Singly ionized ytterbium ion Yb+ has attracted significant attention in recent years due to its potential applications in various fields. It has been extensively studied for developing ultra-precise atomic clocks [1–3], investigating the effects of parity nonconservation [4–6], searching for bosonic dark matter [7], exploring new forces through isotope shift studies [8,9], probing variations of the fine structure constant [10–12], examining violations of local position invariance [13], and investigating local Lorentz symmetry [14–16]. Yb+ also finds applications in quantum computing [17,18], quantum simulation [19], quantum phase transition studies [20,21], and quantum sensors [22]. Precise knowledge of Yb+ properties, especially the transition matrix elements between two atomic states, is indispensable for all these applications.

For instance, the dominant contribution to the differential scalar dipole polarizability of the clock transition 6sS1/22-5dD3/22 stems from the transitions 6sS1/22-6pP21/2,3/2 and 5dD3/22-6pP21/2,3/2 [23], which implies that the accuracy of these transition matrix elements directly determines the uncertainties of the blackbody radiation shift of the clock transition. Additionally, these elements are essential for evaluating the parity-violating amplitude of the 6sS1/22-5dD3/22 transition [4,6]. All of the above mentioned applications are linked to the 6pP1/22 state, which has a cooling termination pathway to the 5dD3/22 state. Meanwhile, having a good understanding of the transition matrix element of 6pP1/22-5dD3/22 is crucial for accurately determining the theoretical limit on state detection of an Yb+ qubit [24].

However, accurate calculation of these transition matrix elements has proven to be challenging due to the abundance of low-lying states within the one-hole-two-particle configurations like 4f135d6s and the strong mixing between different electronic configurations [6,25]. Therefore, it is of utmost importance to perform high-precision measurements of the transition matrix elements. Unlike atomic transition frequencies, measuring transition matrix elements is still notoriously difficult, especially when it comes to achieving uncertainties at the level of 10−3–10−4. However, by combining theoretical information on atomic structure with measurements of “magic-zero” wavelengths in Rb atoms conducted by Herold et al. [26] and with measurements of Stark shifts caused by electronic dipole coupling to Rydberg states in Ba+ conducted by Woods et al. [27], researchers are able to determine the transition matrix elements for the 5sS1/22-6pP21/2,3/2 line in Rb and the 6sS1/22-6pP21/2,3/2 line in Ba+ with high precision, respectively. Nevertheless, due to the complex energy level of Yb+ and Yb, applying these methods to the ytterbium system may not yield accurate results. Hettrich et al. [28] measured the transition matrix element of 4sS1/22-4pP1/22 in Ca+ by comparing dispersive and absorptive light-ion interaction measurements. This methodology could potentially be employed to determine the 6sS1/22-6pP1/22 transition matrix element in Yb+ in a similar fashion. Another approach to obtaining the transition matrix elements involves combining branching fractions with lifetimes [29,30], but the uncertainty of these measurements depends on the precision of the lifetimes and branching fractions.

In this work, we introduce a novel experimental method for accurately measuring the dipole matrix element for the 6pP1/22-5dD3/22 transition in Yb+. Our approach involves using a shelving technique and the excitation pulses with increasing duration with a bin size of 100 ns to determine the shelving time. The associated shelving probabilities are determined by counting the scattered photons from the excited 6pP1/22 state to the ground state 6sS1/22. To ensure accuracy, our protocol includes a real-time background photon correction method that eliminates any background lights and their fluctuations. To minimize detection errors, we employ a highly synchronized sequence for laser control, an intensity-alternating sequence for atomic excitation and photon counting, as well as a power stabilization scheme. The method of counting scattered photons from 6pP1/22 to 6sS1/22 is not sensitive to fluctuations in laser-light polarization, intensity, and magnetic field [31]. Compared to the approach of Hettrich et al. [28], which requires a clock laser for quantum state readout, our method is simpler and relies less on a re-pumping laser [31].

