Abstract
Self-accelerating beams are fascinating solutions of the Schr\"odinger equation. Thanks to their particular phase engineering, they can accelerate without the need of external potentials or applied forces. Finite-energy approximations of these beams have led to many applications, spanning from particle manipulation to robust in vivo imaging. The most studied and emblematic beam, the Airy beam, has been recently investigated in the context of the fractional Schr\"odinger equation. It was notably found that the packet acceleration would decrease with the reduction of the fractional order. Here, I study the case of a general nth-order self-accelerating caustic beam in the fractional Schr\"odinger equation. Using a Madelung decomposition combined with the wavelet transform, I derive the analytical expression of the beam's acceleration. I show that the non-accelerating limit is reached for infinite phase order or when the fractional order is reduced to 1. This work provides a quantitative description of self-accelerating caustic beams' properties.
Highlights
The Schrödinger equation has been at the heart of quantum mechanics for nearly a century
The initial condition does not have a complex phase, so the whole phase dynamics arises as a consequence of the dispersion relation φ(k, t ) = Dα|k|αt
I showed that the previously observed effect of wave-packet splitting at long times is a natural consequence of the linearization of the dispersion relation, which occurs when the fractional order is reduced down to 1
Summary
The Schrödinger equation has been at the heart of quantum mechanics for nearly a century. Only two decades ago was this fundamental equation of physics extended to fractional calculus [1], thanks to Laskin [2–4] He generalized Feynman’s path integral formulation of quantum mechanics to Lévy flights, i.e., beyond Brownian motion or Wiener stochastic processes that are based on usual Gaussian statistics. Fractional derivatives in partial differential equations have a long history of accurately modeling a wide range of physical systems, where their integer-order counterparts failed [5] This includes systems experiencing anomalous diffusion, like fluids in heterogeneous porous media with long-range spatial correlation decaying as a power law [6]. Among the many solutions of the Schrödinger equation, the Airy beams discovered in 1979 by Berry and Balazs [11] share a particular place These intriguing wave-packet solutions notably possess the apparent property of accelerating without external potentials or applied forces.
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