Conclusions

In this chapter we present exact Quantum Monte Carlo results of the ground-state energy and structure of a Bose gas confined in highly anisotropic harmonic traps. Starting from a 3D Hamiltonian, where interparticle interactions are model by a hard-sphere or a soft-sphere potential, we show that the system exhibits striking features due to particle correlations. By reducing the anisotropy parameter $\lambda$, while the number of particles $N$ and the ratio $a_{3D}/a_\perp$ of scattering to transverse oscillator length are kept fixed, the system crosses from a regime where mean-field theory applies to a regime which is well described by the 1D Lieb-Liniger equation of state in local density approximation. In the cross-over region both theories fail and one must resort to exact methods to account properly for both finite size effects and residual 3D effects. For very small values of $\lambda$ we find clear evidence, both in the energy per particle and in the longitudinal size of the cloud, of the fermionization of the system in the Tonks-Girardeau regime.