Mott Domains of Bosons Confined on Optical Lattices

In the absence of a confining potential, the boson-Hubbard model exhibits a superfluid to Mott insulator quantum phase transition at commensurate fillings and strong coupling. We use quantum Monte Carlo simulations to study the ground state of the one-dimensional bosonic Hubbard model in a trap. Some, but not all, aspects of the Mott insulating phase persist. Mott behavior occurs for a continuous range of incommensurate fillings, very different from the unconfined case, and the establishment of the Mott phase does not proceed via a traditional quantum phase transition. These results have important implications for interpreting experiments on ultracold atoms on optical lattices.

Figure 1

The evolution of the local density $n_i$ as a function of position $x$ and increasing the total number of bosons. The trap curvature is $V_c=0.008$, $L=100$, and the onsite repulsion is $V_0=4$. At low fillings the system is in a superfluid phase. Mott insulating behavior appears as the density is increased, but then at yet larger fillings a superfluid begins to form at the center of the insulating region.

Figure 2

Cuts across Fig. 1 show the compressibility profile ${\kappa }_i$ (solid line) associated with the local density $n_i$ (circles). The fillings are $N_b=25$ (a), 33 (b), 50 (c), and 60 (d). ${\kappa }_i$ is very small when $n_i=1$. For the arrows see text.

Figure 3

$N_b$ (number of bosons) as a function of chemical potential, $\mu$ for $V_0=4.5$. No globally incompressible Mott plateau is observed. Inset shows the unconfined case.

Figure 4

The local compressibility as a function of local density for $V_0=4$, $N_b=35$ ($\circ$) and $N_b=80$ ($\diamond$) and for $V_0=4.5$, $N_b=90$ ($△$) and $N_b=141$ ($▽$). $\kappa$ decreases linearly with $n$ as the Mott region, $n=1$, is approached from below or above. This behavior is dramatically different from the unconfined system where $\kappa$ diverges.

Figure 5

The state diagram of correlated bosons in a quadratic confining potential. The solid lines are to guide the eye, and the dashed lines are extrapolations. See text for details.