Introduction

High-precision optical clocks have wide applications in fields such as the new definition of time units1, the test of variations of the fine-structure constant2,3, the constrain of a conjectured fifth force4,5,6,7 and other new physics phenomena8,9. Currently, the singly charged Al+ clock, with an instability of \({\sigma }_{\tau }\approx 1.2\times 1{0}^{-15}/\sqrt{\tau }\), hold a record accuracy of δν/ν ≈ 9 × 10−1910. Here, τ is the averaging time, and ν is the transition frequency. However, due to limited quality factors and systematic uncertainties9,11,12,13, further improvement of the clock performance is challenging. To overcome these difficulties, clocks based on a nuclear transition in the 229Th isomer14 and electronic transitions in highly charged ions (HCIs)15,16,17,18,19,20,21 are put forward. As these systems bear smaller polarizabilities, they are less sensitive to the perturbations caused by black-body radiations and external trapping fields15. While the nuclear clock transition in radioactive 229Th, with tremendous recent progress22,23,24,25,26, has been probed to kHz precision26, a HCI clock based on Ar13+ with a moderate \({\sigma }_{\tau }\approx 3\times 1{0}^{-14}/\sqrt{\tau }\) has already been demonstrated via quantum logic spectroscopy16.

The idea of HCI clocks was first based on hyperfine transitions in hydrogenlike ions27,28. However, ultrastable HCI optical clock transitions usually arise from various level crossings29, namely, between the 4f − 5s29,30,31, 4f − 5p32,33, and 5f − 6p34,35 orbitals. Though only particular ions close to the crossing points would provide optical clock transitions, these transitions are found to be highly sensitive to the variation of the fine-structure constant29, thus attracted significant interest for the search of new physics15. Nevertheless, the proximity of the orbitals still results in significant electric and magnetic dipole polarizabilities36, which could once more introduce considerable systematic uncertainties under external perturbations.

In this work, we discover a large family of HCI clock candidates that contains more than a hundred ultrastable clocks with large intrinsic quality factors that allow for single-ion clocks with instabilities \({\sigma }_{\tau }\approx 1{0}^{-17}/\sqrt{\tau }\). Moreover, as a consequence of large energy gaps with other states, the clock states bear differential electric and magnetic dipole polarizabilities as well as the electric quadrupole moments in the range of 10−12 Hz m2/V2, 103 Hz/T2 and \(1{0}^{-2}\,e{a}_{0}^{2}\) (a0 the Bohr radius), respectively, which are all several orders of magnitude smaller than those in state-of-the-art clocks, enabling the development of ultra-accurate clocks with fractional frequency uncertainty δν/ν < 10−20. In the following, we will first present the properties of these HCIs with focus being on their advantages in constructing ultraprecise clocks, and then discuss consequent opportunities in searching for fine-structure-constant variation, with an outlook in constraining a hypothetical fifth force and in testing of nuclear theories.

Results and Discussion

Level structures

As shown in Fig. 1, the clock transitions arise from fine-structure splittings in ions with one of the open-shell nd4, nd5 and nd6 ground-state configurations. These configurations have tens of terms with total angular momenta ranging from J = 0 and 1/2 to J = 6 and 13/2. For light ions, the energy levels order linearly according to the J values, and thus the lifetimes of the excited states are dominated by M1 transitions to J ± 1 states37. However, for heavy ions which are also highly charged, the strong relativistic influence leads to a reordering of the terms and renders it possible to obtain ultrastable clocks based on high-multipolarity E2/M3/E4 forbidden transitions.

Fig. 1: Clock states in uranium ions.
figure 1

(a) U50+, (b) U52+ and (c) U51+ ions. Blue lines depict the clock transitions as well as the main decay channels of the clock states (shown in red). Similar level structures exist in [Ar]3dm and [Xe]4f145dm (m = 4, 5, 6) ions, but scale to higher and lower energies, respectively.

