Relativistic tunneling through two "transparent" successive barriers

TL;DR

This paper investigates relativistic quantum tunneling through two successive transparent barriers, revealing conditions under which superluminal effects appear and how they depend on particle energy and barrier parameters.

Contribution

It introduces a model for relativistic tunneling through two barriers, showing superluminal effects depend on energy and barrier ratios, extending nonrelativistic tunneling analysis.

Findings

Superluminal behavior occurs above certain energy thresholds.

Superluminal effects depend on barrier length and particle velocity ratios.

Relativistic tunneling differs from nonrelativistic case in transparency limits.

Abstract

In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this "transparent" limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length - free path as function of the ratio group velocity - speed of light.