distributions when considering correlation functions, which makes it a good test model for studying relations between classical and wave operators.

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The source $\hat{\Gamma}_0$ is chosen so that the associated Wigner distribution function takes the form $$\label{WG0} 
W_{\Gamma_0}(x,p)= C \e^{-(1-\cos 2\pi(x-x_0))/(2\pi\sigma_x)^2-(1-\cos 2\pi(p-p_0))/(2\pi \sigma_p)^2},$$ where $C$ is a normalisation constant chosen so that $\Tr\Gamma_0=1$ and $\sigma_x$ and $\sigma_p$ respectively determine the variances in the $x$ and $p$ coordinates. For small enough $\sigma_x$ and $\sigma_p$ this is a nearly Gaussian source distribution in phase space centred on $X_0 = (x_0,p_0)$, and is associated with a corresponding nearly Gaussian source correlation function $\Gamma_0(x_1,x_2)$.

The time evolution of the Wigner transform $W_{K_n}$ of $\hat{K}_n$ is displayed in Fig. \[fig:kr:wig:a\] and is compared with the evolution of a classical phase space density with initial distribution according to (\[WG0\]). The quantum evolution follows the classical dynamics closely for small $n$ as expected from Eq. (\[FP\]) and detailed in \[app:A\]. Once the classical dynamics foliates phase space down to phase space cells of area $1/N$, wave fluctuations take over and wash out the classical phase space structures, see Fig. \[fig:kr:wig:a5\] at $n=10$. Note that, after several interations of the map, the classical density is stretched along the 