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For the case of nd6, optical clock transitions exist throughout the whole isoelectronic sequences up to U68+, U50+ and U18+ for n = 3, 4, 5, respectively, providing nearly a hundred of HCI clocks. While U50+ and U18+ can be obtained within table-top electron beam ion traps (EBITs)38, HCIs such as U68+, with an ionization potential of 8 keV, can be effectively produced by room-sized EBITs39 or by an even larger facility HITRAP40. For nd4 and nd5, tens of optical clock transitions also exist in light elements, but the transition energies scale up to the XUV and soft-x-ray region for heavy elements. Considering that the stability of optical lasers is strongly limited by thermal fluctuations of reference cavities41, clocks based on lasers with shorter wavelengths42,43,44 would be suitable to further improve the clock performance.

To accurately predict the transition energies and lifetimes of these clock states, we employ ab initio fully relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) and configuration interaction (RCI) methods implemented in the GRASP2018 codes45. For the low-lying states in nd6, nd4 and nd5, the scaling of their excitation energies with respect to the atomic number Z are plotted in Figs. 2–4, respectively, where a clear reordering of the levels is observed. These values are obtained via simple calculations that only account for correlation effects within the corresponding ground-state configurations. However, for specific ions where accurate values are needed, such as the ions around the crossing points discussed below and the ions shown in Table 1, large-scale calculations with millions of configuration state functions (CSFs) are performed. Increasing the size of the CSF basis set systematically shifts the values of the excitation energies depicted in Figs. 2–4 by up to 10%46, but has little effects on the crossing behaviour of the levels. The final energies and lifetimes bear a relative accuracy of 0.5% and 20%, respectively. In the following, we will discuss the superiorities of these ions in developing ultrastable and ultra-accurate clocks, with the main focus being on the optical transitions in the nd6 ions.

Fig. 2: Excitation energies of the nd6 5D3,2,1,0 state.
figure 2

(a) Cr- and (b) Mo-like ions as functions of Z. With a ground state of 5D4, term reordering leads to two clock states with J = 2 and J = 0, respectively. Insets are the enlarged regions around the level-crossing points.

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Table 1 Properties for clock transitions in selected 4d6, 4d4 and 4d5ions.
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Clock transitions in n d 6 ions

For ions in this configuration, there are 34 atomic states with a 5D4 term being the ground state. In light ions, the corresponding low-lying excited states are 5D3,2,1,0, respectively, and decay via M1 transitions. Usually, the energies of the fine-structure multiplet scale with Z415, which is the case for J = 1, 3 shown in Fig. 2. However, for heavy elements, a strong relativistic effect leads to large splittings between the nd3/2 and nd5/2 relativistic orbitals (the subscripts are the corresponding single-electron total angular momenta). As a consequence, the wavefunction of states J = 0, 2 is dominated by the \(n{d}_{3/2}^{4}n{d}_{5/2}^{2}\) configuration, and their energies only increase linearly with respect to Z, and become the first two excited states. These two levels decay via slow E2 transitions and thus represent the two clock states illustrated in Fig. 1a.

For Mo-like ions ([Kr]4d6) shown in Fig. 2b, the level reordering of the J = 2 state happens at Ba14+. The transition energy between 5D4 and 3D2 is around 1.70 eV for Ce16+, but only scales up to 3.99 eV for U50+. With a lifetime τ0 equal to 1156 and 125 s for the two ions (see Table 1), these transitions have an intrinsic quality factor Q = 2πντ0 of 3.0 × 1018 and 7.6 × 1017. Assuming sufficiently long laser coherence time, they project a clock instability of \({\sigma }_{\tau }=3.2\times 1{0}^{-17}/\sqrt{\tau }\) and \(4.2\times 1{0}^{-17}/\sqrt{\tau }\), respectively (see Methods). These projected stabilities are several orders of magnitude higher than that projected for currently the most accurate Al+ clock10. Particularly, they are more than 3 orders of magnitude higher than the first HCI clock based on Ar13+16. Thus, this isoelectronic system would contain more than 36 ion candidates suitable for ultrastable optical clocks operating at various wavelengths.

