Manfred (below) posted a complete analysis of the exact setup from Joint Configurations, given in Wikipedia, but failed to put enough exclamation marks on it.
http://en.wikipedia.org/wiki/Hong%E2%80%93Ou%E2%80%93Mandel_effect
That is outstanding! My Xmas present (you decide what you want to substitue for X) from Less Wrong! I understood the prediction, the fact that the result would show up as a dip in the coincidence of arrival times, and when the veil was finally lifted the experiment had been reported in 1987 with the "HOM dip" as the signal result!
It is interesting to note that in some sense you get the opposite result if you use electrons instead of photons. Photons and other bosons give the HOM dip, a suppression of coincidence in the two detectors. Electrons and other fermions give a HOM peak: when two identical fermions pass through the beam splitter you are guaranteed to see one in each detector, they NEVER both head for the same detector! This result is documented , although I don't know if it has been experimentally verified.
So the things that really help me understand and accept the original article are
1) The Wikipedia article, explaining the effect in standard quantum terms and citing an experiment.
2) Understanding that the result applies to Bosons and the opposite to Fermions. It feels like a Boson result, and knowing you get the opposite effect for Fermions relieves the requirement to find your mathematical treatment bulletproof, as it doesn't apply for all particles.
Thanks to everybody who put effort in to clearing this up!
http://en.wikipedia.org/wiki/Hong%E2%80%93Ou%E2%80%93Mandel_effect#!!!!!!
Unfortunately it is not an analysis of the math used in Joint Configurations, which would have been nice. The single-photon case of reflection off a hard boundary reversing phase is obvious from the requirements of meshing with classical physics, but where the pi/2 rotation comes from in the post is unclear.
The following section confused me:
Now - and this is a very important and fundamental idea in quantum mechanics - the amplitudes in cases 1 and 4 are flowing to the same configuration. Whether the B photon and C photon both go straight, or both are deflected, the resulting configuration is one photon going toward E and another photon going toward F.
It looks like you're saying that the result of the experiment is one photon going each way. It took about 3-5 reads to get from that to "the outcome of cases 1 and 4 are identical: one photon going to each detector." I'm not sure if that's just a reading comprehension failure on my part or if there's a way to rewrite the sentence to make it clearer (I might just strike the word "resulting").
The half-silvered mirror is linear, but with complex amplitudes, not with probabilities. So you can shoot a positive amplitude and a negative amplitude through and things will cancel.
Still, I agree that the math used was not justified at all, and a treatment with normal quantum mechanics would have backed it up. I suppose it can be forgiven: the justification is a bit tricky. There is an article on the effect here.
For some reason I hadn't seen the "joint configurations" post before, and would also like to know the answers to your questions.
I am not sure I understand the problem. What would an answer to your question look like? But I'll try to answer your numbered objections and see if that helps.
In your first numbered point, why are you concerned with the phases of the photons? You seem to be thinking that the photons are waves, ie streams of photons, but I believe they are intended to be literal single photons, which do not interact at the mirror. (Indeed you have this in your second numbered point.) Would the experiment be clearer with s/photon/electron? (In which case you need some other device than a half-silvered mirror to get the 50% transmission effect, but that shouldn't be a problem.)
Your second point seems to contain a non-sequitur. You say that the photons do not interact, and therefore any statistical effect must come from the sources. I don't think that follows. This is precisely the point Eliezer is making, that human intuition breaks down in this case and you have to do the math. When you do so, you find that the amplitude for "one photon at each detector" is zero, so you will never observe that result.
On second thought, however, it seems to me that we can explain this in terms of photon interactions, although not at the mirror. Consider that there are two ways to get one photon in each detector: Both deflected, or both went straight. Now look at the path D->E. It `contains' two single-photon amplitudes: A deflected B->D photon, and a straight C->E photon. These two photons have opposite quantum (not electrical!) phases and therefore cancel. Thus the interaction doesn't occur at the mirror, but at the detector.
I don't believe this is EPR at all, it is bog-standard QM with a slightly unusual interaction, namely the rotation by 90 degrees.
So, the standard interferometer setup has the same "photons coming in at right angles" deal. What it looks like EY has set up is the standard interferometer and then added phase difference- so instead of them constructively interfering at one and destructively interfering at the other, they're always balanced.
But I don't think you can actually set up a system like that (though quantum optics was one of my least favorite parts). It seems to me that you would get a 1-1 split on average but be able to get the other two. I would also be more comfortable if the configuration analysis tracked the photons separately- why is deflecting two separate photons the same mathematically as deflecting one photon twice?
[edit] I was confused about EY's post.
EDIT: 1:19 PM PST 22 December 2010 I completed this post. I didn't realize an uncompleted version was already posted earlier.
I wanted to read the quantum sequence because I've been intrigued by the nature of measurement throughout my physics career. I was happy to see that articles such as joint configuration use beams of photons and half and fully silvered mirrors to make its points. I spent years in graduate school working with a two-path interferometer with one moving mirror which we used to make spectrometric measurements on materials and detectors. I studied the quantization of the electromagnetic field, reading and rereading books such as Yariv's Quantum Electronics and Marcuse's Principles of Quantum Electronics. I developed with my friend David Woody a photodetector ttheory of extremely sensitive heterodyne mixers which explained the mysterious noise floor of these devices in terms of the shot noise from detecting the stream of photons which are the "Local Oscillator" of that mixer.
My point being that I AM a physicist, and I am even a physicist who has worked with the kinds of configurations shown in this blog post, both experimentally and theoretically. I did all this work 20 years ago and have been away from any kind of Quantum optics stuff for 15 years, but I don't think that is what is holding me back here.
So when I read and reread the joint configuration blog post, I am concerned that it makes absolutely no sense to me. I am hoping that someone out there DOES understand this article and can help me understand it. Someone who understands the more traditional kinds of interferometer configurations such as that described for example here and could help put this joint configuration blog post in terms that relate it to this more usual interferometer situation.
I'd be happy to be referred to this discussion if it has already taken place somewhere. Or I'd be happy to try it in comments to this discussion post. Or I'd be happy to talk to someone on the phone or in primate email, if you are that person email me at mwengler at gmail dot com.
To give you an idea of the kinds of things I think would help:
1) How might you build that experiment? Two photons coming in from right angles could be two radio sources at the same frequency and amplitude but possibly different phase as they hit the mirror. In that case, we get a stream of photons to detector 1 proportional to sin(phi+pi/4)^2 and a stream of photons to detector 2 proportional to cos(phi+pi/4)^2 where phi is the phase difference of the two waves as they hit the mirror, and I have not attempted to get the sign of the pi/4 term right to match the exact picture. Are they two thermal sources? In which case we get random phases at the mirror and the photons split pretty randomly between detector 1 and detector 2, but there are no 2-photon correlations, it is just single photon statistics.
2) The half-silvered mirror is a linear device: two photons passing through it do not interact with each other. So any statistical effect correlating the two photons (that is, they must either both go to detector 1 or both go to detector 2, but we will never see one go to 1 and the other go to 2) must be due to something going in the source of the photons. Tell me what the source of these photons is that gives this gedanken effect.
3) The two-photon aspect of the statistical prediction of this seems at least vaguely EPR-ish. But in EPR the correlations of two photons come about because both photons originate from a single process, if I recall correctly. Is this intending to look EPRish, but somehow leaving out some necessary features of the source of the two photons to get the correlation involved?
I remaing quite puzzled and look forward to anything anybody can tell me to relate the example given here to anything else in quantum optics or interferometers that I might already have some knowledge of.
Thanks,
Mike