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

94 Bogoliubov excitations of Bloch state condensates
As in the previous chapter 6.2, we have neglected next-neighbour terms of the kind dzf 2 l f 2 l±1
and we have used the definitions of the tunneling parameters δ and δµ (see Eqs.(6.31,6.34)).
Eqs.(7.27,7.28) become
with
¯hω(q) = 2δ sin2 (Uq − Vq)2δ sin 2
qd
=¯hω(q)(Uq + Vq) , (7.35)
2
qd
2(2δµ − δ)sin
2
2 ( qd
2 )+2dgn
dzf 4
(z)+4(δ− δµ)
(7.36)
Comparison of the tight binding expression for the compressibility (6.43) with δ − δµ (6.36)
and n∂δ/∂n (6.38) allows us to rewrite (7.36) in the form
¯hω(q) =
2δ sin 2
qd
2
2
δ +2n ∂δ
∂n
sin 2
The density dependence of this spectrum shows up in three different ways:
qd
+2κ
2
−1
. (7.37)
• in the density dependence of δ discussed in section 6.2 and shown in Fig.6.8 (where the
quantity m ∗ ∝ 1/δ (see relation (6.41)) is plotted);
• in the density dependence of κ −1 which in the tight binding regime can be usually
approximated by the linear law ˜gn (see Fig.5.5);
• a contribution due to the density derivative of δ appears. However its effect in the
Bogoliubov band (7.37) is always small: for small interactions one has n∂δ/∂n ≪ δ;
instead, for larger interactions the inverse compressibility κ −1 dominates both δ and
n∂δ/∂n. Hence, we rewrite (7.37) neglecting this term
¯hω(q) =
2δ sin 2
qd
2δ sin
2
2
qd
+2κ
2
−1
(7.38)
Note that, as is discussed below in section 7.4 and shown in [116, 118] contributions
due to n∂δ/∂n can significantly affect the excitation frequency calculated on top of a
moving condensate.
Fig.7.3 compares the numerical data with the approximate expression (7.38), evaluated
using the quantity κ−1 calculated in section 5.2 and the tunneling parameter δ calculated in
section 6.1. As already found for the lowest Bloch band, for this value of gn, the agreement
with the tight binding expression is already good for s =10.
It is possible to identify two regimes, where the lowest Bogoliubov band (7.38) can be
described by further simplified expressions:
• for very large potential depth, the spectrum is dominated by the compressibility term. In
fact, δ → 0 while κ −1 becomes larger and larger as s increases. Hence, for large enough