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

4.1 Solution of the Schrödinger equation 35
different bands are always separated by energy gaps. Moreover, in the one-dimensional case,
the minimum and maximum values of each band εj(k) are found at k =0and k = π/d.
If the potential V (x) is weak, one can apply first order perturbation theory to calculate the
energy bands εj(k). For this purpose, it is useful to expand the function V in a Fourier series
V (x) = ∞ l=−∞ Vl e il2πx/d . One finds that all bands are shifted by the constant V0 and that
the energy gap between the band j and the band j +1equals to 2|Vj|. In the particular case
of an optical lattice V = sERsin 2 (πx/d), onlyV0 and V1 are nonzero. Hence, the energy
gaps are found to be zero except for the one between the lowest and the first excited band.
Higher order perturbation theory is needed to resolve the gaps between the higher bands. The
opposite extreme of a deep potential is discussed below in section 4.2.
In Fig. 4.1, we plot the first three energy bands for a single particle in the optical lattice
potential V = sERsin 2 (πx/d) for s = 0, 1, 5 as obtained from the numerical solution of
the Schrödinger equation. For s =0we are dealing with a free particle with energy spectrum
ε(k) =¯h 2 k 2 /2m. The corresponding “band spectrum” is obtained by mapping to the jth band
energies with wave numbers k belonging to the jth Brillouin zone ((j − 1)qB ≤|k| ≤jqB).
Fig. 4.1 illustrates that energy gaps become larger while the heights of the bands decrease as
the lattice is made deeper. These effects become most clearly visible by looking at the lowest
band: In fact, the energy gap between first and second band is already large at a lattice depth
of s =1, while the gaps between higher bands are hardly visible. Also, the height of the lowest
band decreases much more rapidly as a function of s than the ones of the higher bands.
ε/ER
8
7
6
5
4
3
2
1
0
−1 0 1
8
a) b) c)
7
6
5
4
3
2
1
0
−1 0 1
¯hk/qB
8
7
6
5
4
3
2
1
0
−1 0 1
Figure 4.1: Lowest three Bloch bands (4.11) in the first Brillouin zone of a particle in the
optical lattice potential V = sERsin 2 (πx/d) for a) s =0,b)s =1and c) s =5.