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Comment on “Photonic tunneling times”
Aephraim Steinberg
To cite this version:
Aephraim Steinberg. Comment on “Photonic tunneling times”. Journal de Physique I, EDP Sciences,
1994, 4 (12), pp.1813-1816. �10.1051/jp1:1994222�. �jpa-00247034�
Classification
Physics Abstracts
42.50 03.658 73.40G
CoDlment on "Photonic tunneling times"
Aephraim M. Steinberg (*)
Department of Physics, U C. Berkeley, Berkeley, CA 94720, U-S-A-
(Received 13 June 1994, received in final form 10 August 1994, accepted 23 August 1994)
Abstract. In [il, some misleading statements were made about [2] and "zero-time" tunnel-
ing. We criticize these daims.
In [Ii, the authors compare their experiments on tunneling of classical microwave puises with
an experiment of ours [2] on single-photon tunneling. They write that in our experiment, "in spite of trie used quantum detection technique, trie optical patin difference itself was measured
in a classical procedure. In addition one should be aware that any medium with a phase velocity larger than c would also induce an apparent superluminal velocity. In contrast, in
our microwave experiment (one can distinguish) between phase and group velocity". These statements are incorrect.
Before proceeding, let us make clear that we are not in disagreement with trie ezperimentai
results of Nimtz et ai. but only with their characterization of our own work, and with their
use of trie wording "trie superluminal propagation of frequency limited signais is possible
It has still to be analyzed, whether a superluminal tunneled frequency limited wave packet represents a signal". Dur experiments, and those of Ranfagni et ai. [3], also find the peak of
a puise or a wave packet to appear on the far side of a barrier sooner than the incident puise
would have arrived had it travelled at the vacuum speed of light c. At the classical level, such effects have long been understood in terms of "puise-reshaping" [4-6]: the leading edge of a puise is transmitted through a barrier with higher probability than trie trailing edge, and thus
although ai no lime is the intensity on the far side of the barrier greater than it would be if the barrier were replaced with an eqmvalent length of vacuum the transmitted peak appears
shifted to earlier times. As tunnehng con be thought of as destructive interference between the varions Feynman patins which spend different lengths of time in the barrier and are ultimately
transmitted [7, 8], this cari be readily understood. While the rising edge of a Gaussian puise (for
a concrete example) is entering the barrier, the amplitude for a partiale to have made multiple reflections before being transmitted is negligible with respect to the amplitude for it to have
(*) Current address: Laboratoire Kastler-Brossel, Université Pierre et Marie Curie, Tour 12, Case 74, 75252 Paris cedex 05, France.
@ Les Editions de Physique 1994
1814 JOURNAL DE PHYSIQUE I N°12
made a single pass, smce the incident intensity is exponentially larger than it was a round-trip
transit time in the past; thus there is not a great deal of destructive interference. After the puise has been mteracting with the barrier for some time, by contrast, there is a significant amplitude for a partiale to have made multiple reflections, and the destructive interference is
more effective [9, loi. It has recently been shown exphcitly that such a picture provides a completely causal explanation of the superlummal appearance of a tunneling peak Ill, 12] by summing over only cattsaiiy retarded Feynman paths, one can reproduce the observed behavior of the peak, thus showing that this behavior does trot imply any faster-than-light dependence of
the output fields on fields at the input. Several other papers have also appeared to demonstrate that the tunneling elfects are perfectly consistent with relativistic causality [13-15], in other words, that they do trot allow communication faster thon c.
The confusion arises because of the tendency to conflate the arrivai of a puise peak with the receipt of a signal, and its emission with the transmission of a signal. But to emit a puise which appears Gaussian, one must begin sending it at least several rms widths in advance. It is with respect to this time that one should calculate a signal delay. Even in the absence of a tunnel barrier, some intensity would impinge on the detectors earlier than the peak; in fact, as discussed in [12-15, 18j (although in the context of superluminal effects related to absorption rather than reflection), this intensity will be greater than the intensity which emerges at the
same time if a barrier is inserted. Since we do not consider this early part of a freely-propagating puise to imply superluminal signal propagation, there is no reason to consider the even lower
intensity which traverses a tunnel barrier to constitute a superluminal signal, merely because mstead of continuing to rise after this early time, the mtensity begins to drop. The important question to ask is the followmg. If an experimenter decides at t
=
0 to send a signal, either a 1
or a 0, but prior to t
=0 his actions were mdependent of this eventual choice, then at what time
can a distant collaborator know whether the signal was a 1 or a 0? In other words, how fast does an abrttpt distttrbance propagate? As is well-known in dassical electromagnetism [17] and also discussed in [12, 13, 14, 15, 18], due to the high-frequency content of such a disturbance,
it will never propagate faster than c. Ail the experiments done on tunneling times so far have
involved smooth, essentially analytic puises, and in no way contradict this statement.
