattice, we have c = ˜gn/m ∗ and hence the depletion scales like ã 1/2 .
This should be compared with the ã and ã 3/2 dependence in the coherent array of 2D discs
(12.62) and in the shallow lattice (12.49) respectively.
Result (12.72,12.73) has a form analogous to the quantum depletion (12.43) of 1D uniform
gas. Yet, recall that strictly speaking (12.73) was obtained in the tight binding regime and
assuming that δ/κ −1 ≪ 1. An interesting difference is given by the fact that the healing
length ξ entering as relevant length scale in the argument of the logarithmic function in the
case of the uniform system is replaced by the interwell spacing d in the case of a deep lattice.