Atomic parity violation (APV) measurements provide a window through which the weak-force interaction between nucleons and electrons at low-collision energies can be viewed. The weak-force interaction perturbs the atomic system, rendering optical transitions that would otherwise be strictly forbidden slightly allowed. The amplitude for these interactions is weak, typically ∼ 10 -11 orders of magnitude weaker than that of the strong D 1 or D 2 lines in cesium, for example. Extracting the weak charge Q w of the nucleus from the transition moment E PNC for the transition requires accurate theoretical models of the atomic wavefunctions.
The most precise value of Q w in any atom to this point is derived from the APV measurements carried out by the Boulder group of Wieman [1] in atomic cesium in 1997. One of the benefits of working in an alkali atom is its 'simple' atomic structure, consisting of a single valence electron outside closed inner shells of electrons. Models of the electronic wavefunctions of this heavy atom have become progressively more refined over the years [2][3][4][5][6][7][8][9][10][11][12][13][14]. To support and enhance these theoretical efforts, laboratory measurements of many atomic parameters, such as electric dipole (E1) matrix elements, have been carried out by several groups [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. E1 matrix elements for transitions between low-lying levels are sensitive to the wavefunctions at moderate distances, comparable to the Bohr radius a 0 , from the nucleus. The precision of many E1 matrix elements for transitions between low-lying states of cesium is now ∼ 0.1%, and the experimental values are in very good agreement with theoretical values. (See Ref. [31] for a compilation of these results.)
Since the weak-force interaction is a contact potential, calculations of E PNC also require precision in the wavefunctions near the nucleus. Theoretical efforts to calculate hyperfine coupling constants are therefore of great interest, since the hyperfine interaction is also sensitive to the electronic wavefunction at the nucleus [32][33][34]. Calculations and measurements of the hyperfine coupling constants A, particularly of J = 1/2 states, are therefore of critical importance to calculations of the weak Hamiltonian, and for gauging their precision.
Recent theoretical work by Ginges et al. [32][33][34] has focused on precision calculations of A for low-lying and intermediate levels of atomic cesium. Their relativistic Hartree-Foch many-body calculation includes effects of core polarization, correlation corrections, quantum electrodynamic (QED) radiative corrections (self-energy and vacuum polarization), and the Bohr-Weisskopf (BW) correction (an accounting of the nonuniform density of the magnetization of the nucleus). In Ref. [33], the authors proposed to use the results of precise measurements of the hyperfine splitting (hfs) of excited ns 2 S 1/2 states to greatly improve the ground 6s 2 S 1/2 state and 7s 2 S 1/2 state hyperfine intervals. (Hereafter, the abbreviated notation ns and np J will be used in place of ns 2 S 1/2 and np 2 P J , respectively.) Their calculations showed that the correlation corrections decreased with increasing principal quantum number n, approaching a constant but nonzero value. They proposed to use measurements of the hfs in high ns states (n > 9) to determine the BW and QED corrections in these states, which can then be scaled for application to the 6s and 7s states. This removes the large uncertainties due to the BW and QED corrections from the hfs calculations. We recently reported measurements [35] of the hfs of the 12s and 13s states of cesium to be used for this analysis. In Ref. [34], Grunefeld, Roberts, and Ginges examined trends in the corrections to the hyperfine coupling constants A, to make predictions of these constants for ns and np 1/2 states of cesium, where 6 ≤ n ≤ 17, which they believe to be accurate at the 0.1% level. Recently, this group has found additional confirmation of the BW correction [36] in historical data on muonic cesium.