Aside from these interpretational differences, Nimtz et ai. make some factual errors in their
description of our experiment. First of ail, the delay-time measurement in [2] relies on a
quantum-mechanical interference effect [19], trot a "classical procedure". One photon traverses the barrier and the other travels in free space. If the two wave packets reach opposite sides of
a single beam splitter simultaneously, their Bose-Einstein statistics lead them to exit the same port, thus making it impossible to observe simultaneous photons at detectors placed at different output ports of the beam-splitter. (In practice, the wave packets do trot overlap perfectly, and while the coïncidence rate displays a dip, it never reaches zero; in [20], we nevertheless observed
a "visibility" (coincidence rate reduction) of 90%, but in [2j, trie setup was optimized for time- resolution instead, and the visibility was only about 60%.) The technique is nonclassical in
that it measures the overlap of single-photon wavepackets. While one might attempt to model
this elfect semiclassically, it has been shown that no such model cari explain visibility greater than 50$lo [21, 22], and that the standard semiclassical model of down-conversion would predict
visibilities far smaller than 1% [23].
More importantly, we have shown both theoretically [24] and experimentally [25] that this technique measures the group velocity and flot the phase velocity. This is because the interfer-
ence is maximized when trie two photon wauepackets reach trie beam-sphtter simultaneously.
If one takes longer than trie other, then it would in principle be possible to distinguish trie
two photons after the beam-splitter, and it is well known that the existence of such weicher
weg information destroys interference (see [20j and references therein). A superluminal phase
velocity would have no effect on our experiment, contrary to the daim of the authors.
Nimtz et ai. also interpret the superluminal group delay as follows: "a delay time is induced
in front of the barriers due to the interference of the incident and the reflected waves. The barrier crossing atone takes place in zero-time, since the measured transition time is indepen- dent of barrier length". This is almost true, but we would make one small correction. While it is correct that the group delay tends to a constant as the barrier thickness tends to infinity,
the authors' interpretation is incomplete. It is possible to apply the method of stationary phase inside the barrier region as well as outside. The wave in the bottier cari be expressed as
the sum of an exponentially decaying "evanescent" comportent and an exponentially growing
"anti-evanescent" comportent. Neither comportent individually accumulates any phase across
the barrier, but there is a phase difference between the two. For an opaque barrier, the evanes- cent component dominates for most of the barrier, but becomes equal in magnitude to the
anti-evanescent component at the for side of the barrier. Thus in the last exponential decay length 1/~ of the barrier region, the phase of the tunneling wave (essentially constant up until
that point) begms to change. About one-half of the total delay for transmission (for a massive
partiale, 2m /h~k; as discussed at length in [26-28j, the electromagnetic problem is analogous to that of tunneling of a partiale with mass m
=
&uJ /c~) comes from this effect within the barrier,
and one-half from the pre-barrier self-interference. Therefore, while the bulk of the barrier does
seem to be traversed in "zero time", the region at the very end is actually traversed at a finite speed (for a massive partiale, approximately the free-space propagation velocity &k/m, as it tutus eut). This is still in agreement with the authors' statement that the measured transition time is independent of barrier length, but we would be cautions about stating that the crossing takes place "m zero-time", since the amount of energy which is transmitted faits exponentially
with the barrier thickness. One cannot directly compare the arrivai times of peaks which tunnel
through different barriers, for these peaks need not have any direct relationship to one another [29j. Reviews of the long controversy over these issues can be found in [30-32j. Despite the
anomalously small tunnehng times, it is inexact to say that the barrier is traversed in zero time. The delay time is finite, but independent of barrier thickness, and can be attributed
in part to propagation within the barrier and in part to self-interference before entering the barrier. These anomalously small delay times are a property of smooth incident puises, arising through convolution of the incident puise with a perfectiy cattsai temporal response function in such a way that no information about the incident puise is conveyed faster than c.
Acknowledgments.
This work was supported by the U-S- Office of Naval Research under grant N00014-90-J-1259.
References
[il Nimtz G., Enders A. and Spieker H, J Phys I France 4 (1994) 565.
[2] Steinberg A.M
,
