Over the past two weeks, the biggest physics news has been the apparent observation of neutrinos (nearly undetectable subatomic particles) moving faster than the vacuum speed of light.  At first glance, this would seem to violate Einstein’s special theory of relativity, which fixes the vacuum speed of light at meters per second, and as a consequence makes it in principle impossible to travel faster than that speed.  The theoretical implications are in fact a bit more subtle, but before we worry too much about those implications the experimental results will need to be checked carefully and independently verified.

While we wait, it is worth noting that in June 0f 2011 a group of researchers performed an experiment to see if light itself could move faster than light!  In particular, the scientists used a little known optical phenomenon known as an optical precursor to see if individual photons might travel faster than while propagating in a material.  In the end, the experiment suggests that these single photons did not in fact violate Einstein’s speed limit, though the results still got a significant amount of press.

The response of many physicists to the news was a collective, “Well, duh!”  The prevailing attitude seems to have been: “What’s so interesting about proving something we already knew?”  In this post I’d like to explore that question a little bit, and explain how some uncertainty remains about the behavior of light in materials.  Along the way, we’ll introduce the fascinating phenomenon of precursors, and see how they can be used both to probe the nature of matter as well as the nature of light.

To start, we need to understand a bit about how light and matter interact with one another, and how this relates to the speed of light in matter.  This is a surprisingly tricky subject, which I’ve covered several times previously on this blog; we’ll review the important aspects, with a few additional details.

Let’s talk about how we measure the speed of an object first.  If we’re looking at the motion of a rigid object, like a speeding car or a thrown baseball, the speed can be determined simply by measuring how much time it takes for an object to travel a distance .  The speed is simply the distance divided by the time:

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There’s a small subtlety to this definition: cars and baseballs are extended objects!  To accurately measure an object’s speed, we have to be consistent in how we define its position.   For a car driving down the road, for instance, we should do all measurements of its position from a fixed position, such as the front bumper, to measure the speed.

But what do we do when the object doesn’t have a fixed position on it?  For example, what is the best way to measure the speed of a hurled bucketful of water?

Water is, of course, not a rigid body, and the volume of water changes shape and spreads out as it travels.  It is strange to realize, after some thought, that a blob of water like this doesn’t really have a well-defined speed.  We arrive at the same problem with a pulse of light traveling in matter: in general, a light pulse has no “fixed point” upon it, and it can change shape as it travels.  There isn’t a single definition for the speed of light in matter that is useful in all cases.

There are three different descriptions of the speed* of light used in physics, each with its own significance and usefulness.  We can introduce each of these by looking at similar definitions for our bucketful of water.

We imagine that our volume of water may be broken up into a collection of well-defined droplets.  The most straightforward and thorough thing to do is to measure the speed of each individual droplet: the velocity of each droplet will be referred to as its phase velocity.  If there is only one or a handful or drops, this set of numbers will be a good description, but for a large number of droplets this collection of numbers will not necessarily tell us anything about the motion of the entire bulk of the water.

A better option is to define the speed of the body of water by its center of mass.  This definition, referred to as the group velocity, works quite well in many cases, but can also be misleading.  If we’re interested, for instance, in the time it takes for water to first reach its target (say a fire), there may be droplets well ahead of the center of mass and the use of the group velocity can overestimate the time.

We may also choose to define the speed of the body of water as the speed of the fastest drop in the collection; this definition is called the signal velocity.  One problem with the signal velocity is the opposite of that of the group velocity: if we need to know when the bulk of a body of water reaches a fire, the signal velocity will underestimate the time.

Physicists use roughly analogous definitions to characterize the speed of light in matter.  Instead of “droplets”, a pulse of light can be decomposed into a collection of waves of different frequencies (colors).  It can be shown that a narrow pulse of light necessarily consists of a very broad spectrum (range of colors), while a very broad, slowly varying pulse has a narrow spectrum.  Also, and significant for our later discussion, a pulse with a very sharp edge necessarily has a very broad spectrum, as well.  This idea is illustrated crudely below:

Atoms respond differently to light of different frequencies, in a phenomenon known as dispersion.  The result of this is that some frequencies of light are absorbed more than others, and in general every frequency of light travels at a different speed.  The term phase velocity is used to refer to the different speeds of each frequency of light.  For a pulse narrow in frequency (like the upper one above), this can sometimes serve as an approximation to the overall speed of light.  However, in many circumstances this phase velocity can be faster than the vacuum speed of light , and therefore does not accurately represent the speed of the pulse.

Because of difficulties with the phase velocity, the group velocity is the standard method of defining the velocity of a pulse in a medium.  It may be considered, in essence, the speed of the “center of mass” of the pulse.  However, if the shape of the pulse is highly distorted as it propagates, the group velocity can lose all useful meaning.  Also, it has been shown that the group velocity can be greater than the vacuum speed of light, a result that generated quite a bit of controversy at first.

