In general, nuclei and electrons in condensed matter and/or molecules are entangled, due to the prevailing (electromagnetic) interactions. As a matter of fact, the "environment" interacting with a microscopic system of interest (e.g. a proton) causes the destruction of the entanglement. This process, called decoherence, has until now prevented experimenters from directly accessing atomic and/or nuclear entanglement effects in real experiments. However, our neutron and electron Compton scattering experiments from protons (H-atoms) in several condensed systems and molecules at ambient conditions demonstrated a new striking effect, i.e. an "anomalous" decrease of scattering intensity from protons, which seem to become partially "invisible" to the neutrons. This effect, which has no interpretation within conventional neutron scattering theory, is proposed to be caused by the non-unitary time evolution (due to decoherence) during the ultrashort, but finite, time-window of the neutron-proton scattering process. Due to the large energy (several eV) and momentum (20-200 Å−1) transfers of these experiments the collisional time between the probe particle and a struck proton is 100–1000 attoseconds long. It is shown that, due to this short timescale, the scattering process must be theoretically treated within quantum dynamics of open quantum systems. New experimental neutron Compton scattering results from a single crystal KHCO3 are presented. They provide the first direct evidence for a connection between the momentum distribution and the "anomalous" scattering intensity of H. Additionally, recent experiments with electron-atom Compton scattering at momentum transfers are mentioned, which reveal the presence of this effect even in scattering from single H2 molecules in the gas phase. Theoretical discussions "from first principles" are presented, also in relation to the quantum Zeno effect, which underline the crucial role of decoherence in the considered experiments. The experimental results and their qualitative interpretation show that epithermal neutrons being available at spallation sources (e.g. ISIS/UK, SNS/USA or ESS/Sweden), and electron spectrometers with large scattering angles, provide tools for investigating new physical and chemical phenomena in the sub-femtosecond timescale.