For Cr-like ions ([Ar]3d6) as presented in Fig. 2a, the J = 2 metastable state starts to emerge from Pd22+. As the electrons are more tightly bound, the transition energies are relatively larger than those in Mo-like ions but are still accessible for optical lasers. Specifically, the corresponding transition energy and lifetime are 3.78 eV and 437 s, respectively, for Cd24+, and scale to 11.4 eV and 18.3 s, respectively, for U68+. This corresponds to quality factors Q > 1018 and clock instabilities \({\sigma }_{\tau } < \, 4\times 1{0}^{-17}/\sqrt{\tau }\) for more than 46 clock ions in this isoelectronic sequence.

Clock transitions in n d 4 ions

As a counterpart of the above-mentioned nd6 isoelectronic systems, the nd4 ions have similar spectroscopic terms but with a reversed level ordering, i.e. with 5D0 being the ground state and 5D1,2,3,4 being the subsequent excited states for light elements. However, as shown in Fig. 3, the energies of these excited states, all scaling relativistically with Z4, do not follow similar level crossings observed in Fig. 2. This is mainly because the ground state of heavy ions is a closed-shell \(n{d}_{3/2}^{4}\) configuration, and all excited states involve at least one excitation to the nd5/2 orbital. Thus, the 5D4 state only gradually becomes lower in energy than the 5D3 and 5D2 states and becomes a long-lived metastable state.

Fig. 3: Excitation energies of the nd4 5D1,2,3,4 states.
figure 3

(a) Ti- and (b) Zr-like ions as functions of Z. Insets are the enlarged depiction around the level-crossing points.

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Through Ti-like Xe32+ to Pm39+ and Zr-like Gd24+ to Ta33+, the lifetime of the 5D4 state is determined by the E2 transition to the 5D2 state and is in the range of 5 hours to 20 days. However, for heavier ions starting from Sm40+ and W34+, respectively, the 5D4 state attains a lower energy than the 5D2 state, with its lifetime being manifested by an E4 decay to the ground state, thus it can serve as an ultrastable clock state. For U70+ and U52+ shown in Fig. 1b, this corresponds to a 206- and 59.8-eV clock state with a lifetime of 49 days (Q = 1.3 × 1024) and 11.7 years (Q = 3.6 × 1025), respectively. While intra-cavity high-harmonic generation has generated photons around 100 eV42, future continuous-wave x-ray free-electron laser oscillators47 could produce intense coherent soft-x-ray sources for such clock applications48. Nevertheless, the projected clock instability will then be in the range of \({\sigma }_{\tau } > 3\times 1{0}^{-19}/\sqrt{\tau }\) (see Methods), which is 10 to 100 times more stable than optical clocks, provided that a coherence time similar to optical lasers can be achieved for the corresponding short-wavelength laser sources.

Clock transitions in n d 5 ions

For V-like Mn2+ and Nb-like Tc2+, the ground-state configuration is already of nd5 configuration. With 6S5/2 being the ground state, there already exists a long-lived metastable 4G11/2 state from the subsequent low-lying 4G11/2−5/2 terms. Nevertheless, for low-charged Ni5+ ion [see the inset of Fig. 4a], even the 4G5/2 and 4G7/2 states have a lifetime of 3.4 and 2.1 hours, respectively, and thus both represent ultrastable clock transitions with intrinsic quality factors around 1019. Though the energies of the excited states still scale with Z4, Fig. 4 shows totally different term-reordering patterns for these half-filled nd5 ions: the strong relativistic effect in heavy elements tends to reverse the order of the low-lying states to be of 4G5/2−11/2, respectively, with 4G11/2 still being lower than the 4G9/2 state for elements up to uranium. Therefore, the lifetime of the 4G11/2 state in these ions is manifested by a fast E2 decay to the 4G7/2 state46. However, for even heavier elements such as U69+ and U51+ shown in Fig. 1c, the M3 decay to the ground state is comparable to or even faster than the E2 decay channel. Thus, with an energy of 197 and 57.2 eV, and a lifetime of 12 hours (Q = 1.3 × 1022) and 80.3 days (Q = 6.0 × 1023), the 4G11/2 state also becomes ultrastable XUV and soft-x-ray clocks with instabilities \({\sigma }_{\tau } \, > \, 1{0}^{-19}/\sqrt{\tau }\). Moreover, the auxiliary fast-decaying 4G5/2 and 4G7/2 states have lifetimes of around 1 μs, rendering them suitable for direct laser cooling as well as for quantum jump detections49.

