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

8.1 Dynamic structure factor 107
The integral mainly depends on the properties of the Wannier function near the center of the
well and can hence be evaluated within the gaussian approximation discussed in section 6.2.
Replacing the integral limits by −∞ and +∞, weobtain
2δ sin
Z1(p) =Ntot
2 (pd/2)
exp
¯hω(p)
− 1 p2
σ2
2 ¯h 2
. (8.14)
This expression for the excitation strength to the lowest band in a deep lattice exhibits all the
characteristics mentioned above with respect to the numeric results:
• The gaussian dependence on the momentum transfer p describes the overall decay of the
excitation strength at large p. This behavior is also present in the absence of interactions
(gn =0). Note that σ is smaller in a deeper lattice, yielding a broader envelope of the
strength (8.14). This reflects that at larger s the excitations have more momentum
components.
• Provided that gn/ER = 0, the ratio between the Bloch and the Bogoliubov dispersion
gives rise to the characteristic oscillations of Z1(p) in the vicinities of p = l2qB where
the excitations contributing to the strength have phonon character: Near the points
p = l2qB, wehave
√ d
Z1(p) ≈ Ntot δκ
2¯h |p − l2qB| exp(− 1
2
σ2 (2lqB) 2
¯h 2 ) , (8.15)
and hence the strength (8.14) vanishes approximately linearly.
If gn/ER =0, Bloch and Bogoliubov dispersion coincide and we are left with the nonoscillating
behavior Z1(p) = exp − 1
2σ2p/¯h 2
. Note that σ → 0 for s →∞and hence
Z1(p) → 1.
• Increasing gn/ER at fixed s or increasing s at fixed gn/ER = 0leads to an overall
decrease of Z1(p): This behavior can be explained by considering the ratio between
Bloch and Bogoliubov dispersion appearing in (8.14). Using Eq.(7.38) for the Bogoliubov
dispersion, we can write
2δ sin 2 (pd/2)
¯hω(p)
=
δκ sin2 (pd/2¯h)
1+2δκ sin2 . (8.16)
(pd/2¯h)
Increasing gn/ER at fixed s leads to a decrease of κ which dominates the slight increase
of δ. So, δκ → 0 and hence the ratio (8.16) and accordingly the strength (8.14)
diminuish. Moreover, increasing s at fixed gn/ER = 0both δ and κ diminish, δκ → 0,
and thus brings about an overall decrease of Z1(p). Hence, from (8.16) it follows that
for increasing gn/ER or increasing s the excitation strength to the lowest band (8.14)
is reduced according to the law
Z1 ∼ √ δκ . (8.17)
Looking at expression (8.11), we note that in the presence of interactions the strength is
suppressed whenever Up ∼−Vp. This must always be the case in the vicinity of p = l 2π/d
where the excitations are phonons. Yet, also the overall suppression of Z1 is a consequence of
such a behavior of the Bogoliubov amplitudes since Up ∼−Vp can be achieved for all p if the
lattice is made sufficiently deep.