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arXiv:2008.06573 · The European Physical Journal D

Interaction of a wave packet with potential structures moving with acceleration

М. А. Захаров, G. V. Kulin, A. I. Frank

Published Feb 1, 2021• 4 citations
arXivHTMLPDF
DOI
10.1140/epjd/s10053-020-00023-1

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American Journal of Physics•Jun 1, 1994•50 citations
The development of ideas about the optical phenomenon called the accelerating matter effect led to the hypothesis of the existence of a very general acceleration effect. Its formulation is that the result of the particle interaction with any object moving with acceleration should be a change in its energy and frequency. The validity of the acceleration effect hypothesis in quantum mechanics has recently been confirmed by numerically solving a number of problems related to the interaction of a wave packet with potential structures moving with acceleration. If these ideas are true, they can be fully attributed to the case of neutron scattering on the atomic nuclei of accelerating matter. Since the time of neutron interaction with the nucleus is very short, the observation of the acceleration effect during scattering by nuclei requires them to move with a very high acceleration. This goal can be achieved if centripetal acceleration is used.
We study the dynamics of an Airy wave packet moving in a one-dimensional lattice potential. In contrast to the usual case of propagation in a continuum, for which such a wave packet experiences a uniform acceleration, the lattice bounds its velocity, and so the acceleration cannot continue indefinitely. Instead, we show that the wave packet's motion is described by relativistic equations of motion, which surprisingly arise naturally from evolution under the standard nonrelativistic Schr\"odinger equation. The presence of the lattice potential allows the wave packet's motion to be controlled by means of Floquet engineering. In particular, in the deep relativistic limit when the wave packet's motion is photonlike, this form of control allows it to mimic both standard and negative refraction. Airy wave packets held in lattice potentials can thus be used as powerful and flexible simulators of relativistic quantum systems.
Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.