In addition to improving our statistical sensitivity, we also took steps to suppress possible sources of systematic error in Gen. II. In the case of the dominantEnrsystematic in Fig.2.7.2,
we both ameliorated the known source of the effect and improved our ability to monitor the experimental imperfections that conspire to produce it. For the other effects, including the nonzero and still-mysterious ΩNE effect and the remaining effects that did not produce a statistically significant shift in the eEDM channel, we made improvements where possible to improve our sensitivity to the physical quantities with which they were correlated. Otherwise, we must trust that better experimental statistics will allow us to average these systematic uncertainties down—or reveal nonzero correlations that will help us identify their cause and fix them.
As discussed in Section 2.7, the Enr systematic was ultimately caused by a conspiracy
of two experimental imperfections: (1) a thermal stress-induced birefringence gradient in the electric field plates caused by laser absorption and (2) a stray, non-reversing electric field at the few-mV/cm level in the interaction region. Patch potentials at the few-mV level are difficult to avoid, so our efforts on imperfection (2) focused on improving our ability to measure Enr regularly and accurately rather than making it go away. Thanks to efforts by
Adam West and others, the microwave pumping Enr measurement apparatus alluded to in
Section 2.7 [11] is now set up on a fairly permanent basis, and the data-taking routine is automated so that mapping the non-reversing E-field can be a part of our regular running routine.
The physical basis for imperfection (1) above is more easily addressable. Nick Hutzler has developed an exhaustive treatment of thermal stress-induced birefringence in glass in his thesis [105]. The crucial observation is that if ϵ is the ellipticity of the state preparation or readout laser, ∆ϵ is the change in ellipticity caused by the field plate birefringence, and wx
is the waist of the laser beam along x (the narrow axis of the elongated beam), then the figure of (de)merit for the deleterious effect is∆ϵ′(wx), or the polarization ellipticity gradient
along the xˆ at a distance of one x-axis beam waist from the center of the laser profile. This is roughly the position where the beam starts to become weak enough for AC Stark shift effects to become more important than optical pumping, as discussed in Section 2.7. Nick finds that this effect has the following dependence on the materials properties of the field plates4:
∆ϵ′(wx)∝KEαVa/κ, (3.2)
where K is the stress-optic coefficient (a constant of proportionality between mechanical stress and optical retardation), E is the elastic modulus, αV is the coefficient of thermal
expansion, a is the optical absorbance of the material at the relevant wavelength, and κ
is its thermal conductivity. This model was corroborated by polarimetery measurements performed by Paul Hess [100, 105]. To minimize this quantity, Adam West and Paul Hess had a new set of field plates made by Thin Film Devices (TFD) out of Corning 7980 glass, which Nick estimated should reduce KEαV/κ by a factor of ≈ 7 relative to the Gen. I
Borofloat glass plates. Since Nick also estimated that the power absorption by the field plates was dominated by the ITO layer, we also had TFD apply an ITO coating with a thickness of 20 nm, 10×thinner than in Gen. I, in order to reduce the absorption coefficient
a. These improvements, combined with the fact that the higher strength of the H →I state transition relative to the H →C transition allows us to reduce the power in the refinement and readout beams by a factor of several, should suppress the Enr systematic by a factor of
a few hundred.
Building on work by Paul Hess and Christian Weber, Vitaly Andreev developed a highly sensitive polarimeter for measuring small fractional polarization gradients across a laser beam [13]. Using this setup, he demonstrated that even with Gen.-I-level laser powers of 2 W at
4The field plate glass and ITO coating both contribute to the laser absorption and birefringence gradient, so for a full treatment, we must sum over both of these sources. See reference [13] for details. Note also that the laser ellipticity gradientitself is the problem, not its (dominant Gen. I) source in the field plates. If other effects, e.g. mechanical or thermal stress-induced birefringence gradients in the vacuum windows, laser optics, etc., cause a similar-sized ellipticity gradient in the laser, these effects could also produce an
Enr-correlated systematic [105]. Vitaly Andreev and Paul Hess have investigated such effects and have not found any at a level of concern for our Gen. II experiment scheme [13,100].
1090 nm, the polarization gradient induced by thermal birefringence in the field plates was suppressed by at least an order of magnitude, to a level consistent with the uncertainty in the measurement. The main result is shown in Fig. 3.2.1.
Gen I field plate Gen II field plate
0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Position mm S I Ellipticity gradients, 2 W 1090 nm Intensity profile a.u.
Figure 3.2.1: Thermal stress-induced polarization ellipticity comparison between ACME Gen. I and Gen. II field plates. The intensity profile of a 2 W 1090 nm laser, elongated so that its waist is30mm alongyˆand1.4mm alongxˆ, is plotted in gray with a Gaussian fit, and the polarization profile of the laser beam after passing through the Gen. I (Gen. II) electric field plates is plotted in blue (orange). S and I are the Stokes parameters characterizing polarization circularity and beam intensity, respectively [29], so that S/I is a measure of the circularly polarized light fraction in the laser beam. Constant ellipticity offsets are unimportant and have been subtracted off. The blue and orange curves are fits to the photo- thermoelastic law discussed in references [100, 105]. For the Gen. II field plates, both the magnitude of ∆S/I and its gradients are suppressed by more than a factor of 10. Figure and results from Vitaly Andreev [13].
As discussed in Section2.7, the mechanism producing an NE˜B˜ =− −+correlated Rabi frequency systematic, ΩNE, is still unexplained. One valuable but rather confounding hint
was that the effect showed a dependence on the direction of laser propagation through the interaction region, i.e. on whether ˆk·zˆ= +1 or−1 [11, 144]. In Gen. I, reversing the laser propagation direction was a time-consuming process requiring realignment of the entire optics breadboard for launching lasers into the interaction region. Therefore, we only performed
the kˆ·zˆ switch once during Gen. I. In Gen. II we would like to perform this switch more frequently to gain a better understanding of its correlations with our experiment phases and to average them down more effectively, so we are building a duplicate optical setup that will allow us to reverse ˆk·zˆsimply by moving a few optical fibers.
Several of the larger systematic uncertainties in Fig. 2.7.2 were related to various B-field imperfections. These were included in our error budget because experiments conceptually similar to ours saw unexplained systematic shifts in the eEDM channel that were correlated with these fields (although we did not see any such effects) [11]. In the Gen. I experiment, the magnetic field was characterized by inserting a fluxgate magnetometer down the beamline once the measurement was complete [11, 144]. In Gen. II, deep pockets intrude into the interaction region so that we can insert an array of fluxgates to monitor the magnetic fields more frequently without breaking vacuum, potentially even while the experiment is running [144]. A more accurate measurement of the B-field imperfections will help to reduce the uncertainty on the correlated systematics.