How Do Phase Velocities Relate to Photons in Classical and Quantum Optics?

[Xian]

So we all learn in E&M and Optics courses that light is an EM wave. We play with simple models of light in which we treat a beam as a single frequency sine wave occupying the entire space. This of course is wrong since all physical signals have finite support (they are non-zero only on a finite interval) and are thus more accurately called wave packets. Anyhow, any wave packet can be turned into a superposition of sine waves over some distribution of frequencies each of which has a well defined phase velocities. So now comes a sequence of questions:

1) Are the component waves physical? Does each of these sine waves have an interpretation in terms of solely photons?

-If YES:
What is the relation between the EM waves phase velocity and the properties of the photons (can EM phase velocity be derived from the photons wave function)?

-If NO:
What is the correct way of decomposing a signal into its physical constituents (photons)? Using this method, can we determine how many photons are in a signal? What does the signal of a single photon look like?

2) Reverse question: If we have a whole ensemble of photons, and we measure it classically as a signal, what would it look like? Assume that we know the states of each photon, and if necessary assume that they are propagating in the same direction.

Its a lot of questions and a tall order, but the reason I ask this is because I want to see the connection between quantum and classical optics. Phase velocity is such an important thing in classical optics that I figure that its crucial that I grasp this concept. Thanks in advance guys!

 

[LostConjugate]

If you have an EM wave packet, first you break down each wave by frequency. Then a photon which is the quanta of that wave has a specific frequency. Photons are not wave packets of the EM wave. They make up the EM wave. EM waves make up packets.

The wave packet that makes up probability functions is another story and not physical.

 

[Xian]

If you have an EM wave packet, first you break down each wave by frequency. Then a photon which is the quanta of that wave has a specific frequency. Photons are not wave packets of the EM wave. They make up the EM wave. EM waves make up packets.

The wave packet that makes up probability functions is another story and not physical.

I understand sinusoids make up the EM wave packet, however this doesn't explain whether the sinusoids are physical things, after all, in general there are infinitely many of them, in fact uncountably many of them (using mathematical parlance). Am I to take this that infinitely many photons are in the wave packet? This can't be right. If we take a quantized approach (with the classical smoothness of EM waves as an approximation) we can assume that all wave packets actually have discrete number of sinusoids making them up, but densely packed in terms of frequencies. Here we could here ascribe each frequency a photon. But this begs the question of what does the Fourier coefficient mean?

 

[LostConjugate]

The collection of sinusoids makes up any function. An EM wave is just one example. Your current position on this Earth over your lifetime is another.

True the wave number for each sinusoid can take on any value. So can numbers. The distance between you and a wall is infinity if you think about it.

Zeno brought up the point many years ago that a man could never walk into a wall because each time he crosses half the distance he must cross half the distance again and an infinite number of processes takes an infinite amount of time.

The calculus shed light on this concept though I am not sure it ever proved him wrong.

Anyways to your question about photons, there is a limited number of photons in an EM wave, a photon in a wave with a specific frequency is the lowest possible detectable energy by matter. This was proved in blackbody radiation experiments.

I don't understand why your connecting the photon with a wave packet that makes up an EM signal.

 

[Xian]

The collection of sinusoids makes up any function. An EM wave is just one example. Your current position on this Earth over your lifetime is another.

True the wave number for each sinusoid can take on any value. So can numbers. The distance between you and a wall is infinity if you think about it.

Zeno brought up the point many years ago that a man could never walk into a wall because each time he crosses half the distance he must cross half the distance again and an infinite number of processes takes an infinite amount of time.

The calculus shed light on this concept though I am not sure it ever proved him wrong.

Anyways to your question about photons, there is a limited number of photons in an EM wave, a photon in a wave with a specific frequency is the lowest possible detectable energy by matter. This was proved in blackbody radiation experiments.

I don't understand why your connecting the photon with a wave packet that makes up an EM signal.

Going off topic for a bit, Zeno's paradox has been resolved with the understanding continuity and limits (concepts in calculus or as known in mathematics, analysis). The amount of time it takes to reach the wall is not infinite because even though you keep summing intervals of times (corresponding to half the distance traveled at each iteration) each interval becomes smaller and smaller (traversing the half of a half of a half takes less time than traversing the first half) and so when you sum all these time intervals, even though there are infinitely many terms, the summations sums to a finite number. There are many infinite series that converge to a finite number and so Zeno's paradox lies in that he didn't understand the finer details of infinite series and automatically assumed adding infinitely many things leads to an infinite answer.

Back on topic:
Photon make up all light signals. Classical EM theory does not have a notion of quanta and so has a special nature. In the analysis of Classical EM we can decompose any EM wave into sinusoids (not any function since Bochner's Theorem clearly shows us the limit of what functions can be expressed as combination of sinusoids). So knowing in what way an ensemble of photons is classically understood is of large importance.

 

[LostConjugate]

There are many infinite series that converge to a finite number and so Zeno's paradox lies in that he didn't understand the finer details of infinite series and automatically assumed adding infinitely many things leads to an infinite answer.

In calculus we take the limit that the infinite series converges towards. I have always wondered how we prove that the series ever reaches this limit though just because it continues to approach it. The only way the series could ever reach the limit is if it takes some sort of quantum jump.