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

70 Stationary states of a BEC in an optical lattice
ε/ER
12
10
8
6
4
2
0
5 10 15 20 25 30
Figure 6.6: Gap between first and second Bloch band of the spectra (6.4) and (6.5) at ¯hk = qB
as a function of lattice depth s for gn =0(solid line) and gn =1ER (dashed and dashdotted
lines respectively). The dotted line indicates the value 2 √ sER given by the harmonic
approximation of the potential well.
where ¯hk is the quasimomentum and j the band index. The corresponding energy functional
yields the “energy per particle”
d/2
εj(k, A) =
−d/2
ϕ ∗
1
jk(z)
Differentiation with respect to A gives
1
¯h
∂εj(k, A)
∂A
2m (−i¯h∂z +¯hA) 2 + sER sin 2 (z)+ 1
2
A=0
s
2
gnd|ϕjk(z)| ϕjk(z)dz.
(6.12)
= i¯h
d/2
dz ϕjk∂zϕ
2m −d/2
∗ jk − ϕ ∗
jk∂zϕjk . (6.13)
It is important that ϕjk need not be differentiated because δε/δϕ =0. The integrand is
Ij(k)/nd where Ij(k) is the current density (6.9). Hence, we have
1
¯h
∂εj(k, A)
∂A
A=0
= Ij(k)
n
. (6.14)
However, the dependence of ε on A is completely fictional. Due to the gauge invariance of
the GP-equation, A can be excluded from the equation by the substitution
φjk = e iAz ϕjk(z,A) =e i(A+k)z ˜ϕjk(z,A) . (6.15)
The function φjk satisfies the usual GP-equation. This means that the energy Bloch bands
for the modified GP-equation (6.10) are the same as for the usual one. However, the function