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arXiv:quant-ph/0601085 · New Journal of Physics

The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities

Herbert G. Winful

Published Jun 13, 2006• 65 citations
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Abstract

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.

DOI
10.1088/1367-2630/8/6/101

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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.
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.
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.
A general relation between group delay and stored electric and magnetic energies is presented for two-port networks. It generalizes the results of Dicke to situations where electric and magnetic stored energies differ. The general result is applied to tunneling evanescent waves in cutoff waveguides. It is shown explicitly that the group delay is equal to the dwell time plus a self-interference delay which is proportional to the net reactive stored energy. The Hartman effect, the saturation of group delay with length in cutoff waveguides, is explained on the basis of saturation of stored energy with guide length. It is pointed out that the anomalously short delays observed in tunneling experiments are not propagation delays and should not be associated with superluminal velocities. A strictly luminal energy velocity is derived and a method is suggested for the measurement of dwell time and energy velocity.
We show that the anomalously short delay times observed in barrier tunneling have their origin in energy storage and its subsequent release. The observed group delay is proportional to the energy stored. This delay is not a propagation delay and should not be linked to a velocity since evanescent waves do not propagate. The "Hartman effect", in which the group delay becomes independent of thickness for opaque barriers, is shown to be a consequence of the saturation of stored energy with barrier length.