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

122 Macroscopic Dynamics
9.5 Small amplitude collective oscillations in the presence of harmonic
trapping
In current experiments, the external potential Vext is provided by a harmonic trap
Vext = m
2
ω 2 xx 2 + ω 2 yy 2 + ω 2 zz 2
(9.34)
to which the optical lattice is superimposed. In the following, we will assume the groundstate
to be adequately described by the LDA developed in section 5.4, implying that the size of
the condensate is much larger than the lattice spacing d (see Eq.(5.20)). In accordance with
the discussion above, this is a necessary condition for the use of (9.17,9.18). Moreover, we
presuppose the chemical potential in absence of the harmonic confinement to exhibit the linear
dependence on average density (see Eq.(5.13))
µopt =˜gn + µgn=0 . (9.35)
The validity of this approximation has been discussed in section 5.2. Within LDA, the groundstate
macroscopic density profile ¯nM is then defined by the relation (see discussion in section
5.4)
µ =˜g¯nM(r)+µgn=0 + Vext . (9.36)
The dynamic equations (9.17,9.18) do not only require the equation of state µopt(nM)
as an input, but also the effective mass m ∗ (nM). As a simplification, we will consider the
effective mass to be density-independent and thus to be given by the single particle effective
mass
m ∗ = m ∗ (gn/ER =0). (9.37)
The applicability of this approximation has been discussed in section 6.1.
With Vext given by the harmonic potential (9.34) and the approximations (9.35,9.37), the
hydrodynamic equations (9.17,9.18) take the form
∂
∂t nM + ∂x(vMxnM)+∂y(vMynM)+ m
m∗ ∂z (vMznM) =0, (9.38)
m ∂
∂t vM
+ ∇ Vext +˜gnM + µgn=0 + m
2 v2 Mx + m
2 v2 My + m m
nM
m∗ 2 v2
Mz =0. (9.39)
These equations are expected to give a correct description of the dynamics occurring on a
length scale of the order of the system size. In particular, we will explore here the limit of
small amplitude collective oscillations which comply with this condition.
Hydrodynamic equations for small amplitude oscillations
Small amplitude collective oscillations are associated with an oscillating perturbation in the
groundstate density ¯nM and a small oscillating velocity field
∆n(r,t) ∝ ∆n(r) e −iωt , (9.40)
∆v(r,t) ∝ ∆v(r) e −iωt . (9.41)