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INTRO. In the parallel slit experiment, single photons ejected from the source at different times still seem to interfere to make a fringe pattern, which in a sufficientliy long exposure time becomes identical to that made by a continuous coherent light source (particle/wawe duality). Holograms are usually made having a continuos laser beam illuminating an object, the image of which is seen by a photographic film. At the same time the film is exposed to a non-diffracted refererence beam of the same frequency but directed at another (fixed) angle so as to form an interference pattern with the object beam. The pattern is recorded on the film as a hologram.

EXPERIMENT. Now suppose we somehow can produce a single photon which "bounces off" the object and "hits" the extremely sensitive film producing a first "object image". Some random time later, we produce another single photon of exactly the same frequency, which we direct at a fixed angle towards the film producing a faint "reference exposure". If we repeat this process umpty-billions of times one should think - in analogy to the double slit experiment - that over time the film will contain the same hologram of the object as if continuos beams were used.

QUESTION. Is this assumption correct, and if so how can it be proved (or disproved)?

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The answer is "yes", a hologram can be made one photon at a time. The experiment has been done. Of course it is a very slow process. The first such experiment I read about used a photographic emulsion to capture the image over a long period of time. A standard 3D hologram recording setup was used. The laser beam was attenuated until an average of less than one photon could be in the holographic setup at a time, and a photographic emulsion recorded the location of each photon at the hologram recording plane. (I'm searching for the paper and will post a link when I find it.)

Modern photon-counting detectors make it a bit easier; and some single-photon-on-demand emitters have been developed in recent years. Some key words you could use in a Google search are "single photon holography" and "photon counting holography"

However, here is where the weirdness of quantum mechanics comes in: In order for a hologram to be recorded, for each photon in the recording it must be impossible to know or even to figure out which path the photon took (the object path or the reference path). If anything is done to control or determine which path each photon takes, there will be no interference pattern. So, if you flipped a mirror very rapidly to direct single photons down the reference or object path alternately (instead of using a beamsplitter for example), then you could not form a hologram because it would be possible to know which path each recorded photon took by detecting its arrival time.

So the answer to the question is a qualified "yes": as long as there is no way to know which path each photon takes, then yes, a hologram will be formed. The question is actually a very good one, because a complete answer would need to explore some fascinating subjects like what a photon is (not the simple answer that it's a packet of electromagnetic energy), how "EPR-based imaging" works, and even this odd paper.

If the question is modified to allow each photon to "make up its own mind" whether to take the object path, the reference path, or both -- and to disallow anything that lets you know which path it took --, then the question still has some interesting content. What if the reference path is a lot longer than the reference path, so we "know" that any interference that occurs is between "somethings" that were emitted at different times?

Well, it turns out that the coherence of, e.g., a Helium-Neon laser is not just the few inches of coherence length described in most specifications. In fact, the coherence is periodic. For a few inches before and after integer multiples of the length of the laser cavity (on the order of a few feet usually), the coherence returns. So photons emitted now are neatly in step with photons emitted at time intervals determined by the laser's cavity length. If the object and reference beams differ in path length by 3 or 6 or 10 times the cavity length, plus or minus an amount smaller than the traditionally calculated coherence length, then it's still possible to record a hologram. In effect, each photon is spread out over a distance of meters -- and over a corresponding length of time.

So, it can be very difficult, even when a laser beam has been attenuated enough that typical calculations would tell us there's only a single photon in the setup at a time, to be sure that the wavefunctions of hundreds of photons aren't present in the setup simultaneously.

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  • $\begingroup$ Do current single-photon-on-demand emitters have narrow enough frequency bandwidths for holography? (There's certainly no theoretical reason why they couldn't, but it might hard experimentally.) $\endgroup$ – Peter Shor May 13 '18 at 0:18
  • $\begingroup$ I'm not sure what the frequency bandwidth (which translates inversely to coherence length) of single-frequency sources is, but I've read that it is in some cases in the MHz range, which would translate to a huge coherence length. See [www2.physics.ox.ac.uk/sites/default/files/2013-11-08/…] . However, note that Gabor made his holograms with a filtered mercury arc source with a coherence length of (presumably) a few microns. His holograms were subject to the same deltaX< coherence length limitation. $\endgroup$ – S. McGrew May 13 '18 at 2:48
  • $\begingroup$ Can you please give a link for ""the experiment has been done". When searching I find recordings of single photons , not a hologram constructed by single photons, which is the OP question. It needs a demonstration that the phases between sequential photons are defined. If they cannot interfere, as your last sentence states, they cannot be carrying information of the object to be holographed, no? $\endgroup$ – anna v May 13 '18 at 4:42
  • $\begingroup$ I'm looking for the paper and will post a link when I find it. My answer does not disagree with your answer. I've edited my answer to make it a bit clearer. $\endgroup$ – S. McGrew May 13 '18 at 5:33
  • $\begingroup$ @anna v: In creating holograms, why do you need the "phases between sequential photons" to be defined? Different photons don't interact with each other except in very non-linear media (like a laser, certainly, but there's no need for the object being holographed to be non-linear, and the film also isn't non-linear). $\endgroup$ – Peter Shor May 13 '18 at 14:19
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If you take an already recorded hologram on film, and pass single photons at a time and accumulate the results on film, i.e. put a film where the hologram forms in space, the pattern recorded on the film will be the same for single photons at a time, or a beam shining through with zillions at a time. It will be a two dimensional record, similar conceptually to the interference pattern of the double slit with single photons at a time.

