The speed that carries no signal

A pulse of light sent through a cell of caesium vapour prepared in the right way emerges from the far window before it has finished entering the near one. This is not a thought experiment: it was measured in 2000, with an advance of sixty-two nanoseconds on a pulse three microseconds long, and the result stood up. Nothing in it violates relativity, and understanding why requires being precise about which of several speeds a wave has, and about what a signal actually is.

Three speeds, of which only one is a speed #

A wave that is not a single frequency has at least three velocities attached to it, and confusing them is the whole difficulty.

The phase velocity $v_p = \omega/k$ is how fast a crest moves. It is not constrained by anything: a crest is not an object and carries nothing, and phase velocities greater than light are entirely ordinary — a waveguide and an ionised gas both have them.

The group velocity $v_g = \mathrm{d}\omega/\mathrm{d}k$ is how fast the envelope’s peak moves, and in an ordinary medium it is a perfectly well-behaved second speed. It comes from expanding the dispersion relation to first order about the carrier, and it is the speed at which energy travels when the expansion is good.

The front velocity is how fast the very first disturbance arrives, and it is the one that is always $c$.

An envelope in its simplest form is two frequencies added: the sum goes in and out of step, and the pattern of loud and silent moves at a speed of its own. That is the whole of what a group is — a superposition whose components drift in phase relative to each other — and everything said about group velocity below is a statement about how that phase relationship changes as the components travel.

In the ordinary case the two speeds simply differ, with nothing pathological about it. Deep-water waves have a group speed half their phase speed and shallow-water waves have the two equal, because the two are different functions of wavelength and the dispersion relation decides both. The factor of two in deep water is what makes a ship’s wake the shape it is, and nobody has ever called it anomalous.

Making the group velocity do something absurd #

The group index is $n_g = n + \omega\,\mathrm{d}n/\mathrm{d}\omega$. In an ordinary transparent medium $n$ rises slowly with frequency, the correction is small and positive, and $n_g$ is a little above $n$. Two things can break that.

Model a medium as one driven oscillator per molecule and its index rises with frequency everywhere except across an absorption line, where it falls steeply. That fall is anomalous dispersion, so called only because it is unusual to be looking at it. The difficulty is where it sits: the absorption peaks exactly where the fall is, so the region with the interesting dispersion is the region that destroys the pulse. Inside it the group index dips below one and goes negative.

That figure states the problem and the obstacle in one picture. A large negative $\mathrm{d}n/\mathrm{d}\omega$ makes $n_g$ small, zero or negative — but a steep fall in the real part of the index is inseparable from a peak in the imaginary part, which is absorption. Anywhere the dispersion is dramatic enough to be interesting, the pulse is being eaten.

The experimental trick is to use gain lines instead of absorption lines, and to sit between two of them. Between the lines the amplification is small and nearly flat, while the two dispersion curves add to give a steep negative slope. That is the medium in the opening figure, and it is why the arrangement is a doublet rather than a single line.

Where the emergent pulse comes from #

The result feels like an error until the mechanism is stated, and the mechanism is not subtle.

Propagation does one thing to each frequency component of a packet: multiplies it by a phase, and by a gain if the medium has any. The emergent pulse is whatever those altered components add up to, and there is no rule saying the sum must resemble the input, or arrive after it — only that each component obeys its own linear equation. Every paradox in this essay lives in the gap between that statement and the intuition that a pulse is an object.

A Gaussian pulse is not a thing that arrives; it is a superposition that has been present, at some level, for ever. Its leading edge — the exponentially small tail hours before the peak — is already in the medium, and because the pulse is analytic, that tail contains the complete information about the shape to come. The medium takes the tail, amplifies its components by frequency-dependent factors, and the components re-add into a pulse whose peak lands early. It is a reconstruction, not a transport.

The test of that account is what happens when the reconstruction is denied its raw material. Give the pulse a genuine front — cut it off so that before some moment it is exactly zero — and the analytic continuation has nothing to work on. Sommerfeld and Brillouin did this calculation in 1914: the front of a truncated wave travels at exactly $c$ in any medium whatever, because the front is carried by the infinite-frequency components and every medium’s index tends to one there. Behind the front come precursors, then the main signal, at speeds that are all at most $c$.

The measurement, and what was actually done #

The experiment worth describing is Wang, Kuzmich and Dogariu’s of 2000, because its design is entirely determined by the obstacle above and every choice in it is forced.

Caesium vapour was prepared with two Raman gain lines about two megahertz apart, both weak. A probe pulse three and a half microseconds long — spectrally narrow enough to sit comfortably between the lines — was sent through six centimetres of the cell. The transmitted pulse emerged with its peak sixty-two nanoseconds earlier than the same pulse through vacuum, having been amplified by a factor of about 1.4 and having kept its shape to within a per cent. The measured group index was −310.

Every number there is a compromise dictated by the same trade. The gain has to be weak, or the amplification distorts; the lines have to be close, or the slope between them is too gentle; the pulse has to be long, or its wings sit on the gain lines and are amplified differently from its centre. What comes out is an advance of about two per cent of a pulse length, and no arrangement of the same physics does dramatically better.

