Quantum temporal probabilities in tunneling systems

TL;DR

This paper develops a quantum framework for analyzing the timing of tunneling particles, introducing a method to define temporal probabilities and resolving the superluminal tunneling paradox.

Contribution

It presents a novel approach to construct temporal probabilities in quantum tunneling, applicable to relativistic particles, and clarifies the nature of tunneling delay times.

Findings

Defined a probability distribution for tunneling times.

Identified two key time parameters: delay time and decay rate.

Resolved the superluminal tunneling velocity paradox.

Abstract

We study the temporal aspects of quantum tunneling as manifested in time-of-arrival experiments in which the detected particle tunnels through a potential barrier. In particular, we present a general method for constructing temporal probabilities in tunneling systems that (i) defines `classical' time observables for quantum systems and (ii) applies to relativistic particles interacting through quantum fields. We show that the relevant probabilities are defined in terms of specific correlations functions of the quantum field associated with tunneling particles. We construct a probability distribution with respect to the time of particle detection that contains all information about the temporal aspects of the tunneling process. In specific cases, this probability distribution leads to the definition of a delay time that, for parity-symmetric potentials, reduces to the phase time of Bohm and Wigner. We apply our results to piecewise constant potentials, by deriving the appropriate junction conditions on the points of discontinuity. For the double square potential, in particular, we demonstrate the existence of (at least) two physically relevant time parameters, the delay time and a decay rate that describes the escape of particles trapped in the inter-barrier region. Finally, we propose a resolution to the paradox of apparent superluminal velocities for tunneling particles. We demonstrate that the idea of faster-than-light speeds in tunneling follows from an inadmissible use of classical reasoning in the description of quantum systems.