Zener Tunnelling in Periodic Two-Dimensional Photonic Lattices with Three-fold Symmetry

We study the interband wave tunnelling (Zener transitions) in two-dimensional hexagonal lattices with superimposed index gradient and derive simple analytical models capturing the essence of the tunnelling phenomenon. We find that the two-dimensional tunnelling in hexagonal lattices occurs at the high-symmetry points (e.g., the M and lceil points), and it involves either three or six Bloch bands being described by the corresponding multi-level Landau-Zener-Majorana (LZM) system. We derive several types of the generalized LZM systems and verify analytical predictions by direct numerical simulations of the full linear Schrodinger equation with tilted periodic potential.