Can c, the speed limit of the universe, the speed of light in vacuum, be exceeded?† In July, 2000, the science-oriented news media were full of reports that pulses of laser light had broken the speed-of-light barrier.† Physicists L. J. Wang, A. Kuzmich, and A. Dogarliu of the NEC Research Institute in Princeton, NJ, had a paper about to be published in the prestigious journal Nature describing an experiment in which not only had laser pulses traveled faster than light, but had actually emerged from the apparatus before they had entered it.
The fact that the paper was published in Nature gave the work considerable credibility, but its implications seemed amazing.† Had the speed of light really been exceeded by light itself?† Has Einstein’s special relativity with its faster than light prohibition just gone the way of Ptolemaic astronomy?† Have the laws of physics as we understand them been overthrown?† Has Einsteinian Causality, the rule that forbids signals or energy from travel faster than the speed of light, been violated?
Contrary to the hopes and wishes of many SF readers and writers, the answer to all of these questions is NO!† All the hoopla in the media boils down to a matter of definition, of what you mean by the velocity of a pulse of laser light.
So let’s start by considering what we mean by the velocity of a wave packet that forms a pulse of light.† The pulse may be described by its time dependence as a pulse with a certain shape, with its centroid occurring at some particular time and with a certain time-width.† From Fourier analysis we know that it can also be described by its frequency dependence, as a collection of waves spanning some band of frequencies.† If we focus on the frequency-domain description of the pulse, we find that each individual frequency component exists at all times from plus to minus infinity.† However, when all of the frequency components are combined, they produce a pulse that is zero in amplitude until a particular time when it rises to its maximum value and then returns to zero.
It's well known in physical optics that a wave packet of light may travel at a different speed from the individual waves that are its frequency components. A light wave having frequency n and wavelength l will travel through space with a speed that is called its phase velocity vp and is given by vp = n l.† The phase velocity is the speed with which the maxima and minima of the light wave move forward.
In vacuum, all frequencies of light have a phase velocity of c, but in a transparent “dispersive” medium like glass or water, a given wave typically move slower than c with a speed that depends on its frequency.† The index of refraction of the medium, a quantity usually greater than 1, is the ratio of the velocity of light in vacuum to the velocity of light in the medium.† However, in some media there are frequency regions, usually near optical resonances, within which the index of refraction is less than 1 and the phase velocity of waves is larger than c.
For example, microwaves are high frequency radio waves (or very low-frequency light) with wavelengths of a centimeter or so.† They are used in applications ranging from aircraft radar to motion detectors and police speed measuring devices.† The phase velocity of microwaves traveling inside a wave guide (a square pipe used in microwave plumbing) is typically larger than c.† This is because there is a cavity resonance in the wave guide at a slightly higher frequency which elevates phase velocities on its upper frequency-slope.
Does that permit faster-than-light signaling?† No.† If, you want to use light or microwaves to send a signal, you must combine a number of such frequency components to form a “pulse” like a dot or dash in Morse code or a binary bit.† A pulse is an energy envelope bounded by zero at its start and finish.† Such a pulse travels with a speed that may be different from the phase velocities of its components and will depend on how frequency varies with wavelength over the transmission band.
The speed of the pulse envelope is called the group velocity vg and is given by the derivative vg = dn/d(1/l).† In other words, the group velocity is the rate at which the wave frequency n changes when the reciprocal of wavelength 1/l is varied.† In free space the group and phase velocities are identical and are both equal to c.† However, in a dispersive medium where the phase velocity depends on wavelength, the group and phase velocities can differ both from each other and from c.
Can the group velocity be greater than c?† Although many textbooks say no, the answer is yes.† In a previous column (Analog, December-1995), I discussed work by the group of GŁnter Nimtz at the University of Cologne, where, through the use of quantum tunneling in a wave guide, the group velocity of a modulated microwave signal was made about 4.7 times greater than c.
Einsteinian Causality was preserved in Nimtz experiment because another velocity limits the speed at which signals can be transmitted.† This is called the front velocity vf and is the speed at which the first rise of the pulse above zero energy moves forward as the electromagnetic pulse moves through the medium.† It was found in the case of microwave tunneling that the front velocity was always less than the velocity of light in vacuum.
