Ultracold Atoms in One-Dimensional Optical Lattices Approaching the Tonks-Girardeau Regime

Recent experiments on ultracold atomic alkali gases in a one-dimensional optical lattice have demonstrated the transition from a gas of soft-core bosons to a Tonks-Girardeau gas in the hard-core limit, where one-dimensional bosons behave like fermions in many respects. We have studied the underlying many-body physics through numerical simulations which accommodate both the soft-core and hard-core limits in one single framework. We find that the Tonks-Girardeau gas is reached only at the strongest optical lattice potentials. Results for slightly higher densities, where the gas develops a Mott-like phase already at weaker optical lattice potentials, show that these Mott-like short-range correlations do not enhance the convergence to the hard-core limit.

Figure 1

Internal energy per site as a function of $U/J$ for a homogeneous model with 128 sites at a temperature $T/J=0.2$. The data points with error bars connected by the full line are the energies obtained by the SSE method. The energies for noninteracting fermions (upper dotted horizontal line) and noninteracting bosons (lower dotted horizontal line) are shown, together with the Bogoliubov approximation for bosons (dashed line on the left) and a first-order perturbation theory for fermions (dashed line on the right).

Figure 2

Local densities $n_i$ in coordinate space as a function of the site index $i$ (left) and the corresponding momentum profiles $n(p)$ as a function of the momentum $p$ (in units $\hslash k$) on the right. The axial optical lattice depths, the ratios $U/J$ and the values of the slope parameter $\alpha$ for soft-core (solid line) and ${\alpha }^{\prime }$ for hard-core bosons (dashed line) are (a) $V_0/E_R=1$, $U/J=1.75$, $\alpha =2.71$, ${\alpha }^{\prime }=1.69$, (b) $V_0/E_R=5$, $U/J=7.85$, $\alpha =1.92$, ${\alpha }^{\prime }=1.38$, (c) $V_0/E_R=9.5$, $U/J=28.6$, $\alpha =1.00$, ${\alpha }^{\prime }=0.78$, (d) $V_0/E_R=12$, $U/J=52.28$, $\alpha =0.72$, ${\alpha }^{\prime }=0.56$, (e) $V_0/E_R=20$, $U/J=258.54$, $\alpha =0.33$, ${\alpha }^{\prime }=0.32$. In each plot the average number of particles is $⟨N⟩\approx 15$, the temperature is $T/J=1$, and the lattice consists of 50 sites.

Figure 3

Density profiles $n_i$ and momentum density profiles $n(p)$ ($p$ in units $\hslash k$) for different fillings $N$ at an optical lattice potential $V_0/E_R=7$ and temperature $T/J=1$, for soft-core (solid line) and hard-core bosons (dashed line).