Parity violation in two-photon J=0-to-1 transitions: Analysis of systematic errors
PParity violation in two-photon J = 0 → transitions: Analysis of systematic errors D. R. Dounas-Frazer, ∗ K. Tsigutkin, D. English, and D. Budker
1, 2 Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: October 31, 2018)We present an analysis of systematic sources of uncertainty in a recently proposed scheme for mea-surement of nuclear-spin-dependent atomic parity violation using two-photon J = 0 → PACS numbers: 32.80.Rm, 31.30.jg
I. INTRODUCTION
We previously proposed a method for measuringnuclear-spin-independent (NSI) atomic parity violation(APV) effects using two-photon transitions betweenstates with zero total electronic angular momentum [1].Recently we proposed another two-photon method, onethat allows for NSI-background-free measurements ofnuclear-spin-dependent (NSD) APV effects [2], such asthe nuclear anapole moment [3]. The latter method, the degenerate photon scheme (DPS) , exploits Bose-Einsteinstatistics (BES) selection rules for J = 0 → E E E M E E J = 0, we extend the anal-ysis to include mixing with J = 2 states as well. II. DEGENERATE PHOTON SCHEME
The proposed method uses two-photon transitionsfrom an initial state of total electronic angular mo-mentum J i = 0 to an opposite-parity J f = 1 fi-nal state (or vice versa ). The APV signal is due tointerference of parity-conserving electric-dipole-electric-quadrupole ( E E
2) and electric-dipole-magnetic-dipole( E M
1) transitions with parity-violating E E ∗ [email protected] E E J = 0 → J (cid:54) = 1 states,such as the NSD weak interaction and, in the presenceof an external static electric field, the Stark effect. Be-cause the NSI weak interaction only leads to mixing ofthe final state with other J = 1 states, it cannot induce J = 0 → B . The optical field is character-ized by polarization (cid:15) , propagation vector k , frequency ω , and intensity I . Because circularly polarized lightcannot excite a J = 0 → ω fi between the ground state | i (cid:105) and an excited state | f (cid:105) of opposite nominal parity. Wework in atomic units: (cid:126) = | e | = m e = 1. The transitionrate is [9]: R = (2 π ) α I | A | π Γ , (1)where α is the fine structure constant, and A and Γ arethe amplitude and width of the transition. Energy eigen-states are represented as | i (cid:105) = | J i IF i M i (cid:105) , and likewisefor | f (cid:105) . Here J i , I , and F i are quantum numbers associ-ated with the electronic, nuclear, and total angular mo-mentum, respectively, and M i ∈ {± F i , ± ( F i − , . . . } isthe projection of F i along the quantization axis ( z -axis),which we choose along B .The transition is enhanced by the presence of an in-termediate state | n (cid:105) of total electronic angular momen-tum J n = 1 whose energy lies about halfway betweenthe energies of the initial and final states (Fig. 1). Fortypical situations, the energy defect ∆ = ω ni − ω fi / R associatedwith the one-photon resonance involving the intermedi-ate state. We assume that the scattering rate from | n (cid:105) to | i (cid:105) is small compared to the natural width Γ f of | f (cid:105) :(Ω R / ∆) Γ n (cid:28) Γ f . In this case, the system reduces to a r X i v : . [ phy s i c s . a t o m - ph ] O c t even odd N S D N S I + N S D FIG. 1. Energy level diagram. Dotted lines indicate APVmixing of opposite-parity states, and upward- and downward-pointing arrows represent two-photon absorption and one-photon fluorescence, respectively. a two-level system consisting of initial and final statescoupled by an effective optical field.The parity-violating E E | f (cid:105) may mix with statesof electronic angular momentum J = 0 ,