Fig. 4: Excitation energies of the nd5 4G5/2,7/2,9/2,11/2 states.
figure 4

(a) V- and (b) Nb-like ions as functions of Z. Insets are the enlarged level structures of the excited states in selected ions.

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Superior clock properties

Besides the high stabilities, another main advantage of these nd6 ions is that their clock states, shown in Table 1, are extremely insensitive to external fields. This is characterized by the small static electric and magnetic dipole polarizabilities defined by αa = ∑k≠a2Ska/3(2Ja + 1)(ℏωka)50,51 Here, \(\left\vert a\right\rangle\) is either the ground state \(\left\vert g\right\rangle\) or the excited clock state \(\left\vert e\right\rangle\), and ℏ the reduced Planck constant. Ska is the line strength of the electric or magnetic dipole transition from \(\left\vert a\right\rangle\) to an intermediate state \(\left\vert k\right\rangle\), and is calculated with the GRASP2018 package45. For state-of-the-art optical clocks, their differential electric and magnetic dipole polarizabilities are ΔαE1 > 10−7 Hz m2/V250 and ΔαM1 > 10 MHz/T251. In the case of Al+ clock at room temperature T = 295 K and magnetic field B = 0.12 mT, this corresponds to systematic shifts such as black-body radiation (BBR) and second-order Zeeman shifts of νBBR/ν = 3 × 10−18 and ν2zm/ν = 9 × 10−16, respectively10. However, the two differential polarizabilities in U50+, with values of 1.61 × 10−12 Hz m2/V2 and 3 kHz/T2, respectively, are smaller by up to 4 orders of magnitude in comparison to current high-performance clocks. As a consequence, the relative BBR shift, AC Stark shift, and second-order Zeeman shift for most of the [Kr]4d6 ions shown in Table 1 are small, rendering them extraordinarily suitable for clock operation with fractional frequency uncertainty δν/ν < 10−20. Such reductions in the differential polarizabilities are mainly a consequence of large separations between the energies of the low-lying clock states and the high-lying intermediate state shown in Fig. 1(a), as well as a result of further cross cancellations between those clock states due to their similarities in electronic configurations. Considering that HCI clocks usually operate at a cryogenic temperature around 4 K16, the corresponding BBR shift will be further reduced by a factor of 3.3 × 10−8 in comparison to the values at room temperature listed in Table 1.

Two other systematic shifts are the first-order Zeeman shift and the shift caused by the coupling between the ion’s electric quadrupole moment QE2 and the gradient of the static electric trapping field. We have thus modified part of the GRASP2018 package to calculate the corresponding Landé g factor and QE2 of these clock states. For the 5D4 ↔ 5D2 clock transition in U50+, as a consequence of cross cancellation between the values of the upper and lower clock states, the differential values give Δg ≈ 0.02 and ΔQE2 ≈ 0.04 a.u., respectively, which are smaller by up to 3 orders of magnitude in comparison to neutral and singly charged atomic systems52. While the Zeeman shift vanishes through probing the transition between the two mz = 0 magnetic levels, the electric quadrupole moment shift can be eliminated via choosing the ‘magic’ angle53 between the magnetic field and the gradient of the electric field. The much smaller ΔQE2 in these HCIs render it possible to reach residual uncertainty δνE2/ν < 10−20.

Systematics due to ionic motion

Besides the systematic effects described in the previous subsection, there are also systematics that do not depend on the internal properties of the HCIs, but on the motional dynamics of the ions in the Paul trap. These external ionic motions would induce first- and second-order Doppler shifts. The first-order Doppler shift can be eliminated via frequency averaging over two counterpropagating probe lasers. For an Al+ clock, this corresponds to a shift of δνD1/ν = (0.0 ± 2.2) × 10−1910, with the uncertainty arising from imperfect alignment of the probe lasers. Similar effects also occur for HCI clock candidates. However, one could further reduce the fractional uncertainty to be below 10−20 via employing two-photon resonant spectroscopy.

