In atomic parity violation (APV) investigations, researchers carry out precise measurements of the strength of optical interactions that are allowed only because of the weak force interaction between the nucleons and the electrons, as mediated by the exchange of the neutral Z 0 vector boson. Measurements of the strength of these extremely weak transitions, combined with precise atomic structure calculations, allow for a precise determination of the weak charge Q w , as well as sin 2 θ w , where θ w is the Weinberg mixing angle. Any discrepancy between the measured value of Q w or sin 2 θ w and the standard model values can be an indication of physics beyond the standard model. It is therefore of great interest to push the accuracy of these measurements to new limits.
Due to the minute oscillator strength of these weakforce-induced optical transitions, direct APV measurements are not feasible. To enable detection, all APV measurements to date are based upon an interference between the amplitude A PNC of the weak transition and the amplitude of a stronger optical interaction, such as a Stark-induced interaction (A St ) or a magnetic dipole interaction (A M1 ). For example, in their APV measurement on the 6s 2 S 1/2 → 7s 2 S 1/2 transition in cesium, which is to date the most precise APV measurement in any element, Wood et al. [1] reported their measurement results in terms of E PNC /β, where E PNC is the weak-forceinduced electric dipole moment for the transition, and β is the vector transition polarizability. A proper evaluation of E PNC therefore requires a precise value for the vector polarizability, β.
Until 2019, the most precise value of β was based upon a theoretical value for the hyperfine-changing component of the magnetic dipole matrix element M 1 hf [2], and a laboratory determination of the ratio M 1 hf /β [3]. The value of β determined in this way is β M1 = 26.957(51)a 3 0 , with a precision of 0.19%. The subscript 'M1' (or later 'α') on β indicates the method of determination.
The alternative method to determine β uses a sumover-states calculation of the scalar transition polarizability α [4][5][6][7][8], combined with a measurement of the ratio α/β [9]. This approach requires precise measurements of or theoretical values for the reduced electric dipole (E1) matrix elements ⟨np J ||r||ms⟩ with m = 6 or 7, n ≥ 6 and J = 1/2 or 3/2. (We will use this abbreviated state notation ms for ms 2 S 1/2 and np J for np 2 P J .) Many of these matrix elements have been measured to great precision over the past thirty years [6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], and in the last six years, our group has undertaken and completed high-precision measurements for six of the eight most significant E1 matrix elements [25][26][27][28]. Prior to 2023, the discrepancy between the two methods for determining β was 0.67%, greater than the sum of their uncertainties [29]. Recent exhaustive theoretical calculations of reduced E1 matrix elements ⟨np J ||r||ms⟩ by Tran Tan et al. [30] along with our recent measurement of the Stark shift of the 7s state [28] reduced the discrepancy between β M 1 [2, 3] and β α [28] to 0.32%, slightly larger than either uncertainty. This determination utilized a mix of high precision measurements of reduced E1 matrix elements along with some theoretical calculations where experimental results are missing or imprecise to determine α. This result combined with the measurement of α/β [9] yields β α = 27.043 (36) a 3 0 [28]. Tran Tan et al. also report a determination of β α = 26.887 (38) a 3 0 [8] utilizing only theoretical calculations of reduced E1 matrix elements for the determination of α and the measured ratio α/β [9].
To investigate a possible cause for this 0.32% discrepancy between values of β, we have carried out a new precision measurement of the ratio α/β for the cesium 6s → 7s transition, which we report in this work. Our technique is based upon two-color, two-pathway coherent control, in which we excite the transition via two coherent optical interactions; two-photon absorption of a laser field at a wavelength of 1079 nm and Stark-induced single-photon absorption of the second-harmonic light at a wavelength of 540 nm. We describe the coherent control process in Sec. II below. In Sec. III, we outline the experimental technique, followed by a discussion of the pumping efficiency measurement of the atomic beam. In Sec. V, we describe possible systematic effects and methods for reducing them below 0.1% of β. We discuss our results in Sec. VI and conclusions in Sec. VII.