1, or 2 accordingto the selection rules for NSD APV mixing [10]. Mixingof the final state with J = 1 states results in a perturbedfinal state with electronic angular momentum 1 that can-not be excited via degenerate two-photon transitions. Weassume mixing is dominated by a single state | a (cid:105) of totalangular momentum J a , and consider the cases J a = 0and J a = 2 separately. A. NSD APV mixing of J = 1 and J = 0 states When | f (cid:105) mixes with a nearby J a = 0 state, only tran-sitions for which F f = I may be induced by the weak in-teraction. Transitions to hyperfine levels F f = I ± J = 0 → A = A PC + A W , (2)where A PC = i Q k − q ( (cid:15) · (cid:15) )( − q (cid:104) F i M i ; 1 q | F f M f (cid:105) , (3)and A W = iζ ( (cid:15) · (cid:15) ) δ F f F i δ M f M i , (4)are the amplitudes of the parity-conserving and weak-interaction-induced parity-violating transitions, respec-tively. Here q = M f − M i is a spherical index, k q is FIG. 2. Field geometry. The propagation vector k mayalternatively be anti-aligned with the magnetic field B . the q th spherical component of ˆ k , (cid:104) F i M i ; 1 q | F f M f (cid:105) is aClebsch-Gordan coefficient, and δ F f F i is the Kroneckerdelta. The quantities Q and ζ are Q = Q fn d ni √
15∆ + µ fn d ni √ , (5)and ζ = Ω fa d an d ni ω fa ∆ , (6)where the reduced matrix elements Q fn = ( J f || Q || J n ), µ fn = ( J f || µ || J n ), and d ni = ( J n || d || J i ) of the elec-tric quadrupole, magnetic dipole, and electric dipole mo-ments, respectively, are independent of F f and I . Here ω fa = ω f − ω a is the energy difference of states | f (cid:105) and | a (cid:105) , and Ω fa is related to the matrix element of the NSDAPV Hamiltonian H NSD by (cid:104) f | H NSD | a (cid:105) = i Ω fa . The pa-rameter Ω fa must be a purely real quantity to preservetime reversal invariance [10]. Note that (cid:15) · (cid:15) = 1 for linearpolarization, whereas (cid:15) · (cid:15) = 0 and hence A = 0 for circu-lar polarization, consistent with conservation of angularmomentum. Hereafter, we assume (cid:15) · (cid:15) = 1.The goal of the DPS is to observe interference of parity-violating and conserving amplitudes in the rate R . When M f = M i , R consists of a large parity conserving termproportional to Q , a small parity violating term (theinterference term) proportional to Q ζ , and a negligiblysmall term on the order of ζ . The interference termis proportional to a pseudoscalar quantity that dependsonly on the field geometry, the rotational invariant : k · B . (7)The form of the rotational invariant follows from the factthat only k ∝ k · B contributes to the amplitude inEq. (3) when M f = M i . Thus the interference term van-ishes if B and k are orthogonal. One way to achievea nonzero rotational invariant is to orient k along B (Fig. 2).We calculate the transition rate when B is sufficientlystrong to resolve magnetic sublevels of the final state,but not those of the initial state. This regime is realisticsince Zeeman splitting of the initial and final states areproportional to the nuclear and Bohr magnetons, respec-tively. In this case, the total rate is the sum of rates fromall magnetic sublevels of the initial state: R → (cid:88) M i R ( M i ) . (8)When the fields are aligned as in Fig. (2), the transitionrate is R ± ∝ Q M f I ( I + 1) ± ζ Q M f (cid:112) I ( I + 1) , (9)where the positive (negative) sign is taken when k and B are aligned (anti-aligned), and we have omitted the termproportional to ζ .Reversals of applied fields are a powerful tool for dis-criminating APV from systematic effects. The interfer-ence term in (9) changes sign when the relative alignmentof k and B is reversed, or when M f → − M f . The asym-metry is obtained by dividing the difference of rates upona reversal by their sum: R + − R − R + + R − = 2 (cid:112) I ( I + 1) M f ζ Q , (10)which