The second-order Doppler shift, also called time-dilation shift, can be separated into three parts54. The first part corresponds to the secular motion and intrinsic micromotion of the ions in the trapping potential, the second part is the excess micromotion under a stray dc field subjected to misalignment and imperfection of the electrodes of the trap, and the third part is the excess micromotion under the residual RF trapping field. The magnitudes of these terms are usually functions of the parameter ξ = qV (see Methods), where q is the charge of the corresponding ion and V the magnitude of the potential applied to the electrodes of the Paul traps. For HCIs, q is much larger than for singly charged ions. However, one only requires a q times smaller potential V to achieve the same trapping confinement for HCIs. Therefore, the HCI clock candidates put forward here would have a similar time-dilation shift in comparison to singly charged ions. For the case where the stray dc field is independent of the trapping potential V, the large q in HCIs would also result in a large signal in monitoring the ion, and thus would lead to a similar uncertainty as that in the singly charged ion clocks in determining the accuracy of the time-dilation shift15. Considering that the time-dilation shift induced by micromotion in an Al+ clock has reached an accuracy of δνD2/ν = 7 × 10−1910, the superior properties of HCI clock candidates discussed in the previous subsection would motivate investigations on better trap design for HCIs to reduce the corresponding uncertainties in this shift15. The vast range of charge-to-mass ratios of the proposed HCIs would also prompt the search of new ions for sympathetic cooling of HCIs.

Sensitivity to α variation

Furthermore, there is a second metastable state (labelled by J = 0) for Mo- and Cr-like ions which lies between 3.23-14.3 eV and 6.75-38.6 eV, respectively, above the ground state. While it decays to the ground state via a highly forbidden E4 transition, its lifetime, ranging from 64 ms to 800 s, is dominated by an E2 transition to the J = 2 excited state. As shown in Table 1, this J = 0 state is much more sensitive to the variation of the fine-structure constant than that of the J = 2 state. Taking the Mo-like Dy24+ as an example, with the sensitivity defined by K = (Δω/ω)(α/Δα), one has a K = 0.07 for the 5D4 ↔ 3D2 transition, but a K = − 1.73 for the 5D2 ↔ 5D0 transition. Such distinct α disproportionality of the two clock transitions in a single [Kr]4d6 ion would largely reduce systematic uncertainties in detecting α variation. For comparison, the clock transitions in Cf15+ and Cf17+ are known to have α sensitivities with ∣K∣ ≈ 609. However, their clock transitions bear polarizabilities with ΔαE1 ≈ 10−9 Hz m2/V2 and ΔαM1 ≈ 104 MHz/T255, representing a relative systematic shift of ν2zm/ν ≈ 1 × 10−13 which are several orders of magnitude larger than those in the [Kr]4d6 ions shown in Table 1, rendering them suffer from more significant systematic uncertainties. Similar dual-clock transitions also exist for the E2 and E3 transitions in Yb+, which provide currently the most stringent constraint of α variation2,3. Nevertheless, the E2 clock state has a lifetime of only 51 ms and both E2 and E3 transitions also suffer from large systematic effects with ΔαE1 > 10−7 Hz m2/V2 and ΔαM1 > 103 MHz/T29 that limits future improvements. One could also propose a Mo-like 229Th48+ clock that combines these electronic clock transitions together with the corresponding 8.34-eV nuclear clock transition K ≈ − 820056), but it requires considerations of hyperfine interactions and nuclear physics beyond the scope of this manuscript.

Potentials in King plot analysis

We have shown that there exist more than 100 ultrastable optical HCI clock candidates arising from fine-structure splittings in the open-shell nd4, nd5 and nd6 (n = 3, 4) ions. The corresponding transitions have intrinsic quality factors, polarizabilities, and quadrupole moments many orders of magnitude better than those in current optical clocks, and thus would allow for more accurate time keeping. Future works will be devoted to show that the above clock transitions also apply to Hf-, Ta-, and W-like 5d4,5,6 ions, respectively, through Hg6+ to U20+, to other nd2,7,8 ions, as well as to open-shell but more complicated 4fm ions. In total, there will be more than 10 clock transitions for a single element.

