What is the correlation between photon events in space and time?

[Scipio]

Hey people!

I'm studying the article "The Quantum Theory of Optical Coherence" by Glauber (see eg. http://prola.aps.org/pdf/PR/v130/i6/p2529_1), and I've a couple of questions.

1. Why are the arguments of E(+) and E(-) in equation 3.6 two different sets of coordinates? We obtained E(-) by taking the complex conjugate from E(+) (eq. 2.15), so wouldn't it make sense to keep the arguments of those two the same?

2. Glauber sets up a correlation function Gⁿ, but what does this function actually describe? I thought it described the correlation between n points in spacetime, and that correlation and coherence where more or less equivalents. However, Glauber argues that if the wave is actually coherent up to the nth order, Gⁿ should factorize, so the photon detection events are independent, ie. not correlated. So.. if the events are correlated, the wave is not coherent? What then is the relation between correlation and coherence?

3. If all those operators should operate independently, and detection involves the destruction of a photon, does that mean that a fully coherent wave (which, according to Glauber, satisfies an infinite order of coherence conditions), should contain infinite photons?

Thank you!

 

[Cthugha]

1. Why are the arguments of E(+) and E(-) in equation 3.6 two different sets of coordinates? We obtained E(-) by taking the complex conjugate from E(+) (eq. 2.15), so wouldn't it make sense to keep the arguments of those two the same?

These are not two different sets of coordinates. Just two different coordinates. He is just emphasizing that the correlation function can be defined between two arbitrary positions and times.

2. Glauber sets up a correlation function Gⁿ, but what does this function actually describe? I thought it described the correlation between n points in spacetime, and that correlation and coherence where more or less equivalents.

No, they are not. First order coherence can more or less be reduced to field correlation, but it does not tell us much about the nature of the light source, just about field coherences. For example you can change the first-order coherence of any light field just by shaping it spectrally and spatially. This does not allow you to distinguish between a light bulb, non-classical light and laser light. This is why you move on to higher order coherence. Second-order coherence is the most common one and describes the intensity correlation instead of field correlation. Naively speaking, it gives you the probability to detect a photon at some point and time, conditioned on a second photon detection at another position and time. Usually this quantity is normalized to the unconditional photon count rates at the two times/positions.

However, Glauber argues that if the wave is actually coherent up to the nth order, Gⁿ should factorize, so the photon detection events are independent, ie. not correlated. So.. if the events are correlated, the wave is not coherent? What then is the relation between correlation and coherence?

Indeed, correlated detections are the trademark of thermal light. Here the light leaves the light source in "bunches" which is a consequence of the light field having Bose-Einstein statistics. Anticorrelated detections are the trademark of non-classical light. Take a single photon source for example. If you detect one photon, the probability to detect another will be zero (otherwise it is not a single photon source). Both of these effects - bunching and antibunching - vanish on a timescale of the coherence time of the light field.

For laser light, the detection of a photon does not change the light field (in contrast to thermal and non-classical light) which is another meaning of g2 factorizing. It is also equivalent to coherent states bein eigenstates of the photon annihilation operator. However, it is important to note that the intensity is the uncorrelated quantity for coherent light. The field is well correlated - which is exactly a consequence of detection events not changing the light field.

3. If all those operators should operate independently, and detection involves the destruction of a photon, does that mean that a fully coherent wave (which, according to Glauber, satisfies an infinite order of coherence conditions), should contain infinite photons?

Hmm, that is a matter of interpretation. There are people that claim that one needs an infinite number of photons to have an infinite degree of coherence. I suppose if you want to detect them exactly at the same time, that is right.

 

[Scipio]

Thank you! That is very helpful. Some other questions:

No, they are not. First order coherence can more or less be reduced to field correlation, but it does not tell us much about the nature of the light source, just about field coherences. For example you can change the first-order coherence of any light field just by shaping it spectrally and spatially. This does not allow you to distinguish between a light bulb, non-classical light and laser light. This is why you move on to higher order coherence. Second-order coherence is the most common one and describes the intensity correlation instead of field correlation. Naively speaking, it gives you the probability to detect a photon at some point and time, conditioned on a second photon detection at another position and time. Usually this quantity is normalized to the unconditional photon count rates at the two times/positions.

