Neutron-state entanglement with overlapping paths

The development of direct probes of entanglement is integral to the rapidly expanding field of complex quantum materials. Here we test the robustness of entangled neutrons as a quantum probe by measuring the Clauser-Horne-Shimony-Holt contextuality witness while varying the beam properties. Specifically, we show that the mode entanglement of the spin and path subsystems of individual neutrons prepared in two different experiments using two different apparatuses persists even after varying the entanglement length, coherence length, and neutron energy difference of the paths. The two independent apparatuses acting as entangler-disentangler pairs are static-field magnetic Wollaston prisms and resonance-field radio-frequency flippers. Our results show that the spatial and energy properties of the neutron beam may be significantly altered without reducing the contextuality witness value below the Tsirelson bound, meaning that maximum entanglement is preserved. We also show that two paths may be considered distinguishable even when the path states significantly overlap. Therefore, we have shown that our experimental results are consistent with the distinguishable subsystem assumption down to a separation of less than 100 nm, proving entanglement and the contextual nature of reality on short length scales. This work is the key step in the realization of the modular, robust technique of entangled neutron scattering, which can extract entanglement information from a sample without the knowledge of the microscopic sample Hamiltonian: only semiquantitative knowledge of the correlation lengths of the relevant degrees of freedom and the timescales of the characteristic dynamics is required.