This is a summary from Physics World of the paper: L J Wang et al. 2000 Nature 406 277-- "Wang and colleagues begin by using a third continuous-wave laser to confirm that there are two peaks in the gain spectrum and that the refractive index does indeed change rapidly with wavelength in between. Next they send a 3.7-microsecond long laser pulse into the caesium cell, which is 6 centimetres long, and show that, at the correct wavelength, it emerges from the cell 62 nanoseconds sooner than would be expected if it had travelled at the speed of light. 62 nanoseconds might not sound like much, but since it should only take 0.2 nanoseconds for the pulse to pass through the cell, this means that the pulse has been travelling at 310 times the speed of light. Moreover, unlike previous superluminal experiments, the input and output pulse shapes are essentially the same."

I realise that the velocity of light in a medium is comprised of the phase velocity, the group velocity, and the front velocity. While the group velocity can exceed the value of c in a vacuum, the front velocity is not supposed to. The way this sounds, the front velocity is exceeding c in a vacuum by 310 times.


blue dot=phase velocity, green=group, red=front (wiki)

The wavepacket seems to exit the cell well before it enters, but the negative refractive index "forward shifts" the leading edge of the pulse. It is argued that although this is superluminal, information cannot be transmitted faster than c. Gauthier and Stenner introduced a jump discontinuity into the waveform.Its max speed was c Here is a popular account from NewScientist . What if you used one photon? What if the photon itself were the information and not a carrier?

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    $\begingroup$ What did you use to create the wave animation? $\endgroup$ Commented Feb 17, 2011 at 14:30

4 Answers 4


The front velocity, defined as the propagation speed of the point where the field first differs (by any arbitrarily small amount) from exactly zero, is always no greater than $c$. (In fact, the front velocity is always exactly equal to $c$, no greater or less.)

The problem here is that a Gaussian pulse extends infinitely in both directions, so it simply does not have a "front" to speak of. Of course the amplitude decays super-exponentially on both sides, but that doesn't matter. There is no causality problem with the pulse emerging arbitrarily early in time, because your input pulse made the field start changing long before that.

As a thought experiment, let's imagine we have a button we can push to start the input pulse going. If any trace of the output pulse comes out before a signal at $c$ has a chance to propagate from when and where the button was pushed, that's a causality violation. But that is impossible, for the following reason:

Since a theoretically perfect Gaussian pulse has no finite start time, but has a nonzero amplitude at arbitrarily early times, it's impossible to create such a perfect Gaussian pulse by pushing a button. Of course, you can get arbitrarily close to perfection, but there will always be some distortion that gets worse and worse as you try to make the pulse fire "faster", that is, try to make a shorter separation in time between the button push and the maximum of the pulse. The theorem that the medium has a causal response to the field guarantees that the response to this distortion will always interfere with the response to the perfect Gaussian to cancel out exactly at times earlier than a signal traveling at $c$.

Of course, in a real experiment, the "button push" (actually some electronic signal to the machine that creates the pulse) happens a relatively long time before any trace of the output pulse is detected.

Here's a great book chapter I just found about this: http://books.google.com/books?id=kE8OUCvt7ecC&pg=PA26 It has lots more math than my answer (though not too high a level) and might clear up a lot of things.

You also ask about single photons, but that opens up a huge can of worms I can't really get into (not least because I don't understand it well enough myself). Let me just say that there is always a minimum amount of noise in any mode / degree of freedom of the electromagnetic field, which is equivalent to half a photon. You could make a pulse of the right shape that's so weak there's only a single photon in it (I'm sure people do things like that all the time), but the problem is if you try to "announce" too early that you've detected that photon, you'll be wrong such a large fraction of the time that you can show statistically that no information is being transmitted. It's really difficult to do quantum-limited measurement in the first place (because you have to severely limit the back-reaction of the measurement apparatus on the field you're measuring), and if you try to do it too fast it becomes literally impossible.


If we accept the summary of the experiment, the laser pulse should emerge from the 6 cm cell in 0.2 ns, but it emerges 62 ns earlier than expected. The claim is that this is over 300 times c. But this is a faulty claim. The expected time is clearly +0.2 ns, while the claimed arrival time is -62 ns. So, even if we accept the claims, the difference in arrival times dictates a negative ratio. Even if we allow infinite propagation velocity, the pulse cannot exit the cell before it enters. The idea that time runs backwards is false.