Precise measurements of the hyperfine coupling constants of the 6p 1/2 and 7p 1/2 states of cesium have been reported previously. The uncertainty in A for these two states is 0.007% [37] and 0.04% [38,39], respectively. For the 8p 1/2 state, however, the measurement uncertainty prior to the present measurement was ∼ 0.2% [40]. (See Ref. [41] for an extensive survey of hyperfine coupling constants in alkali metal atoms.) The goal of this study, therefore, was to determine the hyperfine coupling constant A for the 8p 1/2 state of cesium ( 133 Cs) with reduced uncertainty for direct comparison to the current theoretical results [34], and to facilitate improvements in these theoretical techniques. This measurement is in support of our ongoing investigation towards a high precision determination of the weak charge of atomic cesium [31,42,43]. In addition, we have examined the hyperfine structure of the 8p 3/2 state, and report favorable comparison with prior experimental results, and determined the absolute energies of the 8p 1/2 and 8p 3/2 states with a precision of ∼150 kHz.
The energy levels of the ground 6s and excited 8p 1/2 states of cesium are shown in Fig. 1. The hyperfine interaction splits both states (6s and 8p 1/2 ) into two hyperfine components, of energy (See Ref. [44]),
where F is the total angular momentum (the vector sum of the nuclear I = 7/2 and electronic J = 1/2 angular momenta), A is the magnetic dipole hyperfine coupling constant. and E cg is the center-of-gravity energy of the state. The energy spacing in the ground state is defined to be ∆E 6s /h = 9.192 631 770 GHz, which is equal to 4A 6s .
To measure the hyperfine splitting of the 8p 1/2 state, we measure the absolute frequencies of the individual hyperfine components of the 6s → 8p 1/2 transitions. To achieve this, we drive the electric-dipole transition from the cesium ground state in an atomic beam using a cw narrow-band external cavity diode laser (ECDL), offset phase locked to a frequency comb laser (FCL) source. Precise frequency difference measurements can be made by referencing the driving laser's frequency to the FCL frequency.
A schematic of the experimental configuration is illustrated in Fig. 2. The 778 nm output of the commercial (Toptica) ECDL and tapered amplifier unit is frequency doubled in a lithium tri-borate (LBO) crystal in a singlepass geometry to produce ∼170 µW of light at 388.9 nm. The fundamental beam is separated from the second harmonic with a 40 nm wide band pass filter centered at 400 nm. This filter has an optical density of > 6.6 at the fundamental frequency, and passes 96% of the second harmonic. The 388.9 nm beam is chopped (150 Hz) with a rotating chopper wheel and directed to a vacuum chamber that houses the atomic beam. Excited 8p atoms decay spontaneously via many pathways as they relax to the ground state. The primary detection signal for our measurements is the 852 nm and 894 nm fluorescence of the 6p 3/2 → 6s and 6p 1/2 → 6s decay paths, respectively.
The laser beam is approximately elliptical in shape and Gaussian in intensity, with estimated major ∼5 mm and minor 2 mm axes. The aluminum vacuum chamber is cylindrical in shape, measuring 30 cm in diameter and 45 cm tall. It contains the oven, nozzle, photodetection system, and magnetic field biasing coils, and is pumped to a vacuum level of 5 × 10 -6 torr with a turbomolecular pump. The total magnetic field in the interaction region is reduced to below 10 mG. The atomic cesium beam is generated by an effusive oven fitted with a nozzle composed of an array of stainless steel capillaries (0.58 mm I.D. x ∼1 cm length). The beam then passes through a collimator, consisting of a stack of microscope coverslips (0.17 mm thick) spaced with microscope slides (1 mm thick), which reduces the beam divergence. The laser beam intersects the cesium beam at close to a right angle and is retro-reflected to reduce Doppler shifts. A large area silicon photodiode and long pass (> 700 nm) optical filter lie directly below the interaction region, which is defined by the intersection of the atomic beam and the laser beam. A curved reflector above the photodetector reflects upward-directed fluorescence back down towards the photodetector. The long-pass filter effec-FIG. 2. Experimental configuration for the 8p 1/2 and 8p 3/2 hyperfine spectroscopy. 3.5 W of 778 nm light is generated in a commercial ECDL and tapered amplifer unit and is focused into a lithium tri-borate crystal (LBO) in a single pass configuration to generate second harmonic light at 388-389 nm (170 µW) to excite the 6s → 8pJ transition. This doubled light is directed into a vacuum chamber, through an atomic beam and is retro-reflected to reduce Doppler shifts. We beat 1 mW of the 778 nm beam against the frequency comb laser (FCL) and send 7 mW to the wavemeter for coarse frequency measurement. The combination of these two measurements yields precise absolute frequency measurements. We stabilize and narrow the laser bandwidth at 388 nm by offset phase locking the 778 nm laser to one of the comb teeth. This offset is varied to sweep the frequency of the 388 nm light across the individual hyperfine levels. The following elements are labeled as; M -mirror, BPband pass filter, PD -Photodetector, BS -beam splitter, and LP -low pass filter. The fine red dotted section includes the fluorescence detection and magnetic field canceling coils. The coarse green dotted section illustrates the frequency measurement and stabilization.
