The proposal addresses some important questions, which are at the forefront of particle and nuclear physics and concern the analysis of possible deviations from the Standard Model (SM) of particle physics. As a tool for the test of the SM we propose to use quantum interference experiments with the neutron. The aim of the theoretical part of this project is the analysis of all reliable candidates for physics beyond the Standard Model, which effectively can be measured with a special setup of this experiment as well as the investigations of the obtained results concerning their relation to other fundamental properties of particle physics. These theories can also be selected according to their predictions for the electric dipole moment of the neutron, the appearance of which is a direct consequence of the violation of CP-invariance. A particle, which both is responsible for a CP-violating coupling and a hot-topic dark matter candidate, is the axion. The experiments will therefore search for axion-like interactions at short distances right in the so called axion-window at 10-4 m or below, where the only limits so far are derived from our previous experiments. The newly developed “Gravity Resonance Technique” will improve limits by orders of magnitude, at least two. Best limits can be derived with polarized neutrons, and methods will be developed within this programme. The advantage of the new technique is that an unknown quantity – the energy – is assigned a measured frequency. Any energy change provoked by hypothetical axion fields in the quantum states of neutrons in the gravity potential can now be related to frequency measurements with unprecedented accuracy. A hypothetical axion coupling leads to a potential which is proportional to the 5th force potential and will change the energy states as a function of the range of this coupling. These statements can be generalized: The deviations are expected to be the phenomenological outcome of more fundamental theories, unifying all forces induced shortly after the Big Bang. Such a grand unification is an extension of the SM, containing new symmetry concepts based on supersymmetric and super-gravitational interactions. In turn, supersymmetric theories, describing interactions of point-like particles, can be part of more general theories such as string, superstring and D-brane theories. These theories predict new extra dimensions less than a millimeter in size, which can be picked up in the neutron interferometry q-bouncing experiments, measuring the phase shift of the wave function of the neutron.