We note that, different from previous systems where only a few elements are known to bear ultrastable and ultra-accurate clock transitions9, our findings indicate that superior clock transitions exist in most of the elements in the periodic table. High-precision spectroscopy of the isotope shifts of these clock transitions can be employed to a multidimensional King-plot analysis57. This will have applications in searching for a hypothetical fifth force5,6,7 as well as in extracting accurate nuclear parameters for more than 70 elements and their isotopes ranging from \({{\scriptstyle{{46}}\atop}\!\!\!\!}_{22}{\rm{Ti}}\) to \({{\scriptstyle{{238}}\atop}\!\!\!\!}_{92}{\rm{U}}\) and beyond, and test state-of-the-art nuclear theories7,58. Moreover, for heavy HCIs, these clock transitions also scale up to the XUV and soft-x-ray range, and could enable the construction of ultrastable short-wavelength clocks.

Methods

MCDHF-RCI calculations

The level structures, lifetimes and line strengths mentioned in the main text were calculated via the ab initio fully relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction (RCI) methods implemented in the GRASP2018 code45. Within this approach, the many-electron atomic state function (ASF) is constructed as a linear combination of configuration state functions (CSFs) with common total angular momentum (J), magnetic (M), and parity (P) quantum numbers:

$$\left\vert {{\Gamma }}PJM\right\rangle =\sum _{k}{c}_{k}\left\vert {\gamma }_{k}PJM\right\rangle .$$

Each CSF \(\left\vert {\gamma }_{k}PJM\right\rangle\) is built from products of one-electron orbitals (Slater determinants), jj-coupled to the appropriate angular symmetry and parity, and γk represents orbital occupations, together with orbital and intermediate quantum numbers necessary to uniquely define the CSF. Γ collectively denotes all the γk involved in the representation of the ASF. ck is the corresponding mixing coefficient. The radial wave function for each orbital is obtained via solving the self-consistent MCDHF equations under the Dirac–Coulomb Hamiltonian. Then, the RCI method is applied to account for corrections arising from mass shift, quantum electrodynamic and Breit interactions.

To account for the correlation effect, we expanded the size of the CSF basis set by allowing single and double excitation of electrons from the {ns, np, nd} orbitals to a systematically increasing set of correlation orbitals up to the 8k orbitals. These correlation orbitals are added and optimized with the layer-by-layer approach under millions of CSFs. Therefore, the calculated energies and lifetimes have a relative accuracy of 0.5% and 20%, respectively.

Polarizabilities

The electric and magnetic dipole polarizabilities for clock state \(\left\vert a\right\rangle\) are calculated via the formula50,51

$${\alpha }_{a}=\sum _{k\ne a}\frac{2{S}_{ka}}{3(2{J}_{a}+1)\hslash {\omega }_{ka}}.$$

Here, Ja is the total angular momentum of state \(\left\vert a\right\rangle\). The summation runs over dipole-allowed transitions from state \(\left\vert a\right\rangle\) to all intermediate \(\left\vert k\right\rangle\) states, with Ska and ℏωka being the corresponding line strength and transition energy calculated with the GRASP2018 code. In this study, the intermediate states from the 4p54dm5s, 4p64dm−14f and 4p64dm−15p configurations (m = 4, 5, 6) are considered for the electric dipole polarizability. For the case of magnetic dipole polarizability, the intermediate states from the 4p64dm and 4p64dm−15s configurations are included in the calculations. As states with high-lying orbitals bear larger transition energies, the cutoff employed in our calculations represents a good approximation.

Black-body radiation shift

The interaction of the clock transition with the blackbody radiation (BBR) field gives rise a second-order Stark shift

$${\nu }_{{\rm{BBR}}}=-\frac{1}{2h}{\left\langle {E}^{2}\right\rangle }_{T}{{\Delta }}{\alpha }^{E1}.$$

Here, νBBR is in Hz, h is the Planck constant, \({{\Delta }}{\alpha }^{E1}={\alpha }_{e}^{E1}-{\alpha }_{g}^{E1}\) is the differential static scalar polarizability between the excited state \(\left\vert e\right\rangle\) and the ground state \(\left\vert g\right\rangle\) of the clock transition. The quadratic electric field strength \({\left\langle {E}^{2}\right\rangle }_{T}\) of the black-body radiation at temperature T can be calculated via the formula50

$$\left\langle {E}^{2}(T)\right\rangle ={(831.945\text{V/m})}^{2}{\left(\frac{T}{300\text{K}}\right)}^{4}.$$

For the values presented in Table 1 in the main text, T = 295 K were employed.