We performed measurements on a single Yb174+ ion stored in a miniature four-blade linear Paul trap to determine the electric dipole transition elements. The ablation technique [32], combined with ionization lasers at 399 and 369 nm, was used for ion loading. We specifically selected the Yb174+ ion because it provides easier measurement due to higher fluorescence [33]. The sketch of the experimental setup and relevant transitions are shown in Fig. 1. The trap used to confine the ion is a miniature four-blade linear Paul trap, which is made of 0.3 mm diamond rectangular slices, and the electrodes are constructed by precision laser cutting and gold plating. With this operation, the trap has a more symmetrical structure that can reduce the effects of heating and micromotion. The two blade electrodes were used as radio frequency (RF) electrodes, and the other two blade electrodes were used as opposite RF phase. Two additional electrodes in orthogonal directions were used as compensation electrodes. The center-to-center spacing of adjacent blades is 0.8 mm, and the tip-to-tip spacing between the endcaps is about 8.0 mm, as displayed in Ref. [34].

When the ion was confined in the trap, excessive micromotion of ions could be effectively compensated for by adjusting the voltage at the endcap and the compensation electrodes. Compensation was monitored by position change in a charge-coupled device (CCD) and by using RF photon correlation methods. In addition, in order to reduce the de-excitation caused by the collision between the ions and background gases and ensure a stable measurement environment, the ion trap was installed in an ultra-high-vacuum chamber with a background pressure less than 1.0×10−8Pa. To avoid low-frequency magnetic field noise, the vacuum chamber was surrounded by two layers of μ-metal shielding and a low-noise current source was used to supply stable currents on the three orthogonal magnetic coils. A magnetic field of about 0.4 mT along the trap axis and linear polarization of the laser lights were used to avoid any optical pumping and coherent population trapping. All laser beams were guided through optical fibers and reached the ion, and polarization matching was employed by using quarter and half waveplates in front of the fiber couplers. In order to reduce the influence of beam quality, the three-dimensional precise adjustment frames at each laser beam were used to adjust the laser’s position. Meanwhile, the mechanical irises were used to shape the lights and keep the good beam quality. One 369 nm light overlapping with 935 nm light was used for the optical pumping process with different powers and times. Another 369 nm light with power of 250 μW and beam waist of 60 μm was used for photon detection and Doppler cooling. Both 369 nm lights were power stabilized. The sketch of the experimental setup is shown in Fig. 1(a).

The Yb+ ion was Doppler cooled primarily using the 6sS1/22-6pP1/22 dipole transition at 369 nm because of the short lifetime of the excited state [35]. Due to the fact that the decay channel from 6sP1/22 to 5dD3/22 [24] interrupts the cooling cycle, a re-pumping laser light at 935 nm is used to quickly deplete the 5dD3/22 clock state via the 6sD3[3/2]1/2 state, ultimately returning the ion back to the ground state. This process helps maintain the photon counting rate at an acceptably high level. Additionally, a laser with the wavelength of 760 nm is used to deplete the ion from 6s2F27/2 state via the 6sD1[3/2]3/2 state back to the ground state due to collision with background gases. The relevant transitions related to the 6pP1/22-5dD3/22 and 6sS1/22-6pP1/22 dipole matrix elements of Yb+174 are shown in Fig. 1(b).

In order to achieve multi-period continuous measurement, the 935 nm re-pumping laser’s frequency was referenced to a HighFinesse wavelength meter WS8-2. The 369 nm light was frequency referenced to an ultra-low-expansion (ULE) standard Fabry–Pérot cavity according to the Pound–Drever–Hall scheme, making the linewidth about 300 kHz and drifts about 1.06 Hz/s. The signal detection was realized by the 369 nm fluorescence generated by the spontaneous decay of the ion from 6pP1/22 to 6sS1/22 and was monitored by a photomultiplier tube (PMT) and CCD. The scattered background light was filtered from the fluorescence signal by two narrow-band-pass filters and a pinhole with diameter of 40 μm. The photon collection efficiency was 3.1×10−3 after carefully optimizing the position and focus of imaging and lasers. A computer was used to record all experimental data, and control the photon-counter, all acousto-optic modulators (AOMs), and shutters simultaneously via a field programmable gate array (FPGA) at the accuracy of nanosecond level.

The electric dipole spontaneous emission decay rate Avw can be expressed according to Refs. [5,36–38]: Avw=2.02613×1018λ3(2jv+1)|⟨Ψv∥E1∥Ψw⟩|2,(1)where λ represents the transition wavelength in Å and |⟨Ψv∥E1∥Ψw⟩| is the spatial part of the reduced matrix element of the electric dipole operator.