The control that mattered most was the shape check. If the emergent pulse had been distorted — steepened at the front, say — the “advance” would have been a change of shape misread as a shift, which is what several earlier claims turned out to be. Keeping the shape to a per cent while shifting the peak by many times the timing precision is what makes the result a statement about propagation rather than about pulse reshaping, and it is why the paper spends more space on the shape than on the shift.

Why absorption and anomalous dispersion cannot be separated #

The obstacle named above — that a steep fall in the index is inseparable from absorption — is not an empirical nuisance about the media anybody has tried. It is a theorem, and its premise is causality.

Suppose a medium’s response is linear, so that the polarisation it develops is an integral of the field over past times weighted by some response function. Causality says that weighting is zero for future times: the medium cannot respond to a field it has not yet met. A function that vanishes on half the real line has a Fourier transform whose real and imaginary parts are not independent — each is determined by an integral over the other, and the pair are Hilbert transforms.

Applied to the susceptibility, that says the refractive index and the absorption are two faces of one analytic function. A medium that absorbs somewhere must have a dispersive index everywhere, with a specific shape; and, running it the other way, a steep negative slope in the index at some frequency requires an absorption peak nearby. There is no arrangement of matter that has one without the other, because the constraint comes from the response being causal rather than from what the matter is.

That is why the experiment above is done with gain lines. Gain is negative absorption, and the relations hold with the sign reversed: a gain peak is accompanied by a steep index slope of the opposite sense to an absorption peak’s. Sitting between two gain lines puts the observation point where the two slopes add and the two gains partly cancel, which is the only configuration in which the dispersion can be extreme and the transmission nearly flat.

The relations also give a satisfying closure to the essay’s central claim. Causality is what forbids information from arriving early; causality is also what forces the dispersion that makes a peak arrive early. The same principle produces both the phenomenon and the guarantee that it is harmless.

What “information” turns out to mean #

The lesson is uncomfortable and worth stating plainly: the peak of a smooth pulse carries no information, because it was predictable from the pulse’s own earlier parts.

Information lives at points of non-analyticity — at the moment a signal is switched on, at a discontinuity in slope, at the edge of a modulation. Those are exactly the features that require infinite bandwidth to represent, and they are exactly the features that no dispersive medium can move faster than $c$. A medium can rearrange a smooth envelope freely, because a smooth envelope is redundant; it can do nothing at all to a sharp edge.

Why the constraint matters at all is causal rather than dynamical. If a signal could arrive before it was sent in one frame, a boost puts it into another frame’s past, and two such signals arranged head to tail return a message to the sender’s own past. It is not that superluminal signalling would be fast. It is that it would be a self-contradiction — and the front velocity being exactly $c$, in every medium and at every frequency, is what forbids it.

The other direction: slow light, and the same equation #

The same steep dispersion with the opposite sign makes the group velocity very small rather than very large, and this turns out to be the more useful case.

A steep positive slope of the index within a narrow transparent window — electromagnetically induced transparency — has taken light down to seventeen metres per second in sodium vapour, and to a standstill in a coherent storage scheme. That is the same physics: the group velocity has been engineered, and no information has been slowed either, since the process stores and re-emits rather than transporting.

That trade is the practical ceiling on the whole business. The advance achievable is roughly the pulse duration times the fractional gain difference across its bandwidth, and pushing it up means either a longer pulse, which is a slower signal, or a steeper dispersion, which distorts. Every measured “superluminal” advance is a small fraction of a pulse length for exactly this reason.

At its simplest the object is a single sinusoid: one frequency, extending infinitely in both directions, unchanging, and carrying no information whatever. Nothing about it can encode anything, because nothing about it ever changes. Every question in this essay arises from the fact that a real signal is not one of these, and every velocity discussed is a property of how several of them have been combined.

Why nothing here is peculiar to light #

It is tempting to treat all of this as an oddity of optics, and the arithmetic does not know what medium it is in.

The same construction applies to any wave with a dispersion relation. A pulse on a loaded string near the loading resonance has a negative group velocity; so does a wave in a discrete lattice near the edge of its Brillouin zone; so does an electrical pulse on a transmission line loaded with resonant stubs, which is the version that can be built on a bench for a few pounds and which shows the effect on an oscilloscope. In every case the mechanism is the same reconstruction from the leading edge, and in every case the front cannot be moved.

There is even a version with no wave at all. A row of lamps switched on by a pre-programmed controller can produce a bright spot moving along the row at any speed whatever, including faster than light, and the spot carries no information because the pattern was fixed in advance. The illuminated spot on a wall from a rapidly rotating searchlight does the same thing on a large enough wall. These examples are usually offered as a joke about “not really moving”, but the serious point is that the pulse peak in a gain medium is in exactly the same category: its position is determined by things that happened earlier, and its motion is a bookkeeping fact rather than a transport.

