Isotope dependence of the Zeeman effect in lithium-like calcium

The magnetic moment μ of a bound electron, generally expressed by the g-factor μ=−g μB s ħ−1 with μB the Bohr magneton and s the electron's spin, can be calculated by bound-state quantum electrodynamics (BS-QED) to very high precision. The recent ultra-precise experiment on hydrogen-like silicon determined this value to eleven significant digits, and thus allowed to rigorously probe the validity of BS-QED. Yet, the investigation of one of the most interesting contribution to the g-factor, the relativistic interaction between electron and nucleus, is limited by our knowledge of BS-QED effects. By comparing the g-factors of two isotopes, it is possible to cancel most of these contributions and sensitively probe nuclear effects. Here, we present calculations and experiments on the isotope dependence of the Zeeman effect in lithium-like calcium ions. The good agreement between the theoretical predicted recoil contribution and the high-precision g-factor measurements paves the way for a new generation of BS-QED tests.

B esides hyperfine splitting, isotope shifts of atomic electronic energy levels provide the most common access to nuclear properties 1 . Typically, the dominating nuclear effects contributing to isotope shifts are generated by differences in nuclear masses, also denoted as nuclear recoil shifts (mass shifts), and by differences in nuclear sizes due to different spatial distributions of the nuclear charge (field shift). In absence of the magnetic field, isotope shifts in highly charged ions were first measured in refs 2,3. In particular, relativistic nuclear recoil shifts have been previously probed in experiments on the isotope shifts in the binding energy of boron-like argon 4 and lithium-like neodymium 5 .
As already proposed for different magnesium isotopes 6 , in this paper, we focus on the isotope dependence of the Zeeman effect by studying g-factors of lithium-like calcium isotopes 40 Ca 17 þ and 48 Ca 17 þ . Featuring on the one hand a 20% mass difference and on the other hand almost identical nuclear charge radii 7 , these isotopes provide a unique system across the entire nuclear chart to test the relativistic nuclear recoil shift in presence of a magnetic field.
Most physical effects contributing to g-factors of highly charged ions, for example, the relativistic, radiative, nuclear size or interelectronic-interaction corrections, are calculated using bound-state quantum electrodynamics (QED) in the infinitenuclear-mass approximation. Here, the nucleus is considered as an external Coulomb potential fixed in space. This approach is usually denominated as the Furry picture of QED (ref. 8). However, bound-state QED contributions of the studied nuclear recoil shift require calculations beyond the Furry picture, which are presented in the first part of this paper.
The experimental determination of the tiny g-factor difference Dg g( 40 Ca 17 þ ) À g( 48 Ca 17 þ ), which is in the order of 1 Â 10 À 8 , requires four independent high-precision measurements: the Larmor-to-cyclotron frequency ratios of both calcium ion species as well as their atomic masses. The frequency ratios have been measured successively with a relative uncertainty of about 7 Â 10 À 11 . For this purpose, we studied single ions confined in a dedicated Penning-trap set-up 9,10 . Aiming for atomic masses with relative uncertainties of about 4 Â 10 À 10 , we also improved the atomic mass of 48 Ca by a factor of seven. Here, we used the offline configuration of the Penning-trap mass spectrometer SHIPTRAP (ref. 11) in combination with the novel phase-imaging ion-cyclotron resonance technique (PI-ICR) 12,13 . The finally obtained 1.0s agreement between the predicted and measured g-factor difference decisively confirms relativistic recoil corrections in the presence of strong fields. The reinforced understanding of the interaction between the bound electrons and the nucleus provides the opportunity to extract fundamental constants, namely the fine structure constant a, and nuclear properties via g-factor measurements in heavy atomic systems 14 .

Results
Calculation of the g-factor difference. The theoretical value of the isotope shift in the atomic g-factors is mainly given by a sum of the nuclear recoil and nuclear size contributions. Considering s-states of highly charged ions, the leading order terms scale with Dg rec ¼ m e m À 1 nucl ðZaÞ 2 n À 2 and Dg size ¼ 8ð3n 3 Þ À 1 m 2 e c 2 ' À 2 ðZaÞ 4 r 2 nucl (ref. 15), where n represents the principle quantum number of the valence electron. Further nuclear contributions, for example, nuclear deformation 16 and nuclear polarization 17 are orders of magnitude smaller and at the current level of experimental as well as theoretical precision extraneous to the g-factor difference. For Z ¼ 20 the isotope shift is essentially determined by the mass shift, which in the case of s-states is of pure relativistic origin.
