High precision measurements of the hyperfine structure provides stringent tests for state-of-the-art atomic structure calculations. These calculations play a crucial role in the interpretation of parity nonconservation (PNC) experiments which provide important tests of the standard model at low energy [1]. The accuracy of calculated PNC matrix elements can be assessed by comparing measured hyperfine structure constants with calculated values [2]. In addition, the hyperfine structure provides insight into the nuclear structure of atoms [3].

In this paper we present precision measurements of the hyperfine intervals of the 5D_3/2 manifold of ¹³⁷Ba⁺, which has a nuclear spin I = 3/2. By combining high precision radio frequency (rf) spectroscopy with shelving techniques [4, 5] on singly trapped ions, we measure the hyperfine intervals of the 5D_3/2 manifold to an accuracy below a few Hz. Together with theoretical calculations, this permits a 30-fold reduction in the uncertainty of the magnetic dipole (A) and electric quadrupole (B) hyperfine constants and the first observation of the magnetic octupole moment in ¹³⁷Ba⁺. We determine the magnetic octupole moment, Ω, to an accuracy of 1% percent limited almost entirely by uncertainties in the theory.

Ba⁺ is an excellent candidate for magnetic octupole determination. Long lived metastable D states permit high precision spectroscopy measurements of the hyperfine levels, and high precision calculations are possible [6]. Using the formalism given in [7] it can be shown that the hyperfine intervals, δW_F = W_F − W_F+1, of the 5D_3/2 manifold can be written

The procedure to measure the hyperfine intervals is similar to that proposed in [4, 10] and the relevant level structure is given in Fig. 1. The ion is first Doppler cooled and then optically pumped into the state |F = 2, m_F = 2〉 of the 6S_1/2 level. To measure the hyperfine interval, δW_k, the ion is then shelved to the state |F″ = k, m_F″ = 0〉 of the 5D_3/2 level using a two photon Raman transition similar to the one used in [5]. An rf antenna is then turned on to drive F″ ↔ F″ + 1 transitions. The signal generator used for the rf antenna is synchronized with a GPS-disciplined Rb clock.

 

Fig. 1 Relevant levels of ¹³⁷Ba⁺ for the rf spectroscopy: The ion is prepared in the |F = 2, m_F = 2〉 state of the 6S_1/2 level from where it is shelved to |F″, m_F″ = 0〉 of the 5D_3/2 level with a pair of Raman beams which are red-detuned by ≈ Δ = 2π × 500GHz from the 6P_1/2 level. The rf transition is detected by shelving to the 5D_5/2 level (see text).

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To determine if the hyperfine transition occurred we use a second Raman pulse to transfer the state |F″ = k, m_F″ = 0〉 back to the |F = 2, m_F = 2〉 ground state and then optically pump on the 6S_1/2 to 6P_3/2 transition at 455nm. If the hyperfine transition occurred, the ion will remain in the |F″ = k+1, m_F″ = 0〉 state of the 5D_3/2 level, otherwise it will be shelved to the 5D_5/2 level with ≈ 88% efficiency. Subsequent driving of the 6S_1/2 to 6P_1/2 and 5D_3/2 to 6P_1/2 transitions using the Doppler cooling beams then provides a fluorescence measurement for the probability of driving the hyperfine transition: the ion being bright if the hyperfine transition took place and dark otherwise. The detection efficiency of this scheme is limited to approximately 88% by the branching ratio between the 6P_3/2 and 5D_3/2 which results in unwanted population of the 5D_3/2 level when optically pumping to the 5D_5/2 level. This does not impact on the accuracy at which we can measure the hyperfine transition probability, but only on the amount of averaging needed to achieve a particular level of accuracy.

The experiments are performed in a four-rod linear Paul trap, similar to the ones described in [5, 11]. The trap consists of four stainless steel rods of diameter 0.45mm whose centers are arranged on the vertices of a square with 2mm length of the side. A 5.3MHz rf potential with an amplitude of 170V is applied via a step-up transformer to two diagonally opposing electrodes. A small DC voltage applied to the other two electrodes ensures a splitting of the transverse trapping frequencies and rotates the principle axes of the trap with respect to the propagation direction of the cooling lasers. Axial confinement is provided by two axial needle electrodes separated by 2.4mm and held at 33V. Using this configuration, the measured trapping frequencies are (ω_x, ω_y, ω_z)/2π ≈ (1.7, 1.5, 0.5)MHz.

