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

9.5 Small amplitude collective oscillations in the presence of harmonic trapping 123
To find the frequencies ω, we linearize Eqs.(9.38,9.39) in ∆n(r,t), ∆v(r,t)
∂
m
∆n + ∂x(∆v¯nM)+∂y(∆v¯nM)+
∂t m∗ ∂z (∆vz ¯nM) =0. (9.42)
m ∂
∆v +˜g∇∆n =0,
∂t
(9.43)
where we have made use of (9.36).
inserting (9.43), we obtain
After differentiating (9.42) with respect to time and
∂2
˜g¯nM
∆n − ∂x
∂t2 m ∂x∆n
˜g¯nM
− ∂y
m ∂y∆n
Using (9.36) this equation can be rewritten in the form
∂2
µ − µgn=0 − Vext
µ − µgn=0 − Vext
∆n − ∂x
∂x∆n − ∂y
∂y∆n
∂t2 m
m
− m
˜g¯nM
∂z
m∗ m ∂z∆n
=0. (9.44)
− m
µ − µgn=0 − Vext
∂z
∂z∆n =0.
m∗ m
(9.45)
Its form is affected by the lattice only through the appearance of the factor m/m∗ and the
irrelevant constant µgn=0. This formal difference can be eliminated by replacing the trap
frequency ωz, thez-coordinate and the chemical potential µ by
m
˜ωz =
m∗ ωz , (9.46)
m∗ ˜z = z, (9.47)
m
˜µ = µ − µgn=0 . (9.48)
Using these new quantities we can rewrite Eq.(9.45) in the form
∂2
˜µ − Vext
˜
∆n − ∂x
∂t2 m ∂x∆n
˜µ − Vext
˜
− ∂y
m ∂y∆n
− ∂˜z
˜µ − ˜ Vext
m ∂˜z∆n
where ˜ Vext is a harmonic potential with frequency ˜ωz along the ˜z-direction
˜Vext = m
2
=0, (9.49)
ω 2 xx 2 + ω 2 yy 2 +˜ω 2 z ˜z 2
. (9.50)
At this stage, the differential equation (9.49) is formally identical with the one describing small
amplitude collective oscillations of a condensate with chemical potential ˜µ in a harmonic trap
with frequencies ωx, ωy, ˜ωz without a lattice being superimposed to it.
Frequencies of small amplitude collective oscillations
The above derivation has shown that the condensate in the combined potential of harmonic
trap and lattice oscillates as if there was no lattice and as if the harmonic trap frequency along
the z-direction was ˜ωz = m/m ∗ ωz instead of ωz and the chemical potential ˜µ = µ − µgn=0
instead of µ. This conclusion allows us to apply results obtained for a purely harmonically
trapped condensate to the case when a lattice is added (see [1] chapter 12.2-12.3 and references
therein): We obtain the correct frequency by replacing ωz by ˜ωz and µ by ˜µ.