The study of trapped Bose systems in low dimensions is currently attracting a lot of interest. In particular, 1D systems are expected to exhibit remarkable properties which are far from the mean-field description and are not present in 2D and 3D. The peculiarity of 1D physics consists in the role played by fluctuations, which destroy long-range order even at zero temperature [Sch77,Hal81], and in the occurrence of characteristic effects due to correlations such as the fermionization of the gas in the Tonks-Girardeau regime [Gir60]. Recent experiments with highly anisotropic, quasi-one-dimensional traps have shown first evidences of 1D features in the aspect ratio and energy of the released cloud[GVL+01,SKC+01] as well as in the coherence properties of condensates with fluctuating phase[DHR+01]. From a theoretical viewpoint, the emergence of 1D effects in the properties of binary atomic collisions, by increasing the confinement in the transverse direction, has been pointed out by Olshanii[Ols98]. In the case of harmonically trapped gases, the occurrence of various regimes possessing true or quasi-condensate and the possibility of entering the Tonks-Girardeau gas regime of impenetrable bosons has been discussed in[PSW00].
The ground-state properties and excitation spectrum of a homogeneous 1D system of bosons interacting through a repulsive contact potential have been calculated exactly by Lieb and Liniger long time ago[LL63,Lie63]. For a fixed interaction strength the Lieb-Liniger equation of state reproduces in the high density regime the mean-field result obtained using the Bogoliubov model and in the opposite limit of low density coincides with the ground-state of impenetrable bosons [Gir60]. For 1D systems in harmonic traps, the exact many-body ground-state wave function in the Tonks-Girardeau regime has been recently calculated[GWT01], and the equation of state interpolating between the mean-field and the Tonks-Girardeau regime has been obtained within the local density approximation in[DLO01]. Methods based on local density approximation in the longitudinal direction and on the Gross-Pitaevskii equation for the transverse direction have been recently employed to predict the frequency of the collective excitations [MS02] and the ground-state energy in the 3D-1D cross-over as well as in the 1D mean field - Tonks-Girardeau gas cross-over [DGW02].
In this chapter we present exact Diffusion Monte-Carlo (see Sec. 2.3) results for the 3D-1D cross-over in harmonically trapped Bose gases. As a function of the anisotropy parameter of the trap we calculate the ground-state properties of the system and for highly anisotropic traps we point out the occurrence of important beyond mean-field effects including the fermionization of the gas.