*2.1. Quantum Point Contact*

We employ the established model for a QPC [12] and describe the energy-dependent transmission probability as

$$D(E) = \frac{1}{1 + \exp\left(\frac{-E + E\_0}{\gamma}\right)}\tag{1}$$

This is a step-like function of the energy *E*, see Figure 1b, where *E*0 and *γ* denote the position and width in energy of the step, respectively. For a vanishingly small width, *γ* → 0, the transmission probability reduces to a step function, *D*(*E*) → *θ*(*E* − *<sup>E</sup>*0).

In experiments with 2DEGs, the width or smoothness of the QPC barrier *γ*, typically takes values of the order of 1 meV (corresponding to temperatures of the order of 10 K) [11,32,48,49]. The results presented in this paper are equally valid for different types of conductors, where the transmission function has a (smooth) step-like behavior, such as quantum wires with interfaces or controlled by finger gates. Here, smoothness parameters *γ* of values down to several *μe*V are expected (corresponding to temperatures of the order of 10–100 mK) [29].

**Figure 1.** (**a**) Schematic depiction of the system, with a quantum point contact (QPC) connected to two electronic reservoirs, L and R, with electrochemical potentials *μ*L and *μ*R and temperatures *T*L and *T*R, respectively. (**b**) Transmission probability *D*(*E*) shown as a function of energy, Equation (1), with a step positioned at energy *E*0 and energy-smearing width, or smoothness, *γ*. The solid line shows the limit of vanishing width *γ* → 0.