AC Stark shift

The presence of the clock laser also induces a second-order dynamical Stark shift

$${\nu }_{{\rm{ac}}}=-\frac{1}{2h}{E}^{2}{{\Delta }}{\alpha }_{AC}^{E1},$$

with \({{\Delta }}{\alpha }_{AC}^{E1}={\alpha }_{e}^{AC}-{\alpha }_{g}^{AC}\) and

$${\alpha }_{a}^{AC}=\sum _{k\ne a}\frac{2{S}_{ka}}{3(2{J}_{a}+1)(\hslash {\omega }_{ka}-\hslash \omega )}.$$

Here, ℏω is the clock transition energy. Since ω ≪ ωka for the ions considered in the main text, one has \({{\Delta }}{\alpha }_{AC}^{E1}\approx {{\Delta }}{\alpha }^{E1}\).

Furthermore, E is the corresponding electric field strength of the laser field. To estimate this AC Stark shift, we assumed the clock laser to have a Rabi frequency of Ω/2π = 1 Hz (i.e., a rectangular π pulse with a duration of Δt = 500 ms). Then, the laser field strength is calculated via E = Esτ0/2Δt, with Es and τ0 being the saturation field strength and lifetime of the upper clock state, respectively. However, for the case with lifetime τ0 < 1 s, the saturation intensity with a Rabi frequency of Ω/2π = 1/τ0 was employed.

Second-order Zeeman shift

Similar to the second-order Stark shift, the second-order Zeeman shift is calculated via

$${\nu }_{{\rm{2zm}}}=-\frac{1}{2h}\left\langle {B}^{2}\right\rangle {{\Delta }}{\alpha }^{M1}.$$

Here, \({{\Delta }}{\alpha }^{M1}={\alpha }_{e}^{M1}-{\alpha }_{g}^{M1}\) is the differential magnetic dipole polarizability of the clock transition. A magnetic field with B = 0.12 mT was used to derive the corresponding relative frequency shift ν2zm/ν presented in Table 1 in the main text.

Time-dilation shift

According to reference54, the time-dilation shift of clock transitions due to ionic motion in the \(\hat{i}\) (i = x, y, z) direction in a Paul trap can be written as

$${\nu }_{\,\text{D2}\,}^{(i)}\approx -\frac{\nu }{M{c}^{2}}\left[\frac{{k}_{B}{T}_{i}\left({a}_{i}+{b}_{i}^{2}\right)}{\left(2{a}_{i}+{b}_{i}^{2}\right)}+\frac{4}{M}{\left(\frac{q{b}_{i}{{\boldsymbol{E}}}_{{\rm{dc}}}\cdot \hat{i}}{\left({a}_{i}+{b}_{i}^{2}\right){{{\Omega }}}_{{\rm{RF}}}}\right)}^{2}+\frac{M{\left({b}_{i}R\beta {\varphi }_{{\rm{ac}}}{{{\Omega }}}_{{\rm{RF}}}\right)}^{2}}{64}\right].$$

Here, ν is the transition frequency, M the mass of the ion, c the speed of light. ai and bi are the basic parameters characterizing the motion of the ion in the trap. They are defined as

$$\begin{array}{rcl}{a}_{x}&=&{a}_{y}=-\frac{1}{2}{a}_{z}=-\frac{4qV}{M{R}^{2}{{{\Omega }}}_{\,\text{RF}\,}^{2}},\\ {b}_{x}&=&-{b}_{y}=\frac{2q\tilde{V}}{M{R}^{2}{{{\Omega }}}_{\,\text{RF}\,}^{2}},\quad {b}_{z}=0,\end{array}$$

with q being the charge of the ion. V and \(\tilde{V}\) are the magnitude of the static and radio-frequency (RF) trapping potentials, respectively. ΩRF is the frequency of the RF field, and R the distance from the trap axis to the trap electrodes. For simplicity, we have neglected the geometric factors. As both ai and bi are proportional to the parameter ξ = qV (assuming \(\tilde{V}\propto V\)), the magnitude of the trapping potential V and \(\tilde{V}\) for HCIs is reduced by a factor of q in comparison to that required for singly charged ions.

