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Journal de Physique I

Photonic tunneling times

G. Nimtz, Axel Enders, Harald Spieker

Published Apr 1, 1994• 89 citations
Publisher

Abstract

We report on resonant and non-resonant photonic tunneling experiments. Resonant tunneling of wave packets has revealed a light localization, whereas the non-resonant barrier tunneling has corresponded to a zero-time barrier crossing. For a frequency-limited signal the non-resonant tunneling transmission time yielded superluminal group velocities, whereas in the case of a resonant transition the velocities are much slower than c.

DOI
10.1051/jp1:1994160

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We report on resonant and non-resonant photonic tunneling experiments. Resonant tunneling of wave packets has revealed a light localization, whereas the non-resonant barrier tunneling has corresponded to a zero-time barrier crossing. For a frequency-limited signal the non-resonant tunneling transmission time yielded superluminal group velocities, whereas in the case of a resonant transition the velocities are much slower than c.
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.
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.
Experiments at Berkeley and elsewhere which show that the process of tunneling is apparently superluminal will be reviewed. Conflicting theories for the tunneling time will be compared with experiment. The tunneling particle in the Berkeley experiments was the photon. The measurement of the tunneling time utilized a two-photon light source (spontaneous parametric down-conversion), a Hong-Ou-Mandel interferometer, and a coincidence counter of photon pairs. The tunnel barrier consisted of a photonic-bandgap medium excited at midgap. We find that the peak of a tunneling single-photon wave packet appeared on the far side of the barrier 1.47 ± 0.21 fs earlier than the peak of a wave packet which traveled an equal distance in air. Group velocities which exceed c can also occur in transparent media with inverted atomic populations. Tachyon-like excitations, i.e., collective normal modes of atoms coupled to electromagnetic radiation which possess tachyon-like dispersion relations, occur in these media. However, relativistic causality is not violated by any of these superluminal phenomena.
Photonic tunnelling experiments have shown that i) the Einstein energy relation is violated, ii) the tunnelling process is non‐local, iii) the signal velocity is faster than light, i.e. superluminal, iv) the tunnelling signal is not observable, since photonic tunnelling is described by virtual photons, and v) according to the experimental results the signal velocity is infinite inside the barriers, implying that tunnelling instantaneously acts at a distance. We think these properties are not compatible with the claims of many text books on Special Relativity, see for instance Refs. [1].
Experiments have shown that individual photons penetrate an optical tunnel barrier with an effective group velocity considerably greater than the vacuum speed of light. The experiments were conducted with a two-photon parametric down-conversion light source, which produced correlated, but random, emissions of photon pairs. The two photons of a given pair were emitted in slightly different directions so that one photon passed through the tunnel barrier, while the other photon passed through the vacuum. The time delay for the tunneling photon relative to its twin was measured by adjusting the path length difference between the two photons in a Hong-Ou-Mandel interferometer, in order to achieve coincidence detection. We found that the photon transit time through the barrier was smaller than the twin photon’s transit time through an equal distance in vacuum, indicating that the process of tunneling in quantum mechanics is superluminal. Various conflicting theories of tunneling times are compared with experiment.
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.