The Berry phase: A simple derivation and relation to the electric dipole moment experiments with ultracold neutrons

I derive the effective phase of the spin precession for a neutral particle with spin 1 ∕ 2 moving in a superposition of constant and radio frequency fields. The fields are perpendicular to each other at all times, and the radio frequency field is slowly rotating with angular speed ω . The derivation is accomplished with the help of the exact solution of the Schrödinger equation. I show that the geometrical (or Berry) phase is the linear term in the power series expansion of the effective phase in terms of ω . I also discuss the role of the Berry phase in the experimental search for an electric dipole moment of the neutron with ultracold neutrons.