# Interference of independent electrons

In the Pfleegor-Mandel Experiment, photons from two separate lasers are interfered to produce an interference pattern.

In the experiment, the rate of photons from individual lasers was so low that individual photons from each laser are actually interfering with … nothing! I'm fascinated to get some more insight on this.

Has it been replicated with something other than photons? Electrons, for example?

• Is it possible that it is some property of the measuring system? For instance, it's well established that radio frequency radiation from separate sources will interfere at a receiver, but perhaps that is a property of the antenna's mixing of the source?
• Is it possible to perform the same experiment where the measuring system does not provide a chance for the quanta to mix?
• Does reducing the rate of quanta incident at the detector eventually stop interference, or reduce it, or cause it to break down?

Further links:$_1$Another description of Pfleegor and Mandel's experiment. $_2$On this site: Why can interference from two independent sources be observed? $_3$A review of independent photon interference, if you subscribe to APS Physics.

• – Benjohn Jun 6 '14 at 23:46

I've heard about this experiment, but I haven't studied it. Nor am I as smart as Leonard Mandel. With those declaimers I say: I interpret this result differently. (If a lot of people try to correct my errors in thinking, I'll turn this into a separate question.) And I apologize for not actually answering the question. But I think the premise of the question is flawed, and requires comment.

In my view, each laser is exciting the same mode. Each is increasing the photon number of the same mode. The shape of the mode, which is the thing that determines what the interference pattern looks like, is determined by the wavefunction of the mode. The wavefunction of the mode is determined by the geometry of the situation, as is true of all wavefunctions. In the case of an EM mode, the wavefunction is provided by solutions of the EM wave equation. And like all wavefunctions, it is always populated by a zero-point excitation. That is, even in the absence of an intentional excitation the mode exists and is occupied.

And as we know, the solution of the wave equation for a two-slit experiment is an interference pattern.

Each "photon" (additional excitation to the EM mode) excites the entire mode. It makes no sense to think of photons traveling through one or the other slit. The mode, and the excitation, exists in both slits. When it gets to the detector/screen there is wavefunction amplitude only where the interference pattern is not zero. In accordance with the usual interpretation of wavefunction, there is some probability for the interaction between the mode and the detector in proportion to the squared amplitude of the wavefunction. Detections occur only occur where the interference pattern is non-zero, and when the interaction occurs discrete quantities of energy and momentum are transferred from the mode to the detector/screen, and looks for all the world like a particle hit the screen.

Additionally, in this view interference has nothing to do with "overlapping photons" whatever that might mean. We don't have to have particles that somehow interfere with each other. (I can't imagine what "two particles interfering" might mean.) So it doesn't matter at all that the photon rate in this experiment is slow enough that individual photons never exist at the same time. (What this statement really means is that the rate of discrete detections is low enough that there are no coincident detections. No mention here of "photon particles hitting a detector".)

So I still subscribe to Dirac's statement "Interference between different photons never occurs." Some people reject that statement.

Update

I took a quick look at the Pfleegor-Mandel experiment. They don't measure fringes at all, so a lot of what I say above doesn't apply to their experiment. I'm don't fully understand their experiment yet. It appears that they measure intensity correlations between photons from two different sources. This is not the same as a traditional interference experiment which measures amplitude correlations. They find correlations that oscillate in space in the same way one would expect as from a single source and a beam splitter. The importance of the very slow photon rate is that the photons should be generated with enough temporal separation that any phase relationship between the two sources should have diffused away to zero.

With this (very small) improved understanding I will say that I would have to come up with a different interpretation of P-M that matches my picture. My description above does not apply to the P-M experiment. In fact, the P-F experiment is subtle enough that I think it requires a full quantum mechanical analysis to understand. (I note that even then it's possible that two interpretations can apply to the same theory. For example, the theory of shot noise has more than one interpretation.)

Pfleegor and Mandel themselves stop short of a definitive interpretation. Since then a lot of people have thought about it and done follow-up experiments. The people who did them probably have a good understanding of things. Nonetheless, none of these experiments involve traditional two-slit interference and a single detector/screen. Dirac was certainly not thinking of higher-order correlations when he made his statement, and we (which includes me) shouldn't be applying naive understanding of two-slit interference when trying to interpret these experiments. What Dirac said is probably true as far as it goes, but it may not go far enough.

• I made some previous comments here. I have deleted them as I think I better understand what you are saying. Could you comment on this? You're saying: the structure of the interference pattern exists even if there are no photons. What photons do is perform a kind of sampling of the interference pattern that is already there, and so reveal it? Individual photons don't interfere: not with other photons and not even "with themselves"? This would seem to have some terrifically profound mechanistic implications if its the case, so I can't believe I've got it right. – Benjohn Jun 7 '14 at 7:33

a property of the antenna's mixing of the source

It's not specifically antennas that do this. Using your terms: quanta always mix, inevitably. Indeed, and that's largely the point of the whole business, it's fundamentally impossible to even consider the two sources' photons seperately. Photons can only be distinguished by their frequency / polarisation, but it's just one big Fock Space for all the photons.

(Of course, you can prevent "mixing" by completely seperating the systems. Naturally there won't be any interference whatsoever then.)

That also explains why reducing the rate of photons does not change anything, there will always be interference: because you can't say "that particular photon from laser A is interfering with that one from laser B" anyway.

• If you reduce the incidence sufficiently though, the photons are emitted and cause an observation individually. I presume there is no interference at this point? So when does interference start to happen? Sorry I am misunderstanding. I also wonder if I misunderstand this Mandel experiment – it doesn't produce an image I believe, but rather some other evidence of interference. Is it this form of measurement that stops the systems from being separate? – Benjohn Jun 7 '14 at 0:29
• Again: interference happens no matter how low the incidence, because there is no such thing as individual photons from either source. garyp explained it rather better than I did. — Whether it produces an image or whatever else is not so relevant, it shows the interference, that's what matters. In any case the systems must not be seperate, otherwise each one behaves on its own and there cannot possibly be interference. – leftaroundabout Jun 7 '14 at 0:39
• Okay, sorry. I don't understand "there is no such thing as individual photons from either source". Am I wrong that there are "individual photons", or "from either source", or both? Should I be thinking of the sources as one thing? – Benjohn Jun 7 '14 at 1:04
• Or maybe I misunderstand "the systems must not be separate"? – Benjohn Jun 7 '14 at 1:08
• If the systems were seperate it would mean there's one laser shining at one target, and another shining at another target. Nothing interesting happens. If you want to observe interference, you need to have one connected system. Then, there are still individual photons – that's not the problem – but they can't be assigned to one or the other laser. – leftaroundabout Jun 7 '14 at 9:09