Geometric-phase-induced false electric dipole moment signals for particles in traps

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 $\mathbf{E}$ field are also given and there are results for the averaging of the $\mathbf{B}$ field in the absence of an $\mathbf{E}$ field.

Figure 1

(Color online) The shape of the ${\mathbf{B}}_0$ field lines, when there is a positive gradient $\partial B_{0z}∕\partial z$, shown in relation to an outline of the trap used to store ${}^{199}\mathrm{Hg}$ atoms and UCN’s for the neutron EDM measurements at the ILL. If another field is superimposed having lines that both enter and leave through the sidewalls, like the one on the right-hand side, it will be shown later that it does not affect the false EDM signals that are generated.

Figure 2

(Color online) The Ramsey-Bloch-Siegert Larmor frequency shift $\Delta \omega$ caused by adding a field ${\mathbf{B}}_{xy}$ rotating in the $xy$ plane, plotted against the angular frequency of rotation. The Larmor frequency, before the addition of ${\mathbf{B}}_{xy}$, was ${\omega }_0$. The shift given by Eq. (9) goes to infinity, but that from Eq. (10) has a peak value of ${\omega }_{xy}$.

Figure 3

(Color online) A view of the $xy$ plane of the trap bounded by the circular sidewall. Part of an orbit is shown projected onto the $xy$ plane for a particle undergoing specular reflection. The orbit is characterized by the angle $\alpha$. Vectors $\mathbf{E}$ and ${\mathbf{B}}_{0z}$ point towards the reader and $\partial B_{0z}∕\partial z$ is positive.

Figure 4

(Color online) The false EDM for a particle in a peripheral orbit as its speed is increased. The value of $\partial B_{0z}∕\partial z=+1\mathrm{nT}∕\mathrm{m}$; the holding field $B_{0z}$ is $1\times {10}^{-6}\mathrm{T}$. At the extreme left of the figure, the gyromagnetic ratio $\gamma$ is unimportant and the same results are obtained for any particle. The values on the right are those for a ${\gamma }^2$ factor and a trap radius $R$ $\left(0.25\mathrm{m}\right)$ appropriate to the ${}^{199}\mathrm{Hg}$ atoms in the ILL $n\mathrm{EDM}$ apparatus.

Figure 5

(Color online) The ${\mathbf{B}}_{xy}$ fields (in black) seen by a particle going back and forth close to the $y$ axis. Going towards positive $y$, the ${\mathbf{B}}_{xy}$ field rotates steadily anticlockwise by about 70° as drawn. The first reflection of the particle towards negative $y$ causes an instantaneous anticlockwise rotation by about 110° as drawn. The same two rotations occur on the path to, and at, the second reflection. The size of the rotations depends on the size of $B_{0r}∕B_v$.

Figure 6

(Color online) A transient oscillation following an instantaneous rotation of ${\mathbf{B}}_{xy}$ at a collision. The resulting false EDM settles down to the same value as it would have had if the head of the ${\mathbf{B}}_{xy}$ vector had followed a straight line slowly to the new position.

Figure 7

(Color online) Computer simulation of resonances occurring as a function of trajectory angle $\alpha$ for UCN’s with a constant speed $v_{xy}=5\mathrm{m}∕\mathrm{s}$ and specular reflection, normalized to the value expected from Eq. (29) for a uniform spatial distribution of particles in the trap. The analytic formula is that given in Eq. (78). Away from the resonances $d_{af}$ is independent of $\alpha$.

Figure 8

(Color online) The dependence of the false EDM on the angle $\alpha$ characterising the orbit for the high velocity case $\mid {\omega }_r\mid ⪢\mid {\omega }_0\mid$.

Figure 9

(Color online) Results from computer simulations of the false EDM effect for ${}^{199}\mathrm{Hg}$ atoms, in the cases of peripheral orbit, diameter orbit, and diffuse reflection in 2D and 3D, as a function of velocity. All results are normalized to the analytic result expected for diffuse reflection in the high-velocity limit.

Figure 10

(Color online) The suppression of the false EDM due to collisions with a buffer gas for the regime $\mid {\omega }_r\mid >\mid {\omega }_0\mid$. These simulated data, based on ${}^{199}\mathrm{Hg}$ atoms in the neutron trap at ILL $(R=0.25\mathrm{m})$, indicate that on reducing the mean free path, the suppression amounts to a factor of 2 when the time taken to diffuse across the trap has increased to the point where it is similar to the Larmor period. (b) includes the case of a larger-radius trap. The data are normalized to the expected analytic value of false EDM when there is no buffer gas. The solid lines indicate an overall fit for all of the data within each figure to the function ${[1+{\{b\times 4R^2{\omega }_0∕\left(2\pi v_{xy}\lambda \right)\}}^2]}^{-1}$ where $b$ is a single free parameter.

Figure 11

(Color online) False EDM’s obtained by computer simulation in the $\mid {\omega }_r\mid <\mid {\omega }_0\mid$ case. The results shown are for 2D specular reflection following peripheral and diameter orbits and for 3D diffuse reflection. The analytic result of Eq. (29) is shown as a smooth curve. Other parameters were $\partial B_{0z}∕\partial z=1\mathrm{nT}∕\mathrm{m}$ and $B_0=1\mu \mathrm{T}$.

Figure 12

(Color online) Suppression of the false EDM (relative to the expected value) caused by interparticle collisions in the regime appropriate to ultracold atoms and neutrons. The solid lines indicate an overall fit for all of the data within each figure to the function ${[1+{\{bv∕\left({\omega }_0\lambda \right)\}}^2]}^{-1}$ where $b$ is a single free parameter.

Figure 13

(Color online) A subset of data from the neutron EDM experiment at the ILL, showing the measured false EDM as a function of the measured neutron to mercury frequency ratio. We expect this frequency ratio to be proportional to the magnetic field gradient. (Small, constant vertical offsets have been applied to the data in each plot.)

Figure 14

(Color online) Results of computer simulations for the shift in the ratio $\mid ({\omega }_{Ln}∕{\omega }_{L\mathrm{Hg}})\mid$ with varying gradients $q$ in a small $\mathbf{B}$ field, $B_z=0$, $B_x=qy$, $B_y=qx$, when it is added to a uniform ${\mathbf{B}}_0$ field of $1\mu \mathrm{T}$ aligned with the $z$ axis. The shift is caused by the different averaging of $B_x$ and $B_y$ by the two species. These results are consistent with Eq. (88).

Figure 15

(Color online) Results of computer simulations for the shift in the ratio $\mid ({\omega }_{Ln}∕{\omega }_{L\mathrm{Hg}})\mid$ with varying neutron velocity, when there is a gradient $\partial B_{0z}∕\partial z=20\mathrm{nT}∕\mathrm{m}$ in a ${\mathbf{B}}_0$ field of $1\mu \mathrm{T}$. The shifts are caused by the different averaging, for the two species, of the $B_x$ and $B_y$ field components that must accompany this gradient. The shift at zero velocity is consistent with Eq. (89). When the neutrons travel as fast as the Hg atoms the shift becomes zero.

Figure 16

(Color online) As a particle moves along the path from end 1 to end 2 the ${\mathbf{B}}_{xy}$ vector sweeps the shaded area. On returning from end 2 to end 1 the head of the ${\mathbf{B}}_{xy}$ vector passes along the dotted curve.

Figure 17

(Color online) The angles ${\alpha }_1$ and ${\alpha }_2$, the tangents, and the normals at the ends of a path.

Figure 18

(Color online) A chord path for a particle found at radius $r$. Also shown are the related angles $\rho$ and $\alpha$.