2 well as Δ F = ± 1 transitions. The Δ F = 0 transi- tions, however, present the following two advantages over Δ F = ± 1 transitions: (1) Systematic errors due to mag- netic dipole contributions to E$_{PNC}$ are smaller, and (2) the two frequencies ω$_{3}$ and ω$_{2}$ can be equal, so this mea- surement requires only a single laser source and the ω$_{1}$ beam can be generated by frequency doubling the ω$_{2}$ laser output in a nonlinear crystal. We will consider only Δ F = 0, Δ m = 0 transitions in the following, and write the two-photon transition amplitude of Eq. (4) as A$_{2}$$_{PA}$ = ˜ α ( ε ω 2 ) $^{2}$e 2 iφ $^{$^{ω}$2}$. electric dipole transition slightly allowed as well. The transition amplitude for this interaction is of the form To maintain a constant phase difference between the two-photon amplitude and the linear amplitudes, the op- tical beams must propagate in directions nearly co-linear with one another. Without lose of generality, we de- fine the y -axis along ˆ k , such that ε ω 1 y must vanish for a plane wave or weakly-focussed beam. We allow an ar- bitrary static electric field E = E$_{x}$ ˆ x + E$_{y}$ ˆ y + E$_{z}$ ˆ z , and consider a dc magnetic field B that is primarily in the ˆ z -direction. B will separate the various m -components: Δ E$_{F,m}$ = µ$_{B}$g$_{F}$ mB$_{z}$ , where µ$_{B}$ is the Bohr magneton and g$_{F}$ is 1/4 for F=4 and -1/4 for F=3. The effect of the ˆ x - and ˆ y -components for B , which can be present in an experiment due to imperfect alignment, is to mix the magnetic sublevels | ns $^{2}$S$_{1}$$_{/}$$_{2}$ , F, m 〉 = | ns $^{2}$S$_{1}$$_{/}$$_{2}$ , F, m 〉 (5) + | ns $^{2}$S$_{1}$$_{/}$$_{2}$ , F, m − 1 〉 B$_{x}$ + iB$_{y}$ B$_{z}$ C F,m − 1 F,m −| ns $^{2}$S$_{1}$$_{/}$$_{2}$ , F, m + 1 〉 B$_{x}$ − iB$_{y}$ B$_{z}$ C F,m +1 F,m . A$_{2}$$_{PA}$ ( F, m ; F $^{′}$, m$^{′}$) = e i ( φ $^{ω}$2 + φ $^{ω}$3 ) × (4) {[ ˜ αε ω 2 · ε ω 3 δ$_{F,F}$ ′ + i ˜ β ( ε ω 2 × ε ω 3 )$_{z}$ C F $^{′}$,m′ F,m ] δ$_{m,m}$ ′ + ˜ β [ ± i ( ε ω 2 × ε ω 3 )$_{x}$ − ( ε ω 2 × ε ω 3 )$_{y}$ ] C F $^{′}$,m′ F,m δ$_{m,m}$ ′ ± 1 } , ∑ A = A$_{2}$$_{PA}$ + [{ α E$_{z}$ ε ω 1 z + αE$_{x}$ ε ω 1 x − M ε ω 1 x C F,m F,m − M ε ω 1 z B$_{x}$ B$_{z}$ Δ C (2) } + i { Im ( E$_{PNC}$ ) ε ω 1 z C F,m F,m − βE$_{y}$ ε ω 1 x C F,m F,m + βE$_{y}$ ε ω 1 z B$_{x}$ B$_{z}$ Δ C (2) (6) + βE$_{z}$ ε ω 1 x B$_{y}$ B$_{z}$ Δ C (2) − βE$_{x}$ ε ω 1 z B$_{y}$ B$_{z}$ Δ C (2) } ] e iφ $^{$^{ω}$1}$, where all terms except ε ω 1 x are real, and Δ C (2) = ∑ + / − { ( C F,m ± 1 F,m ) 2 − C F,m F,m ± $_{1}$C F,m ± 1 F,m } is 3/16 for F = 3, m = ± 3, and 1/4 for F = 4, m = ± 4. This factor is small, but not negligible, in comparison to C F,m F,m = ∓ 3 / 4 for F = 3, m = ± 3 or ± 1 for F = 4, m = ± 4. The relative scale of the different amplitudes in Eq. (6) depends on many factors, but for E$_{y}$ < 100 V/cm, the two-photon rate dominates all others, even with cw beam powers where ε ω 2 and ε ω 3 are the amplitudes, and phases φ ω 2 and φ ω 3 , the phases, of the optical waves at frequencies ω$_{2}$ and ω$_{3}$ , respectively, and the coefficients of the two- photon moments, ˜ α and ˜ β , are defined in a form sim- ilar to the Stark polarizabilities. Bouchiat and Bouch- iat [25] noted the relationship between Stark-induced transitions and two-photon absorption. The interference between the two-photon amplitude and the amplitudes that are linear in ε ω 1 can be observed on Δ F = 0 as In each of the previous measurements, a single laser field has been employed, and both interactions (strong and weak) are linear in the amplitude of this field. We now consider this system under the influence of a sec- ond optical field composed of components at frequencies ω$_{2}$ and ω$_{3}$ , where ω$_{2}$ + ω$_{3}$ = ω$_{1}$ , which is capable of driving the 6 s → n $^{′}$s transition via a two-photon inter- action. In order for these amplitudes to interfere, the ω$_{1}$ field component must be phase coherent with ω$_{2}$ and ω$_{3}$ components, as it will be when the former is gen- erated from the latter using a nonlinear optical crys- tal for sum frequency generation. We have previously demonstrated this interference between two-photon ab- sorption and Stark-induced linear absorption on the ce- sium 6 s → 8 s transition [26]. We write the transition amplitude for this interaction in a form similar to that of the Stark-induced transition given by Eq. (1), We include in this expression mixing among magnetic components of the same F, but omit mixing with other F states, an approximation that will be valid for the modest magnetic field strengths characteristic of these measure- ments. We sum the four transition amplitudes, where E$_{PNC}$ is the matrix element for electric dipole transitions due to the state mixing by the PNC inter- actions. To measure A$_{M}$$_{1}$ or A$_{PNC}$ directly is prob- lematic in that their magnitudes are typically well be- low the level of measurement noise. Techniques us- ing an interference between the weak transition and a stronger transition (the Stark-induced transition, for ex- ample) have therefore been developed to effectively am- plify the signal to a detectable level. For example, un- der conditions that allow a strong Stark-induced am- plitude and a weak PNC amplitude on the same tran- sition that add constructively, the net rate scales as W$_{+}$ = | A$_{St}$ + A$_{PNC}$ | 2 ≃ | A$_{St}$ | 2 + 2 | A$_{St}$ || A$_{PNC}$ | . The interference between these various amplitudes can be re- versed by reversing one or more of the fields that in- fluences the sign of the amplitudes, resulting in a rate W − = | A$_{St}$ − A$_{PNC}$ | 2 ≃ | A$_{St}$ | 2 − 2 | A$_{St}$ || A$_{PNC}$ | . A pre- cise measurement of the small difference between W$_{+}$ and W − can then be used to determine | A$_{PNC}$ | .