Exact numerical simulations of a one-dimensional, trapped Bose gas

We analyze the ground-state and low-temperature properties of a one-dimensional Bose gas in a harmonic trapping potential using the numerical density matrix renormalization group. Calculations cover the whole range from the Bogoliubov limit of weak interactions to the Tonks-Girardeau limit. Local quantities such as density and local three-body correlations are calculated and shown to agree very well with analytic predictions within a local density approximation. The transition between temperature dominated to quantum dominated correlation is determined and it is shown that despite the presence of the harmonic trapping potential first-order correlations display over a large range the algebraic decay of a harmonic fluid with a Luttinger parameter determined by the density at the trap center.