DrChinese said:
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.
