Classical patterns in the quantum rainbow channeling of high energy electrons

We are investigating the quantum dynamics of a well-collimated electron beam transmitting through planar channels of the Si crystal. Electron states were represented by wave packets while the electron beam is treated as an ensemble of noninteracting wave packets. We have investigated the relationship between classical caustic pattern and anharmonicity of the potential and analyzed how quantum dynamic depends on the wave packet impact parameter and beam's angular divergence. We found that the extrema of the electron trajectory period, considered as a function of the impact parameter, determine the shape of the caustic pattern. All wave packet probability densities have multiple maxima generated by a self-interference. Their sum, that represents probability density of an ensemble, was found to depends strongly on the beam angular divergence. For small divergence, most peaks of different wave packets are aligned causing wavelike behavior of the ensemble. For moderate angular divergence maxima of some wave packets, are aligned with minima of others, resulting in the emergence of the classical caustic pattern. We have shown and experimentally confirmed that the only indication that the observed caustic pattern is generated by the quantum dynamics is a slight systematic shift of the corresponding caustic maxima.