# Coherence length of a single photon

#### DaTario

Hi All,

Is there any experimental evidence that photons may have attached to itself a coherence length?

Best Regards,

DaTario

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#### mfb

Mentor
Do you have a meaningful definition for such a thing? Something in peer-reviewed publications?
To find an answer the question has to be clear.

#### jfizzix

Gold Member
Theoretically, I would say that due to the number-phase uncertainty relation,
$\sigma_{N}\sigma_{\phi}\geq\frac{1}{2}$,
a true single-photon (Fock) state would have to have a very small (though nonzero) coherence length. Indeed, its phase uncertainty would be maximal, since its photon number uncertainty would be zero.

As one example of experimental research into the coherence properties of single photon states, see:
"Heralded single Photon Partial Coherence"
Phys. Rev. A 82, 023801 (2010).

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#### DaTario

Dear mfb, let me try a sound definition for the coherence length of a single photon.

First, I would say that this coherence length would be a property of an identically prepared ensemble of photons (say photons of an specific atomic transition in a cold atom sample).
Consider a double slit experiment with a very weak beam of photons arriving at the slits plane. So weak that we may count on the fact that there is basically one photon crossing the slits plane at a time.
Now consider the interference pattern generated by this beam after the arrival of several photons to the ecran. This pattern has a transverse extension, which may be related to the number of fringes that show up clearly. The first position to where we can point and say that no interference is happening there defines a path difference from the slits for which no interference appears, revealing what I am trying to think of as a coherence length of this ensemble of identically prepared photons. Thus, coherence length of a photon = smallest path difference for which no interference are observed to occur.
It has an statiscal nature, but it seems that it is a property of each one of the photons emited.

I am trying, as I said before, to build a definition. Does anyone have some further information about it?

by the way, jfizzix, thank you for the reference.

Best Regards,

DaTario

#### jfizzix

Gold Member
So there's one more bit of information to consider. The first order coherence function $g^{(1)}(\vec{x}_1,\vec{x}_2,t_1,t_2)$, is a function that tells you the averaged degree of correlation between the electromagnetic field at one point in space (or time), and another point.

This coherence function is equal to unity, when the two points are on top of each other, and usually goes towards zero, as the two points you're comparing are farther away. A coherence length could be defined as the distance between two points where the coherence function falls below 1/2, 1/e, or some other value.

The coherence function also depends on the quantum state of the electromagnetic field (whether it's attenuated laser light, thermal light, or a true single photon Fock state), since it's that which generates the statistics you measure.

#### DaTario

Very well expressed, in my opinion, jfizzix. I recognize the subtleties of your argument (the smoothness of the decay of the g function and its dependence on the state of the photons).

So, as you said, the coherence length may depend on which ensemble we are choosing to investigate. I would like to focus specifically on the case of our investigating photons produced by spontaneous emission of a cold atoms ensemble (I guess this is the case treated in Weisskopf-Wigner theory for spontaneous emission).

It seems to correspond to a realistic state preparation. Is there any experimental determination of the coherence length of such photons?

Best wishes,

DaTario

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