Small system, medium scattering length

Figure 3.1: Energy per particle as a function of $\lambda$. DMC results: HS potential (solid circles), SS potential (solid triangles), LL Hamiltonian (3.8) (open triangles).Dashed line: GP equation (3.3), solid line: LL equation of states in LDA, dotted line: TG gas, dot-dashed line: non-interacting gas. Error bars are smaller than the size of the symbols.

Figure 3.2: Mean square radius along $z$ as a function of $\lambda$. Dashed line: GP equation (3.3), solid line: LL equation of states in LDA, dotted line: TG gas, dot-dashed line: non-interacting gas. Error bars are smaller than the size of the symbols.

We first consider a system of very few particles ($N=5$) and we consider different values of the ratio $a_{3D}/a_\perp$. Figs. 3.1-3.2 refer to $a_{3D}/a_\perp =0.2$, 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 $\lambda=\omega_z/\omega_\perp$. 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 $\lambda$, showing that for these parameters we are well within the universal regime where the details of the potential are irrelevant. For large values of $\lambda$ the DMC results agree well with the predictions of GP equation. By decreasing $\lambda$ beyond mean-field effects become visible and both the energy per particle and the mean square radius approach the LL result when $N\lambda a_\perp^2/a^2\sim 1$, corresponding to $\lambda\sim 10^{-2}$. Finally, for the smallest values of the anisotropy parameter ( $\lambda\sim 10^{-4}$) 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 $\lambda$ regime can be only obtained by using DMC. A Variational Monte-Carlo (VMC) calculation based on the trial wave function $\psi_T({\bf R})$ described above, would yield results in good agreement with mean-field over the whole range of values of $\lambda$. DMC results using the Lieb-Liniger Hamiltonian $H_{LL}$ 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 $\delta$-potential even for the largest values of $\lambda$. 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 $E/N-\hbar\omega_\perp$ 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 $\lambda$ decreases and one enters the regime $(E/N-\hbar\omega_\perp)/\hbar\omega_\perp \gg\lambda$ where LDA applies. In terms of the mean square radius of the cloud (see Fig. 3.2), the condition of applicability of LDA requires $\langle z^2\rangle^{1/2}$ much larger than the corresponding ideal gas (IG) value.