is maximal when M f is small but nonzero. Rever-sals are sufficient to distinguish APV from many system-atic uncertainties. Nevertheless, there still exist system-atic effects that give rise to spurious asymmetries , whichmay mask APV.We consider two potential sources of spurious asym-metry: misalignment of applied fields, and stray electricand magnetic fields. A stray electric field E may induce E E J = 0 → A S = ξ E − q ( − q (cid:104) F i M i ; 1 q | F f M f (cid:105) , (11)where ξ = d fa d an d ni √ ω af ∆ . (12)When k and B are misaligned ( k × B (cid:54) = ), Stark-induced transitions may interfere with the allowed tran-sitions yielding a spurious asymmetry characterized bythe following rotational invariant:( E × k ) · B ≡ ( k × B ) · E . (13)The resulting Stark-induced asymmetry is1 M f θξ E Q , (14) where θ = | ˆ k × ˆ B | is the angle between the nomi-nally collinear vectors k and B , and E is defined by θE ≡ ( k × B ) · E . The spurious asymmetry (14) maymask the APV asymmetry (10) because both exhibit thesame behavior under field reversals. However, becausethe Stark-induced transition amplitude is nonzero when F f (cid:54) = I , APV and Stark-induced asymmetries can bedetermined unambiguously by comparing transitions todifferent hyperfine levels of the final state.We propose to measure the transition rate by observingfluorescence of the excited, and assume that the transi-tion is not saturated: I < I sat ≡ Γ / (4 πα Q ) , (15)where the saturation intensity I sat is chosen so that R = Γ when I = I sat . In this regime, fluorescence isproportional to the transition rate. The statistical sensi-tivity of this detection scheme is determined as follows:The number of excited atoms is N f = N i R ± t ≡ N ± N (cid:48) , (16)where N i is the number of illuminated atoms, t is themeasurement time, and N and N (cid:48) (cid:28) N are the num-ber of excited atoms due to parity-conserving and parity-violating processes. The signal-to-noise ratio is SNR = N (cid:48) / √ N , or SNR = 8 πα I ζ (cid:112) N i t/ Γ= 2( I / I sat )( ζ / Q ) (cid:112) N i Γ t. (17)The SNR is optimized by illuminating a large number ofatoms with light that is intense, but does not saturate the i → f transition. Although purely statistical shot-noisedominated SNR does not depend on Q , this parameter isstill important in practice due to condition (15). Allowed E E E M Q , which leads to small APV asymmetry. In the oppositecase of forbidden E E E M Q ), an observable signal requires high light intensities,which may pose a technical challenge. B. NSD APV mixing of J = 1 and J = 2 states Mixing of | f (cid:105) with nearby J a = 2 states is qualitativelysimilar to the previous case. Here we make the compari-son explicit. The amplitude of the transition induced byNSD APV mixing of | f (cid:105) and | a (cid:105) is (Appendix A): A W = iζ { (cid:15) ⊗ (cid:15) } , − q ( − q (cid:104) F i M i ; 2 q | F f M f (cid:105) , (18)where ζ = Ω fa d an d ni √ ω af ∆ , (19)and { (cid:15) ⊗ (cid:15) } q is the q th spherical component of the rank-2 tensor formed by taking the dyadic product of (cid:15) withitself [13]. For the geometry in Fig. 2, the transition rateis R ± ∝ C Q ± (cid:112) / C C ζ Q , (20)where the positive (negative) sign is taken when k and B are aligned (anti-aligned), C k = (cid:104) IM f ; k | F f M f (cid:105) for k = 1 ,