In two-pathway coherent control, one exploits the interference between optical interactions that are driven by two distinct, but mutually-coherent, laser fields. We have used this technique previously to measure M 1/β of the 6s → 7s transition in cesium [31,32], where M 1 is the transition magnetic dipole moment. In the present work, we interfere a strong two-photon allowed E1 excitation, driven by an infrared (IR) laser at 1079 nm, with a weaker Stark induced one-photon excitation driven by the second-harmonic radiation at 540 nm. The net excitation rate W of the 7s state is governed by the square of the sum of the transition amplitudes, A 2p for the former and A St for the latter. Then by Fermi's Golden Rule, the transition rate is
where ρ7s (E) is the density of states of the 7s state.
When the laser is tuned to line center and the linewidth is lifetime limited, the transition rate simplifies to
where Γ is the decay rate of the 7s state. Expanding the sum, the transition rate W becomes
(1) The excitation rate consists of d.c. terms due to A 2p and A St by themselves, plus a beat term resulting from the interference between the two amplitudes A 2p and A St . The beat signal can be modulated by varying the phase difference between the amplitudes A 2p and A St .
The two-photon amplitude for the 6s → 7s transition driven by a single-frequency field scales as the square of the field amplitude ε ω of the 1079 nm laser field,
where we define ϕ ω to be the phase of the 1079 nm beam and α is the scalar polarizability for the 2-photon excitation. The phase factor must be included here, as it becomes relevant in the interference effect. The Kronecker δ functions represent the selection rules that only ∆F = 0, ∆m = 0 transitions are allowed [33]. F and m (F ′ and m ′ ) are the quantum numbers indicating the total angular momentum, electronic plus nuclear spin, of the ground 6s state (excited 7s state) and its projection on the z axis, respectively. The Stark amplitude A St is linear in the applied static electric field E and the field amplitude of the 540 nm beam ε 2ω , and depends upon the relative orientation between E and ε 2ω . The amplitude for the Stark-induced transition can be written [34]
α (β) is the scalar (vector) transition polarizability, which parameterizes the transition amplitude when E and ε 2ω are parallel (perpendicular) to one another. We define ϕ 2ω to be the phase of the 540 nm beam, and the coefficients
are derived from Clebsch-Gordon coefficients and tabulated for this transition in Ref. [34]. The relevant coefficients for the present study are C 3,m 3,m = -m/4, which for m = ±3 is ∓3/4.
In our measurements, the two-color (1079 nm and 540 nm) laser field intersects an atomic cesium beam perpendicularly inside a vacuum chamber. Both frequency components of the optical field are linearly polarized. We apply a static magnetic field B ≈ 8.8 G that is closely aligned with k, the direction of propagation of both laser fields, which we use to define the z-axis of our coordinate system. With the laser tuned to a ∆F = 0, ∆m = 0 transition, the Stark amplitude simplifies to
where θ is the angle between the static field E and the polarization direction of the green beam ε 2ω . We scan the relative phase of the 2-photon and Stark laser fields, ∆ϕ = ϕ 2ω -2ϕ ω , and measure the modulation amplitude when the polarization is parallel and perpendicular to the static electric field. The ratio of these measurements yields,
when the polarization is perfectly parallel or perpendicular to the electric field and the population is fully prepared in a single angular momentum substate m. When the atomic preparation is not complete, the ratio of amplitudes can be shown to be
where ⟨m⟩ is the average value of m for ground state atoms in the interaction region. We write the imperfect polarization of the 540 nm beam as
, where ε y represents the primary component of the green (ε 2ω ) beam, ε ′
x a slight rotation of the polarization from the intended direction, and ε ′′
x the slight amount of circular polarization remaining in the ε 2ω beam. The measured ratio R is modified to,
Here the ± in R ± refers to whether the atoms are initially pumped into the m = +3 or -3 Zeeman sublevel. Under an "m" reversal, the second term in Eq. ( 7) changes sign. Although the higher order terms do not change sign and α/β is ∼ 9.9, we can we adjust ε ′ x /ε y and ε ′′ x /ε y to be less than 1 × 10 -3 and 5 × 10 -4 respectively, making these corrections negligibly small. To extract α/β, we carry out successive measurements of R + and R -, and average these results. This is the basis of the measurement reported in this work.
