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I have been searching in the literature for the Wigner function of $|n \rangle \langle m|$. For $n=m$ it can be found in page 120 of Barnett and Radmore's Methods in Theoretical Quantum Optics and it is given by $$ W_n(\alpha) = \frac{2}{\pi} (-1)^n \exp(-2 |\alpha|^2) L_n(4 |\alpha|^2), $$ where $L_n$ is a Laguerre polynomial.

The result I am looking for is used in the QuTiP software package (specifically, in the wigner.py file) to calculate Wigner functions given the matrix representation of state in the Fock basis.

From the code (see line 223 for example) it seen that the Wigner function for $|n \rangle \langle m|$ is related to a generalized Laguerre polynomial, they reference the book Measuring the Quantum State of Light (Cambridge University Press, 1997) by Ulf Leonhardt, but I checked the book and it does not contain the expression that I am looking for.

Thanks in advance for any help/references you can provide.

Cosmas Zachos
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Nicolás Quesada
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    I imagine that Cosmas's answer agrees with the QuTiP implementation, but in any case I would encourage you to raise a GitHub issue detailing this as a documentation bug, to help out the next person to stumble on this one. – Emilio Pisanty Oct 13 '17 at 17:52

1 Answers1

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It is, explicitly, in terms of associated Laguerre polynomials, (actually the Wigner transform of $|n\rangle \langle m|$, given its (idiosyncratic/Moyal) flipped notation, $H\star f_{mn}=E_{n}\,f_{mn}$), $$ f_{mn}=\sqrt{\frac{m!}{n!}} e^{i(m-n) \arctan\left( p/x\right) } \frac{\left( -1\right)^{m}}{\pi\hbar}\left ( \frac{x^{2}+p^{2}}{\hbar/2} \right ) ^ {\left( n-m\right) /2}\!\! L_{m}^{n-m}\!\!\left( \frac{x^{2}+p^{2}}{\hbar/2}\right) \, e^{-\left( x^{2}+p^{2}\right) /\hbar}, $$ equation (74) of this book of ours, as discovered by Groenewold (1946), eqn (5.16), and Bartlett & Moyal (1949), eqn (2.5).

You must heed the normalizations, of course, by taking the n=m limit.

PS Actually, in all fairness to Leonhardt, he does have the formula, albeit split into two parts, (5.105), (5.108).

Cosmas Zachos
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  • Hi Cosmas. Thanks a lot. Your book looks really interesting and seems to be mostly self contained which is great. Just a minor typo correction, $k$ should be $m$ in the Eq. you wrote. – Nicolás Quesada Oct 13 '17 at 17:33
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    @CosmasZachos Was the edit really necessary? You bumped the question into the front page just to add a link to google.books, which is hardly useful anyway. May I ask you please to refrain from making such trivial edits in the future? It seems that you are doing it just to get more votes, which is generally frowned upon here... – AccidentalFourierTransform Mar 11 '18 at 20:58
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    Ahmm I don't mind the misplaced frown, but, yes, the link to Leonhard is salutary, and does not belong to my answer, since the question should not have been asked in the first place, had his book not been thought lacking, if only it were available to the OP. Frankly, I am not interested in game-harvesting votes. You are free to deduct any and all future votes off this question. Deal? A very similar question was asked this moth, of course. – Cosmas Zachos Mar 11 '18 at 21:04
  • Is it known what is the P function of $|n \rangle \langle m|$? in terms of $\alpha$ and $\alpha^*$? – Nicolás Quesada Jan 22 '19 at 20:49
  • Well, yes, indirectly. WP expresses W in terms of P, which you can invert. W(α,α*) is 5.108, 5.105 of Leonhardt's book, which some commenters disapprove of here, but I still think is your best bet. I would not be surprised if Schleich's book has it all worked out. – Cosmas Zachos Jan 22 '19 at 22:43
  • Well, he sort of has, W Schleich 2001, that is. In 12.4.3, 12.4.4, Appendices J.2, J.3. You appreciate it is a monstrous concatenation of derivatives on δ(α)s... Note what happens for n=m... – Cosmas Zachos Jan 22 '19 at 22:56
  • Thank you for this answer. Like the OP I found this in the QUTIP code and wanted to find its source. Interestingly the result seems extremely similar to the "Laguerre Gauss" beam of Optics. https://en.wikipedia.org/wiki/Gaussian_beam – Dast Apr 07 '20 at 10:53
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    @Dast Yes, I have been told of the congruence of the WF formalism to paraxial beam structures, but I have never followed it expertly... – Cosmas Zachos Apr 07 '20 at 11:26
  • ...but here is a relevant answer. – Cosmas Zachos Apr 07 '20 at 13:19