In the course of this master thesis, quantum transport measurements of ultra-cold neutrons (UCNs) traversing an absorbing reflecting mirror system (ARMS) were realised. An ARMS consists of a flat neutron mirror and a second neutron mirror with a mechanically roughened surface (a scatterer). The scatterer is mounted at a distance h above the flat mirror, with its rough surface facing downwards. Within the Earth's gravitational field, UCNs form gravitationally bound states above a flat surface where total reflection occurs. The quantum transport properties of an ARMS are therefore defined by the interaction of these states with the surface disorder and this interaction depends on the characteristics of the surface disorder as well as on the neutron mirrors' material properties. To probe the quantum transport properties of such a system, the transmission through the system was measured in dependence of its slit height h, using different coating materials with different Fermi potentials. To prepare a well-defined phase space for the quantum transport measurements, a second ARMS driven in the classical regime with a slit height of around 200 µm was used as vectorial velocity filter. The phase space preparation was theoretically investigated by three-dimensional classical Monte Carlo simulations and experimentally by performing spatially resolved track detector measurements. The experimental setup was redesigned with respect to a preceding experiment performed in 2011. The new design allowed for a more efficient experimental performance by introducing the ability to change the distance between the bottom and the top mirrors without having to break the vacuum. This decreased the preparation time for every measurement tremendously. Furthermore, within this thesis, a detailed analysis of measurements performed on the rough surfaces of the differently coated scatterers is presented, to enable a better interpretation of the experimental results and to improve future simulations.