Spatial correlations of trapped one-dimensional bosons in an optical lattice

We investigate a quasi-one-dimensional system of trapped cold bosonic atoms in an optical lattice by using the density-matrix renormalization group to study the Bose-Hubbard model at $T=0$ for experimentally realistic numbers of lattice sites. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. For typical parabolic traps we also confirm the widely used local-density approach for describing correlations in the limit of weak interaction. Finally, we note that the superfluid to Mott-insulating transition is seen most directly in the half-width of the interference peak.

Figure 1

(A) Sketch of the state diagram for $v_0=4∕{64}^2$. The insets sketch the shape of the density distribution in the states. (B) Sketch of the phase diagram of the homogeneous system: chemical potential $\mu$ vs $1∕u$. The different symbols in (B) mark the locations of the chemical potential values in the local-density approximation that correspond to the locations in the density profiles marked in (A).

Figure 2

Scaled correlations $C_j\left(r\right)$ [Eq. (2)] for different fixed sites $j$ are plotted as a function of $r$ for different values of $u$. For the coexistence region (b) a shallower trapping potential is chosen, such that the extents of both the incommensurate and the commensurate regions are large enough to allow identification of the algebraic and exponential behavior.

Figure 3

Quasiexact DMRG results for $C\left(j\right)$ (symbols) are compared to Eq. (3) obtained by the hydrodynamical approach [6] (lines). We used $n\left(x\right)=n_0[1-{(x∕R)}^2]$, where $n_0$ and $R$ are determined by fitting to the DMRG results (see insets). The uncertainties are obtained by varying the fit range in the sensible region away from the boundaries.

Figure 4

Interference pattern for the system with (A) open boundaries and with (B) parabolic trap for different values of $u$. Symbols are the results of the DMRG (maximal uncertainty $0.1$) and lines the results of the approximations explained in the text. The insets enlarge the scale of the $y$ axis. For a homogeneous system $u_c(n=1)\simeq 3.37$ is the critical value in the thermodynamic limit according to Ref. 9.

Figure 5

Half-width of the interference peak for the homogeneous (A) and the parabolic (B) system. Arrows in (A) mark the critical value of $u_c$ in the thermodynamic limit (solid and dashed for $n=1$ and $n=2$, respectively) according to Ref. 9. Arrows in (B) mark the three different regimes described in the text. To relate $u$ to the corresponding lattice depth $\left(V_{\text{lat}}∕E_r\right)$ of experiments, we assumed that the depths in the two perpendicular dimensions were fixed to $V_{\text{lat},\perp }∕E_r=50$.