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arXiv:1201.0699 · Physical Review A

Electric-dipole-moment searches: Reexamination of frequency shifts for particles in traps

G. Pignol, S. Roccia

Published Apr 4, 2012• 37 citations
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Abstract

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.

DOI
10.1103/physreva.85.042105

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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\mathbf{E}E field are also given and there are results for the averaging of the B\mathbf{B}B field in the absence of an E\mathbf{E}E field.
It has been shown in an earlier publication [J. Pendlebury et al. Phys. Rev. A 70, 032102 (2004)] that magnetic-field gradients applied to particles in traps can induce Larmor frequency shifts that may falsely be interpreted as electric-dipole moment signals. This study has now been extended to include nonuniform magnetic-field gradients due to the presence of a local magnetic dipole. It is found that, in the high orbit-frequency regime, .the magnitude of the shifts can be enhanced beyond the simple expectation of proportionality to the volume-averaged magnetic-field gradient $⟨\ensuremath{\partial}{B}_{z}∕\ensuremath{\partial}z⟩$.
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-1/21/21/2 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.\mathit{et}\phantom{\rule{0.28em}{0ex}}\mathit{al}.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.
We calculate the Zeeman frequency shifts due to interactions with the light in a far-off-resonance optical dipole trap. These shifts are important for potential use of such a trap to search for an atomic permanent electric-dipole moment (EDM). We present numerical results for Cs and Hg, as examples of paramagnetic and diamagnetic atoms. The vector and tensor light shifts are calculated for a large range of trap optical frequencies, for both red-detuned and blue-detuned traps. We also consider frequency shifts resulting from magnetic dipole and electric quadrupole transitions mixed in by a static electric field. These shifts are particularly important for EDM experiments since they are linear in the electric field. The Zeeman frequency shifts represent a substantial problem for EDM experiments in a dipole trap and must be controlled with care to achieve theoretical sensitivity.
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.