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

8.1 Dynamic structure factor 103
with q lying in the first Brillouin zone and l integer. In fact, since excitations with the same
j and q differing by a multiple of 2π/d are physically equivalent, the quasi-momenta of one
Brillouin zone exhaust all possible excitations. Yet, note that p can not be restricted to one
zone, being the momentum transferred by the external probe.
Inserting (8.5) into (8.1) the dynamic structure factor (8.1) takes the form
S(p, ω) =
Zj(p)δ(ω − ωj(p)) . (8.7)
j
The quantity Zj(p) is the excitation strength to the j-thbandforagivenmomentumtransfer
p.
Zj(p) =N ipz/¯h
dz e u ∗ jq(z)+v ∗
jq(z) ϕ(z)
2 , (8.8)
where q lies in the first Brillouin zone and is fixed by the relation q = p + l2π/d.
The expression (8.7) reveals some interesting properties of the dynamic structure factor in
the presence of an optical lattice:
• When scanning ω for fixed momentum transfer p a resonance is encountered for each
Bogoliubov band. In contrast, when the lattice is switched off only one resonance exists
for a given p. An important consequence is that on one hand it is possible to excite high
energy states with small values of p, and on the other hand one can excite low energy
states, belonging to the lowest band, also with high momenta p outside the first Brillouin
zone. This difference with respect to the uniform case can be understood by noting that
the excitations created by the external probe have well-defined quasi-momentum ¯hq and
accordingly many momentum components ¯hq + l2π/d. The external probe couples to
the component which corresponds to the momentum transfer.
• While the excitation energies ¯hωj(p) are periodic as a function of p, this is not true
for the excitation strengths Zj(p) (see Eq.(8.8)). This reflects the difference between
quasi-momentum and momentum:
To illustrate these two characteristic features of the excitation strengths in the presence of an
optical lattice we plot in Fig.8.1 the three lowest Bogoliubov bands in the first three Brillouin
zones for s =10,gn=0.5ER and indicate the values of the excitation strengths Zj(p) at
specific values of p.
For comparison we plot in Fig. 8.2 the strengths Z1(p), Z2(p), Z3(p) for the uniform system
(s =0) where the Bogoliubov dispersion does not form a band structure. Yet, in order to
facilitate the comparison with s = 0we can formally map excitations for momenta lying in the
j-th Brillouin zone onto the j-th band. The corresponding excitation strengths are given by
(see [1] chapter 7.6)
p
Zj(p) =Ntot
2 /2m
, (8.9)
¯hωuni(p)
with p lying in the jth Brillouin zone and
p
¯hωuni(p) =
2
p2 2m 2m +2gn
(8.10)
the Bogoliubov dispersion of the uniform system. For p lying outside the jth Brillouin zone
the strength Zj(p) is zero.