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I work in the field of synchrotron radiation sources where radiation (often x-rays) is produced from an electron beam going through magnetic fields. The quality of the resulting x-ray beam is determined by a parameter called the brightness, which is formally computed by the Wigner function for the radiation. The Wigner function is claimed to be a representation of the photon flux density in the radiation. It can sometimes be negative, which would be unphysical, but people say this is somehow due to Quantum Mechanics. There is this analogy between light optics and quantum mechanics, where $\hbar$ is replaced by the wavelength of the light. But I don't think that there is actual quantum mechanics involved, though I could be wrong. I think it is just an analogy.

So my question is: When one represents radiation via a Wigner function, is this really quantum mechanics? (A kind of semi-classical approximation?) Can someone point me to good references on understanding this from a slightly deeper perspective? I'm interested if there may be some real mystery here, or if its actually well-understood.

A reference by Kim: http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6202594

Boaz
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    The *formal analogy* (strictly speaking) you are referring to is the brainchild of Bartelt, H. O., K-H. Brenner, and A. W. Lohmann. "The Wigner distribution function and its optical production." Optics Communications 32.1 (1980): 32-38. Bartelt, H., and K. H. Brenner. "THE WIGNER DISTRIBUTION FUNCTION-AN ALTERNATE SIGNAL REPRESENTATION IN OPTICS." Israel Journal of Technology 18.5 (1980): 260-262. Brenner, K-H., and A. W. Lohmann. "Wigner distribution function display of complex 1D signals." Optics Communications 42.5 (1982): 310-314. – Cosmas Zachos Feb 04 '16 at 20:31
  • Thanks for the reference @CosmasZachos. I will try to read it.

    But what distinguishes a formal analogy from true quantum mechanics? Measuring a photon, one finds it at a position in the same sense as for, say, an electron in a hydrogen atom.

    Shouldn't the Wigner function give this probability? I've tried to read some articles on the wave function of the photon as well, but haven't really sorted this issue out yet.

     – Boaz May 22 '17 at 22:28
  • Ummmmm. If you wish to reconstruct wave optics out of quantum mechanics, Wigner functions might be the worst place to start... – Cosmas Zachos May 22 '17 at 23:08
  • I don't want to reconstruct wave optics, I want to understand about the quantum mechanics of photons. The Wigner function is equivalent to the mutual coherence function which is the analogue of the density matrix. In quantum mechanics there is a statistical interpretation of the density matrix. I am wondering what one could say about the statistical interpretation of the Wigner function or mutual coherence function in optics. To what extent is the Wigner function a probability function for finding a photon at a given position and k vector (momentum)? – Boaz May 23 '17 at 05:10
  • Its a relevant question in synchrotron light source beamlines, @CosmasZachos. We have codes for wave optics (one in particular called SRW), and ones for ray tracing (e.g. one called Shadow). An undulator produces a complicated wave front but then one would like to use the ray tracing software by sampling the Wigner function to determine the density of rays. – Boaz May 23 '17 at 05:17
  • The analysis in Schleich's book should be adequate for the Wigner function of photons. I do not believe for a moment the cited Kim talk investigates or accesses quantum correlations of photons. – Cosmas Zachos May 23 '17 at 13:55
  • The Schleich book looks good. But why shouldn't the Wigner function describe the quantum correlations of photons? I understand that the electric field for the radiation field acts as the wave function (putting aside issues of polarization). This is the electric field produced by a single electron. Its essentially representing a photon. So the Wigner function should have the same connection as in other QM contexts.

