Hello everybody!

Maybe someone can give a description within mainstream quantum mechanics of what happens in a radiotelescope?

You know, the big dishes that are 1000km apart, where radioastronomers record the signal received by each antenna, and recombine the signals in a computer to get the angular resolving power of a 1000km wide antenna. Usually called an interferometer.

See where I'm heading to? If not, here's more.

To get get the corresponding angular resolution, photons must be allowed to go through all dishes at the same time in order to interfere.

Even better, as astronomers observe sources that use to have no time coherency (interferometers work perfectly with thermal radiation), I believe it means that each single photon must be allowed to interfere with itself, not just with its colleagues.

However, once the photon has been absorbed by the preamplifier, processed and detected, then its information put on a magnetic tape, sent by plane, lost by the airline company, retrieved, stored in a cabinet, and picked 10 years later to compute a picture, I feel it fair to say that the photon is dead, passed away, finished, adios, bye-bye. And that it ignores the existence of a second magnetic tape, which is meanwhile transferred on a hard disk.

So what?

Is the remembrance of the photon still a bit alive because we have a record of it, including its phase?
Did it go through both dishes because we got many photons and ignore which we detected at which antenna?
Would it be the same with visible light, as we can make coherent detection and observe single photons?

Now THAT'S an interesting question! Others with more expertise than I are welcome to weigh in, of course, but here is how I understand it:

In order to obtain interference years later from recordings made by the two separate telescopes, it is necessary to record the phase information, of course. It is also therefore necessary to receive the light without detecting which photon went into which telescope, as this would necessarily destroy the phase information. (This is essentially why the two slit interference pattern disappears when you detect which slit each photon went through.) However, phase information can only be accurately measured for light intensities in which many photons come in at once as part of the same wavefront. Because of this, it becomes impossible to identify individual photons as arriving at one telescope or the other when the light is strong ("light" is shorthand for electromagnetic radiation of all frequencies).

In some versions of the two slit experiment, detectors of varying degrees of efficiency are stationed at the slits. A detector with 100% efficiency identifies the slit used 100% of the time and destroys 100% of the interference pattern, resulting in the classical two-peak pattern. As the detector efficiency is lowered to 0%, the two peak pattern smoothly morphs into the spatially oscillating two slit interference pattern. From an experimentalist point of view, this shows that a certain fraction of the photons are detected (and contribute to the two hump pattern) while the rest are not (and contribute to the interference pattern). But when many photons come in at once, and since photons are indistinguishable, you cannot identify WHICH photons were detected and which weren't. So you must think of the entire ensemble of photons as being in a partially detected state.

[Note: This may no longer be a pure quantum state, because pure states occur for isolated systems, and this photon ensemble is no doubt entangled with the detectors by now. Entangled systems are typically in a state which does not factor into pure states for each subsystem, which is the mathematical essence of entanglement. This is analogous to the difference between the canonical ensemble and partition function for a closed system (constant energy and particle count), and the grand canonical ensemble and partition function for an open system (variable energy and particle count.)]

As with the uncertainty principle, there are two viewpoints: experimentally, physical limitations prevent the simultaneous measurements of conjugate quantities; theoretically, the lack of commutativity ensures that no possible experiment could measure them simultaneously. The theoretical consideration wins out because it demonstrates a deep reason why every simultaneous measurement must fail. The specific mode of failure in each experiment is then seen as merely the working out of the theorem in a special case. Similarly here, one could note that phase is easily measured when the light is strong, but particle identity cannot be assigned to one receiver or the other, while for very weak light, single photons for instance, it is impossible to NOT see which telescope captured the photon, and it IS impossible to record the phase. Again, between these two extremes, there must exist situations where there is partial phase information and partial identity information. This suggests that there is something analogous to the uncertainty relation here, in which there is a limitation relating phase to particle identity. Probably this is just an expression of the well known uncertainty principle, but it is interesting to look at the situation as an uncertainty relation in its own right.

So the conclusion is that the synthetic aperture method should work when and only when there is a strong enough signal so that there is little chance of identifying which telescope caught which photon. Phase information is then available with sufficient precision to allow interference calculations to be done from stored data.

That's my take on it anyway. Seem reasonable?

--Stuart Anderson

It sounds reasonable to my ears, though I'd need some alternate stirring and settling of the whole.

It certainly would be easier to understand if the light source was a laser (preferable with perfect phase coherence) or a radar. Then "identical photons, ignore which one" would be easy to imagine.

Less easy - ore more difficult - with radioastronomy, where most sources are thermal and spew individual uncorrelated photons. This is, I guess, why you introduce uncertainty relations, so that nonidentical but similar photons can still be taken for another, so that we ignore which one was absorbed at which antenna. Or?

I'm still hoping somehow that our representation with energies/frequencies defined minus an arbitrary value, long disappeared particles that still interfere and countless more peculiarities will be someday replaced by a simpler representation. If it were just for the photoelectric effect and for interferometers, we could imagine some kind of heterodyne properties of a nonlinear vacuum, but now that we have Feynman's diagrams, Aspect's experiments and many more, any such hope looks bad.