Considering the two double magic isotopes 40 Ca and 48 Ca, the nuclear charge radii r nucl ( 40 Ca) ¼ 3.4776 (19) fm and r nucl ( 48 Ca) ¼ 3.4771 (20) fm (ref. 7) are surprisingly similar and by itself subject of present research. In this way, the nuclear recoil shift dominates the g-factor difference of the lithium-like electron configuration to 99.96%.
The lowest order recoil correction, which is non-QED but relativistic, can be derived from Breit equation [18][19][20][21] . The full relativistic theory of the nuclear recoil effect on the atomic g-factor has to be formulated in the framework of QED. So far, a systematic approach has been developed to first order in the electron to nucleus mass ratio m e Á m nucl À 1 and to all orders in Za (ref. 22). As a result, the complete Za-dependence formula for the recoil effect on the g-factor of a hydrogen-like ion has been derived. To zeroth order in Z À 1 , this formula describes also the recoil effect in a few-electron ion with one electron over closed shells, provided the electron propagators are defined for the vacuum with the closed shells included 15 . Generally, this leads to the appearance of two-electron nuclear recoil contributions. However, for the (1s) 2 2s-state of a lithium-like ion, the two-electron contributions vanish, and, therefore, to zeroth order in Z À 1 , one has to evaluate the one-electron contribution only. In the present paper, we evaluate this contribution to all orders in Za for the 2s-state at Z ¼ 20 using the corresponding formula 22 . This result is combined with the radiative and second order in m e Á m nucl À 1 recoil corrections 19,21,23,24 to get the total oneelectron contribution. To evaluate the interelectronic-interaction contribution to the recoil effect of the first and higher orders in Z À 1 , we extrapolated the related results obtained to the lowest relativistic order 25 (Methods section). The uncertainty of this contribution is mainly due to uncalculated higher order relativistic and QED corrections.
To get the total value of the isotope shift, one has also to account for the nuclear size effect. This contribution, being rather small, can be calculated in the one-electron approximation by solving the Dirac equation numerically. Moreover, it can be evaluated using an analytical formula 26 . The root-mean-square nuclear charge radii and their uncertainties are taken from ref. 7. The uncertainty of the nuclear size contribution includes both the nuclear radius and shape variation effects.
The individual contributions of the calculated isotope difference Dg ¼ g( 40 Ca 17 þ ) À g( 48 Ca 17 þ ) are presented in Table 1. It is seen that the QED recoil effect, whose calculation requires using QED beyond the Breit approximation and beyond the Furry picture, is about five times bigger than the total theoretical uncertainty. Measurement concept. For the experimental determination of the g-factor difference, we measured successively the Zeeman splitting of the respective lithium-like ion in a homogeneous magnetic field B using single ions confined in a Penning trap. The Larmor frequency n L , which quantifies the energy difference between the spinup and the spin-down state of the valence electron, is given by: We determine the magnetic field by measuring the cyclotron frequency v c ¼ 1 2p q ion m ion B of the ion with electric charge q ion and mass m ion . In the concluding equation for the g-factor: the magnetic field cancels, if in the ratio Gn L Á n c À 1 both frequencies are probed simultaneously. To obtain the g-factor from the measured frequency ratios G, used in equation (1), the atomic masses of the ions are required. While the masses of 40 Ca m( 40 Ca 17 þ ) ¼ 39.953272233 (22) u with a relative mass uncertainty of dm ion Á m ion À 1 ¼ 0.6 parts per billion (p.p.b.; refs 27,28) and also of the electron with dm e Á m e À 1 ¼ 0.03 p.p.b. (ref. 9) are known with sufficient accuracy, the tabulated value of the mass of 48 Ca is not adequately precise. In the following, we report on high-precision measurements of (i) the 48 Ca mass and (ii) the frequency ratios G( 40 Ca 17 þ ) and G( 48 Ca 17 þ ).
