## 3 Results  
The Wigner function for a two-dimensional scenario, as the one illustrated in Section 2.1, is a function of four variables, \(f_{w}(x,y,k_{x},k_{y})\) and thus needs a five-dimensional space to be visualized. However, useful information can be retrieved from the analysis of the projection of the function onto a two-dimensional space. From a mathematical point of view there are numerous ways to establish such a projection. Two of the variables, called variables of interest, remain active, while the remaining three must be restrained according to some criteria, usually introduced by physical considerations. For example, the value of the Wigner function and of one position and one momentum component can be fixed, or restricted in some limits, or integration on the latter within selected intervals can be performed. As a first experiment we consider a separation of the negative and the positive part of the Wigner function: \(f_{w}^{-}=f_{w}\theta(-f_{w})\) and \(f_{w}^{+}=f_{w}\theta(f_{w})\). Then both quantities are integrated in the whole momentum space to obtain the spatial distribution of both the negative and the positive parts:  
\[f_{w_{xy}}^{-}(x,y)=\iint f_{w}^{-}(x,y,k_{x}k_{y})\;dk_{x}dk_{y};f_{w_{xy}}^{+} (x,y)=\iint f_{w}^{+}(x,y,\;k_{x}k_{y})dk_{x}dk_{y} \tag{5}\]  
We consider as a reference frame the classical (Boltzmann) distribution, characterized by: \(f_{B_{xy}}^{-}(x,y)=0\) and \(f_{B_{yy}}^{+}(x,y)=n(x,y)\) where \(n\) is the classical density. Figure 2a,b show respectively \(f_{w_{xy}}^{-}\) and \(f_{w_{xy}}^{+}\). The quantum density, represented by the sum \(f_{w_{xy}}^{+}+f_{w_{xy}}^{-}\), is shown in Figure 3a. These quantities reflect the following evolution process. In the region below the dopant, denoted by the green 0.15 eV isoline, the electron flux splits into two entangled parts. These maintain the coherence in the region after the dopant, as no dissipation processes are involved. This results in the interference fringes well visible in the upper parts after the dopant of both figures. Remarkably, \(f_{w_{xy}}^{-}\) exhibits firmly negative values in a wide region around the dopant. However, their pattern does not reflect the symmetry of the potential and thus they can't be associated to the nonlocality of the quantum action. The most pronounced negative values are placed after the dopant, in the region of evolution of the two entangled parts. Now, one can connect this negative region with the fact that, quantum mechanically, the scattering process requires a \"collision duration\" [31]. Quantum mechanically, the wave must first build up a correlation with the impurity, and this is then broken up when the collision is completed. The fact that the entanglement exists beyond the impurity is recognition that the event takes a non-zero amount of time to complete. This is true for the impurities as well as for phonons [32]. A similar push of the effect of the impurity potential to the anterior region can be caused by nonlinear screening due to dopants in the contacts [33], although no such screening is used here. In contrast, such behavior is entirely missing in Figure 3b, where the classical density \(n\) is reported: The corresponding region is entirely empty! This shows that the negative excursions of the Wigner function provide a direct indicator for interference effects. It is interesting to see if such analysis can be carried out on other types of projections or is specific to the presented one, where we intentionally separated the negative values of the Wigner function. As a second experiment we consider the Wigner function for two specific points in the momentum space that are (\(k_{x1}\;k_{y1}\)) = (\(-0.26\;\mathrm{nm}^{-1}\), 0.89 \(\mathrm{nm}^{-1}\)) and (\(k_{x2}\;k_{y2}\)) = (0.26 \(\mathrm{nm}^{-1}\), 0.89 \(\mathrm{nm}^{-1}\)). The choice of these two points is related to the fact that they are near to the mean momentum along the \(y\)-direction (0.84 \(\mathrm{nm}^{-1}\)) and symmetrically placed around the initial momentum in the \(x\)-direction.