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

102 Linear response - Probing the Bogoliubov band structure
The effects of interactions on the static structure factor are found to be significantly amplified
by the presence of the optical potential (see section 8.2). Using a sum rule approach,
we prove that the static structure factor vanishes like p/2m∗c for p → 0, where m∗ and c are
the effective mass and the sound velocity at a given lattice depth respectively.
In [133], we have reported the numerical and analytical results for the excitation strengths
of the lowest band and for the static structure factor. The discussion of the excitation strengths
of higher bands is added here.
8.1 Dynamic structure factor
The capability of the system to respond to an external density probe transferring momentum
p and energy ¯hω is described by the dynamic structure factor
S(p,ω)=
〈σ|δ ˆρ †
p|0〉 2
δ (ω − ωσ) . (8.1)
σ
Here, σ labels low energy excitations, |0〉 is the groundstate and δ ˆρp is defined as
δ ˆρp =ˆρp −〈ˆρp〉eq , (8.2)
where ˆρp is the Fourier transform of the density operator ˆn(r)
ˆρp = dre −ip·r/¯h ˆn(r) (8.3)
and 〈...〉eq denotes the expectation value at equilibrium. For a weakly interacting Bose gas,
the matrix elements involved in (8.1) take the form ([1], chapter 5.7 and 7.2)
〈σ|δ ˆρ †
p|0〉 =
dr e ip/¯h·r (u ∗ σ(r)+v ∗
σ(r)) Ψ(r)
, (8.4)
where dr (u∗ σ ′uσ − v∗ σ ′vσ) =δσ ′ ,σ and Ψ is the condensate wavefunction normalized such
that dr|Ψ| 2 = Ntot. The modulus squared of this quantity yields the excitation strength for
the state |σ〉. The response of a harmonically trapped condensate to a density probe has been
theoretically studied in [134, 135, 136, 137].
To study the excitation of the system in presence of an optical lattice, we will assume p
to be oriented along the z-axis. Rescaling the condensate wavefunction according to (5.2)
and using the solutions of the Bogoliubov equations (7.14,7.15) with the orthonormalization
relations (7.8,7.9,7.10) the matrix element (8.4) takes the form
〈σ|δ ˆρ †
p|0〉 = √ N
ipz/¯h
dz e u ∗ jq(z)+v ∗
jq(z) ϕ(z)
. (8.5)
Inserting the Bloch forms (7.11,7.12) for the Bogoliubov amplitudes and noting that
ũ∗ jq (z)+˜v∗ jq (z)
ϕ(z) is periodic with period d, we find that the expression (8.5) is nonzero
only for
q = p ± l 2π
, (8.6)
d