Main Question or Discussion Point
Are photons in the laser output beam mutually independent and can the multi-photon output be described as the tensor product of individual photon states?
No, it does not imply that. The result of a single measurement of the photon number is indeterminate in the sense that it is not predictable. The mean result after many repeated measurements - the mean photon number - is instead well defined. Now in a coherent state the weights of each Fock state in the coherent state are given by a Poissonian distribution around a mean photon number. The result of a single measurement is indeterminate, but the probability to get a certain result is well known and given by a Poissonian distribution.That's not quite what I meant but it's conceptually close. I just don't understand how the photons can be statistically independent.
If it's a sum over Fock states, then the exact number of photons in the output must be indeterminate. But if it's indeterminate, doesn't that imply that we cannot write the output state as a product of photon states?
I never said that. Of course you cannot completely suppress spontaneous emission, but for common lasers the relative contribution of stimulated emission to spontaneous emission is roughly the inverse of the laser beta factor which can be as large as 1000000:1 or above, depending on the kind of laser used. Under these circumstances spontaneous emission is pretty much negligible. It is of course more critical for lasers using a small number of emitters.So if I understood correctly, in a real laser the photons are not completely independent since one cannot suppress spontaneous emission?
Why should it be an entangled state? You get entangled states from parametric processes like parametric down conversion. In this case you will just get a mixture of coherent and thermal light or in other terms light with partial higher-order coherence.If that's the case, should we describe the multi-photon output as an entangled state, or does entanglement only apply when we have a fixed number of photons?
Well, from a purist point of view there will be some mutual dependence. However, you will most likely not find it in the photon pair detection rate I mentioned above. However, you can go to higher orders. So you can also check whether the three-photon, four-photon or n-photon coincidence rate will factorize. Any real light source will show some deviations from complete factorization at some order which will often be very large. Basically this is the same as comparing the emission photon number statistics to Poissonian statistics order by order. You start with the mean and go on with the variance. Then you will compare skewness and kurtosis. In some order you will find a difference. However, for practical purposes the difference usually does not matter."the inverse of the laser beta factor which can be as large as 1000000:1 or above"
That's true. My question was if mutual dependence between output photons exists in principle.
This is highly nontrivial as the exact behavior depends on a lot of parameters. Even if you have just one emitter you can have several relations between spontaneous and stimulated emission depending on the quality of the laser cavity mirrors. It also depends strongly on the exact kind of laser used and the pumping strength. In principle all most common kinds of light (non-classical, coherent, thermal) are possible in such devices. This is still an active area of research today. See for example C. Gies et al., "The single quantum dot-laser: lasing and strong coupling in the high-excitation regime", Optics Express 19, 14370 (2011). There are many other articles on that topic, but this one is freely available here: http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-15-14370"Suppose this number is small (let's say, N=1 emitter to take the extreme case). How would we then write the resulting output state?