Figure 1

Real part of the normalized cross-Wigner (\(\text {Re}[{\mathscr{C}}{\mathscr{W}}]_{\psi ,\psi _0}\)) as a function of x and k for \(t=50\) ms and \(\gamma =0\) in (a,b), while for (c,d) we used \(\gamma =-1\). In (a) and (c), we consider the Gouy phase difference, but in (b,d), we do not consider it. From these plots, it is evident the importance of the Gouy phase in this cross-Wigner formalism for an accurate and complete description of the quantum state. We have similar results for the imaginary component.