Doppler cooling and detection is achieved by driving the 6S_1/2 to 6P_1/2 transitions at 493nm and repumping on the 5D_3/2 to 6P_1/2 transitions at 650nm. The 493nm laser is passed through two electro-optic modulators in order to generate the sidebands required to address all possible transitions between the 6S_1/2 to 6P_1/2 levels. Similarly, to address all the 5D_3/2 to 6P_1/2 transitions, the repumping laser at 650nm is split into four, frequency shifted by acousto-optic modulators, and then recombined into a single fiber. All laser fields are linearly polarized perpendicular to a magnetic field of approximately 1G. This configuration avoids unwanted dark states in both the cooling and detection cycles. Additionally, the field ensures a well defined quantization axis for optical pumping and state preparation.

The magnetic field also gives a second order Zeeman shift of the |F″, m_F″ = 0〉 levels which must be accounted for when measuring the hyperfine intervals. We achieve this by additionally measuring transitions between |F″, m_F″ = 0〉 and |F″ + 1, m_F″ = ±1〉. Half of the difference frequency between the Δm_F″ = +1 and the Δm_F″ = −1 transitions is then given by the linear shift μ_Bg_F B/h̄ with any quadratic shifts cancelled. By measuring the Δm_F″ = 0 transition over a range of magnetic fields, each calibrated by measurements of the Δm_F″ = ±1 transitions, we can map out the full field dependence of the Δm_F″ = 0 transition and interpolate to the desired zero field result. This also yields the second order Zeeman shift coefficient for the Δm_F″ = 0 transitions providing a useful consistency check within our measurements.

An example of a hyperfine splitting measurement for a fixed value of the magnetic field is shown in Fig. 2(a) where the transition probability for |F″ = 0, m_F″ = 0〉 to |F″ = 1, m_F″ = 0〉 is plotted as a function of the external rf frequency. The rf drive power is adjusted to give an on resonant π-pulse time of approximately 10ms resulting in a resonance width (full width half maximum) of approximately 50Hz. The resonance is scanned in 2Hz frequency steps, and the probability of undergoing the hyperfine transition is determined by the average over 200 measurements. The data is fitted [12] via a χ² minimization to the usual Rabi flopping function [13] with additional offset and amplitude parameters to account for imperfect shelving. From the fits we also extract the 68% confidence limits on the fit parameters giving a determination of the center transition frequency with an accuracy of about 1Hz. At this same fixed magnetic field, a similar measurement is also performed for the Δm_F″ = ±1 transitions. For this case we use a shorter π-pulse time of 0.5ms and a larger step of 20Hz for the resonances scans as these transitions are more sensitive to the magnetic field fluctuations and therefore more prone to decoherence effects. This gives a measurement of the resonant frequency for these transitions with an accuracy of about 20Hz which corresponds to a field accuracy of ≈ 50μG.

 

Fig. 2 (a) Transition probability for B ≈ 1G as a function of the external rf frequency for the |F″ = 0, m_F″ = 0〉 ↔ |F″ = 1, m_F″ = 0〉 transition; (b) Measured hyperfine splitting as a function of the magnetic field for the |F″ = 0, m_F″ = 0〉 ↔ |F″ = 1, m_F″ = 0〉 transition.

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The time to perform the Δm_F″ = 0 and field calibration measurements at a particular value of the magnetic field is about 15min. It is therefore necessary to consider the effects of magnetic field drifts which give rise to additional errors in the field calibration. Hence we have monitored the variation of the magnetic field over the course of a day by measuring the first order Zeeman shift of the Δm_F″ = +1 transition. From these measurements we estimate the one standard deviation error in the field calibration to be approximately 450μG which dominates the 50μG error extracted from the Δm_F″ = ±1 field calibration measurements.