The absolute speed of light in a medium is the signal velocity, which is loosely defined as the speed at which information can be conveyed in a medium, or the fastest possible speed that the front of a light signal can travel.  This is generally thought to be no faster than the vacuum speed of light , in agreement with Einstein’s special theory of relativity.

But is the fastest speed of light in matter?  It seems likely, but a few gaps in our knowledge leave it a partially open question.  The quantum-mechanical nature of light-matter interactions leaves open the possibility that some “quantum weirdness” allows Einstein’s speed limit to be broken in a subtle way.

A more concrete concern involves our understanding of how light propagates in matter.  It is well-known that the “causality” of a light signal — and the absolute speed of light — is built into the frequency-by-frequency response of light to matter.  If one knows the speed of light, and the absorption properties of light, for every frequency, one can determine whether or not is the top speed.

But we don’t know these properties for every frequency!  In particular, we don’t know exactly what happens to light in matter for arbitrary high frequencies (frequency approaches infinity) or for arbitrary low frequencies (frequency approaches zero).  There is always the possibility that some small violation of relativity is “hiding” in these extreme frequency ranges.

This is where the idea of optical precursors becomes useful.  Let’s consider the temporal and spectral properties of another pulse, with a square envelope:

This pulse has a large central peak in frequency, but also has long frequency “tails” that stretch out, to some degree, to arbitrarily high frequencies (to infinity) and arbitrarily low frequencies (to zero).  These tails arise due to the instantaneous start and end of the pulse in time: the rapid rise and fall that give the “square pulse” its name.

At the beginning of the 20th century, physicists Arnold Sommerfeld and Léon Brillouin independently performed theoretical work to investigate what happens to such sharp-edged pulses as they propagate a long distance through matter.  In 1914, they each published results** showing that the main body of the pulse (traveling at the group velocity) is preceded by faster-moving waves, now known as “precursors”.  The precursors are general broken into two types: the Sommerfeld precursor, which actually travels at the vacuum speed of light , and the Brillouin precursor, which travels at the speed of light , where is the refractive index of the medium at zero frequency.  A crude illustration of the arrangement of precursors is shown below:

The most remarkable thing about the precursor fields is that they are very weakly absorbed by matter, much less so than the main body of the pulse.  If we send our original square pulse through a very thick piece of absorbing material, the main body will be almost completely absorbed while the precursors will be absorbed only weakly.  This fact has led to some researchers suggesting that precursors could be used to “see” through normally opaque objects, like clouds.

It took quite a few years for Sommerfeld and Brillouin’s theoretical work to be confirmed.  The first observation of precursors was performed in 1969, using microwaves***.  Since then, they have been observed for a variety of wavelengths, and have been observed even in “mundane” materials such as water****.

What is the origin of these precursors?  I wasn’t able to find a simple explanation for them in the literature, perhaps not surprising in terms of their mathematical complexity.  I would like to introduce a (hopefully accurate) description of them, in what I will refer to as the “pendulum slapping” model!

We have all been taught, at some point in our education, the planetary model of the atom.  In this model, an atom consists of a positively-charged nucleus surrounded by one or more orbiting negatively-charged electrons, much like the planets in the solar system orbit the Sun:

With our modern understanding of quantum mechanics, we know that this model isn’t a terribly good one: electrons act much more like “clouds” of negative charge centered on the nucleus, and don’t “orbit” in a well-defined sense.  However, the planetary model does get one thing right: electrons have a characteristic frequency associated with their motion, much like the planets each orbit the Sun with a characteristic frequency (the Earth orbits the Sun with a frequency of 1/365 days).  In a very loose sense, we can picture atoms (and molecules) as collections of oscillating electron pendulums, each vibrating with a characteristic frequency:

What happens when we apply an oscillating electric field (i.e. a light wave) to our pendulum?  The electric field applies a force to the electron, driving its oscillation.

The effectiveness of the electric field in moving the pendulum depends on how close its frequency is to the pendulum frequency.  The pendulum will have the largest swing when the electric field matches its own frequency; this is like a child pumping his legs on a swing to increase his motion.  Any other frequency of the electric field will be less effective in moving the pendulum, but pretty much every frequency will force some oscillation in it.  The energy that is imparted to the pendulum is lost by the electric field, and consequently the light wave; this is the origin of the absorption of light by matter.

There are two extreme limits of frequency in which no energy is transferred to the pendulum: the limit of infinite frequency and the limit of zero frequency.  We can visualize what happens in these cases by imaging that we are “slapping” a pendulum with our hands on either side (or you can try it yourself, if you have a pendulum). When we slap a pendulum at a frequency much higher than its characteristic frequency, a push on the left is almost immediately canceled by a push on the right — the pendulum doesn’t move, and we’ve transferred no energy to it!  In the limit of very low frequency, the pendulum moves, but doesn’t oscillate: we “lift” it to the right with our left hand, then slowly lower it back to its rest position and then “lift” it to the left with our right hand.  At no point does the pendulum oscillate freely, and therefore we have transferred no energy to it.

