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

6.1 Bloch states and Bloch bands 73
depth s reflects the slow-down of the particles by the tunneling through the potential barriers.
Qualitatively, this effect is unaltered in the presence of interactions: The effective mass still
features an exponential increase, yet this increase is weaker than for a single particle. At a given
potential depth, the effective mass is lowered by increasing gn/ER due to the screening effect
of interactions. This is illustrated in Fig.6.9 where we depict the ratio between condensate and
single particle effective mass at different densities. At gn =0.5ER, the condensate effective
mass is about 30% smaller than the one of a single particle at s =15. It is interesting to
note that the ratio between m ∗ (gn) and the single particle effective mass plotted in Fig. 6.9
saturates when s is tuned to large values. This property of the effective mass will become clear
in the next section once we relate m ∗ with the tunneling parameter δ.
m ∗ /m
150
125
100
75
50
25
0
0 5 10 15 20 25 30
Figure 6.8: Effective mass (6.19) as a function of lattice depth s for gn =0(solid line),
gn =0.1ER (dashed line) and gn =0.5ER (dash-dotted line) ( a) s ≤ 30, b)s ≤ 10).
Wannier functions
In analogy to the case of a single particle (see section 4.1), we can introduce the Wannier
functions
fj,l(x) = 1
e −ikld ϕjk(x) , (6.25)
Nw
k
where ϕjk is a Bloch function solution of the stationary GPE (5.4). The inverse relation reads
ϕjk(z) =
fj,l(z)e ikld . (6.26)
l
s