Static-Field Tunneling Ionization in Space-Fractional Quantum Mechanics

TL;DR

This paper develops an analytical tunneling ionization model within space-fractional quantum mechanics, revealing how nonlocal dispersion modifies ionization scaling laws and providing a benchmark for future strong-field studies.

Contribution

It introduces a closed-form fractional tunneling exponent in static fields, extending conventional models to nonlocal quantum dynamics with explicit fractional scaling.

Findings

Derives a fractional tunneling exponent with modified $I_p$ scaling.

Identifies a $ ext{sin}(rac{ ext{ extpi}}{ ext{ extalpha}})$ factor related to nonlocality.

Provides a validation protocol for fractional quantum simulations.

Abstract

Tunneling ionization in static or slowly varying electric fields is a cornerstone of strong-field physics and provides the entry point for semiclassical descriptions of above-threshold ionization and high-harmonic generation. In conventional quantum mechanics, the Perelomov--Popov--Terent'ev (PPT) theory and its Ammosov--Delone--Krainov (ADK) form yield an ionization rate whose defining feature is an exponential dependence governed by an under-barrier (imaginary-time) action. Here we develop an analytical ADK-like tunneling model within \emph{space-fractional} quantum mechanics, where the quadratic kinetic energy is replaced by the Riesz fractional Laplacian of order $1<\alpha\le2$. Working in a static electric field in the length gauge, we derive a closed-form tunneling exponent for a triangular exit barrier. The fractional kinetic operator deforms the conventional $I_p^{3/2}$ scaling to $I_p^{1+1/\alpha}$ and introduces a characteristic $\sin(\pi/\alpha)$ factor encoding the complex-phase structure associated with nonlocal dispersion. We position this benchmark relative to prior tunneling studies in fractional quantum mechanics (primarily scattering through model barriers and fractal potentials) and provide a validation protocol for testing the exponent in time-dependent simulations of the fractional Schr\"odinger equation under a constant field. The result establishes a transparent reference for static-field ionization in nonlocal quantum dynamics and a baseline for strong-field approaches extensions.