tively reduces scattered excitation light (388.9 nm), while efficiently passing longer-wavelength fluorescence (transmission > 97%) at 852 nm (6p 3/2 → 6s) and 894 nm (6p 1/2 → 6s). The photo detection signal is amplified in a transimpedance amplifier and sent to a lock-in amplifier to be demodulated at the chopping frequency.
To stabilize the frequency of the ECDL, a portion of the 778 nm beam is beat against the output of the FCL. This source is a commercial (Menlo Systems) femtosecond 1560 nm fiber laser that is frequency doubled to 780 nm and spectrally broadened in a highly-nonlinear fiber. The frequency comb repetition rate and carrier envelope offset frequency are stabilized to a GPS conditioned oscillator. The beat signal (at frequency ν beat ) is then fed into an analog optical-phase-lock loop. Here we amplify and mix down the beat signal with a stable signal generator to generate an error signal with which we lock the 778 nm source. By sweeping the frequency of the signal generator and counting the beatnote, we carefully control frequency scans across the 8p 1/2 (and the 8p 3/2 in Section III) spectra.
The absolute frequency of the second harmonic beam is given by
where the factor of two accounts for exciting the transition with the second-harmonic beam while beating the fundamental laser against the frequency comb, and N represents the comb tooth number. ν rep and ν offset are the repetition rate and offset frequency of the FCL. N is determined using a wave meter with an accuracy of better than half of the repetition rate of the FCL (∼250 MHz). The sign of the beat note is determined by observing the change in beat note while increasing the laser frequency.
We collect data in the following manner. After the temperature of the oven and nozzle have adequately stabilized to produce a consistent atomic beam density, the signal generator frequency is set to control the offset beatnote. The system pauses for a time 2τ , where τ = 100 ms is the time constant of the lock-in amplifier. One hundred voltage samples are then collected using a 16-bit analog-to-digital converter (ADC) at a rate of one kHz, and are averaged and recorded. Ten sets of 100 voltage samples are collected. This protocol reduces correlation among the ten different data sets. The average and standard error of the mean of the ten voltage sets are then computed and recorded. The frequency of the signal generator is measured with a frequency counter and the beat signal itself is measured with a spectrum analyzer. Both of these frequencies are recorded. Then the signal generator is advanced to the next frequency. We collect a spectrum by stepping up and then back down through the optical transition and carefully search for drifts in the atomic beam density. A scan across the spectrum in both directions takes between four and six minutes, depending on the frequency width of the scan. We collect 15 to 20 spectra for each transition.
We separately measure and record the spectrum of each of the hyperfine components 6s, F → 8p 1/2 , F , where F = 3, 4 (F = 3, 4) is the total angular momentum of the ground 6s (excited 8p 1/2 ) state. We show a single spectrum of the 6s, F = 3 → 8p 1/2 , F = 3 line, as a representative sample, in Fig. 3(a). The spectrum shows the fluorescence signal (lock-in amplifier output) versus the measured beat frequency between the fundamental (778 nm) laser and the nearest comb tooth of the frequency comb laser. The spectra, each consisting of the data as the laser frequency ramps up and back down again, are fit to a Voigt profile using a least-squares fitting algorithm. The fitting parameters include the amplitude, Gaussian and Lorentzian width, center frequency of the peak, and a sloping baseline. The gently (< 1% change) sloping baseline is produced by scattered light, and is present even in the absence of the atomic beam.