    Here's another paper on this by Bazarov: https://arxiv.org/abs/1112.4047

     – Boaz May 23 '17 at 17:52
  • Indeed, but Bazarov takes pains to *contrast* QM and wave optics, but you are arguing for a grand synthesis of quantum optics with coherence theory, all on the basis of phase-space representations, which, here, confuse rather than clarify the points of contact. – Cosmas Zachos May 23 '17 at 18:32
  • Ok, I will have to think about this more. I've been partly motivated in this quest by the following quote by P. Elleaume in “Undulators, wigglers and their applications” p. 54-55: – Boaz May 23 '17 at 21:51
  • "I leave the reader to adopt whichever point of view he prefers […]. I have not found any safer or more elegant alternative than the Wigner function to approximate the distribution of photons in phase space. In the following, I shall keep to this. This discussion may sound strange to a novice. Despite a number of discussions that I have had in the past with various experts, I have been unable to obtain any acceptable consensus on the questions. " – Boaz May 23 '17 at 21:51

4 Answers4

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Citations are from here on Wikipedia.

Is this really quantum mechanics?

In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = |k|sinθ ≈ |k|θ in the small angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference. Seems that answer is "no". While I have a feeling that there is no strict boundary. I usually think of optics as of a "bridge" between quantum and classical mechanics...

It can sometimes be negative

If it becomes negative at any point then simple ray-tracing will not suffice to model the system.

Good references

In the end of the Wikipedia article. For example this one: http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics

Community
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Kostya
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  • Thanks Kostya. I appreciate the answer and the references. So, mostly you can think of the Wigner function as an approximate way to model the physical optics while "expanding around" the geometrical optics. But still, there is some sense in which it represents a distribution of photons? In which case, upon measuring the number of photons, or the transverse extent or divergence, one would get different asnswers each time. In this sense, it might be quantum mechanical? – Boaz Oct 31 '11 at 11:30
  • So, to me, the question still remains as to what is the right "photon wave function". The Wigner function is useful in its own right to capture the physical optics as described in this answer, but can one think of the electric field as like the wavefunction when one does the second quantization?
    Perhaps the probability of finding a photon is proportional to the energy density $E^2+B^2$. Not clear if that reduces to WF in some limit.     – Boaz Nov 01 '11 at 11:45
  • You cannot write a wavefunction in the traditional sense for a photon. The notion of position is meaningless in the case of a photon.I have updated my answer with more articles/information. – Antillar Maximus Nov 01 '11 at 20:58
  • In what sense would position be meaningless for a photon? In the same sense as in quantum mechanics for particles? When we observe a photon it is at a particular position. – Boaz Nov 02 '11 at 09:14
  • @Boaz Perhaps this review might be useful. The negative features of your WDF are due to the *Gabor limit*, not the Heisenberg UP, even though the two are analogous---mathematically formally identical. – Cosmas Zachos May 25 '17 at 15:22
  • Thanks, @CosmasZachos. Both references look interesting. Motl always has sharp things to say, though I do think he can have substantial blind spots. And good to get a different perspective on the uncertainty principle! – Boaz May 26 '17 at 07:28
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You're effectively doing signal analysis. It's neither quantum mechanics nor a semi-classical approximation. Signal processing is (very) often not stochastic (random), unless thermal noise is an issue for accurate modeling of the apparatus. It's often easier to find a way to eliminate the effects of thermal noise than to calculate the effects of thermal noise. If quantum noise has a significant effect, then one has to use quantum mechanics.

The mathematics of signal processing can almost always be presented in terms of Hilbert spaces, just because Fourier analysis of the signals is so important, even when the signals being processed are noise-free, which gives a specific mathematical link between deterministic signal processing and quantum mechanics, however the interpretation of the observables is generally quite different. I would say that the ideas that the "Wigner function is claimed to be a representation of the photon flux density in the radiation", and that negative values are unphysical, are not helpful, that it's better to see the negativity as a consequence of attempting to measure the frequency of a signal over a very small time period, whereas measuring the frequency of a signal precisely in principle requires the signal to be measured as a function of time for all time, because the Fourier analysis of a signal requires us to take the integral over all time, $\int_{-\infty}^\infty f(t)\mathrm{e}^{\mathrm{i}t\omega}\mathrm{d}t$.