An illustration of the experimental configuration is depicted in Fig. 1. The primary laser source for these measurements is a commercial external-cavity-diode laser (ECDL) tuned to the frequency of the two-photon 6s, F = 3 → 7s, F = 3 transition in cesium. This laser generates 70 mW of infrared light at a wavelength of 1079 nm and its frequency is locked, using the Pound-Drever-Hall technique [35], to a resonance of an invar-mounted, 15-cm long, Fabry-Perot cavity to minimize short-term drifts. The cavity length is locked to the Doppler-free two-photon absorption resonance signal produced with a cesium vapor cell and photomultiplier using FM spectroscopy at 600 Hz. We amplify the output of the nm laser in a fiber amplifier to a power of 10 W with a power stability of 0.5 %/8 hr, and frequency double this beam in a periodically-poled lithium niobate (PPLN) frequency doubling crystal. The output power at 540 nm is 1150 mW, with a peak-to-peak variation of < 0.4 %/hr. We separate the 1079 nm and 540 nm beams with a dichroic beamsplitter, phase delay the green beam in a galvanometer-mounted optical window, and recombine the beams on a second dielectric beamsplitter. After recombination, we carefully overlap the two beams, and weakly focus them onto the atomic cesium beam inside a vacuum chamber, crossing at nearly a right angle. The waist diameter of the 540 nm beam as it intersects the atomic beam is 2w ≈700 µm and the waist diameter of the 1079 nm beam is ≈920 µm. These waist diameters equate to maximum beam intensities of 150 W/cm 2 for the 540 nm beam and 225 W/cm 2 for the 1079 nm beam. We estimate that the lateral displacement of the 540 nm beam as the galvo-mounted window scans is 6 µm. This displacement causes an angular shift of the focused beam of 17 µr, which is much less than λ/2w ∼ 0.4 mr, the angular change which would diminish the interference fringe visibility.
The atom beam is generated by an effusive oven, and collimated using a packed array of 0.8 mm inner diameter, 1 cm long stainless steel capillary tubes. These tubes are packed into the nozzle opening, 8 mm high and mm wide. We optically pump the atoms into a single hyperfine component of the ground state, F = 3 and m = ±3, by tuning two preparation laser beams to various hyperfine components of the 6s → 6p 3/2 transition. This technique is well described in [36,37]. The interaction region for this measurement is defined by the intersection of the atomic beam and the two-color laser field. We apply a static electric field to the atoms in the interaction region using an assembly of eight parallel copper rod electrodes that is co-axial with the 1079 nm and 540 nm beams, as illustrated in Fig. 2. The rods are arranged in a ring configuration, with each rod parallel to the laser propagation direction k. Each rod has a diameter of 4.8 mm, and the radius of the ring pattern is 18 mm. Careful choice of the bias voltages applied to each rod allows us to create a uniform electric field in the center of the configuration, which coincides with the interaction region. To rotate the direction of E, we have constructed a switching circuit consisting of solid state relays which rotates the bias voltages applied to each of the electrodes. We show color plots of the electric potential for three configurations of potentials applied to the rods in Fig. 2. We have modeled the electric potential in the region bounded by the electrodes, and find that the electric field, with magnitude 429.7 V/cm, is uniform in direction and magnitude to within 20 mV/cm over the 2 mm diameter volume surrounding the interaction region. We chose this method to vary the angle θ between the laser field ε 2ω and the static field E in order to avoid slight changes in the spatial overlap of the two laser beams and/or variations in the optical power or polarization quality of the green beam that might accompany rotation of the laser polarization ε 2ω .
Atoms that participate in the 6s, F = 3 → 7s, F = 3 transition may decay down to the 6s, F = 4 level. When these atoms reach the detection region, 20 cm downstream from the interaction region, they are excited through a cycling transition via a detection laser tuned to the 6s, F = 4 → 6p 3/2 , F = 5 transition. This process scatters many photons which we collect on a large area photodetector. The photocurrent from this photodetector is amplified with a transimpedance amplifier (TIA) of gain 20 MΩ. To vary the phase difference ∆ϕ, we linearly ramp (12 s period) while slightly modulating (150 Hz) the phase of the single-photon beam using the galvanometer mounted window. This linear ramp causes the relative phase, ∆ϕ, to scan at a rate of Ω = 3.8 Hz. We use the 150 Hz modulation as the reference for the lock-in-amplifier to mix down this low frequency modulation, Ω. The TIA output is sent to the input of the lock-in amplifier and the output of the lock-in amplifier is digitally bandpass-filtered and recorded. The bandpass filter is centered on the phase scanning frequency, Ω, and reduces low frequency drifts and higher frequency noise that is near the modulation frequency, 150 Hz. We scan through > 36 cycles, reset the galvanometer, rotate the orientation of the static electric field, and then scan through > 36 more cycles. We show sample phase scans of the output of the lock-in-amplifier with E ∥ ε 2ω (α, blue) and E ⊥ ε 2ω (β, orange) in Fig. 3.