When the ion is shelved in the 6pP1/22 state by 369 nm light, the fluorescence signal displays an exponential decay. This is evidenced by the population trapping in the 5dD3/22 state, which occurs due to the presence of two channel transitions. The decay probability of a fluorescence signal can be described by the function: P=exp(−bt).(2)Here b=PP1/2γr is the decay parameter with PP1/2 being the fractional population of the 6pP1/22 state, γ is the linewidth of the 6pP1/22 state defined as γ=1/τ, where τ represents 6pP1/22 state lifetime, and r is the branching fraction into the 5dD3/22 state. The population PP1/2, which depends on the power p of the 369 nm light with a constant beam waist and resonant frequency, can be calculated using the following equation [24]: PP1/2=p/psat2(1+p/psat),(3)where psat is the laser’s saturated absorption power of the transition. With this, the decay parameter b can thus be expressed as b=pγr2(p+psat).(4)

Obviously, the electric dipole transition matrix element |⟨Ψv∥E1∥Ψw⟩| can be obtained from the spontaneous decay rate Avw. Avw=γr, and the saturation intensity psat can be extracted from a series of decay parameters b by using Eq. (4), while each b is obtained by Eq. (2) from the measurements of decay probability P at different intensities. Here, the decay rate Avw is APD for the 6pP1/22-5dD3/22 transition.

A simplified diagram illustrating the time sequence and energy levels involved in determining the dipole transition matrix element between the 6pP1/22 and 5dD3/22 states is shown in Fig. 2. This sequence consists of six major steps that are repeated for each measurement. Step 1 involves Doppler cooling of the ion with a duration of 5 ms. Additionally, the 935 nm quenching laser is applied for extra 0.1 ms to excite the ion from the 5dD3/22 state to the 6sD3[3/2]1/2 state. This ensures that the ion is in the ground state. In step 2, a series of laser pulses with increasing duration is used to shelve the ion back to the 5dD3/22 state. This is necessary due to the branching fraction of the ion. Step 3 involves detecting the rate of shelving, ranging from zero to one. If the shelving is unsuccessful, scattered photons can be detected from the 6pP1/22 to 6sS1/22 transition. To induce incoherent shelving to the 5dD3/22 state, two laser pulses with duration of 15 μs and wavelength of 369 nm are applied. These pulses are resonant with the 6sS1/22-6pP1/22 transition. One of the laser pulses is saturated (with a power of 250 μW), while the other is half of the saturated power. The number of photons during the ion is shelved to the 5dD3/22 state with those two pulses denoted as N1 and N2. Here, two pulses at different intensities are assigned to minimize the counting rate error and other errors associated with the PMT dead time [31]. As we have demonstrated, with those two pulses, the ion could be totally shelved to the 5dD3/22 state. In order to subtract the background timely, both pulses are applied again, and the detected photons are denoted as N3 and N4. Because the ion stays in the 5dD3/22 state, no photons are scattered from 6pP1/22 to 6sS1/22, and the N3 and N4 just represent the background information. After background subtraction of N3+N4, the number of detected photons scattered from 6pP1/22 to 6sS1/22 is Ns=(N1+N2)−(N3+N4). To normalize the detected photon counts from 6pP1/22 to 6sS1/22 timely, a similar measurement is performed in step 5 without the shelving process as step 2. The operations performed in step 4 are identical to those in step 1, ensuring that the ion is in its ground state. The number of detected photon counts in step 5 is given by Nb=(N5+N6)−(N7+N8). Here, N5 and N6 represent photons scattered from 6pP1/22 to 6sS1/22 within background, and N7 and N8 only represent background photons. The decay probability into the 5dD3/22 state is then determined by 1−Ns/Nb, where the latter term is the normalized decay probability P=Ns/Nb from 6pP1/22 to 6sS1/22.

To maintain consistent measurement conditions for Ns and Nb and to compensate for any ion heating in the trap, we repeat the arrangement of the Doppler cooling schedule after detecting N4. In the final step (step 6), the cooling and pumping lasers are activated for 5 ms, and the photomultiplier tube (PMT) count, referred to as N9, determines whether the ions remain in the cooling cycle (referred to as “bright”) or transition to the 6s2F27/2 state (referred to as “dark”) due to collisions with background gases. A threshold value was set by σ=Sdark×Sbright [39] to discriminate those two states, and the data in the cycles with N9 below threshold were not recorded. A valid interrogation cycle concludes when the fluorescence signal reappears, signaling the start of the next cooling period.

