# What is the Wigner function of $|n\rangle\langle m|$?

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.

• 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

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).
• 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
• Is it known what is the P function of $|n \rangle \langle m|$? in terms of $\alpha$ and $\alpha^*$? – Nicolás Quesada Jan 22 at 20:49