That makes sense I guess. Could you give a similar application of higher order coherences (ie. higher than 2th)?

Indeed, correlated detections are the trademark of thermal light. Here the light leaves the light source in "bunches" which is a consequence of the light field having Bose-Einstein statistics. Anticorrelated detections are the trademark of non-classical light. Take a single photon source for example. If you detect one photon, the probability to detect another will be zero (otherwise it is not a single photon source). Both of these effects - bunching and antibunching - vanish on a timescale of the coherence time of the light field.

Okay that really helps, but I still don't see what then gⁿ and Gⁿ actually describe. They take photondetection events as their input so I guess they describe correlation in intensity rather than correlation of the field. If gⁿ = 1, that means that both sides in 3.14 are equal if I'm correct? Does that mean that all G's in there are equal to 1? And the expression in 3.11, G⁽¹⁾ (x1,x2), essentially describes the correlation of a point with itself right? Why isn't this always 1?

 

[Scipio]

And two more questions:

Is it correct to say that in the classical picture of light the intensity (of monochromatic light) corresponds to the amplitude of the light wave?

Does the interpretation of E⁺ and E^- as photon creator/annihilator imply that in reality a photon doesn't really travel but is constantly created and annihilated instead?

Thank you for your patience ;)

 

[Cthugha]

That makes sense I guess. Could you give a similar application of higher order coherences (ie. higher than 2th)?

For higher order coherences, the meaning stays similar. g3(r1,t1,r2,t2,r3,t3) gives the normalized probability to detect a third photon at time t3 and position r3, given that you have already detected two photons at t1/r1 and t2/r2. And so on. This is like developing all the conditional probability distributions for detecting photons into a series. If you look at the probability distribution of the photon number, the higher order moments of the correlation function will find their equivalent in the higher moments of the probability distribution. The first order will just depend on the mean photon number, second order coherence depends on the variance of the light field. Third-order coherence depends on the skew, fourth-order coherence on the kurtosis and so on.

Okay that really helps, but I still don't see what then gⁿ and Gⁿ actually describe. They take photondetection events as their input so I guess they describe correlation in intensity rather than correlation of the field. If gⁿ = 1, that means that both sides in 3.14 are equal if I'm correct? Does that mean that all G's in there are equal to 1? And the expression in 3.11, G⁽¹⁾ (x1,x2), essentially describes the correlation of a point with itself right? Why isn't this always 1?

Yes, it is about correlations in intensity, right. For gⁿ = 1, the two sides in 3.14 should be equal. The G's (capital letter) however are not necessarily 1. They are unnormalized and therefore depend on the intensity at the points in question. The g's (small letter) are the normalized ones and should equal 1.

The expression in 3.11 (or better its normalized version g) is not always one because one considers the correlation in terms of an ensemble average. As a visualization think of a light pulse. You repeatedly create several pulses under identical conditions. The expression in 3.11 now depends on whether the light field is intrinsically noisy. If it is, the actual photon number at the same position and the same time (relative to the emission of the pulse from its source) will vary from pulse to pulse. Now you take the ensemble average over all these pulses.

Is it correct to say that in the classical picture of light the intensity (of monochromatic light) corresponds to the amplitude of the light wave?

From a classical point of view, intensity corresponds to the square of the amplitude.

Does the interpretation of E+ and E- as photon creator/annihilator imply that in reality a photon doesn't really travel but is constantly created and annihilated instead?

Interpretations rarely imply much about reality. The creation/annihilation formalism rather tells us that a photon is created at some time, annihilated at some other time, but pretty much nothing about what happens in between. Just as a note: Do not get accustomed to interpret photons as bullets. This might seem intuitive at first, but will make understanding more advanced quantum optics more complicated and you will need to get rid of that picture anyhow.