I Is there a difference between EM waves and photon wavefunctions?

[andresB]

This is the representation in terms of Silberstein vectors mentioned in post #4.

True.

I posted it more for the interpretation the author gives to the equations

 

[Charles Link]

There is no difference between classical and quantum in the case of a single, localized detector.

However, for multi-point detection of correlated electromagnetic fields there is indeed a difference between the classical and the quantum description - not for coherent states where the two descriptions still match but for nonclassical states, for example for squeezed states (used, e.g., to describe parametric down-conversion). This difference is an expression of the fact that electromagnetism in Nature is described by QED and not by classical electrodynamics.

The main difference in the resulting detection statistics of the ionization events is usually described in terms of the notions of bunching and antibunching.

Thank you. This is quite interesting. It is somewhat difficult to stay completely current with all of the new progress that has occurred, but this gives me a good starting point.

 

[vanhees71]

Under natural idealization assumptions that are usually satisfied to good accuracy, the predicted and measured single electron-ionization rate is proportional to the intensity of the electromagnetic radiation field (not the electric field alone). This is the only unambiguous statement. Since this statement is valid both in the semiclassical treatment where there are no photons and in a full quantum field treatment (where - see below - the question of what a single photon is is ambiguous), the interpretation of this in terms of photons, though widely used, is questionable. Photodetection is measuring the intensity of the electromagnetic radiation field by measuring the electron-ionization rate, not by counting the number of emitted photons. If the intensity goes down, one simply needs to wait longer to get a reliable measurement of the intensity to a given accuracy. In my opinion, the detector events have no other meaning than this.

It is in some way analogous to measuring the output flow rate of a water faucet by (observing from far apart - not seeing the details) how many cups are filled in a given time. This works accurately in a short time if a lot of water flows, but if the faucet is only dripping it takes a long time before the number of cups accurately represents the water flow rate. I find this analogy helpful although it is only a classical analogy with a limited explanatory value as it lacks the randomness observed in the quantum case.

The problem with the notion of a single photon in a quantum field setting is that the quantum field typically is in some Heisenberg multiphoton state, and this single Heisenberg state gives rise to an electron-ionization rate rather than a probability. For example, in an ordinary laser the electromagnetic field is in a fixed coherent Heisenberg state, and by waiting long enough this fixed state gives rise to as many electron ionizations as one likes. This even holds when the coherent state has very low intensity. This is quite unlike what is assumed in the typical quantum optical experiments where everything is interpreted as if photons were particles just like nonrelativistic electrons, except that they move with the speed of light. There is a nontrivial interpretation step in going from the former to the latter description, one never analyzed in the literature (as far as I can tell).

The Heisenberg description (QFT) has a straightforward, almost classical interpretation of the state. To go from it to the interaction picture (QM) with the interpretation of an ensemble of particles (prepared or actually being - depending on the interpretation of QM used) in a corresponding state one has to ''invent'' individual photons with ghostlike properties that are completely unobservable until detection events destroy the ghosts and thereby prove their alleged existence. Extremely weird, this assumed picture of individual photons. One is left wondering why these ghosts are completely absent in the semiclassical description (which is quantitatively correct in the case of coherent light) although the detection events still herald their existence (if they herald anything). But if they herald nothing in the semiclassical setting, they also shouldn't herald anything in the quantum case. This then implies that the ghosts are completely unobservable. They can be eliminated without making any observable differences, and using Ockham's razor, they should be eliminated. My language is chosen accordingly.

In short, to make sure to measure single photons you have to prepare single-photon states at the very beginning. A detector click from a laser (coherent state), may it as "dim" as you like, does not necessarily mean one has detected a single photon and only a single photon.

 

[A. Neumaier]

In short, to make sure to measure single photons you have to prepare single-photon states at the very beginning. A detector click from a laser (coherent state), may it be as "dim" as you like, does not necessarily mean one has detected a single photon and only a single photon.

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

 

[f95toli]

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

That depends on the detection method. There are CQED methods that allow you to "sense" the number of photons in a Fock state dispersivly. This can e.g. be done using a system where you have an effective single photon Kerr term.

See e.g. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.107001
for one of the more recent results in this area. This is probably not the best reference,. but all I could think of off the top of my head. .

Thinking of "conventional" detectors such as PN diodes when discussing single photon states is -in my view- just confusing. It is much better both theoretically and experimentally to use some sort of qubit (i..e. non-linear two level) based detection system since you can then distinguish between true single photon states and just "bursts of energy" as mentioned above.
Non-classical states of light is quite a bit research field so there are a lot of references available for those who are interested.

