Mobility edges in bichromatic optical lattices

We investigate the localization properties of single-particle eigenstates in bichromatic one-dimensional optical lattices. Whereas such a lattice with a sufficiently deep primary component and a suitably adjusted incommensurate secondary component provides an approximate realization of the Harper model, the system’s self-duality is broken when the lattice is comparatively shallow. As a consequence, the sharp metal-insulator transition exhibited by Harper’s model is replaced by a sequence of mobility edges in realistic bichromatic optical lattices that do not reach the tight-binding regime.

Figure 1

Wannier function $w_0\left(z\right)$ for the lowest energy band of a monochromatic cosine lattice (3) with depth $V_0∕E_r=3$ (full line), compared to its harmonic-oscillator approximation (9) (dashes). We employ the dimensionless coordinate $z=k_{L}x$, so that $\Delta z=\pi$ corresponds to one lattice period.

Figure 2

Decay of the square of the exact Wannier function $w_0\left(z\right)$ depicted in Fig. 1.

Figure 3

Location of the branch point $h_0∕k_L$, as a function of the depth $V_0∕E_r$ of a monochromatic cosine lattice (full line). Also shown are the estimates (14) (short dashes) and (15) (long dashes).

Figure 4

Plot of $\ln (J_n∕E_r)+n\pi h_0∕k_L$ vs $\ln n\pi$ for $V_0∕E_r=3$ (diamonds), using $h_0∕k_L=0.3661$. The slope $-3∕2$ expected for large $n$ is already showing up for small $n$.

Figure 5

Average extension ${\sigma }_{\mathrm{av}}=\overline{\sigma }$ for the eigenstates of bichromatic optical lattices, as functions of the amplitude $V_1$ of the secondary lattice (4), with incommensurability parameter $g=(\sqrt{5}-1)∕2$. The amplitudes of the principal lattices (3) are $V_0∕E_r=3,4,5,6,8,10,15,20$ (right to left).

Figure 6

Extensions ${\sigma }_{\alpha }$ of individual eigenstates of a bichromatic optical lattice, with depth $V_0∕E_r=3$ of the primary component. As in Fig. 5, we set $g=(\sqrt{5}-1)∕2$ and $N=1000$. The amplitudes of the secondary lattices are $V_1∕E_r=0.25$ (a), 0.293 (b), and 0.35 (c).

Figure 7

Comparison of the estimate (21) for $g=(\sqrt{5}-1)∕2$ (full line) with those values of $V_1∕E_r$ which lead to an average extension of $\overline{\sigma }=0.5$ (dots), as determined from Fig. 5.