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arXiv:quant-ph/0612102 · Physical Review A

Theoretical evidence for the superluminality of evanescent modes

Zhiyong Wang, Cai-Dong Xiong

Published Apr 11, 2007• 15 citations
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Abstract

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.

DOI
10.1103/physreva.75.042105

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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.
Contrary to mechanical waves, the two-slit interference experiment of single photons shows that the behavior of classical electromagnetic waves corresponds to the quantum mechanical one of single photons, which is also different from the quantum-field-theory behavior such as the creations and annihilations of photons, the vacuum fluctuations, etc. Owing to a purely quantum effect, quantum tunneling particles including tunneling photons (evanescent modes) can propagate over a spacelike interval without destroying causality. With this picture we conclude that the superluminality of evanescent modes is a quantum mechanical rather than a classical phenomenon.
Abstract Contrary to mechanical waves, the two‐slit interference experiment of single photons shows that the behavior of classical electromagnetic waves corresponds to the quantum mechanical one of single photons, which is also different from the quantum‐field‐theory behavior such as the creations and annihilations of photons, the vacuum fluctuations, etc. Owing to a purely quantum effect, quantum tunneling particles including tunneling photons (evanescent modes) can propagate over a spacelike interval. With this picture we conclude that the superluminality of evanescent modes is a quantum mechanical rather than a classical phenomenon.
Applying the fact that guided photons inside a waveguide can be treated as massive particles, one can study the superluminality of evanescent modes via showing that a massive particle can propagate over a spacelike interval, which corresponds to quantum tunneling effects. For this purpose, we treat the particle as a quantum reference frame, while attach an inertia observer to a classical reference frame, and then quantize the formulae for the Lorentz transformation between the quantum and classical reference frames, from which we obtain the conclusion that, owing to the Heisenberg's uncertainty relation, the particle can propagate over a spacelike interval.
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.
Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000). Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles.