FIG. 3: Discrimination of the PV effect by E-field modulation under static magnetic field. The Zeeman pattern is shown for the polarization angle θ = π/ 4. 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 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. 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. 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: 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 where 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 where ˜ B and B ′ are reversing and stray non-reversing magnetic fields, respectively. where The magnetic-field imperfections are defined within the same frame of reference by taking analogously 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. 4 B = ˜ B + B $^{′}$, ˜ B = ˜ b$_{x}$ ˆ x + ˜ b$_{y}$ ˆ y + B ˆ z B ′ = b $^{′}$$_{x}$ˆ x + b $^{′}$$_{y}$ˆ y + b $^{′}$$_{z}$ˆ z , B → D B , E → D E , E → D E , and k → D k , (16) D = D ( − α$_{y}$, ˆ y ) D ( α$_{x}$, ˆ x ) . (17) α$_{x,y}$ = ( B − b $^{′}$$_{z}$)( b ′ y,x + ˜ b$_{y,x}$ ) /B $^{2}$. (18) R [1] M [2] M ≡ r$_{M}$ = r (Stark) M + r (APV) M + r (M1) M + r ( φ ) M , (19) ˜ E = ( ˜ E$_{0}$ ˆ x + ˜ e$_{y}$ ˆ y + ˜ e$_{z}$ ˆ z ) cos( ωt ) E ′ = E$_{dc}$ ˆ x + e$_{y}$ ˆ y + e$_{z}$ ˆ z , E = ˜ E + E $^{′}$, E = E (ˆ y sin θ + ˆ z e $^{iφ}$cos θ ) , (15) K = R [1] − 1 [2] − 1 + R [1] +1 [2] +1 − 2 R [1] 0 [2] 0 = ∓ 16 ζ β ˜ E$_{0}$ . (14) IV. PV SIGNATURE: IMPACT OF APPARATUS IMPERFECTIONS