Abstract
We demonstrate optical Bloch oscillation (OBO) and optical Zener tunneling (OZT) in the fractional Schrödinger equation (FSE) with periodic and linear potentials, numerically and theoretically. We investigate in parallel the regular Schrödinger equation and the FSE, by adjusting the Lévy index, and expound the differences between the two. We find that the spreading of the OBO decreases in the fractional case, due to the diminishing band width. Increasing the transverse force, due to the linear potential, leads to the appearance of OZT, but this process is suppressed in the FSE. Our results indicate that the adjustment of the Lévy index can effectively control the emergence of OBO and OZT, which can inspire new ideas in the design of optical switches and interconnects.
Highlights
The convenience of quantum-optical analogies is that they map the temporal evolution of wave functions in quantum phenomena onto the spatial propagation of optical fields in photonic devices
For optical Bloch oscillation (OBO), the beam will be confined to the first Brillouin zone (FBZ) [−π/d0,π/d0], while for optical Zener tunneling (OZT), the beam will escape to the higher Brillouin zone
(i) the momentum of the beam increases linearly from 0 under the action of the transverse force, (ii) the momentum changes its sign and the beam jumps across the FBZ when it undergoes the Bragg reflection at k = π/d0, (iii) the momentum continues to increase linearly until it reaches 0
Summary
The convenience of quantum-optical analogies is that they map the temporal evolution of wave functions in quantum phenomena onto the spatial propagation of optical fields in photonic devices. To the best of our knowledge, Bloch oscillations in the fractional Schrödinger equation were not investigated before. This task is undertaken in this paper. Compared to the standard Schrödinger equation, it features the fractional Laplacian operator instead of the regular one This substitution brings a profound change in the behavior of the wave function. Since the fractional Laplacian causes non-parabolic dispersion, which suggests the possibility of directly modulating the dispersion of a physical system, it is not easy to find real physical systems described by the FSE To overcome this difficulty, a potential link between the FSE and the beam propagation in honeycomb lattice was established, based on the Dirac-Weyl equation[35]. We believe that our research will enrich the OBO and OZT family of phenomena, and inspire new ideas in the research of FSE
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