Determination of the atomic mass of 48 Ca 17 þ . With the Penning-trap mass spectrometer SHIPTRAP 11 , located at GSI Helmholtzzentrum für Schwerionenforschung Darmstadt, the atomic mass of 48 Ca is directly determined by the measurement of the cyclotron-frequency ratio R of the mass doublet of singly charged 48 Ca þ ions and 12 C 4 þ carbon cluster ions: . Instead of using the Brown-Gabrielse invariance theorem n c 2 ¼ n þ 2 þ n z 2 þ n À 2 (ref. 29), both cyclotron frequencies have been determined as the sum of the ion's two radial eigenfrequencies n c ¼ n þ þ n À , where n þ is the modified cyclotron frequency and n À the magnetron frequency. Considering a mass difference of Dm ¼ m( 12 C 4 þ ) À m( 48 Ca þ )E4.8 Â 10 À 2 u we derive a systematic shift of the mass ratio DR o1 Â 10 À 11 caused by possible misalignments and ellipticity of our trap. At the current level of precision, this effect is negligible.
In each measurement cycle, we produce alternately small clouds (r5 ions) of 48 Ca þ and 12 C 4 þ with a laser-ablation ion source 30 and separately transfer them into a preparation trap for cooling and centring via mass-selective buffer-gas cooling 31 (Fig. 1). Then, the particular cyclotron frequency is measured in the measurement trap with the novel PI-ICR (refs 12,13; Methods section). Combining the measured cyclotron-frequency ratio R ¼ 1.00099010175 (35) stat (17) with the known carbon cluster mass m( 12 C 4 þ ) and correcting for the missing electrons and their corresponding binding energies, we obtain the following value for the mass of lithium-like 48 Ca (Methods section): The resulting atomic mass agrees within its uncertainty with the previous less accurate measurements 32,33 .
Measurement of the Larmor-to-cyclotron frequency ratios.
Using a triple Penning trap set-up located at the University of Mainz, and described in detail in refs 34,35, we measured the Larmor-to-cyclotron frequency ratio G of both calcium isotopes. Within a cryogenic (T ¼ 4.2 K) ultra-high vacuum chamber (Po10 À 16 mbar) a miniature electron beam ion source enables the production of highly charged ions. By means of various cleaning routines 35 we remove all unwanted ion species and finally confine a single ion in a five electrode cylindrical Penning trap with an inner radius of r ¼ 3.5 mm. The oscillating ion induces image charges on the electrode surfaces, which we measure to obtain the axial oscillation frequency. In the attached superconducting, tuned axial resonator the induced oscillating currents generate a measureable voltage signal in the order of a few 10 nV. We detect the signal of the thermalized axial motion (T z B5 K) as a minimum ('dip-signal') in the Fourier transform of the thermal noise spectrum of the tank circuit (Fig. 2a). Both radial modes of the ion are thermalized and detected via rf-sideband coupling to the axial resonator generating double dip-signals in the axial frequency spectra. We determine the cyclotron frequency via the Brown-Gabrielse invariance theorem, where eigenfrequency shifts due to trap misalignment and ellipticity cancel 36 .
Simultaneously to the high-precision phase-sensitive measurement of the modified cyclotron frequency 37 , lasting about 5 s, we inject microwaves (MW) at the assumed Zeeman transition frequency (n MW E105 GHz) into the apparatus to induce spin-flips. To assess the success of a spin-flip attempt in our Precision trap (PT), we analyse the electron spin-state before and after the probing in a spatially separated Penning trap, the Analysis trap (AT). Here, a large magnetic bottle (B 2,z ¼ 10(1) Á 10 3 T m À 2 ) couples the magnetic moment to the axial motion, resulting in frequency jumps of the axial oscillation Image of the ion trap radial motions on a position-sensitive MCP detector x axis y axis n z , which are caused by changes of the electron's spin direction. This so-called continuous Stern-Gerlach effect 38 enables the spin-state detection. In case of the 48 Ca 17 þ ion Dn sf z amounts to only 140 mHz at an absolute frequency of n z ¼ 412.4 kHz, which represents a significant experimental challenge. Figure 2b illustrates the distinct detection of a spinflip in the AT. Considering the limiting axial frequency resolution in the AT, we implement a proper cycle weighting to reduce the statistical uncertainty (Methods section).
