Imaginary past of a quantum particle moving on imaginary time

TL;DR

This paper explores the concept of tunnelling time in quantum particles, revealing contradictions in the interpretation of imaginary durations and questioning whether tunnelling is truly instantaneous or has a finite duration.

Contribution

It critically examines the notion of tunnelling time through analytical continuation and the particle's memory, highlighting conceptual issues in defining tunnelling duration.

Findings

Imaginary time in tunnelling does not correspond to a clear finite duration.

Particles show little memory of the field's past, implying near-zero transit time.

The question of tunnelling time is fundamentally ill-posed.

Abstract

The analytical continuation of classical equations of motion to complex times suggests that a tunnelling particle spends in the barrier an imaginary duration $i|\mathcal T|$. Does this mean that it takes a finite time to tunnel, or should tunnelling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards $|\mathcal T|$ being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it "remembers" very little of the field's past behaviour, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill-posed, and we explain why.