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

4.1 Solution of the Schrödinger equation 37
probabilities are obtained from the Fourier expansion of the periodic function ˜ϕjk(x)
˜ϕjk(x) =
ajkl e il2πx/d . (4.13)
l
Inserting this expression into (4.3) yields the expansion of the Bloch function in plane waves
ϕjk(x) =
ajkl e i(l2π/d+k)x . (4.14)
l
Hence the probability for the particle to have momentum ¯h(k + l 2π/d) is given by d|ajkl| 2 .In
the case of the groundstate (j =1,k =0), the more the function ˜ϕjk(x) is modulated by the
presence of the external potential, the more momentum components p =¯hl 2π/d with l = 0
are important.
In Figs.4.3 and 4.4, we plot the probabilities |ajkl| 2 with j =1and ¯hk =0, 0.5qB, qB for a
particle in the optical lattice potential V = sERsin 2 (πx/d) for s =1and s =5respectively.
Note that non-zero values of l become more important as s is increased. When tuning k to
non-zero values, the distribution |ajkl| 2 becomes asymmetric with respect to l =0because ˜ϕjk
can not be written as a real function, indicating the presence of a nonzero velocity distribution
within each well.
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 4.3: Probabilities |ajkl| 2 of momentum components p =¯hk + 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 =1.