A good reference for Wigner functions is Leon Cohen, "Time-Frequency Distributions-A Review", PROCEEDINGS OF THE IEEE, VOL. 77, NO. 7, JULY 1989, DOI: 10.1109/5.30749. Alternatively, a book by the same author, Leon Cohen, "Time Frequency Analysis: Theory and Applications". Sorry to say that either of these will need access to an academic library or cash. I didn't know of any on-line references before I searched for them now --- the Wikipedia page you want is cited on the page cited by Kostya, http://en.wikipedia.org/wiki/Cohen%27s_class_distribution_function; you should certainly read it as well because it is much more relevant to your application. There is a fairly strong sense in which this is well understood, but quantum mechanics, and particularly quantum field theory, is not so well-understood, which makes it correspondingly difficult to say that the relationship between classical signal processing and quantum mechanics is well-understood.

Peter Morgan
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  • Thanks a lot for the answer, Peter. I will have a look at these references. I wasn't aware of this connection to signal processing. – Boaz Oct 31 '11 at 14:32
  • Actually, thinking a little further, its an interesting point about this connection between the Wigner function and the signal processing, but that looks to me to mostly apply to the time structure of the signal. In this application of describing radiation propagation, the Wigner function is used the represent the transverse distribution of the radiation. In this case, the connection isn't so clear to me. – Boaz Oct 31 '11 at 14:42
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    For me there is an emphatic relationship to signal processing. I see all of quantum mechanics and quantum field theory through this lens. – Peter Morgan Oct 31 '11 at 14:51
  • Yes, in the end, we observe things by decoding the signals they produce. Its a new perspective on this topic, and I'll look into it. Thanks! – Boaz Oct 31 '11 at 14:56
  • I think by the "transverse distribution" you mean the polarization properties of a single frequency signal? Do you have a reference that will show me how the mathematics is set up? It isn't quite set up as I'd like to see it on your blog. I'll be looking for the underlying Hilbert space structure and whether the observables correspond to stochastic or deterministic experimental data. If the latter, then it will be essentially deterministic signal processing; if the former, then it may be either stochastic signal processing or more essentially quantum theory. – Peter Morgan Oct 31 '11 at 15:00
  • KJ Kim was one of the people who introduced this concept into the field. The paper at the bottom of the question is an early reference: http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6202594 In equation (4), he defines the brightness in terms of the transverse Electric field for a fixed frequency $\omega$ and bandwith $\delta \omega$.
    Thanks for having a look.     – Boaz Oct 31 '11 at 15:09
  • Right, sorry I didn't look at that reference already, but it was helpful that you pointed me to Eq. (4), which is set up very much in terms of Cohen's type of analysis. The signal is a function of position instead of a function of time, but that's mathematically inessential. This particular paper also "limit[s] our discussion to a single polarization component because, in synchrotron radiation, usually only the horizontal component is important", but adding the vertical component would leave the observables as essentially deterministic signals. Definitely Cohen is relevant for you. – Peter Morgan Oct 31 '11 at 15:32
  • Ok, I found a link to the 1989 IEEE paper by Cohen you reference and will look at it. (I could probably ask the library also, and can get it that way as well.) I do want to understand the math better, but its more in the interpretation I'm confused. For example, we can store a single electron in a synchrotron, and this electron will radiate. (You can actually observe this) In this case, does the Wigner function represent a sort of quantum mechanics like probability distribution for where you will find the photon? That's the kind of thing I'd like to understand. – Boaz Oct 31 '11 at 15:59
  • With that, you've more-or-less moved into QM and a 1-electron synchrotron state. There are correlations between the macroscopic electromagnetic signals in the synchrotron and the electrical signals from CCDs. The latter are engineered to give continuous signals that are mostly flat but that occasionally have sharp rises, so that the signal can be summarized by discrete event times, then one uses probabilities and correlations with the macroscopic electromagnetic signals to model statistics of the discretized events. It's still signal processing, but with a nontrivial discrete structure. – Peter Morgan Oct 31 '11 at 17:20
  • Agreed, but I still don't know if this transverse Wigner function itself tells me the correct QM correlations. The claim in the paper by Kim (and I think by others) is that the Wigner function represents the photon distribution. You suggested maybe that's not right, or so useful. (btw, I'm new to this Stack Exchange, and its telling me to avoid extended discussions in the comments, and to move this to a chat... but I don't have enough points to chat...) Anyway, thanks for engaging on this. – Boaz Oct 31 '11 at 17:53
  • Following answer to other question: http://physics.stackexchange.com/q/437/5928 is a reference to this paper: http://www.cft.edu.pl/~birula/publ/APPPwf.pdf Describing a "photon wave function" and corresponding Wigner function. I'd be curious if this is somehow the same function. Again, long discussion in comments doesn't seem to be what this site wants. – Boaz Oct 31 '11 at 18:23
  • Sometimes the SE rules get in the way, especially for people who are new. The EM field strength is an observable of the quantized EM field at a point, which gives just one statistic (for each point in space-time) that can be generated in a given state of the quantized EM field. When the EM field is relatively large, that's often enough, but when the EM field is small we also want to characterize the correlated fluctuations of the quantized EM field between every 2-, 3-, ..., n-points in space-time. The "photon distribution" is equivalent to all those numbers, not just the EM field strength. – Peter Morgan Oct 31 '11 at 18:24
  • You can't do "chat" until you have more reps, so we will ignore the message! Otherwise we would play nice.

