Beyond Tonks-Girardeau: strongly correlated regime in quasi-one-dimensional Bose gases

We consider a homogeneous 1D Bose gas with contact interactions and large attractive coupling constant. This system can be realized in tight waveguides by exploiting a confinement induced resonance of the effective 1D scattering amplitude. By using a variational {\it ansatz} for the many-body wavefunction, we show that for small densities the gas-like state is stable and the corresponding equation of state is well described by a gas of hard rods. By calculating the compressibility of the system, we provide an estimate of the critical density at which the gas-like state becomes unstable against cluster formation. Within the hard-rod model we calculate the one-body density matrix and the static structure factor of the gas. The results show that in this regime the system is more strongly correlated than a Tonks-Girardeau gas. The frequency of the lowest breathing mode for harmonically trapped systems is also discussed as a function of the interaction strength.