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Tunneling through the Lightspeed Barrier

by John G. Cramer

Alternate View Column AV-75
Keywords: quantum mechanics barrier penetration tunneling superluminal faster-than-light ftl
Published in the December-1995 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 5/2/95 and is copyrighted ©1995 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.

    A controversy is presently raging in certain physics journals and conferences over whether Einstein's speed of light barrier has been breached by light itself. In particular, Prof. Günther Nimtz and his group at the University of Cologne, Germany have published results showing that they used microwaves to transmit what might be interpreted as a signal, Mozart's 40th Symphony, over a path length of 11.4 centimeters at 4.7 times the speed of light. In this column, I want to examine this faster-than-light (FTL) controversy and its implications.

    On of the earliest triumphs of quantum mechanics, when the theory was first formulated in the 1920s, was George Gamow's use of quantum tunneling to explain the phenomenon of alpha decay. It was well known then that certain heavy radioactive nuclei, including radium then widely used for glow-in-the-dark clock dials, would spontaneously and unpredictably spit out an energetic helium-4 nucleus (also called an alpha particle) and be transformed into a new lighter nucleus with two fewer nuclear charges. Alpha radioactivity is often very improbable, and this takes the form of alpha-decay half-lives measured in thousands or even millions of years. The question was, how and why did the alpha particle escape its prison in the nucleus, where it should have been tightly and permanently bound?

    Gamow explained alpha decay as a example of quantum tunneling. He showed that while the particle can freely move with normal positive energy both within the nucleus and far away, there is an intermediate "forbidden zone" in which its net energy is negative. Newtonian mechanics rigorously excludes particles of negative energy, so in that theory the alpha particle could never pass through the forbidden region. Quantum mechanics, however, is somewhat more permissive of negative energy particles . In the allowed regions the alpha particle's motion is described by a quantum wave function that is a complex sine-wave which encodes its direction and momentum. In the forbidden zone the same wave function becomes a "dying" negative exponential, a rapidly decreasing function of penetration distance. This, Gamow, explained, is how the alpha particle escapes its prison in the nucleus. It "leaks" through the forbidden zone as a dying exponential, emerging on the far side of the barrier with a surviving wave function that is tiny but non-zero . Gamow converted this small external wave function to an escape probability and at a stroke explained how the alpha decay process is possible and why it is sometimes so improbable. Gamow's work has become a standard example of the application of the quantum formalism that is presented in most intermediate quantum mechanics textbooks.

    One interesting question not addressed in Gamow's work, however, is how long the alpha particle, or its equivalent, spends in leaking through a barrier. This question was finally addressed in 1955 by Eugene Wigner and his student Leonard Eisenbud, who calculated the time required for the peak of the wave packet to pass trough the barrier. Their conclusion was very strange. They found that under certain circumstances, this transit time reaches a constant value that is independent of the width of the barrier. For a wide barrier and a constant time the corresponding transit velocity, i.e., distance divided by time, can easily become faster than the velocity of light.

    Einstein's principle of causality, arising from the equivalence of inertial reference frames, states that in any reference frame an event occurring as some point in space-time can, at some point a distance L away, have no observable consequences earlier than a time L/c after the original event, where c is the velocity of light. The usual shorthand version of this, which begs the definition of a "signal", is to say that no signal can be transmitted faster than c. At the time of publication the apparent violation of Einsteinian causality implicit in the Eisenbud-Wigner calculation was ignored because it used the non-relativistic Schrödinger formalism. Later Hartman derived the same result from a more rigorous formalism.

    The paradox of FTL barrier transit velocity has not received much interest until recently, when it has been confronted by new experimental work from two independent directions. Work in experimental laser optics performed at the University of California at Berkeley by Raymond Chiao and his group has used interferometry and "down-converting" crystals to perform photon-pair timing measurements at the level of about a femtosecond (10-15 sec). They have been able to clock the passage of single photons of visible light through an optical barrier made of multiply reflecting layers of transparent material acting as a destructive interference filter that selectively absorbs the photons of interest. Most of the photons striking the barrier are absorbed inside and do not emerge, but the few surviving photons traverse the barrier in a time of about 1.5 femtoseconds. The velocity derived from this measurement by dividing barrier thickness by transit time is 1.7 times the velocity of light.

    However, Chiao and his group do not characterize their result as an FTL violation of Einsteinian causality. They point out that the operation of their interference filter requires multiple reflections at the many layer interfaces of their device, and a certain time is required for the destructive interference effect to build up. Thus, their barrier has a higher transmission for the part of the photon wave packet that arrives first, and much stronger suppression of the part of the wave packet that arrives later. This will cause the wave envelope to "advance", with the early part of the wave envelope dominating the transmission process. The photon thus appears to emerge earlier because of the time-variable transmission of the filter.