During the automated measurement process, we probe the Zeeman transition several 100 times at different MW frequencies n MW . Combining the corresponding measured frequency ratios G* ¼ n MW Á n c À 1 with the binary information of the spin-flip, we obtain a G-resonance (Fig. 2c), which depicts the spin-flip probability in the PT versus the measured frequency ratios. With a weighted Gaussian maximum-likelihood fit, we extract the mean value G mean . This value has to be corrected for several systematic shifts (Methods section and ref. 39).

Discussion
Combining the calcium masses with the measured frequency ratios G( 40 Ca 17 þ ) ¼ 4,282.42953545 (30) and G( 48 Ca 17 þ ) ¼ 5,138.83795612 (42), we derive the most precise g-factor values for lithium-like ions from equation (1): The statistical, systematic and ion mass uncertainties are given separately. The absolute values for the g-factors (Table 2) provide a stringent test of many-electron QED calculations in a magnetic field 40,41 . The g-factor difference finally yields the sought-after isotope difference: where the uncertainties of the frequency ratios and the mass measurements are listed separately. Obviously, the uncertainties in the masses of the isotopes dominate the total uncertainty. Since the dominant systematic shifts of the frequency ratios, the image charge shift (Table 3 and Methods section), scales with the mass of the ion, it cancels in the g-factor difference. Consequently, the denoted systematic uncertainty of the frequency ratios is smaller than the quadratically summed statistical uncertainties of the G-ratios given in equations (3) and (4). The comparison of the measured value of the g-factor difference with the theoretical prediction of this work: allows for the first time a direct test of the relativistic interaction of the electron spin with the motile nucleus. Although at present the experiment confirms the calculation only at the 10% level, the uncertainty of the measured frequency ratios is on the level of the QED recoil contribution. Assuming QED calculations are correct within the given error bar, one may use the small uncertainty of the theoretically predicted g-factor difference in combination with the measured frequency ratios and the mass of 48 Ca 17 þ to determine the isotopic mass difference: Dm ¼ m( 48 Ca) À m( 40 Ca) ¼ 7.9899317834 (54) u. The uncertainty of this indirectly obtained mass difference is a factor 5.7 smaller than the directly measured mass difference.
The combination of high-precision measurements of Larmorto-cyclotron frequency ratios, atomic masses of the lithium-like isotopes 40 Ca 17 þ and 48 Ca 17 þ and corresponding g-factor calculations, presented in this paper, enables a variety of fundamental studies. Besides the test of many-electron QED calculations in a magnetic field by considering the absolute values of the g-factors or the indirect determination of the isotopic mass difference, the analysis of the measured and predicted g-factor difference between the calcium isotopes deepens the understanding of the interaction between the bound electrons and the nucleus. A further reduction of the mass uncertainties will enable an even more stringent test of the relativistic recoil predictions in the future. The validation of QED calculations is a prerequisite for further fundamental measurements in atomic physics, for example, the determination of the fine structure constant a via g-factor measurements of heavy, highly charged ions 14 .

Methods
Calculation of the isotope shift. The main contribution to the isotope shift Dg ¼ g( 40 Ca 17 þ )-g( 48 Ca 17 þ ) results from the nuclear recoil effect that must be calculated including the relativistic, QED and interelectronic-interaction contributions. As the nuclear size effect is rather small, it can be evaluated in one-electron approximation by solving the Dirac equation. Consider first the nuclear recoil effect on the atomic g-factor to zeroth order in 1/Z. In this approximation, the m e Á m nucl À 1 nuclear recoil contribution to the g factor of an ion with one electron over closed shells is given in refs 22,23.