The linewidth of this peak is primarily due to the divergence of the atomic beam. Based on the geometry of our collimator and previous measurements with these instruments [45], we estimate an atomic beam divergence of ∼40 mrad. This is in excellent agreement with the measured linewidth of ∼26-27 MHz of the 6s → 8p 1/2 fluorescence peak. (The width of the fluorescence peak is twice that of the spectrum shown in Fig. 3 since the abscissa of this plot, the beat frequency ν beat , is derived from the fundamental laser frequency near 778 nm, while the 8p 1/2 level is excited by the second harmonic of the laser near 389 nm.) The fits to the data, which indicate that the Lorentzian portion of the linewidth is small, typically ∼0.6(1) MHz, are consistent with expected contributions from lifetime broadening, power broadening, transit time effects, and collisional effects. The natural lifetime for the 8p 1/2 (8p 3/2 ) level is 376 ns (320 ns) [46], resulting in a transition linewidth of 0.42 MHz (0.50 MHz). This homogeneous width could be increased slightly by power broadening, but still less than a MHz, as the laser intensity at the center of the beam is a factor of four below the saturation intensity for the 6s → 8p 1/2 transition (I sat ∼ 17 mW/cm 2 , as estimated using data from Ref. [46]) for the highest power measurements. (For the 6s → 8p 3/2 line, the greatest laser intensity used was comparable to the saturation intensity (I sat ∼ 3.7 mW/cm 2 , using data from Ref. [46]), so power broadening is somewhat larger, but still less than 1 MHz.) Transit time broadening, estimated using an atomic velocity of 300 m/s and a beam diameter of 5 mm [47], is expected to contribute less than 50 kHz, and collisional broadening less than 1 kHz [47].
The residuals (the difference between the data and the least-squares-fit result) are shown in Fig. 3(b). The rms value of the residual is ∼0.5% of the peak signal level, and is primarily due to thermal noise in the feedback resistor (50 MΩ) of the low noise transimpedance amplifier. Photon shot noise and amplifier noise are smaller than the thermal noise by a factor of greater than ten.
To adjust for the possible effect of Zeeman shifts, we intentionally apply a magnetic field of one Gauss. No magnetic field shifts or broadening were observed within the resolution of our measurement. Similarly, we studied the effect of a.c. Stark shifts by varying the laser power. A weak dependence (200-650 Hz/µW, varying among the lines) on the laser power was observed. We corrected for this shift by fitting the measured linecenters vs. power to a linear function and extrapolating to zero laser power. This power dependence is illustrated in Fig. 4 for each of the hyperfine components of the 6s → 8p 1/2 transition. (We have not calculated these lineshifts, nor those of the 8p 3/2 lines discussed in Sec. III, as they are complex and not critical for the zero-laser-intensity determination.) Using the zero-power extrapolated peak centers and Eq. ( 1), we calculate the frequency difference between the hyperfine lines when driven from either of the F = 3 or 4 ground state hyperfine levels, and the hyperfine coupling constant (A) for the 8p 1/2 level. These values of A are 42.936(9) and 42.926 (15) MHz when driven from the F = 4 and F = 3 hyperfine level of the ground state, respectively. A weighted average (each data point weighted as 1/σ 2 , where σ is the uncertainty) of these two coupling constants is computed and presented in Table I, along with previous experimental and theoretical values for this hyperfine coupling constant. These data are also shown graphically in Fig. 5 for visual comparison. This measurement of the hyperfine coupling constant (A) for the 8p 1/2 state is in excellent agreement with previous measurements. The 8 kHz uncertainty of our measurement is more than a factor of ten lower than the uncertainty of the previous measurement of Ref. [40]. This measurement agrees well with both theoretical values of Grunefeld et al., 2019 [34]. Results of three other theoretical calculations of A for 8p 1/2 are also presented in Table I The sources of uncertainty for these measurements are recorded in Table II. The uncertainty in the fit includes the statistical uncertainty in the repeated measurements of the center frequency, along with the uncertainty in extrapolating to zero laser power. The uncertainty in the frequency comb laser (FCL) is the uncertainty derived