Single photons bouncing off an object cannot form a hologram. This is because to get a hologram the phases of the photons making up the light beam have to be fixed. There is one huge wavefunction with zillions of photons where the phases are recorded in the hologram. When scattering off the object to generate the hologram, the phases are retained with the extra information coming from the scatter off the object. There is coherence.

A single photon cannot have a fixed phase with a subsequent individual photon following it after scattering on the object. The phase between them is lost by the "single" requirement.

The already recorded hologram has the phase information inscribed, so a photon scattering through it follows the wavefunction that built the hologram.

Edit after comments.

Each photon has its own wave function

enter image description here

Note the imaginary part. All these wavefunctions superposed make the classical electromagnetic wave, building up the real E and B fields of the wave by the total $Ψ^*Ψ$ of the beam. When considering photons, the phases come between the wave functions of individual photons. In a laser beam the phases are fixed and when scattering off an object the absorptions and reflections of individual photons are in step and can be recorded on the film medium, giving a filter ( zillions of "two slits") that will reproduce the image as a hologram.

Thus the phases must not be lost for a hologram to be recorded correctly. It then depends on whether, when one photon at a time is produced, the lasing process is coherent in the time interval so that time between hits makes no difference. In my opinion , there will be differences introduced by the context as discussed in the answer here, , on coherence of single photons.

This is genuine quantum decoherence (of the light state) and it is hard to tell what exactly the experimental outcome will be. If the whole experimental kit is genuinely time invariant, so that the probabilities of each pure state in the mixture are constant, then the answer to your question will be exactly the same as it was when we talked about energy spectral spread in pure quantum states: the probabilities of single detection events at low light levels will reflect the high light level intensities precisely. But in practice this time invariance is very difficult to achieve. You'll find that the notion of coherence length is very, very hard to measure rigorously, because the result will be highly dependent on the integration times, frequency responses, surface areas and so of in your detectors and processing electronics. You'll get a different (probably shorter) coherence length with increasing integration time, but this rule of thumb breaks down when you measure so fast that you begin to approach the "one photon in the kit at a time" régime.

So my answer reflects the latter part above, that in general coherence will be lost at the one photon stage. An experiment would show the limits for creating a hologram one photon at a time.

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  • $\begingroup$ I understand your point about photon phase coherence so I take your answer as a no to my assumption. $\endgroup$ – Jens May 12 '18 at 16:02
  • $\begingroup$ As to illuminating an existing hologram (or double slits) with single randomly phased photons, I want to point out that the hologram only retains amplitude info (which of course depends on the phase difference between the original light scattered from the object and the reference beam), so your argument about the necessity of phase coherence requires that any single photon passing through an existing hologram will immediately attain the right phase relative to the next no matter what their incident phases. Is this correctly understood.? $\endgroup$ – Jens May 12 '18 at 16:03
  • $\begingroup$ A hologram is recorded on a film which doesn't care about the phases of photons. If you send one photon at a time onto this film, it should record exactly the same picture as if you send zillions of photons at once. You just have to make sure that each photon undergoes the proper interference, but there's no need to send zillions of them simultaneously to do this. An arbitrarily weak laser beam (with average photon number much less than 1) should do just as well. $\endgroup$ – Peter Shor May 12 '18 at 16:48
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    $\begingroup$ Recording a hologram of a 3D object is no different from Young's double-slit experiment. If the coherence length of the single photons (don't worry, I'll explain in a minute) is deltaX, it is entirely possible to record a 3D hologram in which the reference and object paths differ by up to deltaX. Now for coherence length of a photon: it makes some people crazy, but in an interferometric measurement of coherence length, if attenuators are used to limit a laser beam to a single photon at a time, the measured coherence length (via photon counting) is identical to that of the unattenuated beam. $\endgroup$ – S. McGrew May 13 '18 at 0:27
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    $\begingroup$ Each photon takes two paths: the reference path and the object path, and interferes with itself on the recording medium. $\endgroup$ – S. McGrew May 13 '18 at 0:29

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