The first term of \({\nu }_{\,\text{D2}\,}^{(i)}\) corresponds to the shift due to secular motion and intrinsic micromotion of the ion. As its value depends on the temperature Ti of the motional states (kB is the Boltzmann constant), it can be reduced via better cooling. The second term is the shift due to an excess micromotion subjected to a uniform static electric field Edc. Though this stray dc field may originate from various mechanisms such as the surface charges and contaminations of an imperfect electrode, one of its dominate contribution is the misalignment of the electrodes of the trap10, with its strength being approximately proportional to the magnitude V of the static trapping field. Therefore, similar to the parameters ai, one also has \(q{{\boldsymbol{E}}}_{{\rm{dc}}}\cdot \hat{i}\propto qV\). The third term stands for the shift due to an excess micromotion under a residual RF field, an effect arising from the phase difference φac between the RF field from two electrodes. Its magnitude depends on bi that scales with qV.

As for the trap parameters ai and bi, all the three contributions to the time-dilation shift are functions of the parameter ξ = qV. Thus, with proper potential V, HCI clocks can experience a similar magnitude of time-dilation shift as singly charged ion clocks. For the case where the stray dc field is independent of the trapping potential V, the large q in HCIs would also result in a large signal in monitoring the ion, and thus would lead to a similar uncertainty as that in the singly charged ion clocks in determining accuracy of the time-dilation shift15.

Projected clock instability

The instability of a single-ion clock presented in the main text is defined via the Allan variance9:

$${\sigma }_{\tau }=\frac{{{\Delta }}\nu }{\pi \nu }\sqrt{\frac{{T}_{c}}{\tau }}.$$

Here, ν is the frequency of the clock transition, Δν the bandwidth of the clock interrogation pulse, τ the average measuring time, and \(\frac{1}{\pi }\) the normalization factor of the probability density function of the spectrum of rectangular pulses. Tc = Δt + td is the duration of a single clock cycle, where Δt represents the duration of the clock laser and td is the total time required for ion cooling, state preparation, state readout and related processes. Following the experimental procedures of an Al+ clock10, we conservatively set td = 0.2 s to provide an upper bound on these effects. For a rectangular probe pulse with a duration Δt, one has Δν = 1/Δt, and the instability can be rewritten as

$${\sigma }_{\tau }=\frac{1}{\pi \nu {{\Delta }}t}\sqrt{\frac{{{\Delta }}t+{t}_{d}}{\tau }}.$$

We note that the upper limit of the pulse duration Δt is set either by the lifetime of the upper clock state, or by the coherence time of the clock laser. Considering that the longest laser coherence time has reached tens of seconds and may continue to increase in the future to 1000 s, we define the projected clock instability by setting \({{\Delta }}t=\min (1{0}^{3}\,\,\text{s}\,,{\tau }_{0})/2\). We also note that, in reality, several clock pulses are required to eliminate the first-order Zeeman shift, electric quadrupole moment shift, first-order Doppler shift, etc. Thus, one usually has an even smaller Δt, and the realistic instability might be larger than the projected στ defined here.

For ions with a clock-state lifetime between 2 and 1000 s, the projected instability can be approximately written as \({\sigma }_{\tau }\approx \frac{2\sqrt{2{\tau }_{0}}}{Q}\frac{1}{\sqrt{\tau }}\), with Q = 2πντ0 being the intrinsic quality factor. Thus, the longer the lifetime, the larger the quality factor and the better the stability. However, for lifetimes much longer than the possible coherence time of optical lasers, the clock stability can be improved via employing XUV or x-ray transitions with higher frequencies, provided similar coherence times could be achieved for XUV and x-ray sources.