The distinction that survives all of them is the one between a pattern and a message. A pattern can be arranged to do anything, because it was arranged — which is also what makes a cone of sound behind a supersonic source a real structure and a searchlight’s spot on a distant wall not one. A message has to be free, and freedom shows up mathematically as non-analyticity — as a place where the future is not determined by the past of the same function.

The product that cannot be beaten #

The slow-light side of the same physics has a use — an optical delay line, a buffer holding pulses in a network without converting them to electricity — and it runs into a limit that is worth stating because it is the same limit the fast-light side hits with the sign reversed.

What a buffer needs is a delay measured in pulse lengths: holding one pulse means delaying by one duration, holding ten means delaying by ten. The relevant figure of merit is therefore the delay multiplied by the bandwidth, which counts how many pulses fit in the line.

Every slow-light scheme bounds that product. The delay comes from a steep index slope, the slope can only be steep over a narrow band, and the band is what sets the shortest pulse the medium can carry without distorting. Pushing the delay up narrows the usable bandwidth in proportion, and the product stays where the medium’s own resources put it — set by an optical depth, a gain-bandwidth product, or the number of atoms available, depending on the scheme.

So a medium is not characterised by how slow it makes light. Seventeen metres a second is a striking number and says nothing on its own; what matters is how many pulse durations of delay were bought and at what bandwidth, and by that measure the dramatic demonstrations and the modest ones are much closer together than the headline velocities suggest.

The fast-light case obeys the mirror image of the same bound, which is why every measured advance is a small percentage of a pulse length. The two are one statement: a medium can rearrange a signal in time only to the extent that it has the spectral resources to do so, and no arrangement gets something for nothing at either end.

The velocity Brillouin could not define #

One honest admission belongs at the end, because the essay has been using a word that does not have a clean definition.

Brillouin, working through the truncated-pulse problem in the 1910s, introduced a signal velocity as the speed at which the disturbance reaches some appreciable fraction of its final amplitude — half, in his treatment. The idea is the right one and the definition is not sharp: which fraction, and why that one? Different choices give different velocities, and near an absorption line they can differ substantially.

The front velocity has no such ambiguity. It is the speed of the very first non-zero disturbance, it is exactly $c$ in every medium, and it needs no threshold. But it is also not measurable, because the field immediately behind a front is exponentially small and no detector responds to it.

So there is a rigorous quantity nobody can measure and a measurable quantity nobody can define unambiguously, and the gap between them is where most of the confusion in this subject lives. What survives all of it is the statement the essay has been building toward, which needs no velocity at all: no measurement made at the far side of any linear medium can determine anything about the input that was not already determined by the input’s history before the light-travel time had elapsed.

That is a statement about information rather than about speed, and it is the only form of the claim that is both exactly true and worth making.

Where the model stops #

Linearity. Every statement here assumes the medium’s response is proportional to the field, so that components propagate independently. At high intensity they do not, and the pulse changes the medium it is travelling through.

A narrow spectrum. The group velocity is the first term of an expansion about the carrier. It describes the peak’s motion only when the higher terms are negligible over the propagation distance, and the figure showing a packet spreading is the second term becoming visible. Where the second term is large the “peak” is not well defined enough for its speed to be a useful quantity.

Steady state. The gain medium is treated as fixed. A real amplifying vapour is depleted by the pulse it amplifies, and the leading edge sees more gain than the trailing edge, which shifts the peak forward by a mechanism that has nothing to do with dispersion at all — and disentangling the two is most of the experimental difficulty.

What the pictures cannot show #

The opening figure draws envelopes, which is what a detector measures, and hides the carrier entirely. The carrier’s phase is doing something quite different from the envelope, and in a negative-index region the two move in opposite directions.

Nor does it show the pulse inside the cell. The envelope there is not a pulse travelling backwards, nor a pulse in two places; the field inside is a superposition whose peak position as a function of distance is a curve that runs backwards, and drawing it would suggest a moving object where there is only a changing sum.

And no figure here shows a front, because a front requires a truncated pulse and the whole effect vanishes for one. That absence is the honest form of the essay’s claim: the advance and the front cannot be drawn on the same axes, because the pulse that shows the advance has no front and the pulse with a front shows no advance.

Where the ladder goes next #

This ladder began with a packet whose crests move at a different speed from its envelope, continued with the spreading that the next term produces, and has now reached the point where the first term stops meaning what its name suggests. The rungs after it are the precursor structure Sommerfeld and Brillouin computed — the shape of what actually arrives first — and the general question of what a dispersion relation is allowed to be, which is where Kramers and Kronig’s relations come from and which turns causality itself into a constraint on the index.

The transferable point is about definitions rather than about waves. The group velocity is not an approximation to the signal velocity that happens to fail in exotic media; it is a different quantity that usually coincides with it. When a quantity is introduced as “the speed of the pulse” and later found to exceed c, the error is almost never in the physics and almost always in the phrase — and the repair is to ask which feature of the pulse the definition actually tracks, and whether that feature could have been predicted from what came before it.