Can the Wigner function be described using coherent states?

Question

The Wigner function for a wave function $\Psi(\vec{r})$ is $$ W(\vec{r},\vec{k}) = \frac{1}{2\pi} \int dy e^{-i \vec{k} \cdot \vec{y}} \Psi^{*}(\vec{r}-\vec{y}/2) \Psi(\vec{r}+\vec{y}/2) . \tag{1} $$ Recently, I've come to learn of the generalized Cahill phase space representation $$ F^{s}(\alpha) = \frac{1}{\pi^{2}} \int d^{2}\xi e^{\alpha\xi^{*}-\alpha^{*}\xi+\frac{s}{2}|\xi|^{2}} \mathrm{Tr}D(\xi)\rho , \tag{2} $$ which is a continuous family of quasiprobability distribution functions with respect to the parameter $s\in\left[-1,1\right]$. Here, $D(\xi) = e^{\xi a^{\dagger} - \xi^{*} a}$ is the displacement operator, where $a$ and $a^{\dagger}$ are the annihilation and creation operators satisfying $[a, a^{\dagger}] = 1$. As $s$ interpolates from $-1$ to $0$ to $1$, it is able to take the form of the Glauber–Sudarshan $P$ function, the Wigner Function and the Husimi $Q$ function.

This means to me that the Wigner function can be represented in a coherent state expansion $W(\alpha)$, but to my knowledge not all states admit to a quantum optics description (like maybe a simple hydrogen atom), therefore can they be presented as such? Is it realizable only mathematically, or does it possess physical meaning in coherent state space?

Answer 1

Coherent states exist for a large number of algebraic systems so it's not hard to define a $Q$-function. The difficulty is that, for other values of $s$, the displacement operator takes on much more complicated expression than the Heisenberg-Weyl algebra, meaning that the "formula" you suggest are not terribly convenient.

There is this recent review (open access) that gives some details on the $SU(2)$ case and the Stratonovich approach to Wigner functions, which would be applicable to $SU(n)$. (The work that "resurrected" this Stratonovich approach is Várilly J C and Gracia-Bondía J M 1989 Ann. Phys. 190 107; see also Dowling, Jonathan P., Girish S. Agarwal, and Wolfgang P. Schleich. "Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms." Physical Review A 49.5 (1994): 4101.)

Additional examples of Wigner functions have been published for various types of potential (Morse, Poeshl-Teller etc): arXiv is full of examples.

So in short, no the WF cannot always be defined as a coherent state although the so-called Wigner kernel is basically just that. There are other bits and pieces to the kernel when you want anything but the $s=1$ quasiprobability distribution.