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

7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes 85
Using the expansion
δΨ(z,t) =
σ
uσ(z)e −iωσt + v ∗ σ(z)e iωσt
, (7.5)
where σ labels the elementary excitations of this stationary state, we find the set of equations
− ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+2dgn| ˜ϕjk(z)|
d
2
− µj(k) uσ(z)+gnd˜ϕ 2 jk(z)vσ(z) =¯hωσuσ(z) (7.6)
− ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+2dgn| ˜ϕjk(z)|
d
2
− µj(k) vσ(z)+gnd˜ϕ ∗2
jk(z)uσ(z) =−¯hωσvσ(z) (7.7)
for the Bogoliubov amplitudes uσ(z), vσ(z) and the excitation energies ¯hωσ.
We require
dz |uσ| 2 −|vσ| 2
=1. (7.8)
Moreover, the Bogoliubov amplitudes must satisfy the orthogonality relations
dz (u ∗ σuσ ′ − v∗ σvσ ′)=0, (7.9)
dz (u ∗ ′ σvσ − vσuσ ′)=0, (7.10)
for σ = σ ′ .
The normalization condition (7.8) implies that for an energetically stable state ϕjk the
spectrum ¯hωσ is positive, while negative values ¯hωσ signify that the stationary state ϕjk is
energetically unstable (see [1], chapter 5.6). The occurrence of complex frequencies ωσ with
a negative imaginary part reveals the dynamical instability of the stationary state ϕjk. The
stability analysis of condensate Bloch states commented on in the previous chapter is thus
based on the solution of the Bogoliubov equations (7.6,7.7).
The presence of a harmonic trap can changes significantly the low energy excitations of the
system. In fact, in chapter 9 we will show that small amplitude collective oscillations occurring
on a length scale of the system size are strongly affected both by the harmonic trap and the
lattice. The excitation spectrum obtained from (7.6,7.7) does not account for the presence of
a harmonic trap. Still, the solution of (7.6,7.7) is relevant also for trapped systems: It yields
a local Bogoliubov band spectrum given by ¯hωσ(nM(r)) where nM(r) is the macroscopic
density profile introduced above in chapter 5.4. This approach is valid for excitations whose
characteristic length scale is small compared to the size of the system.
7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes
Since ˜ϕjk(z) is periodic, we can apply the Bloch theorem as in section 4.1 and write the
Bogoliubov amplitudes in the form
uj ′ q(z) = 1
√ e
Nw
iqz ũj ′ q(z) , (7.11)
vj ′ q(z) = 1
√ e
Nw
iqz ˜vj ′ q(z) , (7.12)