As a Jewish proverb is to say: humans think, and God laughs.

The suggestion you made in your first post was exactly right. There is no difficulty about the interstellar source being thermal and therefore incoherent, because EACH photon fills the entire spherical wavefront. This is the same as the standard interpretation of what the photons do in the two slit experiment, except on a much, much larger scale. However, the principle is the same. Each telescope has access to every photon the star put into the current wavefront, albeit with a very low probability of individual reception. Therefore, the photons need not be coherent in order to interfere, because each photon can interfere with itself, all by itself.

(In fact, the laser itself was puzzling at first, because Dirac had postulated that particles could interfere ONLY with themselves, whereas in the laser, because of the perfect coherence, photons do interfere with each other. For that matter, the coherence itself is an effect of interphoton interference.)

As to the last suggestion I quoted above, you don't need to have nonidentical but similar photons "mistaken" for each other. In fact, I don't think that is even possible. It really does seem to be the case that the photons are so greatly delocalized that each one interacts with both radiotelescopes. Therefore, the incoherence of the source doesn't matter. Of course, the timing is crucial. Because the sources are incoherent, the photons in the process of being emitted at one instant bear no relation to those emitted at a later instant. Unless, that is, the instants are so close together in time that most of the photons being radiated at the first instant are still "in process" at the second instant. This accounts for the phenomenon of "coherence length" which is essentially the correlation time of photon emission multiplied by the speed of light. For most emitters it is on the order of a few hundred wavelengths, while for lasers it is infinite.

For astronomical sources, this means that the travel time lag between the telescopes must be carefully adjusted for before the proper phase relationship appears. This can be done by computing the autocorrelation function of the two light recordings, and finding the location of the peak value. This location gives the time delay between the best-correlated signals, which will almost certainly be when the same wavefront was crossing both telescopes. Adjusting the timing of the records to remove this lag then gives a reconstructed phase relationship, which is enough to calculate the interference.

The core idea here is that if you have a complete recording of the light, including phase, as a function of time, then you have all the physical information necessary to reconstruct the original waves. If you could in principle reconstruct the waves themselves, then certainly in practice you could compute their interference pattern. In a hologram, just this kind of phase information is recorded (as a function of position rather than time) and used to reconstruct the original incident wave, which is how the 3d effect is achieved. Precisely the same considerations apply to the astronomical case.

I'm not trying to monopolize this thread, so others please jump in, especially those better versed in the subtleties than I am (AlphaNumeric -- any comments?).

--Stuart Anderson

Enthalpy, Stuart,

Enthalpy, many situations are concerned with the product of a function f (x) by its shifted version f (x - a). If f (x) is random, the FT of its autocorrelation yields the power spectrum which is of frequent use in noise analysis. In the same field the characteristic function of a random variable may be calculated as the FT of its density probability.

If I have missed the crux of your question, please advise. If not, f (x) is no longer random but deterministic, the same process describes the general problem of interferometry.

One example: In radioastronomy two/2-D array antennas measure the visibility function V (u, v) which is Fourier transformed to get the brightness distribution v (α, δ) of radio sources in the sky (u, v are spatial frequencies, δ and α declination and right ascension).

dawn

Pity, I liked the explanation with the almost-identical photons so much...

But then, as photons emitted by a thermal source are supposed to be completely uncorrelated, photons emitted within a short delay shouldn't be better correlated than if separated by a longer time.

See the trouble coming...?

Correlation works only if you have a good (in fact, an excellent) synchronization between the signals on your magnetic tapes. Introduce a lag, you'll have as many almost-identical photons as before, but no more image.

So should we formulate something like "both recorded signals can interfere because we have enough photons to be allowed to ignore which receiver caught every single one"?

It's getting a bit twisted to my taste...

OK, "correlation within a short delay" does make sense. In fact, "short delay" is related to the time response of the frequency filters at the receivers, and this time response is related to the bandwidth.

So "enough almost identical photons at one time" would mean "more photons as the filter still rings from the previous photon". And then, the simpler formulation could be saved.

I need to meditate more about it.

Actually, I like the formulation in your second-to-last post better. You are right that the frequency filters at the receivers introduce a timescale which you can use to define what a "short delay" is. However, the light source itself also has an inherent timescale, expressed as the "coherence length" of the source. Essentially this means that the phase of the wave is only consistent across a very short time interval. With time delays exceeding this interval, the correlation drops to zero. This approximates the autocorrelation of the "gaussian white noise" stochastic signal; "approximates" because the GWN signal has an ideal delta function as its autocorrelation, while light shows a small but nonzero range of time delays during which the autocorrelation is strong.

If the relative delay between the two receptions of the signal exceeds this time range, then there is nothing that the receiver can do about it. With recorded signals, or course, the signals can be re-synchronized using the autocorrelation function. The time resolution of the receivers doesn't really come into the process much, except that it must be fine enough to resolve the phase of the signals, i.e. the time resolution must be less than half a period (preferably considerably less).