from the fractional uncertainty (10 -12 ) of the GPS conditioned time base used to stabilize the comb and the comb tooth number. The uncertainty in the shift due to the Zeeman effect is determined by the resolution of our measurement and the degree to which we cancel out magnetic fields. We observe no shifts in center frequency for any of the lines when applying a one Gauss field and we zero the magnetic field to within less than 10 mGauss. With this, we estimate that the uncertainty is less than the resolution of the measurement times 1/100. The uncertainty due to beam misalignment is the residual Doppler error due to imperfect retro reflection. This uncertainty is only included in the absolute frequency determinations, discussed in Section IV. The energy levels of the 8p 3/2 states are shown in Fig. 6. The energies of the hyperfine components of the 8p 3/2 state are shifted from the center-of-gravity energy E cg of the 8p state by the magnetic dipole (A), electric quadrupole (B), and magnetic octupole (C) interactions. The coupling constants for these interactions are represented by A, B, and C, as indicated. The energy shifts due to each of these terms are
as given in Refs. [44,53]. The spacing between lines then yield the hyperfine coupling constants A, B, and C, using where ∆ν ij = (E F =i -E F =j )/h is the frequency spacing between the hyperfine peaks. (∆ν ij defined here is distinct from ν F,F defined in Fig. 1, which is the optical frequency of a wave resonant with the 6s, F → 8p 1/2 , F transition.).
The experimental procedure for measuring the hyperfine splitting on the 8p 3/2 line is similar to that for the 8p 1/2 , but varies in a few ways. The wavelength of the transition is similar, 387.7 nm, so the same laser source and LBO crystal are used. The hyperfine splitting on the 8p 3/2 line is less than that of the 8p 1/2 line, allowing us to scan across all of the allowable transitions in a single sweep. The required scan length is larger by 1.5 to 2 times to include each peak in a single scan. Unfortunately, this smaller spectral spacing also means that the individual lines are not completely resolvable (see Figs. 7 and8). The final difference between the two procedures results from the larger transition strength of the 8p 3/2 line in comparison to that of the 8p 1/2 line, by a factor of 5-10. This allows us to lower the power in the second harmonic beam driving the 6s → 8p 3/2 transition without sacrificing the signal-to-noise ratio.
Examples of 8p 3/2 hyperfine spectra driven from the F = 4 and F = 3 hyperfine ground states are illustrated in Figs. 7 and8. We fit each individual 8p 3/2 spectrum with the sum of three Voigt profiles. The fitting parameters include the center frequency for each peak, a single Gaussian and Lorentzian width (all peaks were constrained to the same widths in a single spectrum), amplitudes for each peak, and a sloping baseline. The peaks were also allowed to have a slight asymmetry to account for imperfect alignment through the atomic beam. This asymmetry is of the form Voigt×(1 + α(ν -ν 0 )), where α is a small parameter (α ≈ -5.8(3) × 10 -3 MHz -1 ), and ν 0 is the peak frequency. The relative peak heights of the various hyperfine components in spectra such as Figs. 7 or 8 are within 10% of calculated values based on Wigner 6j symbols at the lowest optical intensities [54].
We adjusted for the effect of a.c. Stark and Zeeman shifts on the 8p 3/2 lines. Within the resolution of the measurement, no observable shifts were observed when applying a 1 Gauss magnetic field. When varying the power from 50 µW to 170 µW, a slight shift in hyperfine peak spacing ∆ν ij was observed. The power dependence of each of the hyperfine lines varies in the range of 200-700 Hz/µW. We fit these individual splittings versus power and extrapolate to zero laser intensity. These values are reported in Table III. The uncertainties of these values are determined from the distribution of the fitted center frequencies and their extrapolation back to zero laser intensity. Two values of ∆ν 43 are reported here, one for excitation out of the F = 3 component of the ground state, the other for excitation out of the F = 4 ground state. These values differ by somewhat more than their combined uncertainties. We use the weighted average (weight = 1/σ 2 ) of these two values, with the uncertainty expanded by χ 2 red = 2 [55], where χ 2 red is the reduced chi squared of the data.
