Measurement of the bound-electron g-factor difference in coupled ions – Nature

Mixing and preparing the coupled state

After determining the spin orientations of the individual ions, one ion is excited to a magnetron radius r₋ ≈ 600 μm to prepare for the coupling of the ions. They are now transported into electrodes next to each other, with only a single electrode in between to keep them separated. Subsequently, this electrode is ramped down as quickly as experimentally possible, limited by d.c. filters to a time constant of 6.8 ms to keep any voltage change adiabatic compared with the axial frequencies of several 10 kHz. The potentials are also optimized to introduce as little axial energy as possible during this mixing. Subsequently, both ions are brought into resonance with the tank circuit one at a time by adjusting the voltage to repeatedly cool their axial modes. Once thermalized, the axial frequency is automatically measured and adjusted to the resonance frequency. From the observed shift in axial frequency compared with a single cold ion, the separation distance d_sep of the ions can already be inferred, without gaining information about the common mode. At this point, both ions are cooled in their respective cyclotron motions via sideband coupling¹⁵. The common-mode radius r_com of the coupled ions can be measured by applying a C₄ field contribution, causing the axial frequency to become dependent on the magnetron radius. With the amplitudes of the axial and reduced cyclotron motion being small, this frequency shift allows for the determination of the root-mean-square (r.m.s.) magnetron radius of each ion. If the common mode is large, the modulation of the magnetron radius, owing to slightly different frequencies of separation and common mode, will lead to visible sidebands owing to the axial frequency modulation.

For small common mode radii, we will simply measure half the separation radius for each ion. In combination with the known separation distance, the common mode radius can now be determined; however, owing to limited resolution of the axial frequency shift and the quadratic dependency, ({r}_{{rm{com}}}approx sqrt{{r}_{{rm{r}}{rm{.m}}{rm{.s}}.}^{2}-{left(frac{1}{2}{d}_{{rm{sep}}}right)}^{2}}), a conservative uncertainty after the ion preparation of r_com = 0(100) μm is assumed. For consistency, we have prepared the ions in the final state and again excited the common mode to a known radius that could be confirmed using this method. In case of a large initial common mode, we first have to cool it. Unfortunately, addressing it directly is complicated, as the separation mode will always be cooled as well. However, using the method described in ref. ²⁰, we are able to transfer the common-mode radius to the separation mode. This requires a non-harmonic trapping field with a sizeable C₄, combined with an axial drive during this process. The axial frequency will now be modulated owing to the detuning with C₄ in combination with the modulated radius owing to the common mode. As the ion will be excited only when close to the drive, we gain access to a radius-dependant modulation force, which finally allows the coupling of the common and separation modes.

Finally, with the common mode thus sufficiently cooled, we directly address the separation mode, cooling it to the desired value. Owing to the strong axial frequency change during cooling, scaling with ({d}_{{rm{sep}}}^{3}) and typically being in the range of Δv ≈ 150 Hz, the final radius cannot be exactly chosen but rather has a distribution that scales with the power of the cooling drive used. Therefore, one can chose to achieve more stable radii at the cost of having to perform more cooling cycles, ultimately increasing the measurement time. We choose a separation distance d_sep = 411(11) μm, with the uncertainty being the standard deviation of all measurements as an acceptable trade-off between measurement time and final separation distance distribution. Furthermore, although a smaller separation distance directly corresponds to a decreased systematic uncertainty (Methods), the increased axial frequency shift as well as a deteriorating signal quality of the coupled ions result in a practical limit around d_sep = 300 μm.

Measurement sequence

Before irradiating the microwave pulses, the cyclotron frequency is measured via the double-dip technique using ²²Ne⁹⁺. This measurement is required to be accurate to only about 100 mHz, which corresponds to a microwave frequency uncertainty of about 400 Hz, which is neglectable considering a Rabi frequency of over 2 kHz for a spin transition. The microwave pulse is applied at the median of the Larmor frequencies of ²²Ne⁹⁺ and ²⁰Ne⁹⁺ and therefore detuned from each Larmor frequency by about 380 Hz. This detuning is taken into account when calculating the required time for a π/2 pulse.