Here, ' ¼ c ¼ 1, eo0, m B is the Bohr magneton, m a is the angular momentum projection of the state a, p k ¼ À ir k is the momentum operator, A cl ¼ [B Â r]/2 is the vector potential of the homogeneous magnetic field B directed along the z axis, is the transverse part of the photon propagator in the Coulomb gauge. The tilde sign indicates that the related quantity (the wave function, the energy and the Coulomb-Green functionG o ð Þ) must be calculated in presence of the homogeneous magnetic field B directed along the z axis. As we consider an ion with one valence electron over the closed shells, the Coulomb-Green function is defined asGðoÞ ¼ Pññ j iñ h j o ÀẼ n þ iZẼ n ÀẼ F À Á Â Ã À 1 , whereẼ F is the Fermi energy and Z-0. In equation (7), the summation over the repeated indices (k ¼ 1,2,3), which enumerate components of the three-dimensional vectors, is implicit. Formula (7) incorporates both one-and two-electron nuclear recoil contributions to zeroth order in 1/Z. For the (1s) 2 2s-state of a lithium-like ion, the (1/Z) 0 two-electron contribution is zero and, therefore, we restrict our consideration to the one-electron contribution only. For the practical calculations, the one-electron contribution is conveniently represented by a sum of low-order ('non-QED') and higher order ('QED') term, Dg ¼ Dg L þ Dg H : where V(r) ¼ À (aZ)/(r) is the Coulomb potential induced by the nucleus and n ¼ r Á r À 1 . The low-order term can be derived from the relativistic Breit equation, while the derivation of the higher-order term requires using QED beyond the Breit The Dirac value, as well as the QED, interelectronic-interaction and screened QED corrections cancel in the g-factor difference. The two predicted g-factors agree with the measured values. approximation. For this reason, we call them the non-QED and QED contributions, respectively. The low-order term Dg L can be evaluated analytically 42 : where E is the Dirac energy and k ¼ ð À 1Þ j þ l þ 1 2 j þ 1 2 À Á . To the two lowest orders in aZ, we have As follows from this formula, for an s-state (k ¼ À 1) the non-relativistic contribution to Dg L vanishes and the low-order term comes from pure relativistic (B(aZ) 2 ) origin. The calculation of the higher order term, Dg H , is a much more difficult task. For the 1s-state it is calculated in ref. 42. In the present paper we performed the corresponding calculation for the 2s-state. Details of this calculation and the corresponding results for other ions will be published elsewhere.
In addition to the main one-electron nuclear recoil contribution, we have to consider the radiative (Ba) nuclear recoil correction and the (m e /m nucl ) 2 nuclear recoil correction. To the lowest order in aZ, these corrections were evaluated in refs 19,21,23,24. We need also to account for the interelectronic-interaction effects of the first and higher orders in 1/Z. To evaluate these effects we extrapolate the lowest order relativistic results from ref. 25. The uncertainty of the interelectronic-interaction contribution is mainly due to uncalculated higher-order relativistic and QED corrections.
To get the total value of the isotope shift, we also evaluate the nuclear size correction. The root-mean-square nuclear charge radii and their uncertainties are taken from ref. 7. The uncertainty of the nuclear size contribution includes both the nuclear radius and shape variation effects. The individual contributions to the isotope shift of the g-factor for 40 Ca 17 þ and 48 Ca 17 þ are presented in Table 1.
In Table 2 we list the various contributions to the g-factor of 40 Ca 17 þ and 48 Ca 17 þ . The Dirac value, as well as the QED, interelectronic-interaction, and the screened QED corrections 17 cancel out in the isotope difference. The finite nuclear size and nuclear recoil corrections lead inherently to the isotope shift.
The PI-ICR measurement scheme. After the transfer of the ions from the preparation trap into the centre of the measurement trap (Fig. 1), the coherent components of their magnetron and the axial motions are damped via 1 ms dipole rfpulses at the corresponding motional frequencies to amplitudes of about 0.01 and 0.4 mm, respectively. These steps are required to reduce a possible shift in the ratio of the 48 Ca þ and 12 C þ 4 ions due to the anharmonicity of the trap potential and inhomogeneity of the magnetic field to a level well below 10 À 10 (see ref. 13 for details). After this preparatory step, the radius of the ion cyclotron motion is increased to a radius of 0.5 mm to set the initial phase of the cyclotron motion. Then, two excitation patterns, called in this work 'magnetron-motion phase' and 'cyclotron-motion phase', are applied alternately to measure the ion cyclotron frequency n c . In the 'magnetron-motion phase' pattern the cyclotron motion is first converted to the magnetron motion with the same radius. Then, the ions perform the magnetron motion for 100 ms accumulating a certain magnetron phase. After 100 ms have elapsed, the ions' position in the trap is projected onto a positionsensitive detector by ejecting the ions from the trap towards the detector 43 . In the 'cyclotron-motion phase' pattern the ions first perform the cyclotron motion for 100 ms accumulating a certain cyclotron phase with a consecutive conversion to the magnetron motion and again projection of the ion position in the trap onto a position-sensitive detector. The angle between the ion-position images corresponding to two patterns with respect to the trap centre image is proportional to the ion cyclotron frequency n c . Pulse patterns are applied for a total measurement time of B5 min. On this measurement scale the 'magnetron-motion phase' and 'cyclotron-motion phase' can be considered to be measured simultaneously. Data with 45 detected ions per cycle are not considered in the analysis to reduce a possible shift in the ratio of the 48 Ca þ and 12 C 4 þ ions due to ion-ion interaction. To eliminate a possible cyclotron-frequency shift, which arises due to incomplete damping of the coherent component of the magnetron motion, the time between the damping of the magnetron and axial motions and the excitation of the ion cyclotron motion is varied over the period of the magnetron motion. The positions of the magnetron motion and cyclotron motion phase spots are chosen such that the angle between the phase spots, calculated with respect to the centre of the measurement trap, do not exceed few degrees. This is required to reduce the shift in the ratio of the 48 Ca þ and 12 C 4 þ ion masses due to the possible distortion of the ion-motion projection onto the detector to a level well below 10 À 10 (ref. 13).
