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We first consider a system of very few particles () and we consider different
values of the ratio
. Figs. 3.1-3.2 refer to
, and we present results for the energy per particle and the
mean square radius of the cloud in the longitudinal direction as a function of the
anisotropy parameter
. Results from the GP equation
(3.3) and from the Lieb-Liniger equation of state in LDA are also shown. We
find that the HS and SS potential give practically the same results even for the
largest values of
, showing that for these parameters we are well within
the universal regime where the details of the potential are irrelevant. For large
values of
the DMC results agree well with the predictions of GP equation.
By decreasing
beyond mean-field effects become visible and both the energy
per particle and the mean square radius approach the LL result when
, corresponding to
. Finally, for the
smallest values of the anisotropy parameter (
) we find clear
evidence of the Tonks-Girardeau gas behavior both in the energy and in the shape of
the cloud. It is worth stressing that beyond mean-field effects occurring in the
small
regime can be only obtained by using DMC. A Variational Monte-Carlo
(VMC) calculation based on the trial wave function
described above,
would yield results in good agreement with mean-field over the whole range of values
of
. DMC results using the Lieb-Liniger Hamiltonian
of Eq.
(3.8) are also shown and coincide with the results of the 3D Hamiltonian
(3.1). This shows that the 3D interatomic potential is correctly described by
the 1D
-potential even for the largest values of
. In fact, due to
the small number of particles, the density profile of the cloud in the transverse
direction is correctly described by the harmonic oscillator ground-state
wave function (see Fig. 3.9). The 1D character of the system is also
evident from Fig. 3.1 which shows that
is always
smaller than the transverse confining energy. Deviations of DMC results from the LL
equation of state arise because of finite size effects. These effects become less
and less important as
decreases and one enters the regime
where LDA applies. In terms
of the mean square radius of the cloud (see Fig. 3.2), the condition of
applicability of LDA requires
much larger than the
corresponding ideal gas (IG) value.