Separation of ions

The strong magnetic bottle, or B₂ contribution, that is present in the AT gives rise to a force that is dependent on the magnetic moment of the ion. The main purpose is to allow for spin-flip detection via the continuous Stern–Gerlach effect. In addition, this B₂ can be utilized to create different effective potentials for the ions depending on their individual cyclotron radii r₊. These give rise to the magnetic moment ({mu }_{{rm{cyc}}}={rm{pi }}{nu }_{+}{q}_{{rm{ion}}}{r}_{+}^{2}), which then results in an additional axial force in the presence of a B₂. To use this effect to separate the coupled ions, one of them is pulsed to r₊ ≈ 800 μm at the end of the measurement in the PT. Subsequently, both ions are cooled in their magnetron modes, resulting in a state where one ion is in the centre of the trap at thermal radii for all modes while the other is on the large excited cyclotron radius. We verify this state by measuring the radii of both ions to confirm the successful cooling and excitation. Now, we use a modified ion-transport procedure, with the electrode voltages scaled such that the ion with r₊ > 700 μm cannot be transported into the AT but rather is reflected by the B₂ gradient, whereas the cold ion follows the electrostatic potential of the electrodes. The hot ion is transported back into the PT and can be cooled there, leaving both ions ready to determine their electron spin orientation again, completing a measurement cycle. This separation method has worked flawlessly for over 700 attempts.

Rabi frequency measurement

To determine the required π/2-pulse duration, a single ion, in this case ²²Ne⁹⁺, is used. We determine the spin orientation in the AT, transport to the PT, irradiate a single microwave pulse and check the spin orientation again in the AT. Depending on the pulse duration t, the probability of achieving a change of spin orientation follows a Rabi cycle as

Here, Ω_R is the Rabi frequency and ΔΩ_L is the detuning of the microwave drive with respect to the Larmor frequency. With a measured Rabi frequency of Ω_R = 2,465(16) Hz, we can irradiate the mean Larmor frequency of the two ions, with the difference being about 758 Hz. Thereby, we are able to use a single pulse simultaneously for both ions while accounting for the detuning to achieve a π/2 pulse of 101.1 μs for both ions simultaneously. The corresponding data are shown in Extended Data Fig. 1. The fit includes a shot-to-shot jitter of the microwave offset δΩ_L to account for the uncertainty of the magnetic field. The measurement is performed on a magnetron radius of r₋ = 200(30) μm to achieve similar conditions to those in the coupled state.

Determination of charge-radii differences

We would in principle be able to improve on any charge-radii differences, where this is the limiting factor for the theoretical calculation of g. This holds true for most differences between nuclear spin-free isotopes, as well as differences between different atoms, provided they are either light enough for theory to be sufficiently precise or close enough in nuclear charge Z such that the corresponding uncertainties are still strongly suppressed.

Fitting function for the Larmor frequency difference

To derive the fitting function of the correlated spin behaviour of the two ions, we first assume that both ions have been prepared initially in the spin-down state, indicated as ↓. The probability to find each ion individually in the spin-up state (↑) then follows the probability of a Rabi oscillation with the frequency of the difference between the ion’s Larmor frequency ω_L1 or ω_L2, respectively, and the common microwave drive frequency ω_D. The probability of finding both ions after the measurement sequence in the spin-up state follows as

Similarly, the probability of finding both ions in the spin-down state can be written as

Both cases, where either both ions are in the spin-down or the spin-up statehave to be considered, as we cannot perform a coherent measurement of the individual Larmor frequencies with respect to the microwave drive. As a result, the information about the Larmor frequency difference is encoded only in the common behaviour of the spins. Therefore, we have to look at the combined probability of both ions either ending up both in the spin-up or spin-down state (case 1; Extended Data Fig. 2), or the complimentary case, where the two spins behave differently, with one ion in the spin-up state and the other ending in the spin-down state. The joint probability is given by

where, owing to the loss of coherence with respect to the drive frequency, the term in the middle line of equation (8) averages to 1/2. The same formula can be derived for any known initial spin configuration.

