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

9.5 Small amplitude collective oscillations in the presence of harmonic trapping 125
the absence of the lattice [43] by simply replacing
ωx → m/m∗ωx ,
(9.54)
ωy → m/m∗ωy , (9.55)
where we have assumed the lattice two be set up in the x, y plane and ωx, ωy
are the harmonic
trapping frequencies in these directions. If ωz >> ωx m/m∗ ,ωy m/m∗ , the lowest energy
solutions involve the motion in the x − y plane. The oscillations in the z-direction are instead
fixed by the value ωz of the harmonic trap. These include the center-of-mass motion (ω = ωz)
and the lowest compression mode (ω = √ 3ωz). The frequency ω = √ 3ωz coincides with the
value obtained by directly applying the hydrodynamic theory to 1D systems [139, 140] and
reveals the 1D nature of the tubes generated by the 2D lattice. If the radial trapping generated
by the lattice becomes too strong the motion along the tubes can no longer be described by
the mean field equations and one enters into more correlated 1D regimes associated with a
modification of the equation of state which affects the frequencies of the collective oscillations
[141]. It is interesting to note that the study of the macroscopic behavior of a Bose-Einstein
condensate in a two-dimensional lattice and the investigation of one-dimensional Bose gases can
be merged: The equation of state entering the hydrodynamic equations for a two-dimensional
lattice can be chosen such as to reflect the one-dimensional nature of the gas strongly confined
in the tubes of the lattice. This allows to identify signatures of strong correlations in the
macroscopic 3D properties (density profile, collective modes) of the sample [142].
Comparison with experiment
The predictions (9.51) and (9.52) for the center-of-mass oscillation and the axial breathing
mode in an elongated condensate have been confirmed experimentally in [73, 75]. In these
experiments gn ≈ 0.2ER at the center of the harmonic trap at s =0. For this relatively low
value of the parameter gn/ER it is reasonable to work with the approximations (9.35,9.37)
neglecting the density dependence of m∗ and supposing the chemical potential to be linear in
the density.
Fig. 9.1 displays both the experimental data reported in [73] and the theoretical curve
(9.51) for the dipole frequency as a function of lattice depth. Note that experimentally it is
difficult to measure this frequency in a very deep lattice since the system enters a regime of
nonlinear oscillations (see discussion below in section 9.6). For this reason, oscillations with
the frequency (9.51) can be observed only if the oscillations amplitude is kept very small. In
fact, it has turned out to be difficult to excite measurable harmonic center-of-mass oscillations
at lattice depths s>9.
In [75] experimental data for the frequencies of the axial breathing mode and the dipole
mode have been compared with each other. By plotting ωQ as a function of ωD (see Fig. 9.2)
the ratio of the two frequencies is determined to be 1.57 ± 0.01 in good agreement with the
value 5/2 predicted based on the results (9.51,9.52). Furthermore, assuming the theoretical
prediction to be true, the value of the effective mass m ∗ at a given lattice depth s can be
extracted and compared to the calculated value. The results are depicted in Fig. 9.3 which
also includes data obtained from the GP-simulation reported in [143] for the two types of
oscillations. In contrast with the s-dependence of ωD and ωQ, the frequency of the transverse