# Coherence of Single Photons

Gold Member
I thought that it would be possible to get a single photon to a double slit which would be sufficiently coherent to create an interference pattern. You would use entangled photon pairs, which I realize is not coherent light to start with. And use the Alice stream to herald the Bob stream. But cannot you take the Bob stream and run it through some magic box with a pinhole exit (causing it to no longer be entangled - and of course I am kidding about a magic box) and then end up with a coherent source? You would not be obtaining any information about Alice at that point.

So I guess I am saying I don't suitably understand how a single photon is incoherent if you can diffract it through a point source and send to a double slit. I can visualize that you cannot have destructive interference from the source, and I guess to some extent that means no uncertainty in the source location.

So the basic idea is to build up an interference pattern and proving it is done one photon at a time. I swear I have seen references to just that.

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naima
Gold Member
I am not sure that i understand very well what you write about single photons, but it recalls me one thing:
Coherent states are not Fock states and single photons are Fock states.

Gold Member
I am not sure that i understand very well what you write about single photons, but it recalls me one thing:
Coherent states are not Fock states and single photons are Fock states.
Yes, and I see that (sorta). However, I know that an out of the box entangled photon does not produce an interference pattern in a double slit. But presumably once it is no longer sharing a state with its partner, it can be considered a single photon and placed in a Fock state. Or not?

Gold Member
I guess I usually think of coherence as it relates to some set of photons (and their relationship to each other), and this is obviously not the same thing - since there is just one at a time.

Cthugha
So I guess I am saying I don't suitably understand how a single photon is incoherent if you can diffract it through a point source and send to a double slit. I can visualize that you cannot have destructive interference from the source, and I guess to some extent that means no uncertainty in the source location.
One has to look into the history of physics for a bit to make sense of the terminology when it comes to coherence. In classical optics coherence works on the field level. Loosely speaking it is a measure of "if I know my field here and now, how well can I predict its value at a different position and time". Measuring that is pretty easy. The standard experiments done to achieve that are the Michelson interferometer (time) and the double slit (space). Varying the path length distance or the slit distance will tell you something about your light source and you will get some characteristic coherence time and length.

However, these are not yes/no quantities. Coherence time can be long or short, but there is no absolutely incoherent light in this sense. Also, if you have a look at what determines coherence time or length, it is pretty simple. Coherence time is governed by the Fourier transform of the power spectral density of your light field. Or simply speaking: If you have a narrow spectral line, coherence time will be very long. If you have a broad line, the coherence time will be pretty short. For (transversal) coherence length a similar reasoning shows that the relative angular size of your light source determines your spatial coherence. The more point-like your light source is, the larger the coherence length will be.

However, these relations have a simple consequence: If coherence solely depends on the spectrum and the size of your light source, you can make any light field have the same coherence properties just by filtering. Take some light from the sun, put it through a narrow spectral filter and a pinhole for spatial filtering and it can be as coherent as laser light (usually not as bright, though). The same applies for single photons. They can be as coherent or incoherent as your light source and filtering allow. Therefore: Yes, you may or may not see an interference pattern using single photons. That depends.

With the invention of the laser, people realized that there are some deeper properties to coherent laser emission which cannot be explained by the simple field coherence relation given above. Laser light has some properties which you cannot mimic by just filtering light from the sun. These occur in second order coherence. This really requires considering intensities and photon numbers. The idea was that a coherent state should be as classical as possible. Classically we can achieve zero uncertainty. This is of course not possible in quantum physics - coherent states are just states of minimal uncertainty. Still, there is some other property to classical measurements: They are considered non-invasive. You can measure a classical system again and again and will get the same result. In quantum physics this is still not possible, but coherent states give you the closest thing possible: the expectation value of photons present in the light field does not change if you detect one.

That sounds odd as every photon detection event will destroy a photon, but one can see that such a state must exist by going to the extremes. Consider a Fock state with well defined photon number. If you detect a photon, the number of photons in the field will be reduced and so will be the expectation value. Now consider a very noisy light field which contains no photons at all most the time, but fires short bursts containing many photons within a short time window. During these short bursts the photon number present will be way above the average photon number. Also the probability to detect a photon will be very high. So if you detect a photon at some time,actually the probability to detect another one must increase, although that sounds counterintuitive as you just removed one photon from the field. Surprisingly, light from the sun is of this kind.