Calculation of the g-factor difference

The g-factor difference can be directly calculated from the determined relation given by

Although the input parameters of mass and cyclotron frequency of one of the ions are still required, the precision relative to Δg is only of about 7 × 10⁻⁵, strongly relaxing the need for ultraprecise masses and a cyclotron frequency determination. In contrast, when measuring absolute g factors, the precision of the mass and cyclotron frequency typically limit the achievable precision to the low 10⁻¹¹ level.

Setting constraints on new physics

Measuring the g factor allows for high-precision access to the properties of very tightly bound electrons, and hence to short-range physics, including potential new physics (NP). Bounds on NP can be set with isotopic-shift data on the g factor of hydrogen-like neon. The Higgs-relaxion mixing mechanism, in particular, involves the mixing of a potential new (massive) scalar boson, the relaxion, with the Higgs boson. It has been proposed as a solution to the long-standing electroweak hierarchy problem¹³ with the relaxion as a dark-matter candidate⁴⁰. Constraints on this proposed extension of the standard model can be set with cosmological data, as well as with particle colliders, beam dumps and smaller, high-precision experiments (see, for example, ref. ⁴¹ and references therein).

The most common approach in atomic physics is to search for deviations from linearity on experimental isotopic-shift data in a so-called King plot analysis^{29,35,41,42,43,44}, which can be a sign of NP, although nonlinearities can also happen within the standard model^12,24,37,44, which limits the bounds that can be set on NP parameters. The King plot approaches also suffer from strong sensitivity on nuclear-radii uncertainties²⁵. Here we present constraints on NP from data on a single isotope pair. The influence of relaxions (scalar bosons) on atoms can be expressed^29,41,43,44 by a Yukawa-type potential (often called the ‘fifth force’) exerted by the nucleus on the atomic electrons:

where m_ϕ is the mass of the scalar boson, α_HR = y_e y_n/4π is the NP coupling constant, with y_e and y_n the coupling of the boson to the electrons and the nucleons, respectively, A is the nuclear mass number and ħ is the reduced Planck’s constant. Yukawa potentials naturally arise when considering hypothetical new forces mediated by massive particles. The corresponding correction to the hydrogen-like g factor is given by¹²

where (gamma =sqrt{1-{(Zalpha )}^{2}}). The mass scale of the hypothetical new boson is not known⁴¹, apart from the upper bound ({m}_{varphi } < 60,{rm{GeV}}). In the small-boson-mass regime ({m}_{varphi }ll Zalpha {m}_{{rm{e}}}), the contribution to the g factor simplifies to

In the large-boson-mass regime ({m}_{varphi }gg Zalpha {m}_{{rm{e}}}), we obtain

We can set bounds on the NP coupling constant by comparing the measured and calculated values of the g-factor isotopic shift (see refs. ^12,45 for an implementation of the same idea with transition frequencies in atomic systems). Uncertainties from theory are a source of limitation in this approach. The standard-model contributions to the isotopic shift of the g factor of hydrogen-like neon are given in Extended Data Table 1, as calculated in this work based on the approaches developed in the indicated references. As can be seen, the largest theoretical uncertainty comes from the leading finite nuclear-size correction, and is due to the limited knowledge of nuclear radii (the uncertainty on the finite nuclear-size correction owing to the choice of the nuclear model is negligible at this level of precision). We note that the standard source for these nuclear radii is data on X-ray transitions in muonic atoms²³.

In the NP relaxion scenario, the energy levels of these muonic atoms are also corrected by the relaxion exchange. Another source of r.m.s. charge radii and their differences is optical spectroscopy. The electronic transitions involved are far less sensitive to hypothetical NP than muonic X-ray transitions. The radius difference between ²⁰Ne and ²²Ne extracted from optical spectroscopy⁴⁶ agrees with the one determined from muonic atom data within the respective uncertainties, which shows that NP need not be taken into account to extract nuclear radii from these experiments at their level of precision. To conclude, for our purposes, hypothetical contributions from NP do not interfere with the interpretation of muonic atom data for the extraction of nuclear radii.