Edit: Note also that is is possible to perform tomography of photon states using "classical" detectors by using methods borrowed from radio engineering (IQ mixing) . It is is of course an indirect method since you can't "see" individual events, but as long as you are able to average over many events you can re-construct the state of the photons generated by your source.

 

[A. Neumaier]

There are CQED methods that allow you to "sense" the number of photons in a Fock state dispersivly.

How do they respond to dim coherent laser light?

 

[vanhees71]

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

Exactly. Now we completely agree :-)).

 

[A. Neumaier]

Blais, Alexandre, et al. "Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation." Physical Review A 69.6 (2004): 062320.

These systems can be used to create and manipulate just about any photon state you can imagine, including obviously simple Fock states.

I was talking about photons in a beam extending over a laboratory desk, and their detection. Your setting is quite different.
How do you get the photons from a photon beam into the cavity to be detected by your solid state devices?

 

[A. Neumaier]

Note also that is is possible to perform tomography of photon states using "classical" detectors by using methods borrowed from radio engineering (IQ mixing) . It is is of course an indirect method since you can't "see" individual events, but as long as you are able to average over many events you can re-construct the state of the photons generated by your source.

This can be done for the state of any source - one doesn't need any assumptions on whether or not it contains ghostlike particles.

The density matrix of an $n$-level system has $n^2-1$ independent degrees of freedom, and it is not difficult to find this many independent observables such that, from the measurement of their mean (by repeated observation of individual systems emitted by the source), the density matrix can be reconstructed by solving a linear system of equations.

 

[Elemental]

Sciencejournalist00,

On your original question,

Or is the electromagnetic wave considered to be the wavefunction of the photon?

consider the following. One performs a "box quantization" of the electromagnetic field by treating each cavity mode of the electromagnetic field as a harmonic oscillator. Here though you are actually starting with cavity modes of the electromagnetic field rather than modes of the Schrödinger wave function but it goes the same way. Singling out a particular mode, from Maxwell's equations it is found one can treat the electric field as the generalized coordinate and the magnetic field as the conjugate momentum. (Of course other approaches are possible including treating the magnetic field as the coordinate or more commonly quantizing the vector potential, but this approach illustrates the relation of wave function to EM field.)

You write the Hamiltonian for the EM field energy, then re-express it in terms of annihilation/creation operators exactly as is done for the harmonic oscillator. Presto, you obtain EXACTLY the same Schrödinger equation and equations of motion as you get for the harmonic oscillator. This has a wave function whose allowed modes have the same frequency as those of the electromagnetic field you started with! One can then define coherent states like you get from a laser, Fock (number) states, squeezed states, etc.

So the modes of the wave function sure look a lot like electromagnetic field modes. But what about its interpretation? For the harmonic oscillator, the magnitude squared of the wave function gives the probability of the position (coordinate) of the particle. For the quantized electromagnetic field as derived here, it does not give position probability, it gives the probability that the electric field will have a particular value. Not the same thing. However I've noticed in many quantum optics papers that the difference in interpretation between the Schrödinger wave function and the electromagnetic filed is glossed over.

 

[vanhees71]

One should stress that there is no consistent interpretation of relativistic wave equations in the sense of the "first quantization formalism", except in certain low-energy cases, where the physics is close to non-relativistic dynamics, i.e., only for massive quanta (e.g., bound states, where the binding energy is small compared to the masses of the bound quanta).

For massless quanta, and photons are with amazing accuracy observed as massless quanta, no such "wave-function interpretation" makes any sense. Since photons are described as massless spin-1 fields, they do not admit the definition of a position operator, and thus the position of a photon is not describable as an observable. So the localization of a photon is not even definable.

Another reason is that photons are very easily produced and destroyed. This leads to the description of relativistic quanta in terms of a quantum many-body, where quanta can produced and destroyed, and thus the number of quanta is not conserved. This is precisely what quantum field theory does in a very natural way and as explained in the previous posting #40.

The "box quantization" described in this posting is not only an interesting real physical situation in "nano physics", where one investigates single photons or states with a small number of photons in cavities ("cavity QED"), but also a convenient mathematical technique to quantize the (completely gauge fixed, using the radiation gauge) electormagnetic field canonically. In this case, you rather use periodic than rigid boundary conditions, because then you have a well defined momentum operator for the photons in the box-regularized version. At the end of the calculation you can take the infinite-volume limit, leading to the usual continuous momenta of photons in free space.