    Some of those links will get you thinking about it in what I think is the right way. You see them talking about the quantized EM field. But it's treacherous ground on which we all have to try to find as firm a foundation as we can.

     – Peter Morgan Oct 31 '11 at 18:32
  • Ok, yes, I guess these different n-point corellations are what the QFT of the electromagnetic fields would give you. So, its not clear whether this Wigner function somehow approximates this. – Boaz Oct 31 '11 at 18:32
  • Thanks. I have to stop commenting for this evening anyway (I'm in Europe). I'd also like to stay on firm ground with this stuff, but in my field, its just claimed that the Wigner function does this, but I don't really know how well founded this is. – Boaz Oct 31 '11 at 18:35
  • Indeed, the application of the WF description is a signal processing analogy similar to the time-frequency ones adduced above, by din't of the similarity, *not identity!, of the propagation scalar paraxial optics equation involved to the Schroedinger equation, cf[http://www.ejournal.unam.mx/rmf/no486/RMF48612.pdf cf.]. Connecting such second quantization type* macroscopic wave equations to the Schr equation is extraordinarily fraught with logical misconceptions and throwing the phase space extension of the WF on the fire to boot cannot help things, rather than obscure them! – Cosmas Zachos Feb 04 '16 at 20:23
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The Wigner function is used to describe joint probabilities between two sets of observables that do not commute. I could elaborate a bit more, but there is a lot of literature available where the authors explain this better than I can.

A couple of useful articles from arXiv:

Probabilistic aspects of Wigner function

Negativity of the Wigner function as an indicator of nonclassicality

Photon viewed from Wigner Phase Space

Google Book result of the above paper

What is a photon?: OPN-Trends special series by experts in Quantum Optics -pdf file

