# Validity of the 'photon wavetrain' model of coherence

In some optics textbooks, coherence is introduced with the "photon wavetrain" model. In this model, we consider light generated by many atoms each making the same transition. During each transition, a photon is released, and we model its electromagnetic field as a small wavepacket. These wavepackets all have the same frequency (up to the uncertainty principle) but unrelated phases, making the whole wave incoherent.

This is a nice and very intuitive picture, but it doesn't really make sense: if we're considering single photon emissions, we have to treat the electromagnetic field as a quantum field. A quantum field can only have classical values in the many-particle limit, yet here we're talking about the (classical) electromagnetic field of a single (quantum) photon. That can't be valid.

In what sense, if any, does this picture survive in QED? How is it corrected?

• just to clarify: are you asking how coherence can exist in the classical/many-particle limit? Or how the model makes sense for few photons? – Wolpertinger May 29 '16 at 15:51
• @Numrok The latter. – knzhou May 29 '16 at 15:54
• It doesn't have to be generated by many atoms. It could be just one Electron. All the photons don't have to be coherent either but the ones with the same frequency are if emitted from the same location and time. There wouldn't even be a so called wave or electromagnetic field without millions or billions of single photons. – Bill Alsept May 29 '16 at 20:03
• Right, there is no such thing as "small wavepackets;" localized blobs flying through space like billiard balls. Atoms emit 3D spherical wavefunctions, or classical EM sphere-waves. "Small wavepackets" seems a misguided attempt to eliminate QM issues and instead have the atoms firing off little classical bullets. ("Small," yes the wavetrain may be short for broad spectral linewidth. But where are the "small wavepackets" in an atom's sphere-wave which may come from a distant star and be light-years wide?) – wbeaty Feb 11 '17 at 9:15

A comment about coherence in general

I find the definition of coherence as some sort of "unrelated phase" problematic for a couple of reasons:

• This formulation somewhat implies that coherence is discrete, i.e. there is incoherent and coherent. Of course that is not true, you can have a continuum partially coherent states. But what quantity are you going to write down to quantify the degree of coherence? The "unrelated phase"-quantity?
• It does not emphasize that coherence is really about statistical correlations of observables.

Coherence is really defined by the following quantity:

$$g(r_1,t_1; r_2,t_2) = \frac{\langle E^*(r_1,t_1)E(r_2,t_2)\rangle}{\left[ \langle|E(r_1,t_1)|^2\rangle \langle|E(r_2,t_2)|^2\rangle \right]^{1/2}}$$

(see also wikipedia). So it quantifies the correlations of observables at different points in time and space of some statistical ensemble/time-dependent field.

IMO this removes all conceptual difficulties of coherence, but might be difficult to deal with in practice. The model that is described in the question is a perfect example: to compute the degree of coherence we would have to find the average in the expression above. For classical electromagnetic waves this involves an average over the ensemble that produces the field (1), for QFT or even just quantum mechanics it would also involve the expectation value of the the quantity measured (e.g. electromagnetic field).

Relation to the Question

In this picture there is no conceptual problem for few photon systems. Of course it might be practically hard to actually evaluate the averages, in particular write down the expectation values for the multiple photon states.

(1) For non-ergodic ensembles one runs into difficulties here and has to use a time average instead, which might be even more difficult to do in practice. I talk about the ensemble here because I find it more intuitive, but there is no difference with regards to the question.