On causality, apparent 'superluminality' and reshaping in barrier penetration

TL;DR

This paper analyzes quantum tunneling through barriers, showing how reshaping and complex shifts can create apparent superluminal effects without violating causality, by examining the role of the barrier as a beam splitter and the resulting pulse modifications.

Contribution

It introduces a detailed analysis of tunneling-induced pulse reshaping and complex shifts, clarifying the nature of apparent superluminal phenomena in quantum tunneling.

Findings

Barrier acts as a beam splitter creating shifted pulse copies

Reshaping can cause an apparent reduction and shift of the initial envelope

Superoscillatory transmission amplitude leads to an effective advancement

Abstract

We consider tunnelling of a non-relativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backwards relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac $\delta$-function, the transmission amplitude is superoscillatory for finite momenta and tunnelling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analysing apparent 'superluminality' in terms of spacial displacements helps avoid contradiction associated with time parameters such as the phase time.