The semi-classical Floquet-Markov master equation for Monte Carlo spin integration

Abstract: In designing an experiment to measure a neutron electric dipole moment (nEDM), it is often necessary to determine the behavior of an ensemble of spins under time-dependent and randomly fluctuating magnetic fields. This is particularly relevant for the proposed nEDM@SNS experiment, which features ultra-cold neutrons (UCNs) and helium-3 atoms occupying the same measurement cell, as well as a sinusoidal dressing field. In this work, we investigate a new technique to calculate the frequency shifts arising from magnetic field inhomogeneities and motional magnetic fields, particularly in the case where the spins in question are subjected to a time-periodic magnetic field. The method is based on Floquet theory, a general framework for analyzing periodic linear differential equations, and Redfield theory, which governs the time evolution of the density matrix in the presence of weak couplings to an environment. We benchmark the results against the analytical results derived for the static magnetic field case, as well as against the results of a conventional Runge-Kutta integrator, and find agreement with both. We further study the performance of the method, and find order-of-magnitude improvements in runtime over the conventional integrator.