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

6.1 Bloch states and Bloch bands 69
ε/ER, µ/ER
12
11
10
9
8
7
6
5
4
3
2
a) b)
−1 −0.5 0 0.5 1
¯hk/qB
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
−1 −0.5 0 0.5 1
¯hk/qB
Figure 6.5: Bloch band spectra εj(k) (6.4) (dashed lines) and µj(k) (6.5) (dash-dotted lines)
at s =5for gn =1ER. Solid lines: Single particle Bloch band spectrum at s =5(gn =0
where εj(k) =µj(k)). In b) the groundstate value has been subtracted for each data set.
Current, group velocity and effective mass
A stationary state is characterized by a spatially uniform, time-independent current. In the
following, we will show that the current density
Ij(k) =nd i¯h
∂
ϕjk
2m ∂x ϕ∗jk − ϕ ∗ ∂
jk
∂x ϕjk
(6.9)
associated with a certain condensate Bloch state is determined by the energy band structure
in the same way as in the single particle case. Let us consider the modified GP-equation
− ¯h2
2 ∂
+ iA + V (z)+g |Ψ|
2m ∂z 2
Ψ=µΨ , (6.10)
which is obtained by replacing the momentum operator −i¯h∂/∂z by −i¯h∂/∂z +¯hA, where
A is a constant. The presence of A does not violate periodicity so we can look for solutions
of Bloch form
Ψjk(z,A) =e ikz ˜ Ψjk(z,A) , (6.11)