Chaos and band structure in a three-dimensional optical lattice

Classical chaos is known to affect wave propagation because it signifies the presence of broken symmetries. The effect of chaos has been observed experimentally for matter waves, electromagnetic waves, and acoustic waves. When these three types of waves propagate through a spatially periodic medium, the allowed propagation energies form bands. For energies in the band gaps, no wave propagation is possible. We show that optical lattices provide a well-defined system that allows a study of the effect of chaos on band structure. We have determined the band structure of a body-centered-cubic optical lattice for all theoretically possible couplings, and we find that the band structure for those lattices realizable in the laboratory differs significantly from that expected for the bands in an “empty” body-centered-cubic crystal. However, as coupling is increased, the lattice becomes increasingly chaotic and it becomes possible to produce band structure that has behavior qualitatively similar to the “empty” body-centered-cubic band structure, although with fewer degeneracies.

Figure 1

The directions of the polarization angles relative to the Cartesian axes containing the laser beams.

Figure 2

The 3D potential energy surface for one unit cell of the 3D lattice for different energy intervals: (a) $a=0.1$ and $V(x,y,z)<21.64\text{d.u.;}$ (b) $a=0.1$ and $48<V(x,y,z)<72\text{d.u.;}$ (c) $a=0.5$ and $V(x,y,z)<15.33\text{d.u.;}$ (d) $a=0.5$ and $80<V(x,y,z)<120\text{d.u.;}$ (e) $a=1.0$ and $V(x,y,z)<3.0\text{d.u.;}$ (f) $a=1.0$ and $120<V(x,y,z)<180\text{d.u.}$.

Figure 3

The potential energy surface $V_{\text{IM}}(y,z)$ for one unit cell the invariant manifold $(x=0,p_x=0)$ for coupling constants (a) $a=0.1$, (b) $a=0.5$, and (c) $a=1.0$. The energy scale of the shadings is shown in dimensionless units.

Figure 4

Poincare surfaces of section of $p_z$ versus $z$ for intersections condition $y={cos}^{-1}\left(\frac{-a}{1+a}\right)$ on the invariant manifold $p_x=0,x=0$: (a) $a=0.1,E=31\text{d.u.}$; (b) $a=0.1,E=41\text{d.u.}$; (c) $a=0.1,E=51\text{d.u.}$; (d) $a=0.5,E=18\text{d.u.}$; (e) $a=0.5,E=24\text{d.u.}$; (f) $a=0.5,E=64\text{d.u.}$; (g) $a=1.0,E=1.0d.u.$; (h) $a=1.0,E=44\text{d.u.}$; (i) $a=1.0,E=84\text{d.u.}$.

Figure 5

The Lyapunov exponents for trajectories on the InM (stars) and in the 3D unit cell (diamonds) for coupling constants and energies: (a) $a=0.1$ and $E=51\text{d.u.}$; (b) $a=0.5$ and $E=64\text{d.u.}$; (c) $a=1.0$ and $E=84\text{d.u.}$ All energies are above the high saddle so trajectories are free to move throughout the optical lattice, regardless of whether they are on the InM or off of it.

Figure 6

(a) The Brillioun zone; (b) The $\Gamma \text{−}H$ lowest six “empty lattice” energy bands; (c) the $\Gamma \text{−}N$ lowest six “empty lattice” energy bands, for a body centered cubic lattice (all in d.u.).

Figure 7

(a) The lowest six bands, for the $\Gamma \text{−}H$ direction of the Brillioun zone, for an optical lattice with periodicity $40\pi$ in each direction: (a) $a=0.1$; (b) $a=0.5$; (c) $a=1.0$ (all in d.u.).

Figure 8

(a) The lowest six bands, for the $\Gamma \text{−}N$ direction of the Brillioun zone, for an optical lattice with periodicity $40\pi$ in each direction: (a) $a=0.1$; (b) $a=0.5$; (c) $a=1.0$ (all in d.u.).

Figure 9

The Lyapunov exponents for trajectories in the 3D unit cell (diamonds) for coupling constants and energies: (a) $a=0.1$ and $E=13\text{d.u.}$; (b) $a=0.5$ and $E=12\text{d.u.}$; (c) $a=1.0$ and $E=5\text{d.u.}$ All energies are above the high saddle so trajectories are free to move throughout the optical lattice, regardless of whether they are on the InM or off of it.