# Interpretation of Wigner function in optics

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

• 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 4 '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 5:10
• 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 5:17

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

• 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 1 '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 1 '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 2 '11 at 9:14
• @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

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.

• 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
• 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

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

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

• +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 1 '11 at 6:09
• 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 1 '11 at 6:18
• 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 1 '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 1 '11 at 21:04

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