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1 SUPPLEMENTARY INFORMATION doi: /nature1306 Statistical measurement uncertainty The statistical uncertainty of the experimentally determined ratio of Larmor- and cyclotron frequencies, which we denote as Γ 0 ν LL ν cc, results from the confidence interval for the estimated parameters of the fit to the recorded spin-flip resonances (Fig. 1). As each individual measurement only yields a boolean information of a possible change of the spinstate after a microwave excitation rather than a self-contained Larmor-frequency information, only an ensemble of a few hundred measurements can be used to extract the Larmor frequency with a line fit. The individual events of these resonances, consisting of a measured cyclotron frequency and the corresponding microwave excitation frequency, are randomly distributed. In order to circumvent artefacts and bias, the resonances are fitted without any binning using the maximum-likelihood method. The binned data shown in Fig. 1 is used solely for presentation purposes. Due to the significantly lower ion temperature and magnetic field inhomogeneity in the Precision Trap compared to previous experiments [1], the line-width of the resonance is not any more limited by the Boltzmann distribution of the axial energy, but rather by the random magnetic field jitter during one measurement cycle. While the cyclotron frequency, determined with the pulse and amplify (PnA) technique, yields the average magnetic field during the measurement, the spin dynamics follows a more complicated averaging procedure. During the microwave excitation, which takes typically between 5 and 10 seconds, the magnetic field jitter, along with the finite stability of the frequency standards, renders the fast (~105 GHz) Larmor oscillation incoherent. As a result, for small microwave drive amplitudes the Larmor (spin-flip) resonance PP SSSS (ν MMMM ) can be modelled in the rotating-wave approximation of the Rabi oscillation equation (neglecting sidebands resulting from a rapid modulation of the magnetic field with the ion s motional frequencies): PP SSSS (ν MMMM ) = 1 Ω 0 Ω +(ν MMMM ν LL0 (1+αα EE EE EE + δδδδ ee )) BB0 EEEE kk BB TTEE ff(δδδδ)dddddd ddδδδδ, (S1) where ν MMMM is the microwave drive frequency, ν LL0 the unperturbed Larmor frequency, ΩΩ the Rabi frequency, which depends on the drive strength, k B the Boltzmann constant, T z the axial temperature, B 0 the homogeneous part of the magnetic field and αα zz = BB ( ππ) BB 0 mm iiiiii ν describes the shift of the Larmor (as well as the cyclotron) frequency as a result EE of a finite axial energy (see Eq. (S3)). In our case, the expectation value of this shift ( δν LL ν LL compared to the typical variations of the magnetic field during the measurement ( δν LL ν LL ) is small ). For random variations of the magnetic field ff(δδδδ), the resulting line-shape is symmetric and shifted slightly by ν LL0 αα zz dd zz. Since simultaneously also the cyclotron frequency shifts by the same relative amount, the ratio is invariant with respect to 1

2 RESEARCH SUPPLEMENTARY INFORMATION simultaneously also the cyclotron frequency shifts by the same relative amount, the ratio Γ is invariant with respect to the axial frequency to very good approximation (see Eq. (S6)). The relative full width at half maximum (FWHM) of the resonance in our setup is about Each resonance contains roughly 400 individual data points, which allows determining the resonance centroid with a precision of about 10% of the standard width of the resonance. Various resonances have been recorded for different ion temperatures to verify the validity of the model of systematic shifts. The extrapolation to zero energy of the modified cyclotron motion yields Γ 0 with a relative precision of , which is subsequently corrected for systematic shifts. Systematic shifts Many of the inherent systematic uncertainties typically associated with precision Penning-trap mass measurements cancel to a good extent for bound electron g-factor measurements. Here, the particle of interest (electron) and the reference (ion) are naturally at the same location. By determining Larmor- and cyclotron frequency ν L and ν c simultaneously, both temporal and spatial variations of the magnetic field cancel in the ratio Γ 0 of these frequencies. However, the techniques used to determine these frequencies result in a number of subtle effects that have to be considered. Despite careful optimisations, the Penning-trap apparatus designed for this measurement has a number of residual deviations from the ideal Penning trap, mostly the finite size of the trap electrodes and the finite energy in the motional modes of the ion. The considered effects include, amongst others: Interactions with the trap electrodes While the ideal Penning trap features a quadrupole potential extending into infinity, any real-world apparatus needs an electrode structure with finite size to reproduce this potential. The trapped ion will induce both image currents and charges into the electrodes. The image current creates a voltage drop across the electrodes depending on the impedance. Consequently, this voltage drop becomes significant only close to the resonance frequencies of the detection tank circuits. While the interaction with the axial tank circuit is accounted for already in the lineshape used for the fit to the dip signal, the cyclotron tank circuit would cause potentially serious systematic shifts. During the measurement, the tank circuit frequency was detuned by about 1.8 full linewidths, limiting the size of the cyclotron frequency shift to only.0(5) ppt. The strength of the image charge interaction on the contrary is independent of the frequency. This effect was recently re-evaluated by our group []:

3 SUPPLEMENTARY INFORMATION RESEARCH δδνν cc mm = 1.9 iiiiii, (S) νν cc 8 ππ εε 0 BB aa 3 where m ion is the mass of the ion, ε 0 the vacuum permittivity, B 0 the magnetic field strength and a the inner radius of the trap electrodes. The limitation for the precision of this correction comes from the uncertainty of the ion s position with respect to the trap electrodes due to patch potentials and machining imperfections. While we have very little information of the ion position in the Precision Trap, we can derive a decent estimate from measurements in the geometrically identical Analysis Trap (AT), where the known magnetic bottle reveals an offset of no more than 90 µm. From numerical calculations we assign an uncertainty of 5% to the cyclotron frequency shift, which corresponds to an offset of less than 00 µm which seems reasonable. Still, this uncertainty sets a limitation to the achievable accuracy of the experiment. However, the strong scaling with the trap radius will allow reducing this effect significantly in future experiments. Ion temperature During the measurement the axial motion is continuously in thermal equilibrium with the tank circuit. The cyclotron motion in contrast is normally decoupled from the environment and is cooled by sideband-coupling to the axial motion. The development of a new cryogenic amplifier has enabled us to reduce the effective temperature of the axial tank circuit considerably compared to previous experiments, i.e. down to 5.5(1) K. During the cooling process, noise feedback is used to reduce the effective temperature of the tank circuit even below the 4. K of the environment. Nonetheless, the finite motional amplitudes cause the ion to scan a certain range of the residual magnetic inhomogeneity and electrostatic anharmonicity. Residual axial energy The finite axial oscillation amplitude acts to shift both the measured cyclotron as well as Larmor frequency. The shifts due to B can be deduced from the matrix [3]: δδδδ + δδ + δδδδ zz δδ zz δδδδ = δδ δδδδ LL δδ LL BB BB 0 mm iiiiii ωω EE δδzz δδ dd + 1 δδzz δδ + 1 dd zz dd. (S3) The instantaneous shift of the Larmor frequency due to the axial energy is thus given by: δδδδ LL BB δδ LL = E BB 0 mm iiiiii ωω z. EE (S4) At the same instant, the calculated cyclotron frequency shifts to very good approximation by 3

4 RESEARCH SUPPLEMENTARY INFORMATION δδδδ cc δδcc = BB BB 0 mm iiiiii ωω ωω + ωω EE ωω cc Thus, in the instantaneous frequency ratio Γ(E + ) most of the shifts cancel: δδγ Γ = BB BB 0 mm iiiiii ωω EE 1 ωω + E zz = δδδδll δδ LL ωω + ωω ωω 50 ppt. (S5) cc ωω ωω cc E zz δδδδll δδ LL 0.04 ppt. (S6) However, the axial energy changes continuously during the measurement process due to the thermalisation with the tank circuit. While the determined cyclotron frequency reflects the expectation value of the energy, the spin-flip probability is distributed according to the axial energy distribution, leading to a slightly asymmetric resonance shape. However, in our setup this asymmetry is completely masked by the magnetic field jitter, which makes the resonance Gaussian. The centroid of this resonance is then shifted according to equation (S6), as long as a non-linear deformation of the resonance due to saturation can be neglected, which is the case for all recorded data sets. Residual magnetron energy The magnetron mode is cooled by radiofrequency sideband coupling to the thermalised axial motion. The resulting temperature of TT = ωω ωω EE TT zz is sufficiently low such that the shift according to Eq. (S3): δδδδ + δδ + = BB ωω BB 0 mm iiiiii ωω3 EE dd zz (S7) can be safely neglected. Beyond that, the shift again cancels to good extent in the frequency ratio Γ. Residual cyclotron energy Even though the PnA detection technique [4] helps to keep the cyclotron energy during the measurement low, it still gives the dominant contribution to the energy dependent systematic shifts. While this shift is very small for the lowest energies used, several resonances with different cyclotron energies have been recorded to allow an extrapolation to zero energy and a cross-check of the model of systematic shifts. The finite cyclotron radius can lead to sizable systematic shifts due to the residual magnetic inhomogeneity and special relativity. The shift of the Larmor- and modified cyclotron frequency due to magnetic inhomogeneity is given by the shift matrix (S3): δδδδ LL δδ LL = δδδδ+ δδ + BB E BB 0 mm iiiiii ωω + +,hot. (S8) The calculated free cyclotron frequency in addition has a contribution from the residual cyclotron energy E +,cold during the axial frequency measurement. δδδδ cc δδcc BB BB 0 mm iiiiii ωω + E +,cold E +,hot. (S9) Thus, this component survives in the ratio Γ and generates a small shift: 4