Taking (Delta {g}_{{rm{theo}}}^{A{A}^{{prime} }}=1.1times {10}^{-11})  as the theoretical error on the isotopic shift, it can be seen from equation (12) that this corresponds to an uncertainty of Δy_e y_n ≈ 7.1 × 10⁻¹⁰ (and a 95% bound on y_e y_n twice as large as this) in the small-boson-mass regime ({m}_{varphi }ll Zalpha {m}_{{rm{e}}}), which is weaker than the current most stringent bounds coming from atomic physics (H–D 1S–2S, ref. ³⁶). In the large-boson-mass regime ({m}_{varphi }gg Zalpha {m}_{{rm{e}}}), our bound remains weaker, but becomes more competitive and is more stringent than those of ref. ³⁵, owing to two favourable factors. First, the nuclear charge Z in equation (13) is larger than the screened effective charge perceived by the Ca⁺ valence electron, and larger than the charge of the hydrogen nuclei, which also enter the scaling of the bound obtained with these respective ions²⁹. Second, when carrying out a King analysis as done in ref. ³⁵, one works with two different transition frequencies, and the leading term in the hypothetical NP contribution in the large-boson-mass regime, which is the equivalent of the right-hand side of equation (13), is cancelled out in the nonlinearity search, owing to its proportionality to the leading finite nuclear-size correction²⁹, leaving the next term, which scales as ({({m}_{varphi }/(2Zalpha {m}_{{rm{e}}}))}^{-1-2gamma }), as the first non-vanishing contribution.

In the present case, the g factor of a single electronic state is considered (for a single isotope pair), and this cancellation does not occur. This leads to competitive bounds in the large-boson-mass regime with the simple g factor isotopic shift of hydrogen-like ions, as shown in Fig. 3 (where we used the exact result, equation (11)). We compare our bounds on the coupling constant y_ey_n = 4πα_HR, to the bounds obtained in refs. ^35,36, through isotopic-shift measurements in Ca⁺ (see the curve Ca⁺ IS-NL in Fig. 3) and H (with nuclear radii extracted from muonic atom spectroscopy), as well as to the bounds obtained through Casimir force measurements³³, globular cluster data³⁴ and a combination³⁵) of neutron scattering^47,48,49,50 data and free-electron g factor¹ (g − 2)_en.

We also reproduce the preferred range for the coupling constant obtained in ref. ³⁷, through isotopic-shift measurements in Yb⁺ (Yb⁺ IS-NL). This range was obtained by assuming that the observed King nonlinearity in the experimental isotopic-shift data is caused by NP. By contrast, all nuclear corrections to the g factor that are relevant at the achieved experimental precision were taken into account in our approach, allowing for an unambiguous interpretation of the experimental data.

In Fig. 3, we also indicate projected bounds that could be obtained from isotopic-shift measurements of the g factors of both hydrogen-like and lithium-like argon. Combining both measurements allows the approximate cancellation of the leading finite nuclear-size corrections through considering a weighted difference^11,27 of hydrogen-like and lithium-like g factors. On the basis of our earlier discussion of the domination of theoretical uncertainties by the uncertainty on the leading finite nuclear-size correction (Table 1), the interest of this approach is readily understood. Our calculations indicate that argon is in the optimal range for setting bounds on α_HR with this approach.

The weighted difference approach is not preferred in the large-boson-mass regime, however, because of strong cancellations of the NP contribution. A similar approach based on a weighted difference of the g factor and ground-state energies of hydrogen-like ions should yield even more stringent bounds²⁸. Both these weighted-difference-based approaches are insensitive to uncertainties on the nuclear radii, as such, the bounds that they can generate are fully independent of any assumptions on NP coupling to muons.

Calculation of the initial phase difference

As our method relies on a single external drive for this specific measurement, used to drive both spins simultaneously, the drive has to be applied at the median Larmor frequency. This results in an additional phase difference that is acquired during the π/2 pulses. We have determined this phase to be Φ_init = 35.8(50)° using a numerical simulation. Here we use the knowledge of the Rabi frequency as well as the uncertainty of the magnetic-field determination, which leads to an effective jitter of the microwave drive from cycle to cycle. The simulation is performed for different evolution times, extrapolating to the phase that would be measured for zero evolution time. Although the phase that we can extract from the measured data as a cross-check is consistent with this prediction, we still assign an uncertainty of ±5° to the simulation.