With this background, we can consider the faster-than-light laser pulses reported by the NEC Research Institute.† The NEC physicists were able to create a pulse of light from a tuned laser that passed through a gas of cesium atoms most of which had been put in an excited atomic state by optical pumping.† This created a condition of atomic population inversion like that used to produce laser action, but in this case the population-inverted gas acted as an optical amplifier for certain frequencies, increasing the number of photons in the pulse as it passed through the gas.† The frequency of the entering light pulse was arranged to be between two optical resonances in the cesium gas, where there existed a region of "anomalous dispersion" in which the usual dependence of frequency on inverse wavelength was reversed.† In the limited frequency region occupied by the pulse, the phase velocity and frequency decrease as the reciprocal wavelength increases, so that the value of group velocity is a negative number.† The observed group velocity that the NEC group reports is vg = -c/310, i.e., the group velocity is a small negative fraction of the speed of light.† How can one interpret this negative group velocity?† It means that the energy envelope of the wave moves a positive distance in a negative time.† In other words, the pulse envelope emerges from the cesium gas region before it enters.
The NEC group reported that their laser pulses, which were about 4 microseconds wide, were found to “advance” by about 62 nanoseconds, i.e., to arrive early by that time interval.† Since light in vacuum travels at a speed of about a nanosecond per foot, the peak of the exiting pulse was therefore about 62 feet beyond the apparatus at the time when the peak of the input pulse was just entering the cesium gas.
This is an interesting result, but does it represent a world-shaking discovery, as some media reports suggested?† Actually, no.† Even in this dramatic situation, the front velocity of the laser pulse through the apparatus was less than c.† The NEC experiment could lead to a means of sending information at closer to the speed of light, but it does not provide a way of sending information faster than c.
Moreover, the NEC experiment was not the first time a negative group velocity had been observed.† In fact, three years ago the same effect was produced with electrical pulses in a simple electronic circuit.† In 1997, Mitchell and Chao published a paper showing how to construct a circuit using bandpass amplifiers that gave entering pulses a negative group velocity.† In their paper they show a single-sweep oscilloscope trace of a 40 millisecond wide Gaussian (bell-shaped) electronic pulse that is advanced by about 10 milliseconds after passing through their circuit, i.e., with the pulse center emerging at the output of the circuit 10 milliseconds before it arrives at the input.
However, Mitchell and Chao showed both mathematically and electronically that causality was not violated by this device, and no faster-than-light messages can be sent with it.† The centroid the pulse is advanced while its overall Gaussian shape is preserved, but the “front” of the input pulse is not advanced.† In particular the front velocity vf of the pulse is less than the velocity of light.† In other words, the initial rise-above-zero of the input pulse always happens before the initial rise-above-zero of the output pulse.† Since no signal can be sent faster than the velocity of the front of the pulse, causality is preserved and any signals will travel slower than c.
The same argument applies to the NEC laser pulses.† The light pulse has been reshaped by the cesium gas so that its centroid is advanced, but the front of the laser pulse, the first few photons, are not advanced and do not emerge from the cesium gas before they arrive.
This leads to the next interesting question.† Is the front velocity of an electrical or laser pulse always less than the velocity of light?† Perhaps not.† Prof. GŁnter Nimtz of the University of Cologne, whose microwave tunneling work was mentioned above, has new results that involve the tunneling of “evanescent” microwaves across a gap between two nearly-touching plastic prisms through which they are conducted.† These recent results indicate that in this tunneling environment both the group velocity and the front velocity of a microwave pulse can exceed the speed of light.
Does this result mean that using this effect, a signal could be transmitted faster than the speed of light?† Nimtz has argued that it does not.† He points out that a signal must both rise above the zero baseline and return to zero, and that the superluminal effect that he observed was sufficiently small that with pulses of a width compatible with his apparatus, there was no possibility of having a rise-and-fall signal pulse that could violate Einsteinian causality.
The law of causality thus has suffered some very serious assaults, but so far it has survived.† However, the old rule that I learned in graduate school, requiring that the group velocity must always be less than c, has been broken definitively.† The new rule that the front velocity must always be less than c may also have been broken.† But Einstein’s rule that the signal velocity must be less than c remains in place.† At least, for the moment …
AV Columns On-line: Electronic reprints of over 100 "The Alternate View" columns by John G. Cramer, previously published in Analog, are available on-line at: http://www.npl.washington.edu/av .
FTL Laser Pulses:
“Gain-Assisted Superluminal Light Propagation”, L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (20 July, 2000); see also the authors' explanation of their work at http://www.neci.nj.nec.com/homepages/lwan/gas.htm
Negative Group Velocity in Electronic Circuits:
“Causality and Negative Group Velocity in a Simple Bandpass Amplifier”, Morgan W. Mitchell and Raymond Y. Chiao, American Journal of Physics 66, 14-19 (1988).
Evanescent Wave Tunneling:
“Demonstrating Superluminal Signal Velocity”, H. Aichmann, A. Haibel, W. Lennartz, G. Nimtz, and A. Spanoudaki, Proceedings of the International Symposium on Quantum Theory and Symmetries, 18-22 July 1999, (ISBN 9-8102-4237-9), 605-611 (2000).