These limits, in which no energy is transferred to the pendulum, are the origins of the precursors.  We have seen that a square wave has components of arbitrarily high frequency and arbitrarily low frequency, and these components are only very weakly absorbed by the medium.  The high frequency components of the wave result in the Sommerfeld precursor, and the low frequency components of the wave result in the Brillouin precursor.

For tests of the absolute speed of light, the Sommerfeld precursor is of particular interest, because (a) it tests the whether a violation of relativity is “hiding” at high frequencies, and (b) the speed of a precursor is theoretically supposed to be equal to , making “faster than light” violations relatively easy to spot.

All of this brings us back at last to the June paper in Physical Review Letters on the “Optical precursor of a single photon”, by a research group in Hong Kong.  A photon is a single quantum particle of light; all of the discussion of precursors up to this point have concerned pulses consisting of many, many photons.  Two questions arise when considering precursors and single photons:

(1) Do precursors even exist for a single photon?  One would naturally be inclined to say “yes”, but it may be that, on a quantum level, precursors inherently involve the interaction of many photons at once.

(2) Can single photons travel faster than ?  One would be inclined to say “no” in this case, but again the behavior of single photon precursors (if they exist) might be subtly different than a group of many photons.

The Hong Kong researchers investigated these possibilities by producing coupled pairs of photons using the following experimental configuration (adapted and simplified from the article):

The fundamental components of the system are a pair of magneto-optical traps (MOT) containing Rubidium atoms.  Magneto-optical traps use light and magnetic fields to localize a group of atoms and cool them to a low temperature, forming a low-density medium.  The first MOT is excited by a pair of lasers — a pump beam and a coupling beam — and through a complicated light-matter interaction pairs of photons are produced.  The physics of this production is too complicated to discuss here, but the result is a higher frequency “Stokes photon” and a lower-frequency “anti-Stokes photon”.  These photons are produced at the same time and are therefore correlated in time.

The Stokes photon is picked up by detector 1 and sends a signal to a function generator, which triggers an electro-optic modulator that the anti-Stokes photon passes through.  This modulator “chops off” the front end of the anti-Stokes photon signal, producing a sharp-edged pulse necessary for precursor generation*****.  This sharp-edged pulse propagates into the second MOT, in which its waveform is expected to take a precursor shape.

It is essential for this experiment that a single anti-Stokes photon be measured at a time; after the second MOT, the light signal is split by a beam-splitter and sent to a pair of detectors, detectors 2 and 3.  Because a single photon can only arrive at a single detector, the simultaneous tripping of both detectors implies the presence of more than one photon and the event is thrown out.  The time of arrival of the single anti-Stokes photons can be tallied, producing a profile of the average behavior of the photons.

The coupling laser illuminating the second MOT could modify the optical properties of the Rubidium atoms into one of two configurations.  In the first configuration, the absorption properties of the medium can be completely suppressed, in a technique called electromagnetic induced transparency (EIT).  With EIT, both the main body of the pulse and the precursor signal make it through the MOT and can be measured by the detectors.   Furthermore, the main signal is slowed considerably in speed and is separated in time from the optical precursor.  In the second configuration, the group velocity of the main signal is faster than , i.e. the group velocity is “superluminal”.  However, this main signal is highly absorbed.

What were the results of the experiments?  The researchers measured the speed of the main body of the pulse and the precursor for both MOT configurations.  The precursor was clearly visible in both cases, and its speed was found to be exactly , regardless of the MOT behavior.  Therefore no true “faster than light” behavior was observed, even when the group velocity was greater than the vacuum speed of light.

An important aspect of this result is a partial answer to a debate in quantum information theory — how fast does a single photon transmit information?  As we’ve noted, there are multiple definitions of the speed of light in matter, and it has remained a bit of a mini-mystery (at least to some) which of these described the speed at which information is transmitted.  The observation that a photon does have a precursor signal suggests that single photons can travel at the vacuum speed of light in matter, at least under the right conditions.

The result isn’t as earth-shattering as the possible discovery of superluminal neutrinos, but it does highlight a number of unusual aspects of optics, and further strengthens our understanding of both relativity and quantum mechanics!

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* In physics, “velocity” is used to refer to the vectorial motion of an object: not only how fast it is going, but in what direction.  “Speed” is used to refer to the magnitude of velocity, or simply how fast the object is going.  We use the terms interchangeably here in the sense of “speed”.

** A. Sommerfeld, Ann. Phys. (Leipzig) 349 (1914), 177.  L. Brillouin, Ann. Phys. (Leipzig) 349 (1914), 203.

*** P. Pleshko and I. Palócz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22 (1969), 1201.

**** S-H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92 (2004), 193903.

***** Actually, the anti-Stokes photon will already have a precursor signal due to its propagation in MOT 1!  This precursor is also chopped off by the EOM.

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Zhang, S., Chen, J., Liu, C., Loy, M., Wong, G., & Du, S. (2011). Optical Precursor of a Single Photon Physical Review Letters, 106 (24) DOI: 10.1103/PhysRevLett.106.243602