−4 −2 0 2 4 0 1 2 3 4  E$_{z}$ / Im(E$_{PNC}$)C$_{F,m}$ F,m K(E z ) / K(0) (a) −4 −2 0 2 4 −0.5 0 0.5  E$_{z}$ / Im(E$_{PNC}$)C$_{F,m}$ F,m (E z )/ (b) 3 scale of the variation in E$_{z}$ necessary to carry out these measurements is determined by | Im ( E$_{PNC}$ ) /α | . Using Wieman’s results for E$_{PNC}$ /β [18] on the 6 s → 7 s tran- sition, and for α/β [27], we estimate | Im ( E$_{PNC}$ ) /α | to be ∼ 0.158 mV/cm. A precise measurement as described above will therefore require controlled application of E$_{z}$ at this level, and cancelation of stray fields to a level significantly lower. This can be achieved using precision voltage sources (which can control biases of field plates to the sub- µ V level). It is not necessary to calibrate the other factors that appear in Eq. (9), i.e. A$_{2}$$_{PA}$ or ε ω 1 z , in that one only needs to determine the relative varia- tion in K ( E$_{z}$ ) vs. E$_{z}$ . The sign of Im ( E )$_{PNC}$ /α can be determined from observations of δφ vs. E$_{z}$ . FIG. 1: The (a) amplitude and (b) phase of the modulation signal vs E$_{z}$ . and modest focussing. By carefully selecting beam po- larizations and controlling static electric and magnetic fields, one can select a very limited number of contri- butions to the interaction. For example, to determine the ratio Im ( E$_{PNC}$ ) /α , one can align the polarization of the harmonic beam ( ω$_{1}$ ) to the ˆ z direction, apply a low- magnitude dc field in the ˆ z direction, and zero the field components B$_{x}$ , B$_{y}$ , E$_{x}$ , E$_{y}$ , and ε$_{x}$ . In this geometry, the surviving transition moments sum to Stray fields B$_{x}$ , B$_{y}$ , E$_{x}$ , E$_{y}$ , and ε ω 1 x can affect the amplitude of the modulation signal, and these must be minimized in order to make reliable measurements. We examine the individual terms of Eq. (6) in order to esti- mate their magnitude and suggest means for their reduc- tion. We first consider those terms that contribute to the imaginary part of this signal, i.e. those that will add to the Im ( E$_{PNC}$ ) ε ω 1 z C F,m F,m term directly. We expect that the dominant term of this type that must be addressed is a magnetic dipole contribution, − M ε ω 1 ′′ x C F,m $_{F,m}$, where ε ω 1 ′′ is the amplitude of the imaginary component of the harmonic field, ε ω 1 = ε ω 1 ′ + iε ω 1 $^{′′}$. We note that the Boulder experiments were also affected by a magnetic dipole contribution, which they countered by employing counter-propagating laser beams [18]. In their experi- ment, the ε ω 1 ′′ x field component was essential to their measurement, while in the scheme addressed here, it is not. To reduce this component to the level of the PNC contribution requires ε ω 1 ′′ x ∼ 0 . 5 × 10 − 4 ε ω 1 z . Reduction to lower levels than this is, of course, highly desirable. While this will clearly be an experimental challenge, re- cent progress using high-Q traveling-wave optical cavi- ties to produce linearly-polarized light of high purity has been reported [28]. The degree to which this magnetic dipole term is reduced can be determined experimentally by reversing the direction of B and repeating the mea- surement of K ( E$_{z}$ ). The spatially-varying ac Stark shifts present in the Boulder measurements will not be a factor in the present scheme. Stray fields can also contribute to the apparent PNC signal through Stark-induced terms such as βE$_{y}$ε ω 1 x C F,m $_{F,m}$. We estimate that this effect can be reduced to the 10 − 3 level by reducing E$_{y}$ to ∼ 1 mV/cm and the polariza- tion purity to ε ω 1 ′ x /ε ω 1 z < 10 − $^{3}$. There are, of course, no detectors available to measure field strengths at these low levels; for this the methods described by Wood [29] for nulling stray fields in their measurements should be applicable to the present measurements as well. For ex- ample, referring to Eq. (6), increasing B$_{x}$ to elevate the iβE$_{y}$ε ω 1 z ( B$_{x}$/B$_{z}$ ) e iφ $^{$^{ω}$1}$Δ C (2) term, allows one to null E$_{y}$ , at which point signal modulation as a function of Δ φ is minimized. Other stray field components can be nulled in a similar manner. To this point, we have discussed stray field contribu- This modulation is a result of the interference between the two-photon absorption from the ω$_{2}$ beam and the sum of the various weak amplitudes (each of which are linear in the amplitude of the ω$_{1}$ beam). We show the normalized amplitude K ( E$_{z}$ ) /K (0) and the phase shift δφ = tan − $^{1}$( αE$_{z}$/Im ( E$_{PNC}$ )) as a function of E$_{z}$ in Fig. 1. Measurements of the modulation amplitude as a function of field E$_{z}$ can therefore be used to deter- mine the ratio | Im ( E$_{PNC}$ ) /α | . At E$_{z}$ = 0, the observed modulation is due solely to the interference between two-photon absorption and the PNC-induced amplitude, since the Stark-induced transition is absent in this case. As E$_{z}$ is increased, the amplitude of the modulation in- creases due to the growing contribution of the Stark- induced transition. The Stark- and the PNC-induced moments add in quadrature in Eq. (9) since these ampli- tudes differ in phase by π/ 2, as shown in Eq. (7). The where we have omitted the terms that are second-order in ε ω 1 x , and define Δ φ = 2 φ ω 2 − φ ω 1 , the controllable phase difference between the optical fields. The total transition rate of Eq. (8) has a large dc component that represents the two-photon absorption rate alone, and a weak sinusoidal modulation of amplitude K ( E$_{z}$ ) = 2 | A$_{2}$$_{PA}$ | √ ( αE$_{z}$ ) 2 + ( Im ( E$_{PNC}$ ) C F,m $_{F,m}$) 2 ε ω 1 z . (9) W = | A$_{2}$$_{PA}$ | 2 + K ( E$_{z}$ ) sin(Δ φ + δφ ( E$_{z}$ )) , (8) ∑ A = A$_{2}$$_{PA}$ + ( αE$_{z}$ + iIm ( E$_{PNC}$ ) C F,m F,m ) ε ω 1 z e iφ $^{$^{ω}$1}$. (7) The transition rate scales as W = | ∑ A | $^{2}$, yielding a transition rate of the form