2, and we have omitted a term proportional to ζ .For simplicity, we focus on the case F f = I + 1 (the cases F f = I, I − R ± ∝ F f − M f ( I + 1)(2 I + 1) (cid:34) Q ± ζ Q M f (cid:112) I ( I + 2) (cid:35) , (21)and the asymmetry is R + − R − R + + R − = 2 M f (cid:112) I ( I + 2) ζ Q , (22)which is maximal when M f = I . In the case of maximalasymmetry, the SNR isSNR = 8 παC I I ζ (cid:112) N i t/ Γ= 2 C I ( I / I sat )( ζ / Q ) (cid:112) N i Γ t. (23)where C I = (cid:112) I/ [2( I + 1)( I + 2)] is a numerical coeffi-cient and I sat is given by Eq. (15).Static electric fields may induce a J = 0 → | f (cid:105) and | a (cid:105) , giving rise to systematiceffects that may mimic APV. When J a = 2, the ampli-tude of Stark-induced transitions is (Appendix A): A S = ξ [ E − q − (cid:15) · E ) (cid:15) − q ]( − q (cid:104) F i M i ; 1 q | F f M f (cid:105) , (24)where ξ = d fa d an d ni √ ω af ∆ . (25)The spurious asymmetry due to Stark mixing is charac-terized by the rotational invariant( k × B ) · [ E − (cid:15) · E ) (cid:15) ] . (26)Unlike for the J a = 0 case, both the Stark effect and theweak interaction may induce transitions to F f = I, I ± | f (cid:105) when J a = 2, eliminating the pos-sibility of using APV-free transitions to control system-atic effects. However, the Stark- and weak-interaction-induced asymmetries have different dependence on M f : R + − R − R + + R − = 2 M f (cid:112) I ( I + 2) ζ Q (cid:124) (cid:123)(cid:122) (cid:125) APV + (2 I + 1) M f F f − M f θξ ˜ EQ (cid:124) (cid:123)(cid:122) (cid:125) Stark , (27)where ˜ E is defined by θ ˜ E ≡ ( k × B ) · [ E − (cid:15) · E ) (cid:15) ]. ThusAPV can be distinguished from spurious asymmetries byanalyzing the Zeeman structure of the transition, e.g. , TABLE I. Available atomic data for application of DPS toSr and Ra. Here a is the Bohr radius.Transition (1 → d / ( ea )Sr a s S → s p P b s S → s p P s p P → s d D s d D → s p P a Ref. [14] b Ref. [15] by comparing transitions to sublevels M f = I and M f = I − | f (cid:105) mixes with J = 2 states:( k · B )( (cid:15) · B ) . (28)This rotational invariant describes APV interference intransitions for which M f = M i ± III. APPLICATIONS OF DPS
We now turn our attention to the two-photon 462 nm5 s S → s p P transition in Sr ( Z = 38, I =9 / s p P state (∆ = 34 cm − ), and the parity-violating E E s p P and 5 s s S states ( ω af = 184 cm − ). Weused expressions presented in Ref. [16] to calculate theNSD APV matrix element: Ω fa ≈ κ s − , where κ is adimensionless constant of order unity that characterizesthe strength of NSD APV. The width of the transition isdetermined by the natural width Γ = 1 . × s − of the5 s p P state [14]. Other essential atomic parametersare given in Table I. Resolution of the magnetic sublevelsof the final state requires a magnetic field larger than2Γ /g ≈
10 G, where g ≈ . F = I hyperfine level of the 5 s p P state. We estimatethat d an ≈ ea , Q fn /d an ≈ α/
2, and µ fn (cid:28) Q fn . Thenthe APV asymmetry associated with this system is about4 κ × − .Spurious asymmetries due to stray electric fields canbe ignored when θE (cid:28) I + 1) (cid:112) I ( I + 1)( ζ /ξ ) ≈ θ < . ◦ .Regardless of misalignment errors, APV can be discrim-inated from Stark-induced asymmetries by comparingtransitions to the F (cid:54) = 9 / s p P state.To estimate the SNR, we consider experimental pa-rameters similar to those of Ref. [18]: N i ≈ atomsilluminated by a laser beam of characteristic radius0 . I = I sat ≈ × W/cm . In this case, Eq. (17)yields SNR ≈ κ × − (cid:112) t/ s. The saturation intensitycorresponds to light power of about 2 kW at 462 nm.High light powers may be achieved in a running-wavepower buildup cavity. With this level of sensitivity, about300 hours of measurement time are required to achieveunit SNR. The projected asymmetry and SNR for theSr system are to their observed counterparts in the mostprecise measurements of NSD APV in Tl [19].Another potential candidate for the DPS is the 741nm 7 s S → s p P transition in unstable Ra( Z = 88, I = 3 / t / = 15 days). This systemlacks an intermediate state whose energy is nearly halfthat of the final state; the closest state is 7 s p P (∆ ≈ − ). Nevertheless, it is