Geometric phases in electric dipole searches with trapped spin-1/2 particles in general fields and measurement cells of arbitrary shape with smooth or rough walls

Abstract

The important role of geometric phases in searches for a permanent electric dipole moment of the neutron, using Ramsey separated oscillatory field nuclear magnetic resonance, was first noted by Commins [Am. J. Phys. 59, 1077 (1991)] and investigated in detail by Pendlebury et al. [Phys. Rev. A 70, 032102 (2004)]. Their analysis was based on the Bloch equations. In subsequent work using the spin-density matrix, Lamoreaux and Golub [Phys. Rev. A 71, 032104 (2005)] showed the relation between the frequency shifts and the correlation functions of the fields seen by trapped particles in general fields (Redfield theory). More recently, we presented a solution of the Schr\"odinger equation for spin-$1/2$ particles in circular cylindrical traps with smooth walls and exposed to arbitrary fields [A. Steyerl et al., Phys. Rev. A 89, 052129 (2014)]. Here, we extend this work to show how the Redfield theory follows directly from the Schr\"odinger equation solution. This serves to highlight the conditions of validity of the Redfield theory, a subject of considerable discussion in the literature [e.g., M. P. Nicholas et al., Prog. Nucl. Magn. Reson. Spectrosc. 57, 111 (2010)]. Our results can be applied where the Redfield result no longer holds, such as observation times on the order of or shorter than the correlation time and nonstochastic systems, and thus we can illustrate the transient spin dynamics, i.e., the gradual development of the shift with increasing time subsequent to the start of the free precession. We consider systems with rough, diffuse reflecting walls, cylindrical trap geometry with arbitrary cross section, and field perturbations that do not, in the frame of the moving particles, average to zero in time. We show by direct, detailed, calculation the agreement of the results from the Schr\"odinger equation with the Redfield theory for the cases of a rectangular cell with specular walls and of a circular cell with diffuse reflecting walls.