The Hartman effect and weak measurements "which are not really weak"

TL;DR

This paper explores the Hartman effect through the lens of weak measurements, showing that tunnelling delay measurements are inherently weak and can be made precise despite low transmission probabilities.

Contribution

It demonstrates that wavepacket tunnelling acts as a 'self measurement' of delay, revealing the Hartman effect as a divergence of weak values and showing the possibility of precise measurements with narrow wavepackets.

Findings

Weak measurements in tunnelling are inherently weak due to low transmission probability.

The Hartman effect corresponds to diverging weak values as post-selection becomes unlikely.

Precise delay measurements are possible with narrow wavepackets despite wide barriers.

Abstract

We show that in wavepacket tunnelling localisation of the transmitted particle amounts to a quantum measurement of the delay it experiences in the barrier. With no external degree of freedom involved, the envelope of the wavepacket plays the role of the initial pointer state. Under tunnelling conditions such 'self measurement' is necessarily weak, and the Hartman effect just reflects the general tendency of weak values to diverge, as post-selection in the final state becomes improbable. We also demonstrate that it is a good precision, or 'not really weak' quantum measurement: no matter how wide the barrier d, it is possible to transmit a wavepacket with a width {\sigma} small compared to the observed advancement. As is the case with all weak measurements, the probability of transmission rapidly decreases with the ratio {\sigma}/d.