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We derive a two-band finite-dimensional model for the description of the condensate tunnelling in an accelerating optical lattice, taking into account the fine Bloch band structure. The model reveals a very strong dependence of the final band populations on the initial populations and phases. Most importantly, additionally to the known asymmetric dependence on the nonlinearity, there is also a notable asymmetry in the sensitivity of the tunnelling probability to the nonlinearity-induced initial population of the Bloch band to which the tunnelling takes place. This fact can explain the experimentally observed unexpected independence of the upper-to-lower tunnelling probability on the nonlinearity. Finally, we compare the predictions of the two-band model with that of the well-known nonlinear Landau-Zener model and find disagreement when the two bands are initially populated. The disagreement can be qualitative and reveals itself even for a negligible nonlinearity. However, the two models agree remarkably well if just one band is populated initially.

We show that propagation of optical beams in periodic lattices induces power oscillations between the Fourier spectrum peaks, with the indices related by the Bragg resonance condition. In the spatial coordinates, this is reflected in the beam position oscillations. A simple resonant theory explains the phenomenon. The effect can be used for controlled generation of the Floquet-Bloch modes in photonic lattices.

We study Zener tunneling in two-dimensional photonic lattices and derive, for the case of hexagonal symmetry, the generalized Landau-Zener-Majorana model describing resonant interaction between high-symmetry points of the photonic spectral bands. We demonstrate that this effect can be employed for the generation of Floquet-Bloch modes and verify the model by direct numerical simulations of the tunneling effect.