In experiments searching for a nonzero electric dipole moment of trapped particles, frequency shifts correlated with an applied electric field can be interpreted as a false signal. One such effect, referred to as the geometric phase effect, is known to occur in a magnetic field that is nonperfectly homogeneous. The increase in sensitivity of experiments demands improved theoretical description of this effect. In the case of fast particles, like atoms at room temperature and low pressure, the validity of established theories was limited to a cylindrical confinement cell in a uniform gradient with cylindrical symmetry. We develop a more general theory valid for an arbitrary shape of the magnetic field as well as for arbitrary geometry of the confinement cell. Our improved theory is especially relevant for experiments measuring the neutron electric dipole moment with an atomic comagnetometer. In this context, we have reproduced and extended earlier numerical studies of the geometric phase effect induced by localized magnetic impurities.
Theories are developed to evaluate Larmor frequency shifts, derived from geometric phases, in experiments to measure electric dipole moments (EDM's) of trapped, atoms, molecules, and neutrons. A part of these shifts is proportional to the applied electric field and can be interpreted falsely as an electric dipole moment. A comparison is made between our theoretical predictions for these shifts and some results from our recent experiments, which shows agreement to within the experimental errors of 15%. The comparison also demonstrates that some trapped particle EDM experiments have reached a sensitivity where stringent precautions are needed to minimize and control such false EDM's. Computer simulations of these processes are also described. They give good agreement with the analytical results and they extend the study by investigating the influence of varying surface reflection laws in the hard-walled traps considered. They also explore the possibility to suppress such false EDM's by introducing collisions with buffer gas particles. Some analytic results for frequency shifts proportional to the square of the
E field are also given and there are results for the averaging of the
B field in the absence of an
E field.
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.
Pendlebury
etal. [Phys. Rev. A 70, 032102 (2004)] were the first to investigate the role of geometric phases in searches for an electric dipole moment (EDM) of elementary particles based on Ramsey-separated oscillatory field magnetic resonance with trapped ultracold neutrons and comagnetometer atoms. Their work was based on the Bloch equation and later work using the density matrix corroborated the results and extended the scope to describe the dynamics of spins in general fields and in bounded geometries. We solve the Schr\"odinger equation directly for cylindrical trap geometry and obtain a full description of EDM-relevant spin behavior in general fields, including the short-time transients and vertical spin oscillation in the entire range of particle velocities. We apply this method to general macroscopic fields and to the field of a microscopic magnetic dipole.
The interaction of the electric and magnetic dipole moments of a particle with the electromagnetic field is investigated in an approach that deals with four-dimensional (4D) geometric quantities. The new commutation relations for the 4D orbital and intrinsic angular momentums and also for the 4D dipole moments are introduced. The expectation value of the quantum 4-force, which holds in any frame, is worked out in terms of them. In contrast to it the whole calculation in [1] ([1] J. Anandan, Phys. Rev. Lett. \textbf{85}, 1354 (2000)) has been made only in the rest frame of the dipole. It is proved that, e.g., the expression for the 3D force
f in [1] is not relativistically correct and that the quantum 4-force is not zero in the experiments proposed in [1]. This means that the phase shifts that could be observed in such experiments are not topological phase shifts.