Measurements for the first hyperfine interval, δW₀, are shown in Fig. 2(b). For each hyperfine interval we have taken two sets of data on separate days. For each set, we measured the transition frequency at 10 values of the field: 5 with the field above zero and 5 below. The inset highlights two points taken at a similar field setting on separate days. Confidence limits from the fits to the resonance scans determine the vertical errors, which are smaller than the thickness of the lines shown in the inset. We fit the data for each hyperfine interval to a quadratic form, α_kB²− δW_k, using a χ² minimization. Since the second order Zeeman shift coefficients, α_k are determined by the three values of δW_k, we fit all three quadratic forms simultaneously. However, we may still determine a χ² statistic for each data set. Since the α_k are only weakly dependent on the δW_k, the minimization procedure is equivalent to three independent single parameter fits with the α_k fixed to values consistent with the fitted values of δW_k. In Table 1 we give the δW_k along with the reduced χ² for each fit. Errors reported here are again the 68% confidence limits extracted from the fits. For the field calibration and calculation of the α_k we have used values of g_J and g_I reported in [14] and [15] respectively.

Table 1. Measured hyperfine intervals, δW_k, for the 5D_3/2 manifold of ¹³⁷Ba⁺.

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It is worth noting that the quadratic form used for the fitting is only an approximation. As the field strength increases, higher order terms become important, which can shift the zero field value extracted from a quadratic fit. The smaller the hyperfine splitting, the larger the effect. It is for this reason we have restricted our field values to ∼ 1G. Over the range of field values we have used, we estimate that this effect could be as large as ∼ 0.5Hz for δW₀. For field values in the range 1 ∼ 2G this systematic error would be comparable to the uncertainty in our current measurement.

Additional systematic errors could arise from three possible sources: off-resonant coupling of the rf drive to other Zeeman levels, coupling of the ion trap rf field to the ion via micromotion, or AC Stark shifts due to leakage light from the 650nm lasers used in the experiment. In Table 2 we give estimates for these shifts. For the rf drive field, the largest effect comes from the field calibration measurements since the Rabi rate for these measurements is significantly larger than for the Δm_F″ = 0 measurements. In this case, the Stark shifts vary with the magnetic field and can alter the quadratic form in Fig. 2(b) causing a shift in the zero-field point. This effect is largest for the F = 0 ↔ F = 1 transition and the estimate given in the table assumes equal power available for the Δm_F″ = 0, ±1 transitions. The effects of micromotion were estimated using the results in [16]. Additionally, we shifted the ion position to induce micromotion and saw no measurable shift in the resonance. For the 650nm shelving beam, it was found that 1μW of leakage light was enough to shift the resonances by ∼ 60Hz consistent with calculations. The estimate given in Table 2 is based on this measurement and the additional attenuation used to remove the shift. For the repumping beams we attempted to measure the leakage light level by measuring the rate at which population was removed from the D_3/2 level. As there was no statistically significant population loss over 200ms, the estimate given is an upper bound.

Table 2. Estimates of the possible systematic errors.

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The hyperfine coupling constants inferred from the measured values of δW_k are given in Table 3. To account for the possible systematic errors we have added 0.5Hz to the errors given in Table 1. The correction terms were calculated from the expressions given in [7], using the matrix elements 〈5D_3/2||T₁||5D_5/2〉 = −995(10)MHz/μ_N and 〈5D_3/2||T₂||5D_5/2〉 = 339(5)MHz/b [8]. The magnetic moment 0.937365(20)μ_N and quadrupole moments 0.246(1)b were taken from [17] and [9] respectively.

Table 3. Hyperfine coupling constants.

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The final values obtained for A and B are within three standard deviations of the results reported in [18] with a 30-fold reduction in the uncertainty. In addition, our value for C has an accuracy of 0.2% and is the first observation of the magnetic octupole moment in ¹³⁷Ba⁺. Using Eq. (4) we calculate an octupole moment of

In conclusion we have performed high precision measurements of the hyperfine structure in ¹³⁷Ba⁺. Our measurements have greatly reduced the uncertainty of the currently available hyperfine structure constants and have provided an estimate of the nuclear octupole moment, Ω, accurate to 1%. We note that our inferred value of Ω has the opposite sign compared to the estimate given in [6]. This estimate is based on the shell model for which such discrepancies have also been reported for Cesium [19] and Rubidium [20]. Thus our measurements may present an interesting test case for comparison with nuclear-structure calculations. Approximately 10% of the estimated value for Ω is given by the correction factor, ζ. Future planned measurements on the 5D_5/2 manifold of the same ion will allow the dependence on this factor to be eliminated. This would also provide additional consistency checks between measurements and calculations.