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Slow and Fast Light (SL)

The meaning of group delay in barrier tunneling: A reexamination of superluminal group velocities

Herbert G. Winful

Published Jan 1, 2006• 2 citations
Publisher

Abstract

It is widely believed that photons tunnel through bandgaps with group velocities exceeding c. Here we examine the experimental evidence and show that the measured group delays are photon lifetimes as opposed to transit times.
DOI
10.1364/sl.2006.tud1

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We show that the group delay in tunnelling is not a traversal time but a lifetime of stored energy or stored probability escaping through both ends of the barrier. Because it is a lifetime associated with both forward (transmitted) and backward (reflected) fluxes, it cannot be used to define a group velocity for forward transit in cases where a wavepacket is mostly reflected. For photonic tunnelling barriers the group delay is identical to the dwell time which is also a property of an entire wavefunction with reflected and transmitted components. Theoretical predictions and experimental reports of superluminal group velocities in barrier tunnelling are re-interpreted.
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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.
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.
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.