FIG. 2.

The current I versus time t for the (a) occupied and (b) unoccupied $\mathcal{S}$ states using an initially empty and filled $\mathcal{P}$, respectively, offset to frequency μ. Here, ${\omega }_{\mathcal{S}}=0.1{\text{ms}}^{-1}$ (a typical cold-atom tunneling frequency), ${\omega }_{\mathcal{P}}=0.1{\text{ms}}^{-1}$, and the total lattice length is N = 64. (c) The LDOS for $\mathcal{S}$ determined from Eq. (4) using the average current in the region $t\in \mathcal{T}=[2\text{ms},32\text{ms}]$ [demarcated by dashed lines in (a) and (b)], which minimizes transient and edge effects. The error bars (shaded regions) indicate the standard deviation of the current in the region $\mathcal{T}$ combined with the broadening error $2{\omega }_{\mathcal{P}}$. The Fermi level, ω_F, is found from the point where the occupied and unoccupied states cross over. (d) Integrated error of the LDOS versus total lattice size and (e) averaging time [for N = 32 (blue solid line) and N = 512 (green dashed line)]. The baseline error [when N → ∞, the dotted line in (d) and (e)] is set by the non-zero ${\omega }_{\mathcal{P}}$, which broadens the actual LDOS [see Eq. (A5)]. For short lattices and times, the error in the LDOS is already only a few percent. Thus, even modestsized systems or calculations can effectively reconstruct the LDOS.