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Journal de Physique I

On superluminal barrier traversal

Axel Enders, G. Nimtz

Published Sep 1, 1992• 238 citations
Publisher

Abstract

We report on microwave measurements of barrier traversal time of electromagnetic wave packets in an undersized waveguide. This time was significantly shorter than the ratio $L/c$, with $L$ the barrier length and $c$ the velocity of light in vacuum. Consequently the transport velocity for the wave packet inside the barrier has to be superluminal, i.e. larger than $c$.

DOI
10.1051/jp1:1992236

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We report on microwave measurements of barrier traversal time of electromagnetic wave packets in an undersized waveguide. This time was significantly shorter than the ratio L/cL/cL/c, with LLL the barrier length and ccc the velocity of light in vacuum. Consequently the transport velocity for the wave packet inside the barrier has to be superluminal, i.e. larger than ccc.
The tunneling of particles is in direct analogy with the transmission of evanescent electromagnetic waveguide modes as has been shown quite recently. We compare experimental data of an electromagnetic wave packet traversing an evanescent waveguide region with theoretical values derived for particle tunneling through a rectangular potential barrier. The transmission time was deduced by transformation of the experimental frequency data to the time domain. The data are in agreement and reveal superluminal wave-packet velocities for opaque evanescent regions.
The mechanism of superluminal traversal time is investigated from the viewpoint of interference between multiple finite wave packets, due to the multiple reflections inside the well or barrier. In the case of potential well traveling that is classically allowed, each of the successively transmitted constituents is delayed by a subluminal time. When the thickness of the well is much smaller in comparison with a characteristic length of the incident wave packet, the reshaped wave packet with superluminal traversal time in transmission maintains the pro. le of the incident wave packet. In the case of potential barrier tunneling that is classically forbidden, though each of the successively transmitted constituents is delayed by a time that is independent of the barrier thickness, the interference between multiple transmitted constituents explains the barrier thickness dependence of the traversal time for thin barriers and its barrier thickness independence for thick barriers. This manifests the nature of the Hartman effect. Copyright (c) EPLA, 2008.
Using a two-photon interferometer, we have measured the time delay for a photon to tunnel across a barrier consisting of a 1.1-\ensuremath{\mu}m-thick 1D photonic band-gap material. The peak of the photon wave packet appears on the far side of the barrier 1.47\ifmmode\pm\else\textpm\fi{}0.21 fs earlier than it would if it were to travel at the vacuum speed of light c. Although the apparent tunneling velocity (1.7\ifmmode\pm\else\textpm\fi{}0.2)c is superluminal, this is not a genuine signal velocity, and Einstein causality is not violated. The measured tunneling time is consistent with the group delay (``phase time''), but not with the semiclassical time.
The time delay connected with the transmission of an electromagnetic wave across a barrier is investigated. The stationary phase method implies that the effective group velocity of the transmitted wave, inside the barrier, can exceed the velocity of light in vacuo. Consequently the transmission of a signal with a sharp front is studied to determine the upper bound for the signal velocity. The approximation employed does not yield a conclusive answer in general, while for the special case of critical incidence the proof is given that the effective signal velocity does not exceed the velocity of light in vacuo.
Though both theoretical and experimental investigations have revealed the superluminal behavior of evanescent electromagnetic waves, there are many disputes about the physical meaning and validity of such superluminal phenomenon, which is due to the fact that the traditional investigations are based on the theory of tunneling time and concerned with the problem of what the group velocity of evanescent waves means. In this paper, by studying the quantum probability amplitude for photons to propagate over a spacelike interval along an undersized waveguide, we present theoretical evidence for such superluminality.