Edit: 11/2/2011

Maxwell Wavefunction of a Photon Photon Wavefunction

Interaction between Light and Matter, a wavefunction approach-pdf file

Antillar Maximus
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  • +1, but note for the future that it's preferable to link to the arXiv abstract pages, not directly to the PDF. Some people prefer the postscript versions, some people don't have fast connections, or perhaps one can go to the published versions. – Peter Morgan Oct 31 '11 at 23:25
  • Copy on the links. I will update my post with a few more references. – Antillar Maximus Nov 01 '11 at 06:09
  • Thanks for the answer and the links AM. These look like good discussions of the probability aspects of the Wigner function and what the negativity means. However, in both these papers, the Wigner function is defined in terms of the wavefunction $\Psi$ which is already in the framework of quantum mechanics. In the optics case, one starts with the electric field which is classical. I think the answer given by Kostya is largely correct in that the "quantum" behavior is about failure of ray optics. The question is whether there's a further sense of the WF describing the distribution of photons. – Boaz Nov 01 '11 at 06:18
  • Boaz, you can always quantize the Electromagnetic field and consider the "classical limit", where you are interested in the mean value of the field (or field squared, which is what you detect experimentally). However, if you choose to work in the classical regime, the question of non-commuting observables does not arise, so using the Wigner representation seems pointless. By distribution of photons, do you mean photon counting? If so,the classical picture is inadequate to describe experimental results (except for, say, a coherent state). – Antillar Maximus Nov 01 '11 at 20:54
  • Just added a very interesting article series (pdf file) to the main answer post above. Hope this helps. – Antillar Maximus Nov 01 '11 at 21:04
  • Looks like an interesting paper on photons, AM. Thanks you, and I will look at it. Regarding quantizing the EM field, the formalisms I know about are (1) wave functional for the field configuration and (2) S-matrix to compute transitions between in states and out states. Neither of these formalisms make it seem easy to think of propagating wave packets for the photon(s). There is this Wigner function, but I don't know if there's any relation to the QED or some kind of QM. Anyway, I'm definitely learning and refining my question. Thanks to you and the others! – Boaz Nov 02 '11 at 09:09
  • I looked quickly over the OPN trends paper- quite interesting, and leads me to believe there may still be some mysteries regarding light and photons to discover. The last article by Mack and Schleich does say that one can represent photons and their quantum properties by a Wigner function. Still, the Wigner function is defined in eqn 19 in terms of the wave function, $\psi$ whereas in the paper by KJ Kim, its defined via the electric field (horizontal component). So the question would be whether we're talking about the same thing here! – Boaz Nov 02 '11 at 10:52
  • I remember asking this and my teacher said that Wigner proved that you cannot define a position operator for anything beyond a spin 1/2 particle. But since then attempts have been made to introduce a wavefunction (I have updated my post with more references). As for the question of the measurement process itself, I have to think a bit more about it. I actually like David Finklestein's article, but that is because I am biased in favor of Group theory. – Antillar Maximus Nov 02 '11 at 13:51
  • The reason you don't hear much about the photon wavefunction is because the more interesting question is to study interactions of the photon field with the electron field. The Wigner function is intimately related to QM because it describes measurements that are incompatible (i.e position and momentum). This, as far as I am aware of does not exist in classical field theory. I don't know much about S-Matrix theory, but I was under the impression that it describes particle-particle collisions only. – Antillar Maximus Nov 02 '11 at 13:55
  • This stuff about the photon wave function is quite interesting. In some sense the fields themselves are the wavefunction, though I still have to understand this issue of the position operator not existing. So perhaps the Wigner function as defined by Kim and as used to compute Brightness in synchrotron light sources is also describing the localization and angular properties of the photons. In the light source case, the radiation is produced in an undulator that can be several meters long.One can define an image plane at the center however, and its here that one computes the Wigner function. – Boaz Nov 02 '11 at 14:24
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The Wigner function is an equivalent representation of the quantum state used in the Wigner & Moyal formulation of quantum mechanics. Sometimes named the phase space formulation.

In fact the Wigner function is more general than the wavefunction, because also represents quantum mixed states whereas wavefunctions only can represent pure states. This recent formulation is rather popular in quantum optics because the equation of motion takes a simple form for quantum harmonic oscillators (a quantum field is a collection of such oscillators) and because of other reasons.

Yes, the Wigner function is a pseudo-probability distribution and can take on negative values. The negative values are a consequence of quantum interference.

Therefore the answer to your question is yes when one represents radiation via a Wigner function this is just quantum mechanics.

Any advanced textbook in quantum mechanics (e.g. Ballentine) discusses the Wigner function. A classic reference in quantum optics using the Wigner function is http://www.amazon.com/Quantum-Optics-Phase-Wolfgang-Schleich/dp/352729435X

juanrga
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