Design and Monte Carlo Simulation of a Phase Grating Moir\'e Neutron Interferometer to Measure the Gravitational Constant

The gravitational constant (G) is the least precisely known fundamental constant of nature, with persistent and significant discrepancies between measurement methods. New techniques for measuring G with systematic effects different from commonly applied pendulum methods are required. Neutrons are convenient probes of gravitational forces as they are both massive and electrically neutral, properties that allowed a single-crystal neutron interferometer (NI) to achieve the first experimental demonstration of gravitationally induced quantum interference. Despite this, the limitation of single-crystal NIs to monoenergetic beams significantly reduces neutron flux, making precision gravitational measurements unfeasible. A new NI design called the phase-grating moir\'{e} interferometer (PGMI) has been shown to increase neutron flux by orders of magnitude while allowing grating separation that maintains similar interferometer area to previous NI devices. Here, we propose and describe an experiment to measure G using the PGMI to a precision comparable to measurements from the CODATA 2022 evaluation. A Monte Carlo model for incorporating nonlinear potentials into a PGMI is introduced. This model is used to evaluate sources of systematic uncertainty and quantify the uncertainty in G arising from these effects. The effect of lunar gravitation on a torsion pendulum experiment from CODATA 2022 is calculated, and the need for possible correction factors is demonstrated. This work demonstrates that a neutron PGMI can be used to measure G to $150$ parts-per-million in the near term with the potential to achieve greater precision in future experimental designs.