In each cycle of the shelving process, the duration of the shelving pulse, denoted as ti, was varied while keeping the laser power constant at a certain value, denoted as p. A total of 60,000 clock cycles were performed, with ti ranging from zero to N times 0.1 μs, increasing in steps of 100 ns. The maximum range of ti values was chosen to ensure that the final 5dD3/22 population rate approached one, allowing for the determination of the decay parameter b. The experimental data points and the corresponding fitting using Eq. (2) are plotted in Fig. 3(a). Note that performing the measurement at the resonant frequency of the 6pP1/22-6sS1/22 transition is challenging due to ion heating causing photon counts to decrease or even ion loss. To deal with this issue, we referenced the 369 nm laser’s frequency to a ULE cavity with an accuracy of approximately 300 kHz. Additionally, to reliably determine the spontaneous decay rate at 6pP1/22-6sS1/22 resonant frequency, especially over long periods of measurement time, we select a series of red frequency detunings from −54 to −6MHz, and determine the decay rate APD at resonant frequency by extrapolation.

In our experiments, we varied the power of the 369 nm laser from 0 to 550 μW, with a beam waist of approximately 60 μm. The resulting experimental data were then fitted to Eqs. (2) and (4) using two fitting parameters psat and APD=γr. Figures 3(a) and 3(b) give an example of the experimental data and its fitting of decay parameter b and decay rate APD at −6MHz frequency detuning from resonance, where the residuals between the data points and the curve are also given. Similarly, such operations are repeated at different frequency detunings, and the obtained decay rates are plotted in Fig. 3(c). The data points have been fitted with an exponential curve (shown in red), and the decay rate at the resonance frequency is determined to be 0.62796(81) μs−1.

It should be emphasized that the ascertained decay rate APD was intricately linked to the intensity of the shelving laser, and the implementation of laser power stabilization was carried out in this regard. In order to mitigate any potential linear fluctuations during the measurements, data collection was performed in a randomized manner. This randomness is reflected in the fact that the laser power and pulse time ti do not change continuously from small to large or from large to small.

In accordance with our adopted scheme, the dispersed photons from the 6pP1/22 state to the 6sS1/22 state remain impervious to fluctuations in both light intensity and frequency. This immunity has been previously established, not only with regard to the strength and direction of the magnetic field [31], but also through experimental examination as in our prior work [40]. The error resulting from collisions between individual ions and the ambient gases is negligible, given the maximum collision rate of 2.45(94)×10−4s−1, which is orders of magnitude lower than the decay rate APD. Consequently, it has no impact on the decay parameter b.

Furthermore, we disregard the decay from the metastable 5dD3/22 state in this context, as its lifetime of τD3/2=54.83±0.18ms [40] is orders of magnitude greater than both the shelving and measurement times. The ion in 5dD3/22 state only has one transition channel, 5dD3/22-6sS1/22. The decay rate ADS from 5dD3/22 state to 6sS1/22 state is ADS=1/τD3/2=1.824(6)×10−5μs−1. Compared to the decay rate APD=0.62796(81)μs−1 from 6pP1/22 state to 5dD3/22 state, the lifetime of 5dD3/22 nearly has no influence on APD.

For the residual birefringence in the detection optics, we checked this effect by adopting the same way as Ref. [31]. We varied the direction of the applied magnetic field and observed no variation in the decay rate APD within the statistical uncertainty of the measurement. Considering magnetic field induced states mixing and the mixing states modified decay rate, we changed the magnetic field to a relatively large value, but we did not find the difference of the measured decay rate.

Ion trap heating and ion’s motion may be another factor affecting the detected photons, further affecting decay information. The micromotion of ions could be effectively compensated for by adjusting the voltage at the endcap and the compensation electrodes. The compensation was monitored by a position change in a charge-coupled device (CCD) and by using RF photon correlation methods. After good micromotion compensation, the detected photons at high and low RF powers have no significant differences. The heating due to the residual micromotion can be reflected in the rise time for the fluorescence intensity to return to its original level after a prolonged blocking of the cooling laser beam. No delay was observed during the period of ≥1s blocking time, which was three orders of magnitude longer than the experimental pulse duration we applied. That means the influence of heating and ion’s motion is very small and can be ignored.

