Tunneling times with covariant measurements

TL;DR

This paper demonstrates that tunneling always reduces the probability of a signal arriving before a deadline, countering the idea that tunneling can enable faster communication despite the Hartman effect.

Contribution

It proves that the trade-off between tunneling speed and intensity favors no faster signals, using a covariant measurement framework for various initial states and detectors.

Findings

Tunneling decreases the probability of early arrival for all states and detectors.

The Hartman effect does not translate into faster signals in terms of detection probability.

The results hold for arbitrary positive, compactly supported potentials and detection schemes.

Abstract

We consider the time delay of massive, non-relativistic, one-dimensional particles due to a tunneling potential. In this setting the well-known Hartman effect asserts that often the sub-ensemble of particles going through the tunnel seems to cross the tunnel region instantaneously. An obstacle to the utilization of this effect for getting faster signals is the exponential damping by the tunnel, so there seems to be a trade-off between speedup and intensity. In this paper we prove that this trade-off is never in favor of faster signals: the probability for a signal to reach its destination before some deadline is always reduced by the tunnel, for arbitrary incoming states, arbitrary positive and compactly supported tunnel potentials, and arbitrary detectors. More specifically, we show this for several different ways to define ``the same incoming state'' and ''the same detector'' when comparing the settings with and without tunnel potential. The arrival time measurements are expressed in the time-covariant approach, but we also allow the detection to be a localization measurement at a later time.