    The experimental measurements of Nimtz and his co-workers at the University of Cologne operate in a very different domain. They use 8.7 GHz microwaves (free space wavelength 3.4 cm) traveling in a rectangular wave guide that contains a "barrier" section of reduced dimensions, in which the incoming microwaves are strongly attenuated. Nimtz and his group have performed both time-domain and frequency-domain measurements demonstrating that their experimental configuration extends well into the region predicted by Eisenbud and Wigner and by Hartman where the barrier transit time becomes constant. In particular, Aichmann and Nimtz have recently transmitted Mozart's 40th Symphony as frequency modulated microwaves through an 11.4 cm length of barrier wave guide at an FTL group velocity of 4.7 c, receiving audibly recognizable music from the microwave photons that survived their barrier passage. The transit time through the barrier was about 81 picoseconds and was observed to be constant for barriers with widths varying from 4.0 cm to 11.4 cm.


    The work of the Nimtz group raises the question of whether Einsteinian causality has in fact been violated and has spawned a controversy. The players in it, as is characteristic of careful scientists, have engaged in a careful tableau of discussion of various definitions of "velocity" and "causality" that skirt any claim of the fall of Einsteinian causality. One contingent has suggested that the FTL speed in the Nimtz experiment, like that of the Chiao group, might result from time-varying transmission probability in the barrier waveguide. The other argues that the filter advance of the Chiao group is peculiar to their filter type and does not apply to the Nimtz results.

    What is meant by a signal has also been a matter of debate. For example, Mozart's 40th Symphony, while it is certainly a signal in some sense, does not contain modulation envelopes or switching edges that rise in 80 picoseconds and could thus place Einsteinian causality under stress by conspicuously arriving too early. Further, since any increase in barrier thickness brings with it a corresponding and exponentially increasing attenuation of any signal, it is not feasible to increase the barrier thickness to distances large enough that the causal implications of a constant barrier transit time become more apparent.

    It seems to me, however, that this is no longer a matter of theories or definitions, but an experimental question that should be treated as such. The Nimtz apparatus can be viewed as one element of a longer multi-stage device that could reach the goal of directly testing Einsteinian causality. Such a device would be constructed of many stages providing long Nimtz barrier transmission elements alternating with short fast amplification and pulse restoration elements for transmitting frequency-modulated digital signals of the high bandwidth. A digital signal would be sent through a barrier element, received and cleaned up to restore its digital pulse structure and remove the effects of attenuation and noise, then amplified and transmitted through the next barrier element. This should be possible with no loss of information from stage to stage.

    Provided each such element-pair provided a net FTL group velocity, such pairs could be iterated to obtain very long transmission path lengths. It should thus be possible, given sufficient funds, to construct a device even kilometers long to provide a definitive test of Einsteinian causality and perhaps an unambiguous demonstration of FTL signal transmission. Or perhaps there are underlying problems that make this goal impossible. If so, I don't see them.


    Since this is a science fiction magazine, let me now turn to the SF implications of FTL signal transmission. Through the works of Ursula Le Guin and Orson Scott Card, many readers of SF should be familiar with the concept of the ansible, a device whose operating principle is never described but which provides instantaneous FTL communication between widely separated star systems and starships. It is a convenient device for connecting distant parts of an interstellar plot together without postulating FTL space travel. To a physicist, however, the ansible raises a plethora of questions. How does it work? How does it resolve the simultaneity ambiguities implicit in special relativity? If it does work, why isn't it used it to find the glaring holes that must be present in relativity and to provide guidance to a better theory? Why don't they also use it to send signals backward in time? And so on.

    A ground-based FTL signal transmitter using Nimtz barrier penetration would raise many of the same questions. Einsteinian causality arises from the democratic and even-handed way in which relativity treats all inertial reference frames. This democracy is possible because no directly comparisons are possible of the times of events in widely separated space locations. With FTL communication such comparisons become possible, reference-frame democracy is destroyed, and a preferred reference frame for the universe will emerge. Special relativity would have to be modified to accommodate the new results, becoming a new and different theory.

    The other interesting SF implication of FTL communication is back-in-time communication. This is more difficult but possible. One must have two FTL communication lines moving in carefully timed paths in opposite directions at near light speed. The transmitted signals must be handed off from a ground-based transmitter to the input of FTL transmission system 1 moving to the right. The output of FTL transmission system 1 is then handed off to the now-adjacent input of FTL transmission system 2, which is moving to the left. Finally the output of FTL transmission system 2 is handed back to the ground station as it moves by. If this is done properly, the signal will arrive earlier than it leaves, and back-in-time signal transmission will have been achieved. I leave the reader to ponder the full implications of this, but clearly lotteries, race tracks, and the stock market will be severely impacted.


References:

Quantum Barrier Penetration Time:
T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).

Single Photon Barrier Penetration:
A. M. Steinberg, P. G. Kiwat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).

Microwave Barrier Penetration:
W. Heitmann and G. Nimtz, Phys. Lett. A196, 154 (1994);
A. Enders and G. Nimtz, Phys. Rev. E 48, 632 (1993).


SF Novels by John Cramer:  my two hard SF novels, Twistor and Einstein's Bridge, are newly released as eBooks by Book View Cafe and are available at : http://bookviewcafe.com/bookstore/?s=Cramer .

AV Columns Online: Electronic reprints of about 177 "The Alternate View" columns by John G. Cramer, previously published in Analog , are available online at: http://www.npl.washington.edu/av.


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