For a very detailed explanation, including a very good and intuitive introduction to the necessity for the field-quantization approach for relativistic particles, of this approach to photons, see Landau&Lifshitz, Course of Theoretical Physics, vol. IV.

 

[tade]

This is not correct. it is well-known that photon location cannot be described by a probability distribution

What about the double-slit experiment, and the interference fringe pattern?

Aren't the bright fringes bright because photons have a higher probability of being there, and the dark fringes are dark because the photons have a low probability of being there?

What about double-slit experiments where they reduce the intensity to one photon at a time, yet there is still an interference pattern?

 

[A. Neumaier]

What about the double-slit experiment, and the interference fringe pattern?

The point where a beam hits a screen determines only the transversal position, not the longitudinal one. Transversal position is reasonably well-determined in the paraxial approximation.

 

[tade]

The point where a beam hits a scree determines only the transversal position, not the longitudinal one. Transversal position is reasonably well-determined in the paraxial approximation.

I'm sorry, I don't understand this terminology.

How does this relate to probability waves, or that photons cannot be described by a probability distribution?

 

[A. Neumaier]

I'm sorry, I don't understand this terminology.

If a beam goes into the z-direction, the transversal position are the x,y coordinates only, while z is the longitudinal position. The paraxial approximation defines what a beam is in term of a field.

How does this relate to probability waves, or that photons cannot be described by a probability distribution?

There are commuting operators for transversal position but not for longitudinal position.

 

[tade]

If a beam goes into the z-direction, the transversal position are the x,y coordinates only, while z is the longitudinal position. The paraxial approximation defines what a beam is in term of a field.There are commuting operators for transversal position but not for longitudinal position.

Ok, is that considered a transversal probability distribution?

My understanding so far is described by this Wikipedia passage:

Sending particles through a double-slit apparatus one at a time results in single particles appearing on the screen, as expected. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (see the image to the right).

This demonstrates the wave-particle duality, which states that all matter exhibits both wave and particle properties: the particle is measured as a single pulse at a single position, while the wave describes the probability of absorbing the particle at a specific place of the detector. This phenomenon has been shown to occur with photons, electrons, atoms and even some molecules, including buckyballs.

This is the electron build up over time. What's the fundamental difference between electrons and photons in this scenario?

I posed a question about wavefunction and EM waves in my thread: https://www.physicsforums.com/threa...ld-and-probability-amplitude-of-waves.869530/
which is what led me to this thread.

 

[Nugatory]

Aren't the bright fringes bright because photons have a higher probability of being there, and the dark fringes are dark because the photons have a low probability of being there?

No, and the problem is in the words "being there" - wording it that way is already assuming that the photon has a position and hence a probability distribution for that position. There's no substitute for actually learning and using quantum electrodynamics, but if you want a non-rigorous and intuitive sort of understanding, you could say that the bright fringes are bright because there is a higher probability that the electromagnetic field will deliver energy to those areas, or you could say "appearing there" instead of "being there".

 

[tade]

No, and the problem is in the words "being there" - wording it that way is already assuming that the photon has a position and hence a probability distribution for that position. There's no substitute for actually learning and using quantum electrodynamics, but if you want a non-rigorous and intuitive sort of understanding, you could say that the bright fringes are bright because there is a higher probability that the electromagnetic field will deliver energy to those areas, or you could say "appearing there" instead of "being there".

I see.
Am I right to say that a photon cannot be described by a scalar wavefunction? (a wavefunction similar to the wavefunctions of the Schrödinger equation) That only a vector field can describe it?

 

[A. Neumaier]

I see.
Am I right to say that a photon cannot be described a scalar wavefunction?

The wave function is given by the Silberstein vector field.
You may wish to browse my theoretical physics FAQ.

 

[tade]

The wave function is given by the Silberstein vector field.

Ahhh, thank you.

 

[A. Neumaier]

Ahhh, thank you.

See http://arnold-neumaier.at/ms/optslides.pdf

 

[tade]

See http://arnold-neumaier.at/ms/optslides.pdf

Thanks.

By the way, can you help with me these problems?

https://www.physicsforums.com/threads/what-is-the-theoretical-basis-for-kramers-law.869641/

https://www.physicsforums.com/threads/plancks-1st-derivation-of-plancks-law.869497/