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

38 Single particle in a periodic potential
Group velocity, current and effective mass
A further analogy between quasi-momentum and actual momentum is revealed by calculating
the mean velocity ¯vj(k) of a particle in the Bloch state ϕjk(x). One finds
¯vj(k) ≡〈ϕjk|ˆ˙x|ϕjk〉 = ∂εj(k)
. (4.15)
¯h∂k
This quantity is referred to as the group velocity. According to relation (4.15) the particle
remainsatrestonaveragewhenk = lπ/d since at these values of the quasi-momentum
the energy bands exhibit local minima or maxima. Furthermore, the particle mean velocity
decreases as the potential is made deeper due the flattening of the bands. The dependence of
the group velocity on the quasi-momentum for different potential depths is illustrated in Fig.
4.5 for a particle in the optical lattice potential V = sERsin2 (πx/d).
A closely related quantity is the current density associated with a certain Bloch state
Ij(k) = i¯h
∂
ϕjk
2m ∂x ϕ∗jk − ϕ ∗ ∂
jk
∂x ϕjk
. (4.16)
d|ajkl| 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3 −2 −1 0 1 2 3
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3 −2 −1 0 1 2 3
l
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3 −2 −1 0 1 2 3
Figure 4.4: Probabilities |ajkl| 2 of momentum components p = l 2π/d in the state with band
index j =1and quasi-momentum ¯hk =0(left), ¯hk =0.5qB (middle), and ¯hk = qB (right)
for a particle in the optical lattice potential V = sERsin 2 (πx/d) with s =5.