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

110 Linear response - Probing the Bogoliubov band structure
Z3(p)/Ntot
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−6 −4 −2 0 2 4 6
p/qB
Figure 8.6: Excitation strength to the third Bogoliubov band Z3(p) (8.8) for gn =0.5ER at
lattice depth s =5(solid line) and s =0(dashed line).
factor. Notice that in the absence of two-body interactions (gn =0), S(p) =1for any value
of p (see dash-dotted lines in Figs.8.7 and 8.8). As we will see, S(p) is strongly affected by
the combined presence of two-body interactions and optical lattice.
A second important sum-rule obeyed by the dynamic structure factor is the model independent
f-sum rule
p
¯hωS(p, ω)dω = Ntot
2
.
2m
(8.19)
Another important sum-rule is the compressibility sum-rule corresponding to the low-p limit of
the inverse-energy weighted sum-rule
S(p, ω)
dω
κ
¯hω = Ntot ,
p→0 2
(8.20)
where κ is the compressibility (5.11) (see section 5.2). As discussed above in section (7.4)
the compressibility of a condensate loaded in an optical lattice is naturally expressed in terms
of the sound velocity c, characterizing the low-q phononic behaviour of the dispersion law
(¯hω(q) =c¯hq), through the relation (see Eq.(7.48))
κ = 1
m∗ . (8.21)
c2