The first 2.6 s of the scan are removed to allow the band-pass filter time to stabilize. We cut the resultant F,m = -3/4. The larger (thin blue) trace demonstrates the α interference with an electric field parallel to the static polarization and smaller (thick orange trace) illustrates β interference. The inset plot (dotted red section) is horizontally stretched to highlight the phase difference between the α and β interference. This shift is consistent with a negative value for the ratio of α/β.
36 cycle scan into 12 sections to avoid the effects of small phase variations present during long scans. Each section is fit to a sinusoid to determine its amplitude. The average amplitude of each fit in a single scan is recorded as well as the standard error of the fitted amplitudes. The α and β (i.e. the blue and orange traces, respectively, in Fig. 3) interference amplitudes and their respective uncertainties are combined to obtain a ratio and an uncertainty. This process is repeated while reversing the Zeeman pumping to change the sign of the C F ′ ,m ′ F,m coefficient and cancel out systematic errors due to the small circular polarization contribution to the β signal. We make ≈ 160 ratio measurements and combine these ratios weighted by the inverse of their uncertainties squared (1/σ 2 ) to attain a final ratio and uncertainty.
As introduced in the previous section, the ground state cesium atoms are optically prepared prior to entering the interaction region. We find that typically ≈ 99% of the ground state population is in the selected initial state, (F, m) = (3, ±3). Since preparation into a single m level is not perfect, careful determination of the average value of the m, denoted ⟨m⟩, is critical. (See Eq. ( 6) or ( 7).) We discuss the measurement of ⟨m⟩ and evaluate its uncertainty here.
We use Raman spectra like those shown in Fig. 4 to determine the fractional population, denoted f m , in each m level. We collect these spectra using a pair of 852 nm beams tuned to the Raman transition that couples the two hyperfine lines of the ground state. For the purposes of this discussion, we illustrate the process for the case in which we prepare the ground state atoms in the state (F, m) = (3, 3), but the technique is applicable to other initial states as well. The two lasers driving the Raman interaction are tuned to 852 nm, detuned from the 6s → 6p 3/2 transition by ∆ ∼ 2π×1 GHz. The two lasers are phase-locked to one another, and frequency stabilized to a saturated absorption resonance in a cesium vapor cell. The beams are combined in a polarization maintaining fiber patchcord and are collimated, linearly-polarized, and then circularly polarized. This polarization state drives ∆m = 0 transitions, (F, m) = (3, m) → (4, m). The Raman and 540 nm beams are aligned such that they are spatially overlapped on the atomic beam, but with the Raman lasers blocked during ratio measurements and with the 540 nm and 1079 nm beams blocked during pumping efficiency measurements. This allows direct detection of the population at the location of the intended interaction. We drive the individual transitions by scanning the frequency difference between the Raman lasers and observe population that has been driven to the F = 4 hyperfine level. We average fifteen scans across the transition. We show sample Raman spectra collected during the measurements in Fig. 4.
Due to the requirement for a precise determination of ⟨m⟩, our measurements of α/β are restricted to the F = 3 → F ′ = 3 hyperfine line. When pumping atoms to the (F, m) = (4, ±4) ground state, a ∆m = ±2 Raman transition is necessary to determine ⟨m⟩. The transition strength of the ∆m = ±2 transitions are around 50 -300 times weaker than that of the ∆m = 0 transition. To determine ⟨m⟩ with better that 0.1% uncertainty, the polarization extinction ratio of the two Raman beams has to be beyond feasible to avoid driving ∆m = 0 transitions. Since the primary contribution of uncertainty in this measurement of α/β comes from the determination of ⟨m⟩ and ∆m = 0 transitions would introduce a large systematic error, we chose to limit our measurements of the ratio α/β to the 6s, F = 3 → 7s, F ′ = 3 transition.