a good candi-date for the DPS, partly due to the presence of nearly-degenerate opposite-parity levels 7 s p P and 7 s d D ( ω af = 5 cm − ). In this system, NSD APV mix-ing arises due to nonzero admixture of configuration7 p in the 7 s d D state [20]. Numerical calcula-tions yield Ω fa d an /ω af ≈ κ × − ea [21] and Γ =2 . × s − [15]. Other essential atomic parameters aregiven in Table I. Like for the Sr system, we estimate that Q fn /d an = α/ µ fn (cid:28) Q fn , yielding an approxi-mate asymmetry of 7 κ × − . Laser cooling and trappingof Ra has been demonstrated [22], producing about N i ≈
20 trapped atoms. When I = I sat ≈ W/cm ,Eq. (23) gives SNR = κ × − (cid:112) t/ s. For a laser beamof 0.3 mm, the saturation intensity corresponds to lightpower of about 300 kW at 741 nm. These estimates sug-gest that unit SNR can be realized in under 3 hours ofobservation time. Compared to the Sr system, the Rasystem potentially exhibits both a much larger asymme-try and a much higher statistical sensitivity. IV. SUMMARY AND DISCUSSION
In conclusion, we presented a method for measur-ing NSD APV without NSI background. The proposedscheme uses two-photon J = 0 → Sr and
Ra that arepromising candidates for application of the DPS.
ACKNOWLEDGMENTS
The authors acknowledge helpful discussions withD. P. DeMille, V. Dzuba, V. Flambaum, M. Kozlov, andN. A. Leefer. This work has been supported by NSF.
Appendix A: Derivation of transition amplitudes
In this appendix, we derive amplitudes for induced E E J = 0 →
1. Bose-Einstein statistics selection rules
Here we provide a brief review of BES selection rulesfor J = 0 → ω = ω ≡ ω ), co-propagating ( k = k ≡ k ) photons [4].Since the only transitions of relevance are of this type,they are referred to as simply “degenerate transitions”without cumbersome qualifiers. We ignore hyperfine in-teraction (HFI) effects by assuming that there is zeronuclear spin.It must be possible to write the absorption amplitude A for a degenerate transition in terms of the only quan-tities available: the polarizations of the two photons, (cid:15) and (cid:15) ; the final polarization of the atom in its ex-cited J = 1 state, (cid:15) e ; and the photon momentum k .With the requirement of gauge invariance of the photons( (cid:15) , · k = 0), only three forms of A are possible: A a ∝ ( (cid:15) × (cid:15) ) · (cid:15) e ; (A1a) A b ∝ ( (cid:15) · (cid:15) )( (cid:15) e · k ); (A1b) A c ∝ [( (cid:15) × (cid:15) ) · k ]( (cid:15) e · k ) . (A1c)Amplitudes A a and A c are odd under photon inter-change, and hence vanish because photons obey BES.However, amplitude A b is even and may yield a nonzeroabsorption amplitude. In the case of degenerate tran-sitions between atomic states of the same total parity, A b vanishes because it is odd under spatial inversion.Hence degenerate transitions between like-parity statesare forbidden by BES selection rules . However, degen-erate transitions may be allowed when the initial andfinal states are of opposite parity.When (cid:15) = (cid:15) ≡ (cid:15) , as would be the case if the photonswere absorbed from the same laser beam, the degeneratetransition amplitude reduces to A b ∝ ( (cid:15) · (cid:15) ) k · (cid:15) e . There-fore, the amplitude of a degenerate transition betweenopposite parity states is A b = i ( (cid:15) · (cid:15) ) Q k − M ( − M , (A2)where k − M ( − M is the projection of ˆ k onto the spin ofthe excited atom and the factor of i ensures time reversalinvariance. Parity-conserving perturbations, such as the Zeeman effect orthe hyperfine interaction, may induce degenerate transitions be-tween like-parity states via two mechanisms: splitting of the in-termediate state into non-degenerate sublevels, and mixing of thefinal state with nearby like-parity J (cid:54) = 1 states [23].