The vanishing of correlation outside a small time range is due to the nature of the emission process. The photons themselves are emitted in a finite time interval, so they are time localized. Each photon lasts for a certain time, and so if your two signal receptions are close enough together, most of the photons are common to the two times of reception, and therefore, both receivers are looking at the same exact set of photons (plus or minus a few that started or ended between the two reception times). So it is not a case of the photons being merely similar; they are actually the same ones. Of course, if the reception times get too far apart, all the photons that were available at the first reception time have already passed, and the second reception is looking at a later generation of photons. Thus the correlation vanishes, because the later photons do not inherit any phase information at all from the previous ones; after all, they are emitted by entirely different atoms, which became energized after (or while) the earlier set of radiating atoms were emitting. In that case, you might as well be looking at two entirely different stars -- no correlation will be present.

In this second case, the photons are just as similar as in the first case, but this time there is no correlation of phase. Therefore, similarity is not the determining factor; instead, correlation requires that the selfsame set of photons be available at both receivers. I say "available" because of course each photon ultimately must be absorbed by one of the receivers, but you cannot know which one without destroying the phase information that allows the correlation.

So I strongly feel that the physics here is identical to that in the two slit experiment, except that the distances and times are vastly expanded, which defeats our human intuition based on local causality. There is one difference: whereas in the two slit experiment, you have a single detector, so that it is possible to detect each photon without also detecting which slit it came through, in the radiointerferometry observation, each scope has its own detector and recorder. The two slit experiment can be done with such low light levels that photons come through singly, and you still get the interference pattern; with two radio telescopes, it is hard to see how you could NOT know which scope the photon came through in the single photon case. Here in order to make things work you must have enough photons come through that each individual one is "lost in the crowd" so reception without identification can proceed.

Well, that's how things look to me, at least!

--Stuart Anderson

Yes, time coherency of the photon source...

But on the other hand, as an electrical engineer, you can just neglect the photon nature of radiowaves and compute the time resolution necessary to your correlator based on the bandwidth of your filters. I did it for goniometric radiolocation, including from thermal noise, it simply works. I don't expect a µs coherency from a thermal source.

[Radioastronomy is more fun in that the photons may have been emitted a billion years ago by a source that has disappeared meanwhile and has no chance to adapt its radiation to our receiver.]

Now, one may argue that slow correlation works because the received waves contain slowly varying components which are correlated over a long time compatible with narrow filters.

But then, it would mean that photons decide at the receiver to be coherent over a long time, depending on the needs of the receiver, provided the radiation contains the necessary power in a narrow bandwidth.

As the filters usually act late in the receiver chain, I dislike such an interpretation - a photon going through 100 transformations in the receiver's parts before it decides what its own nature is... Such a representation doesn't help me much.

Hi Enthalpy,

I think we've been talking about two different things here (correct me if I'm wrong). The coherency of a thermal source is on the order of a few hundred cycles, while filter bandwidth considerations only come into play for time resolutions less than 1 cycle. Say you want to detect the phase of the strongest component of a signal that is bandlimited to f<F Hz. Then to get, for instance 1/8 wave precision, you would need a time resolution of 1/(8F). (Stop me if that's not what you meant!) Certainly you need good time resolution to phase correlate two signals (much better than 1/8F of course, that was just an example), but the time RESOLUTION wasn't the main issue with the delayed interference of radio signals. The main trouble is to get the two recorded signals synchronized to within the time coherency of the source, so that the interference pattern will be meaningful.

So there are two distinct times which are characteristic of this system: the resolution time, which depends on the BW, and the coherence time, which depends on the source. For any practical system, the resolution time will be much less than one cycle and the coherence time much greater than one cycle.

The whole process for recorded signals will have three steps: First, from timestamps on the data, roughly synchronize the record signals. Second, with low time resolution on the order of 1/2 cycle, cross correlate the two signals and look for the peak value, which gives the amount to offset one signal to get the best correlation. Third, after making the adjustment in the second step, do a fine time resolution cross correlation to get the phases as well synchronized as possible. This produces signals with only slight phase variations, which encode the parallax information that allows you get a sharp image from the synthetic aperture method.

Of course, I'm talking as if this were all taking place in the time domain. For recorded data, frequency domain methods may be much easier. I haven't thought about how one might do that, though Dawn mentioned it earlier in this thread. Of course, besides the time synchronization, the signals also have two spatial degrees of freedom, so once you have time synched them, you still have the spatial variation (which may be in spatial frequency domain, as Dawn mentioned), and these are what are actually used in the synthetic aperture calculations.

In earth based (non-astronomical) synthetic aperture radar, none of these timing issues comes up, since the coherence time is essentially infinite for electronically generated signals. All you need is sufficient time resolution to get accurate values for the observed phase as functions of angular position.
Does that make sense?

--Stuart Anderson