coherent states $|\bm{c}\rangle= |\alpha,\beta\rangle$ of such modes and define the corresponding Husimi-Q function as $$Q(\alpha,\beta) = \frac{1}{\pi^2} \langle \alpha,\beta | \rho | \alpha,\beta\rangle.$$ This Husimi function will also satisfy a quantum Fokker-Planck equation of the form (\[FP\]). The correspondence table (\[Jp\_orbital\])–(\[Jz\_orbital\]) now becomes

[rCl]{} \[corr\_Jp\]\[J\_+, \] && \_+(Q) = (\^\* \_[\^\*]{} - \_) Q,\
\[corr\_Jm\]\[J\_-,\] && \_[-]{}(Q) = (\^\* \_[\^\*]{} - \_) Q,\
\[corr\_Jz\]\[J\_z, \] && \_z(Q) = (\^\* \_[\^\*]{}+ \_- ) Q.

For a single spin-1/2 system, the most general density matrix may be written as $$\label{rho_spin12}
\rho = \frac{1}{2} (1 + \bm{\tau} \cdot \bm{\sigma}),$$ where $\sigma_i$ are the Pauli matrices and $\tau_i = \operatorname{tr}(\rho \sigma_i)$. In this case, it follows that the corresponding Husimi-Q function is given by the particularly simple form [@Scully1994] $$\label{Q_spin12}
Q(\alpha,\beta) = \frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2} \;  \bm{c}^\dagger \frac{(1 + \bm{\tau} \cdot \bm{\sigma})}{2} \bm{c},$$ where $\bm{c} = (\alpha,\beta)$ is to be interpreted as a two-component spinor. For an arbitrary spin, we write instead $$\label{Q}
Q(\alpha,\beta) = \frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2} \; V(\alpha,\beta),$$ where $$V(\alpha,\beta) = \sum\limits_{m,m'}  \frac{\rho_{m,m'} (\alpha^*)^{J+m} (\beta^*)^{J-m} \alpha^{J+m'} \beta^{J - m'}}{\sqrt{(J+m)! (J-m)! (J+m')! (J-m')!}}.$$ One may verify that $V(\alpha,\beta)$ is a homogeneous function of degree $2J$ in $\alpha$ and $\beta$. Thus, using Euler’s theorem for homogeneous 