Analysis of systematic shifts of Δg of coupled ions

Here we evaluate the total systematic shift and its uncertainty for this method, specifically for the measurement case of ²⁰Ne⁹⁺ and ²²Ne⁹⁺. For this approach, we consider only a separation distance and no common mode. For small common-mode radii r_com ≤ 100 μm, which we give as an upper limit, the systematic effects discussed here are actually further reduced²⁰. We have to consider multiple individual measurements performed with single ions to characterize these frequency shifts and experimental parameters. More explanation on the methods used can be found in ref. ⁶, and the individual frequency shifts are derived in ref. ⁵¹. We define our electric potential, and specifically the coefficients C_n as

with applied ring voltage V_r, the characteristic trap size d_char and the Legendre polynomials P_n. The magnetic-field inhomogeneities B₁ and B₂ are defined as

where z is the axial position with respect to the electrostatic minimum of the trap. First, we consider the two main axial frequency shifts that depend on the magnetron radius of an ion:

If the shift of v_z is measured to be zero for any radius r₋, these two shifts cancel and we can conclude that ({C}_{4}=frac{3}{4}frac{{C}_{3}^{2}}{{C}_{2}}). As it is typically not feasible to tune this for arbitrary radii, especially as higher orders will have to be considered as well for larger radii, we allow a residual ({eta }_{{rm{el}},{r}_{-}}), which includes both the residual observed shift and all neglected smaller contributions. This is a relative uncertainty, scaling with r²:

Similarly, we consider all frequency shifts that depend on the cyclotron radius r₊ of an ion:

The electrostatic contributions are identical to those for the magnetron mode, and per the assumption above will also combine to the same ({eta }_{{rm{el}},{r}_{+}}), scaling with the cyclotron radius. However, we have to consider the additional terms that stem from the magnetic-field inhomogeneities, which are sizeable in this mode owing to the significantly higher frequency:

In addition, for large cyclotron excitations, we have to consider the relativistic effect of the mass increase, which also slightly shifts the axial frequency:

The combined shift depending on magnetic inhomogeneities can be expressed as

Although we cannot currently tune these contributions actively (which could be implemented by using active compensation coils²²), we can slightly shift the ion from its equilibrium position to a more preferable position along the z axis to minimize the B₂ coefficient. Doing so, we have achieved frequency shifts of v_z close to zero for any cyclotron excitations as well, which means these terms have to cancel as well. We will still allow for another residual error from higher orders, as well as a small residual shift, defined as η_mag. The observed difference in the frequency shift between cyclotron and magnetron excitations ({eta }_{{rm{mag}}}+{eta }_{{rm{el}},{r}_{+}}-{eta }_{{rm{el}},{r}_{-}}) can be used to cancel the identical electric contributions ({eta }_{{rm{el}},{r}_{+}}) and ({eta }_{{rm{el}},{r}_{-}}) when measuring at the same radius. If we solve this combined equation for C₃, we are left with only the magnetic-field-dependent terms B₁ and B₂, which is what the Larmor frequency difference is sensitive to

where ξ summarizes the shifts depending on the radial modes of the ion. Now, instead of looking at frequency shifts of individual ions, we consider the effects on coupled ions. Owing to their mass difference, the coupled state is not perfectly symmetrical but slightly distorted owing to the centrifugal force difference. In the case of the neon isotopes, this leads to a deviation of δ_mag = 0.87%, with the definition of ({r}_{1}={d}_{{rm{sep}}}frac{(1-{delta }_{{rm{mag}}})}{2}) and ({r}_{2}={d}_{{rm{sep}}}frac{(1+{delta }_{{rm{mag}}})}{2}), when choosing ion 1 to be ²⁰Ne⁹⁺ and ion 2 as ²²Ne⁹⁺. Consequently, the frequency difference ({nu }_{{{rm{L}}}_{1}}-{nu }_{{{rm{L}}}_{2}}) will be positive, as the g factor (and therefore the Larmor frequency) for ²⁰Ne⁹⁺ is larger than for ²²Ne⁹⁺. We now consider the axial position shift as a function of the slightly different ({r}_{-}^{2}). This is given by

Now we want to express all frequency shifts in terms of v_L, which is to a very good approximation dependent on only the absolute magnetic field, first considering only the effect of B₁ and all shifts along the z axis:

The difference in the shift for the individual ions can then be written as

We have now the additional uncertainties all summarized in the term scaling with the above-defined factor ξ. The final shift to consider is the same radial difference as mentioned before in the presence of B₂. This leads to additional individual shifts in the v_L of the ions as

As a relative shift with respect to the measured Larmor frequency difference, this can be written as

Combining these shifts, ({nu }_{{rm{L}},{B}_{2}}^{{rm{rel}}}) and ({nu }_{{rm{L}},{B}_{1}}^{{rm{rel}}}), results in

We find that, in the ideal case where neither magnetron nor cyclotron excitations produce shifts of the measured axial frequency v_z, the final difference of the Larmor frequency is also not shifted at all. Here we use the worst case, with a measured combined relative shift for (frac{{eta }_{{rm{mag}}}}{{r}_{+}^{2}}approx frac{125,{rm{mHz}}}{560},.) This corresponds to a systematic shift of (frac{Delta (Delta {nu }_{{rm{L}},{rm{tot}}})}{Delta {nu }_{{rm{L}},{rm{tot}}}}=6times {10}^{-13},,) which we did correct for in the final result. This has been confirmed by performing two measurements on different separation distances, of d_sep = 340 μm and d_sep = 470 μm. Both measurements have been in agreement after correcting for their respectively expected systematic shift. The uncertainty of this correction of 5 × 10⁻¹³ has been evaluated numerically by combining the uncertainties of η_mag and the radii intrinsic to its determination, an uncertainty of δ_mag and the potential of a systematic suppression of the systematic shift by a residual common-mode radius.

Different axial amplitudes

The measurement is performed by first thermalizing the ²⁰Ne⁹⁺, then increasing the voltage to bring the ²²Ne⁹⁺ into resonance with the tank circuit. This will slightly decrease the axial amplitude of the ²⁰Ne⁹⁺, which nominally has the larger amplitude when cooled to the identical temperature, compared at the same frequency owing to its lower mass. The residual difference in amplitude will lead to a further systematic shift in the presence of a B₂, which has been evaluated to about 3 × 10⁻¹⁴ and can therefore safely be neglected at the current precision.

g-factor calculation

In Extended Data Table 1, the individual contributions to the g factors of both ions are shown. The main uncertainty, the higher-order two-loop QED contribution, is identical for both ions and does cancel in their difference and can therefore be neglected for the uncertainty of Δg. The finite nuclear size (FNS) correction gives the dominant uncertainty in Δg, which in turn is determined by the uncertainty of the r.m.s. radius²³. The next error comes from the nuclear polarization correction, which sets a hard limit for a further improvement in the determination of the r.m.s. radius. The difference in the spectra of photonuclear excitations of ²⁰Ne and ²²Ne defines the contribution of the nuclear polarization to Δg. As the dominant contribution to the nuclear polarization of ^20,22Ne comes from the giant resonances, one has to estimate the isotope difference of this part of the spectrum. The measurements of the absolute yields of the various photonuclear reactions are reported in refs. ^52,53 for ²⁰Ne and in refs. ^54,55 for ²²Ne. On the basis of these data, we conclude that the integrated cross-section for the total photoabsorption between ²⁰Ne and ²²Ne differs by less than 20%, which we take as the relative uncertainty of the nuclear polarization contribution to Δg. The hadronic vacuum polarization (see, for example, ref. ⁵⁶) corresponds to the small shift of the g factor by the virtual creation and annihilation of hadrons and is largely independent of the nuclear structure. In the g-factor difference of ²⁰Ne⁹⁺ and ²²Ne⁹⁺, the QED contribution to the nuclear recoil can be resolved independently from all common contributions. A test of this contribution by means of an absolute g-factor measurement is possible for only the small regime from carbon to silicon and for only stable isotopes without nuclear spin. For smaller Z ≤ 6, the QED contribution is too small to be resolved experimentally, and for Z > 14, the uncertainty of the two-loop QED contribution is too large to test the QED recoil. In addition, such an absolute g-factor measurement would also require the ion mass to similar precision, which is not the case for the approach via the direct difference measurement performed here.