Single-Atom Density of States of an Optical Lattice

We consider a single atom in an optical lattice, subject to a harmonic trapping potential. The problem is treated in the tight-binding approximation, with an extra parameter $\kappa$ denoting the strength of the harmonic trap. It is shown that the $\kappa \to 0$ limit of this problem is singular, in the sense that the density of states for a very shallow trap ($\kappa \to 0$) is qualitatively different from that of a translationally invariant lattice ($\kappa =0$). The physics of this difference is discussed, and densities of states and wave functions are exhibited and explained.

Figure 1

Numerically determined wave functions for a trap with $\kappa a^2=t/100$. Each wave function has been offset along the $y$-axis by its energy, in units of the hopping integral. The numerical calculation uses a lattice with $N=130$ sites. For the energies considered, there are no noticeable numerical artefacts. The lower and upper solid curves correspond to the classical and Bragg turning points, Eqs. (2),(3).

Figure 2

Finite-size scaling of the binned DOS for 1D lattices with finite strength of the trapping potential: $\kappa a^2/t=5·{10}^{-6}(+)$, ${10}^{-5}(\times )$, ${10}^{-4}(*)$, and ${10}^{-3}(\square )$. In all cases the numerical cutoff energy, ${ϵ}_N$, lies outside the energy range showed in the plot. This is illustrated in the inset: it shows the DOS for fixed $\kappa a^2=t/1000$, but calculated using lattices with different numbers of sites: $N=300(\square )$, $350(*)$, $420(\times )$, and $2000(+)$.

Figure 3

DOS of a single atom in an optical lattice for (a) $d=1$ and (b) $d=2$. Curves: continuum-limit results obtained within the WKB approximation. Histograms: numerical diagonalization for finite-size systems with trap strengths $\kappa =5\times {10}^{-6}t/a^2$ (a) and $\kappa =1.5\times {10}^{-2}t/a^2$ (b), using lattices with $N=5000$ and 10 000 sites, respectively.