In light of recent work by Xiao et al. [38], who evaluated corrections to the scalar and vector polarizabilities, as well as the magnitude of a tensor term, due to hyperfine coupling, and determined that these corrections are not observable at the current level of measurement sensitivities, the value of α/β is expected to be independent of which hyperfine line is used in the measurements. The fractional population f m is proportional to A m /S m , where A m is the peak area and S m is the calculated Raman line strength. We have determined that the weak ∆m = 0 transitions are well below saturation levels, such that the peak area grows linearly with the square of the Rabi frequency for the transition, around 1 -1.5% of the atoms are excited. We observe reasonable agreement, < 7% disagreement, between the calculated line strengths, S m , and the observed non-Zeeman pumped spectra. Here the atoms are only prepared into a single hyperfine level and are not deliberately pushed into the extreme Zeeman levels. Disagreement between calculated line strengths and the observed peak areas either originate due to the calculation of line strengths, from an uneven distribution of atoms in the Zeeman sublevels originating from the oven, and/or a weak Zeeman pumping effect due to the hyperfine pumping. It is not critical to know the individual line strengths to 0.1% if the pumping quality is sufficiently high such that the contribution of adjacent Zeeman levels is small. We quantify ⟨m⟩ as
where A m /S m are normalized to achieve a total fraction of one. We measure ⟨m⟩ = 2.981 (2), where ⟨m⟩ = 3.000 would indicate perfect preparation in the m = 3 state. The deviation in ⟨m⟩ between preparation into the m = +3 or m = -3 is below 0.09%.
It is critical to identify and reduce systematic effects in precision measurements of weak transitions. The largest systematic effect is that of the alignment of the polarization along the static electric field, discussed in Sec. II. To mitigate this effect, we measure and compare the modulation amplitudes with the electric field rotated ±45 • from the vertical. If the polarization alignment is slightly rotated toward the +45 • direction, for example, then the signal amplitude will be slightly larger when E is rotated to +45 • than -45 • . We rotate the polarization ε 2ω of the green beam to equalize the signal modulation with E rotated to +45 • or -45 • to reduce ε ′
x /ε y to less than 10 -3 , where ε ′
x represents the real portion of the laser field perpendicular to the static electric field and ε y is the laser field parallel to the static electric field. This reduces the systematic uncertainty to below 90 ppm. See Eq. ( 7). An unwanted imaginary portion of the laser polarization, ε ′′
x , produces a systematic error that is first order in ε ′′
x /ε y , but changes sign under an m reversal. Using a crossed polarizer, we measure an extinction ratio of 2.5 × 10 -7 , indicating that ε ′′ x < 0.5 × 10 -3 ε y . The effect of this circular polarization component ε ′′
x causes a ±0.8% deviation in the ratio measurements under ⟨m⟩ reversal. This deviation is consistent with the measured circular polarization of the beam. We reduce this systematic effect by changing m and averaging the ratios.
Other high order optical transitions must be minimized through careful alignment of static magnetic and electric fields along with polarization. The ratio of moments for the magnetic dipole (M1) transition to the Stark vector transition is M 1/β ≈ 29.5 V/cm [3,31]. We apply a static electric field of 430 V/cm and align the static electric and magnetic field such that the magnetic dipole transition drives a ∆m = ±1 transition while the vector Stark and two-photon transitions each drive a ∆m = 0 transition. Perfect alignment of B along k would eliminate all two-photon/M1 interference and would not contribute to the systematic uncertainty. We measure the transverse magnetic field components B x and B y , by observing the Raman spectra, to be less than 10 mG, compared to B z ≈ 8.8 G. This factor, along with the applied electric field and the ratio M 1/β ≈ 29.5 V/cm, reduces the M1 contribution to below 80 ppm of the vector Stark amplitude. The electric quadrupole (E2) transition is ≈ 230 times smaller than the M1 transition [39] and could potentially produce a 0.03 % contribution. In addition to being so small, this contribution is further reduced by averaging the results determined from m = +3 and m = -3. The β contribution for m = ±3 reverses sign, while the E2 contribution does not. Since we measure the modulation amplitude, this relative sign change allows cancellation of the E2 moment. Therefore the E2 transition, like the circular polarization error, averages out to zero under an ⟨m⟩ reversal.