Data sets for the ion cyclotron-frequency ratio R. The cyclotron frequencies n c of the 48 Ca þ and 12 C 4 þ ions are measured alternately for several days. The total measurement period is divided in 45 B1-h periods. In addition, each 5 min measurement is divided in 10 30-s periods. For each of the 45 1-h periods the ratio R 1 h of the cyclotron frequencies 48 Ca þ and 12 C 4 þ ions is obtained along with the inner and outer errors 44 by fitting to the 12 C þ 4 frequency points a polynomial of fifth order P 2 (t) with constant coefficients a 0 , a 1 , a 2 , a 3 , a 4 and a 5 and to the 48 Ca þ frequency points a polynomial P 1 (t) ¼ R 1 h Â P 2 (t). The final cyclotron-frequency ratio R mean is the weighted mean of the R 1 h ratios, where the maximum of the inner and outer errors of the R 1 h ratios are taken as the weights to calculate R mean (Fig. 3). The difference between the inner and outer errors does not exceed 10%. The final frequency ratio R with its statistical and systematic uncertainties is R mean ¼ 1.00099010175 (35) stat (17) syst . The systematic uncertainty in the frequency-ratio determination originates from the anharmonicity of the trap potential, the inhomogeneity of the magnetic field and the distortion of the ion-motion projection onto the detector 13 .
The atomic mass of 48 Ca 17 þ . The mass of a C 4 þ cluster is calculated by considering the dissociation energy: E diss ¼ 18.0(17) eV (ref. 45), the ionization energy: E ion ¼ 11.0(7) eV (ref. 46) and the missing electron: The atomic mass of neutral 48 Ca. For completeness, we also specify the atomic mass of neutral 48 Ca. Correcting for the mass of the missing electron and its : The PT-weight finally has to be included in the maximum-likelihood function: which is used, to extract the final mean value G mean . In comparison to the common cut-analysis, we improve the relative uncertainty of G mean by 20 p.p.t.
Data sets of the C-resonances. Various G-resonances are recorded at different modified cyclotron energies during the phase-evolution time of the modified cyclotron mode and the simultaneous probing of the Larmor frequency n MW in the PT. In Fig. 5 the mean values from the maximum-likelihood fit, see equation (19), are plotted for 40 Ca 17 þ (a) and 48 Ca 17 þ (b). The slope is given mainly by the relativistic mass shift in the cyclotron frequency. The Larmor frequency is far less susceptible to relativistic shifts owing to the slow Thomas precession of the electron, which is bound to the heavy ion, leading to a suppression by a factor n L =n c . From linear extrapolations to zero modified cyclotron energy we derive our statistical G-values: which have to be corrected by systematic shifts.
Systematic shifts and uncertainties of C( 40 Ca 17 þ ) and C( 48 Ca 17 þ ). The systematic shifts of the Larmor-to-cyclotron frequency ratios and the corresponding uncertainties are listed in Table 3. The dominant systematic shift and uncertainty is given by the image charge shift. Here, the induced image charges at where r is the inner radius of the Penning trap. Due to the r À 3 -scaling, this shift can be reduced in future experiments by increasing the size of the Penning trap. All other systematic shifts, which are at least one order of magnitude smaller than the image charge shift, are explained in refs 10,35.