Breakdown of Dirac dynamics in honeycomb lattices due to nonlinear interactions

We study the dynamics of coherent waves in nonlinear honeycomb lattices and show that nonlinearity breaks down the Dirac dynamics. As an example, we demonstrate that even a weak nonlinearity has major qualitative effects on one of the hallmarks of honeycomb lattices: conical diffraction. Under linear conditions, a circular input wave packet associated with the Dirac point evolves into a ring, but even a weak nonlinearity alters the evolution such that the emerging beam possesses triangular symmetry, and populates Bloch modes outside of the Dirac cone. Our results are presented in the context of optics, but we propose a scheme to observe equivalent phenomena in Bose-Einstein condensates.

Figure 1

(a) Perfect honeycomb lattice that has two sites in a unit cell. (b) The first Brillouin zone with the high symmetry points. (c) The first two bands. (d) The vicinity of the intersection (Dirac) point.

Figure 2

(a) Input beam from the vicinity of $K_{+}$. (b) The output of the linear propagation. (c) The output of the nonlinear propagation with focusing nonlinearity. (d) The output of the nonlinear propagation with defocusing nonlinearity.

Figure 3

The Bloch mode decomposition of the input wave packet from the vicinity of the Dirac point. (a)–(d) The Bloch distribution of the first band $|g_1(\mathit{q})|{}^2$. (e)–(h) The Bloch distribution of the second band $|g_2(\mathit{q})|{}^2$. In each figure we write the corresponding population. Clearly, focusing nonlinearity transfers the energy to the first band, and defocusing nonlinearity transfers the population to the second band.

Figure 4

(a) The population in the first two bands during the propagation in the presence of defocusing nonlinearity, where $max\{|n_2\left|\right|\psi |{}^2\}=4\times {10}^{-5}$. (b) The final population imbalance as a function of the strength of the nonlinear refractive index.

Figure 5

(a) Calculation of ${\mathcal{H}}_{\mathit{cl}}(z)$ and ${\mathcal{H}}_L(z)$ in the presence focusing nonlinearity. (b) The nonlinear contribution to ${\mathcal{H}}_{\mathit{cl}}(z)$, which decreases rapidly during the propagation.

Figure 6

Contour plot of the first two bands. The Dirac points are denoted by $K_{\pm }$. The black arrows point in the directions where the group velocity is maximal.