To search for and eliminate systematic contributions due to the applied electric field, we perform reversals of the applied static electric field as well as a rotation of the laser polarization by 90 • . We see no variation with reversal of the electric field by 180 • , and report the average of these results. We also make measurements of the signal amplitude ratio upon reversal of the electric field, 0 • → 180 • from vertical. A ratio of one is expected in the absence of any stray fields. We see no significant deviation among the electric field reversals. The measurements with the laser polarization rotated by 90 • tests for any small ellipticity of the electrode ring pattern and resulted in a 0.15% deviation, as discussed in the next section.
The relative uncertainty in the measurement of the ratio R after 160 scans (1 hour) is typically 0.15 -0.2% and the average value among the different data sets for the reduced chi squared, χ 2 red is 1.18. For any data set for which χ 2 red > 1, we multiply the uncertainty by the square root of χ 2 red [40]. The primary contribution to this uncertainty is due to shot noise in the measurement and relative phase fluctuations between the 540 nm and 1079 nm beams.
To search for any possible ac Stark shifts that affect the ratio of α/β, we collect scans at several different powers of the 1079 nm and 540 nm, and extrapolate to zero intensity. We observe a slight dependence in the ratio of α/β on the 1079 nm beam intensity (See Fig. 5.) and no deviation from the 540 nm beam intensity. We also carry out measurements with the polarization of the laser beams rotated from vertical to horizontal and see a minimal de- TABLE I. Sources and magnitudes of uncertainty for the determination of the ratio α/β. The primary sources of uncertainty originate from the fit and the determination of ⟨m⟩.
We add the errors in quadrature to obtain the total uncertainty.
viation, α/β = -9.894 (9) for vertical polarization and α/β = -9.909 (7) for horizontal. This difference could originate from a slightly elliptical shape of the electrode pattern, as discussed in Sec. V, or from statistical variations in the measurement of R. Due to the similar uncertainties and the reasons above, we report the unweighted average of these ratios, α/β = -9.902( 6) stat (7) sys , as our final result, where the subscript 'stat' indicates the uncertainty due to statistical fluctuations and the subscript 'sys' indicates the systematic uncertainty, primarily due to the ⟨m⟩ determination. The primary sources of uncertainty for these measurements result from statistical uncertainties of the leastsquares fits to the sinusoidally-varying data, and from the determination of the average value of the magnetic sub-level ⟨m⟩. We tabulate these uncertainties and their magnitudes, as well as other less significant uncertainties, in Table I.
We show past theoretical and experimental determinations of α/β in Fig. 6. The theoretical calculations use a sum-over-states approach to determine α and β independently, and divide the results. Due to a large cancellation between terms of opposite sign in the calculation of β, the relative uncertainty in β is much larger than in α. For this reason, the theoretical calculations have difficulty attaining the same precision as experimental determinations, and a measurement of α/β is critical for the determination of β. Fig. 6(b) shows the present result and the previously accepted value of α/β by Cho et al. [9] on an expanded scale. Our measurement technique differs in several regards from that of Ref. [9], including smaller influence of a.c. Stark shifts, our use of two-color coherent control, and our use of much lower optical intensities and strictly linear field polarization. The two measured values are in excellent agreement. The pink line in (b) shows the weighted average of these two results, which we suggest as the recommended value, α/β = -9.903 (6), and the shaded blue region indicates this recommended value's uncertainty. Using this recommended value of α/β and the sum-over-states calculation of α [28], we arrive at a new value for β α = 27.048 (26) a 3 0 . We show a plot of past and present values of the vector polarizability β in Fig. 7.
In this paper, we have described our new measurement of the ratio of the scalar to vector transition polarizability, α/β. This precision measurement reaffirms the previously accepted value [9] and indicates a need to search along other avenues for the cause of the discrepancy between the two techniques that determine β. This discrepancy, β α -β M 1 = 0.091 (57) a 3 0 , must be resolved since β is the moment to which the weak-force-induced electric dipole moment is scaled.