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counterparts R [2] M and combine the results in the follow- ing way: 

Here we take θ = ± π/ 4, which are optimal polarization angles for the PV measurements (see next section). This method has the advantage that K is independent of E$_{dc}$ , so that the E-field bias may be chosen based on technical requirements of the experimental apparatus. 

While the current Yb-APV apparatus has been de- signed to minimize systematic effects, the PV mimicking systematics may be a result of a combination of multiple apparatus imperfections. In order to understand the im- portance of these effects, the electric and magnetic field misalignments and stray fields were included in a theoret- ical model of the transition rates as well as the excitation light’s deviations from linear polarization. In addition, we relax the assumption that δk = 0 and include the effects of the residual M1 transition. 

The quantization axis is defined along ˆ z , and following the ideal case model, the axis of the standing light wave is collinear with ˆ x . We added a small ellipticity to the light polarization by taking 

where φ is a small phase. For | φ | ≪ 1, the ellipticity of the light is 2 φ sin(2 θ ). The electric field imperfections are included by taking 

where 

are the AC and DC components of the electric field. It is assumed that the y- and z-components of the AC field are in phase with the leading oscillating E-field. The impact 

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## IV. PV SIGNATURE: IMPACT OF APPARATUS IMPERFECTIONS 

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of the out-of-phase AC components was analyzed within a complete model of the systematics and found to be negligible. The AC components are due to misalignments of the applied E-field with respect to the light wave axis as well as to the quantization axis ˆ z . The DC components arise due to a misalignment of the DC-bias field and also due to stray electric fields in the interaction region. 

The magnetic-field imperfections are defined within the same frame of reference by taking analogously 

where 

where ˜ B and B ′ are reversing and stray non-reversing magnetic fields, respectively. 

Equations (1) through (4) apply when the quantiza- tion axis is along the magnetic field, thus a rotation D is applied to each of the vectors E , B , E , and k such that D B ∝ ˆ z . That is, we take 

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where 

Here the matrix D ( α, ˆ n ) represents a rotation about an axis ˆ n through angle α . The angles α$_{x}$ and α$_{y}$ are given by 

Thus, the electric field E and the polarization vector E acquire additional components after the rotation (be- sides, for example, e$_{y}$ and ˜ e$_{y}$ ). 

Due to the imperfections, the normalized-rate modu- lation amplitudes now include additional terms besides the Stark- and the PV effects: 

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where r (Stark) M is the Stark contribution due to the DC- bias and the field imperfections, r (APV) M is the PV-Stark- interference term, r (M1) M is the M1-Stark-interference con- tribution, and r ( φ ) M is a contribution due to the distorted linear polarization of the light (which is a Stark contri- bution, but we explicitly separate the contribution linear in φ ). Expressions for the lowest-order terms are sum- marized in the Table I. 

The normalized-amplitude combination (14) has been chosen to determine the PV asymmetry. Since the M1 and ellipticity terms have opposite signs for M = ± 1, their contributions to K cancel, while the contributions from r (APV) M add. 

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