Here's the explanation. Everybody knows that if you keep adding energy to a particle, it will speed up less and less for the same increment. The limit to velocity turns out to be c. However, that does NOT make it an absolute limit. Observable, measurable velocity is the cosine projection of a faster velocity. It is known as Proper velocity. Although it is the ratio of Proper length to Proper time, Proper length is measured in the co-moving frame while Proper time is measured in the relatively moving frame. That makes the physics definition of Proper velocity an improper derivative, something which is disallowed in mathematics. Physics is less logical, and only downgrades Proper velocity to a mathematical trick or convenience that facilitates some calculations.

Nevermind that Proper velocity is the 3-velocity term in 4-velocity, which is Lorentz Transformable, or that Proper velocity times invariant mass is ALWAYS momentum. It is Newtonian momentum at low speeds, but relativistic momentum at high speeds. It is valid for all velocities, and Proper velocity ranges from -infinity to +infinity. We will dispense with the false premise that Proper velocity is not real. Technically, that's actually true, but not in the sense that it is not physical. It is not real, because it is complex, both real and imaginary components. It does exist, and it is more than real. We can only measure cosine projections of time and distance. Since they both transform the same way, their ratio seems to be invariant in magnitude for both the stationary and moving observer. Truth is, even though both observers agree on the magnitude of relative velocity, they are both wrong. What they observe and measure is the cosine projection of Proper velocity.

As Proper velocity approaches infinity, its cosine projection asymptotically approaches c. The limit of the cosine projection is just c. But as the cosine projection that maps to infinite Proper velocity, it is hardly an absolute limit. However, the mapping between Proper velocity and its Newtonian projection is an isomorphism, and every element in each set is uniquely paired with one element from the other set. Every Proper velocity from -infinity to +infinity maps to an observable velocity from -c to +c. It is impossible to travel faster than c, because there is no corresponding Proper velocity faster than infinity. Similarly, time dilation asymptotically approaches zero as Proper velocity approaches infinite. There is no Proper velocity greater than infinity to push the time backwards. Even at infinite Proper velocity, time just freezes.

The summary states that the index of refraction varied sharply with frequency. This appears to be a resonance effect, which results in a negative index of refraction. It concerns me that the velocity of the rotating frame is quite likely non-relativistic in magnitude, but the relativistic correction factor, although small, is missing. c+v is problematic, but γc+γv is not. Because γc+γv is the sum of the 4-velocity components (γc-γv is the complement, the difference). These two combinations are the coordinates in eigenvector spacetime. It is a property of the eigenvector coordinates that their product is a relativistic invariant. (γc+γv)(γc-γv) is γ²c²-γ²v² = γ²(c²-v²) = γ²c²(1-β²) = c², the same relativistic invariant as in Minkowski spacetime.

The individual factors can also be expressed as γc+γβc and γc-γβc, which are (γ+γβ)c and (γ-γβ)c which are the same as e^w c and e^-w c. Clearly, their product is also c². The exponential is an eigenvalue of the Lorentz matrix, and the modified velocity is the result of a squeeze mapping. The Lorentz matrix is a pure diagonal matrix with the two eigenvalues on the main diagonal, and the velocity vector is the transpose of [c,c]. It is the nature of eigenvectors that a squeeze mapping scales each coordinate by inverse factors, so their product is invariant with respect to w, the rapidity. It also means that the column vector itself is also invariant with respect to w. Eigenvector spacetime preserves the area defined by a point on a hyperbola and perpendiculars to the two eigenvector axes. Since the eigenvectors are defined by Σ=ct+r and Δ=ct-r, they represent the worldlines of photons. Coordinates in eigenvector spacetime are measured by light rays, and are ALL invariant with respect to relative velocity. All Lorentz transforms are then the product of eigenvalues and invariant coordinates. In eigenvector space, this experiment confirms that the speed of light is invariant in two directions simultaneously.

Since there is no mention of eigenvalues, eigenvectors or complex geometry, I seriously doubt the conclusions offered by the experimenters.


You couldn't use only one photon, because the experiment depends on the light being in a pulse. A pulse is a sum of light waves with a range of different frequencies, which all experience different indices of refraction (i.e. phase velocities) when they travel through the medium. This is called dispersion. It is the precise combination of the frequencies in the pulse, and the dispersion and group-velocity dispersion in the caesium cell, that make this experiment possible.

