Phase Coherence, Visibility, and the Superfluid–Mott-Insulator Transition on One-Dimensional Optical Lattices

We study the phase coherence and visibility of trapped atomic condensates on one-dimensional optical lattices, by means of quantum Monte Carlo simulations. We obtain structures in the visibility similar to the kinks recently observed experimentally by Gerbier et al. [Phys. Rev. Lett. 95, 050404 (2005); cond-mat/0507087]. We examine these features in detail and offer a connection to the evolution of the density profiles as the depth of the lattice is increased. Our simulations reveal that, as the interaction strength $U$ is increased, the evolution of superfluid and Mott-insulating domains stall for finite intervals of $U$. The density profiles do not change with increasing $U$. We show here that in one dimension the visibility provides unequivocal signatures of the melting of Mott domains with densities larger than 1.

Figure 1

(a) Visibility $\mathcal{V}$ and $S_{\max }$ as functions of the on-site interaction $U/t$. Initially, $\mathcal{V}$ decreases as $U/t$ increases. After $U/t\approx 6.3$, its rate of reduction decreases due to the freezing of the density profiles (see text). The fast decrease after $U/t\approx 7$ is related to the formation of the central Mott core. (b) Density profiles at four different values of $U/t$ and in the TG regime. The profiles for $U/t=6.3$ and 6.8 virtually coincide. The system under consideration has 40 bosons on an 80-site chain and a trapping potential $V_T{a}^2=0.01t$. Error bars on the data in this and all subsequent figures are smaller than the symbol sizes.

Figure 2

(a) Trapping ($E_T$) and interaction ($E_P$) energies as functions of $U/t$. (b) Ratio $\gamma =|E_p/E_k|$ of potential to kinetic energy, and the chemical potential ($\mu$) needed to maintain $N_b=40$. The results are for the system in Fig. 1.

Figure 3

(a) Visibility $\mathcal{V}$ as a function of the on-site interaction $U/t$, for $N_b=60$ and $V_T{a}^2=0.06t$, parameters which allow both $n=1$ and $n=2$ Mott regions to exist. For comparison, results for pure Mott-insulating phases with $n=1$ and $n=2$ in open lattices without a trap are also given. In the inset, the straight lines show the perturbative results of Ref. 11 in 1D (see text). (b) Integrated density over 20 lattice sites around the center of the trap.

Figure 4

Density profiles corresponding to the points signaled by arrows in Figs. 3, 5. The continuous line in (b) is the result in the TG regime. $N_b=60$ and $V_T{a}^2=0.06t$.

Figure 5

Ratio $\gamma =|E_p/E_k|$ of potential to kinetic energy, and the total energy of the system ($E_S$) vs $U/t$, for the system in Figs. 3, 4. $N_b=60$ and $V_T{a}^2=0.06t$.