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

9.6 Center-of-mass motion: Linear and nonlinear dynamics 129
9.6 Center-of-mass motion: Linear and nonlinear dynamics
The result
ωD =
m
ωz
(9.56)
m∗ for the frequency of small amplitude dipole oscillations in the presence of the harmonic potential
(9.34) has been obtained in the previous section 9.5 under the condition that the chemical
potential without harmonic trap can be approximated by the linear law µopt =˜gnM + µgn=0.
Now, we will show that in the particular case of the dipole, no knowledge about the density
dependence of the chemical potential is required in order to derive the result (9.56). As
previously, we require the effective mass to be density-independent m ∗ = m ∗ (gn =0).
The center-of-mass motion in the lattice direction is associated with a uniform macroscopic
superfluid velocity in the z-direction
vMz =¯hk(t)/m , (9.57)
associated with a quasi-momentum which depends only on time, but not on postion. The
hydrodynamic equations for small currents (9.17,9.18) with this choice for vMz, vMx = vMy =
0, andVext given by the harmonic potential (9.34) read
∂
∂t nM + m
m∗ ¯hk
m ∂znM =0, (9.58)
¯h ∂
∂t k + mω2 zz =0.
These equations can be recast to describe the motion of the center-of-mass
(9.59)
Z(t) = 1
N
dr (znM) , (9.60)
yielding
∂ m
Z −
∂t m∗ ¯hk
=0,
m
(9.61)
¯h ∂
∂t k + mω2 zZ =0. (9.62)
These equation allow for a solution Z ∝ eiωDt with
m
ωD =
m∗ ωz , (9.63)
in agreement with (9.51). This result is independent of the equation of state of the system
since the equations (9.58,9.61) do not contain µopt(nM) at all. Actually, it is a general property
of the center-of-mass oscillation in a harmonic potential not to be affected by the interactions
between particles. This reflects the fact that this particular type of oscillation does not involve
a compression of the sample.