In the same issue as Wang's paper, Nature printed a "News and Views" article about it. There, they say:

They use smooth, well-defined light pulses, so that the peak of the pulse at the output results from the forward rising edge of the input pulse, which occurs far earlier in time, making it consistent with causality. An abrupt feature in the light pulse would not be able to travel faster than c. This means that even if the 'effect' appears to precede the 'cause', you still can't send useful information — such as news of an impending accident — faster than c.

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    $\begingroup$ This doesn't make sense. One photon is not an "abrupt feature". It's just one quantum excitation of an oscillator mode - or quantum superposition of many oscillator modes - of the electromagnetic field. For any field configuration you can have, such as a laser pulse of arbitrary shape, you can have a single photon with that "wavefunction" too. (I only say "wavefunction" in scare quotes because there are problems with treating a photon as a classical particle with a wavefunction.) It sounds like you're thinking of a photon as a sharp spike in the field, which is just incorrect. $\endgroup$ Commented Feb 17, 2011 at 13:44
  • $\begingroup$ Ok, I know that a photon is not a sharp spike in the field, and I have removed the part about one photon being an "abrupt feature". However, if you have a single photon detector at the other end of the cesium cell, you certainly won't get a click before sending the photon into the cell. i.e. I still maintain that the experiment depends on the light being in a pulse. $\endgroup$
    – ptomato
    Commented Feb 17, 2011 at 14:07
  • $\begingroup$ I don't understand why you say that you can have a single photon with any possible features of a laser pulse of arbitrary shape. Can you explain that in more detail? Or, if I'm completely on the wrong track, then answer the question so that I can learn too? $\endgroup$
    – ptomato
    Commented Feb 17, 2011 at 14:10
  • $\begingroup$ Well, as I hastily explain in the last part of my answer, you actually can get a click before you might think you'd be able to, but the problem is you'll also get a lot of clicks from the unavoidable quantum noise, so statistically you don't get any information faster than $c$. $\endgroup$ Commented Feb 17, 2011 at 14:32
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    $\begingroup$ A photon in a definite-momentum state is a state with a single excitation of a plane-wave mode of the EM field: $a^\dagger(k)|0>$. But you can also have a superposition of these states, for example $(a^\dagger(k_1)|0> + a^\dagger(k_2)|0>)/\sqrt{2}$. This is an eigenstate of the photon number operator with eigenvalue 1, so there's definitely one photon there, but its momentum is uncertain. Similarly, you can have $\int d^3k f(\mathbf{k}) a^\dagger(\mathbf{k}) |0>$, where $f$ is some function whose square integrates to one. This is a single photon with $f$ as its "wavefunction". $\endgroup$ Commented Feb 17, 2011 at 14:38

I think : Mechanics( quantum) of light :

The light is the limit(the border,of things,the edge of the matter) Over the limit are no more things,( there is no matter anymore ) If you want to pass( break, to outrun ) the limit, you should use : not matter anymore, ( not particles anymore) (Light speed ) It can't be reached with mechanical movement( mechanical particles movement). Mechanical movement( motion) is for things under the light's limit,( things with mass, volume etc... For matter,)light is the "wall"( limit) between matter and " No matter " . Don't waste time to break( to outrun) the light's limit with particles ( like neutrino, tachyon etc..) In this state( situation) you don't have to do with particles , because you want to go over( beyond ) the light ( over the particles) therefore is wasting of time . Another parallel thought :

The possible outcome : C= constant means : The light doesn't move,( doesn't travel ) The photons doesn't move, therefore ( speed of) light is constant, because they don't move , is just as to turn the page, light just display it( unveils it) , but doesn't move ( doesn't travel)

The book 91:3

This is a new point of view

So you should make another math system, another physical system ( over{ beyond} the matter) , should find new sources .

If you will approach to the limit( to the border), the limit doesn't have sense anymore, if you pass the limit, the things will change, the system doesn't work anymore, a new system will be applied( will be working) , for example : if you want to become smaller and smaller and smaller , until you become a string, that is the edge, there is nothing beyond it, if you want to go beyond it, you should leave the entire system, and take a new( different) system . It is like to travel with a car, you can travel with the car until in front of you there is the sea, now you should leave the car and take a new vehicle ( a ship) , because the car doesn't work beyond the edge ( beyond the limit ,the land ), there is different environment beyond the land, the car doesn't work there, same is your system, you want to go there but your system doesn't work there, need a new system, new variables, "new tools "... 🦩

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