{'fraction_non_alphanumeric': 0.024676125848241828, 'fraction_numerical': 0.0, 'mean_word_length': 5.21455938697318, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Study of the single-band Bloch oscillation effect with the pseudo-spectral method

Using the pseudo-spectral method for wave-packet evolution, we formulate and implement a new theoretical approach for studying the Bloch oscillation effect in a general number of dimensions. For this purpose, a novel recursive formula is obtained in the representation of Wannier functions for the infinitesimal time evolution of an electronic wave packet within the single-band approximation. This formula is valid for general position-dependent applied electric fields and arbitrary initial wave-packet shapes and crystal structures, and can also be generalized for time-dependent fields. For a tight-binding simple `cubium' band, the infinitesimal time evolution can always be expressed in terms of Bessel functions of integer order; and for the `empty-lattice model' band it can be expressed in terms of Fresnel functions for arbitrary electric fields also. As a further analytical application of our formalism, we derive an exact (finite-time) wave-packet evolution for the homogeneous and static electric field in the simple `cubium' band. For this case, an estimate of the numerical error of the pseudo-spectral method is obtained for the mean position of the wave packet. As an illustrative example of the numerical implementation of our theoretical formalism, we present a simple one-dimensional numerical simulation for a Gaussian wave packet moving in a parabolic potential well. Finally, a general proof of the unitarity of the predicted time evolution is also presented, and different properties of our formalism pointed out.