The error caused by the Poissonian noise of the PMT is negligible, at the level of 10−6 [41]. The errors resulting from the PMT dead time and spontaneous decay during the detection periods of Ns and Nb have the same undercounting rate and decay rate within the same time interval, effectively canceling each other out and leaving the photon detection unaffected. Without a pumping process with the 369 nm laser, using our photon detection scheme Ns/Nb=(N1+N2−N3−N4)/(N5+N6−N7−N8) we determined the photon detection probability to be 1.000(1). The results demonstrated that this scheme can eliminate or at least largely reduce the effect of PMT dead time, which leads to undercounting of both Ns and Nb with the same error rate.

Following the statistical and systematic analysis, we have determined the final result for the Yb174+6pP1/22-5dD3/22 decay rate at resonance frequency to be APD=0.62798(82)μs−1.

Using Eq. (1) in conjunction with the measured decay rate APD and its corresponding transition wavelength of λ=24384.2Å [42], we are able to determine the matrix element for the 6pP1/22-5dD3/22 transition to be 2.9979(20) ea0 with an uncertainty of 0.07%, representing an order of magnitude improvement in precision of 2.97(4) ea0 [6,24]. In addition, by combining this result with the known lifetime of the 6pP1/22 state of 8.12(2) ns [35], we are able to extract the transition matrix element for 6sS1/22-6pP1/22, which is found to be 2.4703(31) ea0. It is worth noting that our method achieves a comparable level of accuracy to the 4sS1/22-4pP1/22 matrix element measurement of 2.8928(43) ea0 in Ca40+ [28], while also offering a smaller uncertainty compared to the measurement of the electric quadrupole reduced matrix element of 12.5(4) ea02 for the 6sS1/22-5dD5/22 transition [30] in Yb+171.

The relationship between γ and the lifetime τ is given by γ=1/τ. By using the lifetime of the 6pP1/22 state mentioned above [35], we can determine the branching fraction r for the 6pP1/22-5dD3/22 transition to be 0.00510(2). This represents an order of magnitude improvement in measurement precision compared to the previous study by Olmschenk et al. [24], which used ultrafast pulses and a time correlated single-photon counting technique. Figure 4 illustrates a comparison of the experimental and theoretical branching fractions [4–6,24,25,43–45]. Our result shows good agreement with the measurement of 0.00501(15) by Olmschenk et al. [24], and it is also in close proximity to all the theoretical calculations using different methods. However, there are some discrepancies in the dipole matrix elements and branching fractions between our value and the theoretical results, which may arise from the fact that all the theoretical calculations do not take into account the core-excited 4f136s2F2 configuration in Yb+. These discrepancies highlight the need for the development of high-accuracy many-body theoretical methods.

In summary, we present a simple experimental scheme to accurately determine the transition matrix element from the 6pP1/22 state to the 5dD3/22 state in a single Yb+174 ion. The dipole matrix element is found to be 2.9979(20) ea0, which is an order of magnitude improvement over the existing results [6,24]. By combining our result with the lifetime of the 6pP1/22 state, we are able to extract the branching fraction and the matrix element for the 6sS1/22-6pP1/22 transition. These transition matrix elements are crucial for accurately evaluating the black-body radiation shift of the 6sS1/22-5dD3/22 clock transition and the amplitude of the related parity-violating transition. Moreover, these highly accurate transition matrix elements can be used as probes to test quantum electrodynamic effects [46]. Our method can also be readily applied to precisely determine the matrix elements for the npP1/22-(n−1)dD3/22 transitions in Ca+(n=4), Sr+(n=5), Ba+(n=6), and Ra+(n=7) at a precision level of 10−3–10−4. Together with the available branching fractions of these ions [31,47–49], one can also obtain the lifetimes of the npP1/22 state and the nsS1/22-npP1/22 transition matrix elements with a precision of 10−3–10−4. Finally, our method can serve as a guide for active atomic clock research [50].

Acknowledgment. We thank Z.-C. Yan, T.-Y. Shi, Y.-J. Cheng, J.-G. Li, J. Jiang, and M. S. Safronova for fruitful discussions.

Author Contributions.H. Shao, Y.-B. Tang, and H.-L. Yue carried out the device design and fabrication and drafted the manuscript. H. Shao, H.-L. Yue, Z.-X. Ma, and Y. Huang carried out the measurements. H. Guan and K.-L. Gao conceived the project and coordinated the research. Y.-B. Tang, F.-F. Wu, and L.-Y. Tang supported in theory and helped draft and edit the manuscript. All authors analyzed the measurement results and gave their final approval for publication.