We take the fitted zero power hyperfine splittings from Table III and calculate hyperfine coupling constants A, B, and C using Eq. ( 4). We derive the uncertainty in A, B, and C using standard error propagation techniques, such as described in Ref. [55], using Eq. ( 4) and the fitted uncertainties of ∆ν ij . A summary of these results and those of previous measurements are reported in Table IV. Our result for A and B for the 8p 3/2 level is compared graphically with previous experimental and theoretical results in Fig. 9. Our value for A differs by about two standard deviations from those of Refs. [56] and [57].
The difference from Refs. [58][59][60] is larger. The uncertainty of our measurement is comparable to, but slightly larger than that of Refs. [56] and [57]. Prior values of the electric quadrupole coupling constant B from previous measurements have not been consistent, with uncertainties comparable to the values themselves. Our result of B = -0.005 (40) MHz, while less than its uncertainty, is within the distribution of prior experimental and the theoretical results. Remarkably, our measurements lead to a value of the magnetic octupole constant C for this level.
There have been no previous reports of this constant for the 8p 3/2 state.
Along with high precision determinations of the hyperfine coupling constants for the 8p 1/2 and 8p 3/2 states, our measurements also provide high precision absolute frequency determinations for these states. We use the fitted peak centers from each of the hyperfine lines in Eqs. ( 1) and (3) to calculate the center of gravity frequency for both the 6s → 8p 1/2 and 6s → 8p 3/2 transitions, respectively. We estimate the maximum angular deviation between the forward and retroreflected beam to be 0.21 mrad based on the geometry of our test setup. Due to this possible beam misalignment, we estimate the uncertainty in the linecenter frequency is 150 kHz, which is added in quadrature with the respective line center uncertainty due to the fitting procedure, the frequency comb uncertainty, and the uncertainty in the Zeeman shift. We report these absolute frequency measurements in Table V. Our values for the line center frequencies agree well with the measurements by Kleiman [62] for both lines. The uncertainty of our measurement, however, is significantly smaller. The value for the 6s → 8p 1/2 linecenter frequency by Liu and [52] is in poor agreement with (15) This work
our measurement and that of Kleiman.
To validate our absolute frequency measurements, we measure the wavelength of the 778 nm ECDL, when driving the 6s → 8p 1/2 transitions, with two separate commercial wavemeters. Both of these commercial wavemeters are based on stabilized single-frequency helium neon interferometry and claim an accuracy of 60 MHz. For instance, when driving the 6s, F = 3 → 8p 1/2 , F = 3 transition, the two wavemeters agree to within 17 MHz of one another. As a further consistency check, we have stabilized another ECDL at λ = 852 nm to the 6s, F = 4 → 6p 3/2 , F = 5 transition using saturated absorption spectroscopy and measured its frequency using one of the wavemeters above. This value agrees to within 12 MHz of the precise value specified in Ref. [63]. The combination of these frequency measurements give us confidence in the correct determination of the comb tooth number, N , and the laser frequency determined using Eq. (2).
In this work, we have reported a new, high precision measurement of the hyperfine coupling constant A = 42.933 (8) MHz for the 8p 1/2 state in atomic cesium-133. This value is in support of theoretical efforts towards high precision values of electronic wavefunctions, which are a critical component in characterizing the parity nonconserving weak interaction. We also report values for the hyperfine coupling constants A, B, and C for the 8p 3/2 state as well as absolute frequency measurements of both the 8p 1/2 and 8p 3/2 states.
The primary limitation to the uncertainty in these measurements originates from the divergence of the atomic beam, resulting in Doppler broadening of the spectral peaks. Therefore, improvements of the results discussed here could be achieved by increasing the collimation of the atomic beam. This would require a new, redesigned nozzle and collimator. Since collimation tends to reduce the beam density, and therefore the signal size,