Introduction

The study of quasi-1D Bose gases in the quantum-degenerate regime has become a very active area of research. The role of correlations and of quantum fluctuations is greatly enhanced by the reduced dimensionality and 1D quantum gases constitute well suited systems to study beyond mean-field effects [PSW00]. Among these, particularly intriguing is the fermionization of a 1D Bose gas in the strongly repulsive Tonks-Girardeau (TG) regime, where the system behaves as if it consisted of noninteracting spinless fermions [Gir60]. The Bose-Fermi mapping of the TG gas is a peculiar aspect of the universal low-energy properties which are exhibited by bosonic and fermionic gapless 1D quantum systems and are described by the Luttinger liquid model [Voi95]. The concept of Luttinger liquid plays a central role in condensed matter physics and the prospect of a clean testing for its physical implications using ultracold gases confined in highly elongated traps is fascinating [MLE98,RFZZ03a].

Bosonic gases in 1D configurations have been realized experimentally. Complete freezing of the transverse degrees of freedom and fully 1D kinematics has been reached for systems prepared in a deep 2D optical lattice [MSKE03,TOH+04]. The strongly interacting regime has been achieved by adding a longitudinal periodic potential and the transition from a 1D superfluid to a Mott insulator has been observed [SMS+04]. A different technique to increase the strength of the interactions, which is largely employed in both bosonic and fermionic 3D systems [IAS+98,OHG+02] but has not yet been applied to 1D configurations, consists in the use of a Feshbach resonance. With this method one can tune the effective 1D coupling constant $g_{1D}$ to essentially any value, including $\pm\infty$, by exploiting a confinement induced resonance [Ols98,BMO03]. For large and positive values of $g_{1D}$ the system is a TG gas of point-like impenetrable bosons. On the contrary, if $g_{1D}$ is large and negative, we will show that a new gas-like regime is entered (super-Tonks) where the hard-core repulsion between particles becomes of finite range and correlations are stronger than in the TG regime. In this Chapter we investigate using Variational Monte Carlo techniques (Sec. 2.2) the equation of state and the correlation functions of a homogeneous 1D Bose gas in the super-Tonks regime. We find that the particle-particle correlations decay faster than in the TG gas and that the static structure factor exhibits a pronounced peak. The momentum distribution and the structure factor of the gas are directly accessible in experiments by using, respectively, time-of-flight techniques and two-photon Bragg spectroscopy [SMS+04]. The study of collective modes also provides a useful experimental technique to investigate the role of interactions and beyond mean-field effects [MSKE03]. Within a local density approximation (LDA) for systems in harmonic traps we calculate the frequency of the lowest compressional mode as a function of the interaction strength in the crossover from the TG gas to the super-Tonks regime.