2. Wigner-Eckart theorem
We use the following convention for the Wigner-Eckarttheorem (WET). Let T k be an irreducible tensor of rank k with spherical components T kq for q ∈ { , ± , . . . , ± k } .Then the WET is [24] (cid:104) J IF M | T kq | J IF M (cid:105) == ( J IF || T k || J IF ) √ F + 1 (cid:104) F M ; kq | F M (cid:105) , (A3)where ( J IF || T k || J IF ) is the reduced matrix elementof T k and (cid:104) F M ; kq | F M (cid:105) is a Clebsch-Gordan coeffi-cient. If T k commutes with the nuclear spin I , then itsreduced matrix element satisfies [24]( J IF || T k || J IF ) √ F + 1 = ( − J + I + F + k ( J || T k || J ) ×× (cid:112) F + 1 (cid:26) J F IF J k (cid:27) , (A4)where ( J || T k || J ) is the reduced matrix element of T k inthe decoupled basis, and the quantity in the curly bracesis a 6 j symbol. E - M and E - E transition amplitudes In the following, summation over the magnetic sub-levels M n of the intermediate state is implied. The E M E E A b = 2 (cid:104) f | [( k × (cid:15) ) · µ ] | n (cid:105)(cid:104) n | ∆ ( (cid:15) · d ) | i (cid:105) = i (cid:18) µ fn d ni √ (cid:19) k − M ( − M (cid:104) F i M i ; 1 q | F f M f (cid:105) , (A5)and A b = (cid:104) f | [ i { k ⊗ (cid:15) } · Q ] | n (cid:105)(cid:104) n | ∆ ( (cid:15) · d ) | i (cid:105) = i (cid:18) Q fn d ni √ (cid:19) k − M ( − M (cid:104) F i M i ; 1 q | F f M f (cid:105) , (A6)respectively. Here µ , d , and Q are the magnetic dipole,electric dipole, and electric quadrupole moments of theatom.To derive Eqs. (A5) and (A6), we have assumed (cid:15) · (cid:15) = 1, as is the case for linear polarization, and we haveomitted a common factor of ( − I − F f . Equations (3) and(5) follow from the definition A PC ≡ A b + A b .
4. Induced E - E transitions E E | f (cid:105) and | a (cid:105) due to both the weak interaction and Stark effect. The final state of the transition is the per-turbed state | f (cid:105) + χ ∗ | a (cid:105) , where χ is a small dimensionlessparameter that depends on the details of the perturb-ing Hamiltonian. The amplitude for the induced E E A E E = χ (cid:104) a | (cid:15) · d | n (cid:105)(cid:104) n | ω ni − ω (cid:15) · d | i (cid:105) . (A7)Here d is the electric-dipole moment of the atom andsummation over the hyperfine levels and magnetic sub-levels of the states | n (cid:105) and | a (cid:105) is implied.In Eq (A7), the quantity (cid:104) a | · · · | i (cid:105) is the amplitude ofthe allowed degenerate two-photon i → a transition. Itcan be expressed as the contraction of two irreducibletensors: (cid:104) a | (cid:15) · d | n (cid:105)(cid:104) n | ω ni − ω (cid:15) · d | i (cid:105) = (cid:88) k,q { (cid:15) ⊗ (cid:15) } ∗ kq (cid:104) a | T kq | i (cid:105) , (A8)where { (cid:15) ⊗ (cid:15) } kq = (cid:88) µ,ν (cid:104) µ ; 1 ν | kq (cid:105) (cid:15) µ (cid:15) ν , (A9)is the tensor of rank k = 0 , (cid:15) with itself, and T kq is a tensor whose matrix ele-ments we wish to express in terms of those of the dipolemoment d . Neglecting hyperfine splitting of the interme-diate state, T k commutes with I . Then, since J i = 0, wehave (cid:104) a | T kq | i (cid:105) = ( − I − F a + k ( J a || T k || J i ) √ J a + 1 (cid:104) F i M i ; kq | F f M f (cid:105) , (A10)for k = J a and q = M a − M i . Using the WET to simplifythe left-hand side of Eq. (A8), we find( J a || T k || J i ) = 1 √ d an d ni ∆ , (A11)where d fa = ( J f || d || J a ) is the reduced matrix elementof the electric dipole operator. When k (cid:54) = J a , the ma-trix element (cid:104) a | T kq | i (cid:105) vanishes. Therefore, only the ten-sor { (cid:15) ⊗ (cid:15) } k of rank k = J a contributes to the i → a transition. Note that the tensor of rank k = 1 satisfies { (cid:15) ⊗ (cid:15) } q ∝ ( (cid:15) × (cid:15) ) q ≡
0, and hence the J i = 0 → J a = 1transition has zero amplitude, consistent with more gen-eral selection rules for degenerate two-photon transi-tions [4].When the mixing of | f (cid:105) and | a (cid:105) is due to the weakinteraction alone, the perturbation parameter is given by χ = χ W , where χ W = (cid:104) f | H NSD | a (cid:105) ω fa ≡ i Ω fa ω fa , (A12)for F a = F f and M a = M f . Equations (4)and (18) followfrom the definitions ζ k ≡ (Ω fa /ω fa )( J a || T k || J i ) / √ k + 1for k = J a = 0 , E , the pertur-bation parameter becomes χ = χ W + χ S , where χ W isgiven by Eq. (A12) and χ S = (cid:104) f | H S | a (cid:105) ω fa , (A13)where H S = − d · E is the Stark Hamiltonian. In thiscase, A E E = A W + A S , where A W ∝ χ W and A S ∝ χ S are the amplitudes of the transitions induced by the weakinteraction and Stark effect, respectively.For a general J = J i → J f transition, the Stark-induced E E { (cid:15) ⊗ (cid:15) } ∝ ( (cid:15) · (cid:15) ) or { (cid:15) ⊗ (cid:15) } with E . Thereare four such tensors: one each of ranks 2 and 3, and twoof rank 1. However, for J = 0 → { E ⊗ { (cid:15) ⊗ (cid:15) } } q = − √ (cid:15) · (cid:15) ) E q (A14)and { E ⊗ { (cid:15) ⊗ (cid:15) } } q = (cid:114)
115 [ E q − (cid:15) · E ) (cid:15) q ] . (A15)Stark mixing of | f (cid:105) with | a (cid:105) gives rise to a Stark-inducedamplitude whose dependence on applied fields is de-scribed by either the tensor in Eqs. (A14) or the onein Eq. (A15) depending on whether J a = 0 or J a = 2.The corresponding amplitudes are given by Eqs. (11) and(24), and the parameters ξ and ξ can be expressed interms of the reduced dipole matrix elements by apply-ing the WET to Eq. (A7) with χ = χ S . This procedureyields Eqs. (12) and (25). [1] D. R. Dounas-Frazer, K. Tsigutkin, D. English, andD. Budker, Phys. Rev. A , 023404 (2011).[2] D. R. Dounas-Frazer, K. Tsigutkin, D. English, andD. Budker, To be submitted to 2011 PAVI conferenceproceedings.[3] W. C. Haxton, C.-P. Liu, and M. J. Ramsey-Musolf,Phys. Rev. C , 045502 (2002).[4] D. DeMille, D. Budker, N. Derr, and E. Deveney, AIPConference Proceedings , 227 (2000).[5] M. Gunawardena and D. S. Elliott, Phys. Rev. Lett. ,043001 (2007).[6] J. Gu´ena, D. Chauvat, P. Jacquier, E. Jahier, M. Lintz,S. Sanguinetti, A. Wasan, M. A. Bouchiat, A. V. Pa-poyan, and D. Sarkisyan, Phys. Rev. Lett. , 143001(2003).[7] A. D. Cronin, R. B. Warrington, S. K. Lamoreaux, andE. N. Fortson, Phys. Rev. Lett. , 3719 (1998).[8] D. English, V. V. Yashchuk, and D. Budker, Phys. Rev.Lett. , 253604 (2010).[9] F. H. M. Faisal, Theory of Multiphoton Processes (Plenum Press, 1987).[10] I. B. Khriplovich,