Obviously there must be some kind of intermediate state with just a little bit of noise such that the two effects mentioned - reduction of the photon number by destroying the first one and increasing the photon number because of the noise present - will cancel. This is the case for coherent states. It can be shown that their photon number distribution must be Poissonian (just the right amount of noise) and any detection of aphoton will give you exactly the same expectation value for photons present as you had before. This also makes it clear why coherent states are eigenstates of the photon annihilation operator.

As one can see, single photons are of course incoherent in this sense - if you detect them you are left with zero photons - but they can be as coherent if you like them in the first sense as you can filter them just like any other light source. If you start from entangled photons you can even filter them non-locally which has been used intensively in some pretty cool publications....and sorry for making this reply a bit long.

1 person
DrDu
I think what is meant with "one photon at a time" is at most one photon at a time, i.e. you use not single photon eigenstates but coherent states formed basically from no photon vacuum, a little bit of one photon state and negligible contribution of multi photon states.

naima
Gold Member
When each second a laser sends a Fock pulse with N =1 to the detector screen we can say that the pattern is built by successive single photons.
Now each second a laser sends a coherent state to the detector screen. How does the pattern appear? with a hit each second or with possible sets of hits?
I suppose that in the first case no interference pattern appears with a 2 slit setup.

Cthugha
So the basic idea is to build up an interference pattern and proving it is done one photon at a time. I swear I have seen references to just that.
Yes, but these are very recent experiments: "Time-resolved double-slit interference pattern measurement with entangled photons" by P. Kolenderski et al., Scientific Reports 4, 4685 (2014).

Scientific Reports should be open access:

http://www.nature.com/srep/2014/140415/srep04685/full/srep04685.html?WT.ec_id=JA-SREP-20140507

naima
Gold Member
a coherent state is a superposition of Fock states. Could you, please, describe a setup which measures the occupation number? (with a screen?)

Cthugha
When each second a laser sends a Fock pulse with N =1 to the detector screen we can say that the pattern is built by successive single photons.
Now each second a laser sends a coherent state to the detector screen. How does the pattern appear? with a hit each second or with possible sets of hits?
I suppose that in the first case no interference pattern appears with a 2 slit setup.
This really does not matter. For similar spectral and angular emission properties any light source will give you the very same interference pattern, no matter whether you use a laser, single photons, a light bulb or sunlight.

a coherent state is a superposition of Fock states. Could you, please, describe a setup which measures the occupation number? (with a screen?)
Doing that adequately is a problem as one needs to take detector efficiencies into account to get absolute occupation numbers. If you just want to know whether a state is coherent or something else you rather try to measure normalized quantities which do not depend on the occupation number exactly. One such example is the ratio of the variance of the photon number distribution to its mean. In practice you measure photon pairs on two detectors (or twice with the same detector) and normalize this quantity by the product of the mean photon count rates during the same time window. Distributions with larger variance will give you more photon pairs. Fock states will give you less photon pairs. As quantum efficiency of the detectors enters twice into the photon pair count rate and twice into the mean count rates taken for normalization, these terms just cancel out.

The simplest experiment used for measuring that is the Hanbury Brown-Twiss setup consisting of a beam splitter and two photo diodes with single photon sensitivity (http://en.wikipedia.org/wiki/Hanbury_Brown_and_Twiss_effect).

Other possible choices of detectors include Superconducting nanowires instead of photodiodes ("Measuring intensity correlations with a two-element superconducting nanowire single-photon detector", Phys. Rev. A 78, 053826 (2008), http://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.053826) (good quantum efficiency, not so great detector dead time), using non-linear processes like two-photon absorption ("Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors", Nature Physics 5, 267 - 270 (2009), http://www.nature.com/nphys/journal/v5/n4/abs/nphys1218.html) (ultrafast, but not sensitive of the single photon level) or a streak camera in single photon counting mode ("Direct observation of correlations between individual photon emission events of a microcavity laser", Nature 460, 245 (2009), http://www.nature.com/nature/journal/v460/n7252/full/nature08126.html) (pretty fast and in principle sensitive on the single photon level, but pretty low quantum efficiency of the detector).