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

74 Stationary states of a BEC in an optical lattice
m ∗ (gn)/m ∗ (gn=0)
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0 5 10 15 20 25 30
Figure 6.9: Ratio between the effective mass m ∗ (gn) and the single particle effective mass for
gn =0.5ER (solid line) and gn =0.1ER (dashed line) as a function of lattice depth s.
‘The condensate Wannier functions (6.25) form a complete orthonormal set and fulfill the
relation
fj,l(x) =fj(x − ld) . (6.27)
As in the single particle case, the Wannier function fj,l of a condensate is localized at site l
and spreads less and less over other sites the deeper the lattice. For convenience, we choose
the Wannier functions to be real in the following.
Momentum and quasi-momentum
The momentum distribution of a Bloch state is obtained in complete analogy to the single
particle case (see chapter 4.1). In fact, we find that apart from a slight screening effect
interactions do not have a strong effect for typical values of gn.
Swallow tails
For gn ≥ s one encounters Bloch state solutions which lead to loops (“swallow tails”) at the
edge of the lowest energy Bloch band [107, 106, 108, 109, 110, 111]. The appearance of these
states goes along with a non-zero group velocity of the Bloch state at k =¯hqB, insharp
contrast with the typical behavior of a single particle. For smaller values of gn > 0 such loops
can also be found at the center of the first excited band [107]. For the values of gn we have
s