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arXiv:2205.15375 · Physical Review A

Superluminal tunneling times without superluminal signaling: Fading of the MacColl-Hartman effect at early times

Randall S. Dumont, Tom Rivlin

Published May 19, 2023• 5 citations

Abstract

A curious feature of quantum tunneling known as the MacColl-Hartman effect results in the numerical observation that particles can traverse a barrier with effective superluminal speed. However, because tunneling is never certain, any attempt to use this effect to send a signal faster than light would require sending many particles. In this work, we consider sending---in parallel, without interactions between particles---sufficiently many particles to ensure at the least one of them tunnels. In this case, in spite of the time advance of the mean time for a single tunneling particle, the mean time to send one bit of information is larger for tunneling particles than for the same number of free photons. This removes any possibility of superluminal signaling. We show that the mean time to send one bit using $N$ particles is determined by the early-time tail of the distribution of tunneling times for one particle and that, when this early-time tail is highly accurately modeled using steepest descent, the MacColl-Hartman effect is seen to fade away.

DOI: 10.1103/physreva.107.052212

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A curious feature of quantum tunneling known as the MacColl-Hartman effect results in the numerical observation that particles can traverse a barrier with effective superluminal speed. However, because tunneling is never certain, any attempt to use this effect to send a signal faster than light would require sending many particles. In this work, we consider sending---in parallel, without interactions between particles---sufficiently many particles to ensure at the least one of them tunnels. In this case, in spite of the time advance of the mean time for a single tunneling particle, the mean time to send one bit of information is larger for tunneling particles than for the same number of free photons. This removes any possibility of superluminal signaling. We show that the mean time to send one bit using

N

particles is determined by the early-time tail of the distribution of tunneling times for one particle and that, when this early-time tail is highly accurately modeled using steepest descent, the MacColl-Hartman effect is seen to fade away.

Experiments done with single photons in the early 1990's produced a surprising result: that single photons pass through a photonic tunnel barrier with a group velocity faster than the vacuum speed of light. This result has stimulated intense discussions related to causality, the speed of information transfer, the nature of barrier tunneling and the meaning of group velocity. The superluminality of tunneling photons is now textbook material, although the authors note that controversy still remains. Another paradoxical result, known as the Hartman effect, is that the tunneling time of the photons becomes independent of barrier length in the limit of opaque barriers. In this paper we examine the meaning of group velocity in the context of barrier tunneling. We ask whether a single tunneling photon can be described by a group velocity and whether the short group delays imply superluminal group velocity. We resolve the paradox of the Hartman effect and show that the predicted and measured group delays are not transit times but photon lifetimes.

Abstract This is a review of studies of quantum tunneling, which is described by the one-dimensional Schrödinger equation, and electromagnetic tunneling, where “superluminal” velocities and times of tunneling are considered. Integral and integrodifferential equations have been presented to describe tunneling. According to these equations, superluminal motion is impossible. The paradoxical Hartman effect has been discussed and explained. It has been shown that the velocity of passage of a particle in a beam through a barrier in the case of steady-state and time-dependent quantum tunneling is equal to the velocity of its incidence on the barrier and quasiphotons inside any layer of matter carry the energy always at a subluminal velocity. However, the tunneling time of a single particle or a photon is meaningless.

I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.

Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.

We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.

A unified approach to the time analysis of tunnelling of nonrelativistic particles is presented, in which Time is regarded as a quantum-mechanical observable, canonically conjugated to Energy. The validity of the Hartman effect (independence of the Tunnelling Time of the opaque barrier width, with Superluminal group velocities as a consequence) is verified for all the known expressions of the mean tunnelling time. Moreover, the analogy between particle and photon tunnelling is suitably exploited. On the basis of such an analogy, an explanation of some recent microwave and optics experimental results on tunnelling times is proposed. Attention is devoted to some aspects of the causality problem for particle and photon tunnelling.