5 SUPPLEMENTARY INFORMATION RESEARCH δδγ BB Γ E BB 0 mm iiiiii ωω +,cold 1.4 ppt. + (S10) Special relativity acts differently on the cyclotron and Larmor frequencies. While the cyclotron frequency shifts according to the relativistic mass increase: δδδδ cc δδcc = EE + mm iiiiii cc, (S11) the Larmor frequency experiences only a much smaller shift due to the Thomas precession: δδδδ LL δδ LL = δδ cc δδll EE + mm iiiiii cc. (S1) While for a free electron these two shifts would cancel in the ratio, for the much heavier ion one obtains δδγ Γ δδδδ cc δδcc. (S13) By combining the equations (S9) and (S13) it becomes possible to calibrate the cyclotron energy. Finally, the cyclotron energy causes a motional magnetic field resulting from the Lorentz transformation of the electrostatic trapping field dd : This contribution is negligible for our energies. Characterisation of the residual field imperfections Magnetic field δδδδ ββ cc dd. (S14) The dominant secular contribution to the ion s eigenfrequencies in the PT results from the quadratic portion of the residual magnetic field imperfections (B or magnetic bottle ). Several techniques are applied to determine this value. The most direct method uses asymmetric offset potentials at the endcap electrodes in order to shift the equilibrium position of the ion in several steps of roughly 50 µm. For each position first the electrostatic potential is made harmonic by tuning the correction voltages and the magnetic field is determined by a measurement of the cyclotron frequency. The magnetic field plot as a function of the equilibrium position, which can be calculated from the applied potentials, allows to determine B using a polynomial fit. However, in the g-factor trap the main source of inhomogeneity is the ferromagnetic ring in the AT. As a result, the magnetic field inhomogeneity in the PT is predominantly a linear gradient, which complicates the precise extraction of the small quadratic term: B = (0.56±1.30) T/m. A more precise value can be achieved by comparing the shift of the axial and the cyclotron frequency following an excitation of the cyclotron motion. While the axial frequency is shifted predominantly by the magnetic bottle, the cyclotron frequency has an additional contribution of, in our case, similar size by the relativistic mass increase (S3 and S11). The ratio of the frequency shifts thus allows extracting B = (0.97±0.13) T/m. 5

6 RESEARCH SUPPLEMENTARY INFORMATION Electrostatic We optimise the electrostatic field by measuring the axial frequency shift following a magnetron excitation with varying amplitudes for different values of the correction voltages. The resulting array of curves can be fitted with the wellknown expressions for the electrostatic frequency shifts. The resulting optimal correction voltage setting is eventually corrected for the small B shift. Previous world data average of the electron mass The current (010) CODATA value comprises 4 individual results. The systematic error analysis done for this work has revealed two flaws in the analysis of the 004 data of Verdú et al. [5]. Firstly, the interaction with the image charges was not included and secondly, the correction of the axial energy shift was incorrect. Re-evaluating the data causes the result to shift by more than one standard deviation. Following the procedure outlined in [6], it is possible to compile a corrected world average value. According to our calculation, the resulting value of the electron mass is about 0.6 standard deviations smaller than the current CODATA [6] value and thus in decent agreement with the new, much more precise value. Measurement Measurement sequence sequence Spin-state Detection in AT Microwaves Transport to PT Wait Double-Dip (10s) Dip PnA (5-10s) Microwaves Dip Transport to AT FIG. S1: Overview of the measurement sequence 6

7 SUPPLEMENTARY INFORMATION RESEARCH Definition of symbols: Symbol Meaning ν L ν MW ν c ν + ν z ν - E + E z E - a q m ion ε 0 B 0 B Γ β c k B Table SI: Definition of used symbols Spin precession (Larmor) frequency Microwave drive frequency Free space cyclotron frequency Modified cyclotron frequency Axial frequency Magnetron frequency Modified cyclotron energy Axial energy Magnetron energy Trap radius Ion charge Ion mass Vacuum permittivity Magnetic field, homogeneous component Magnetic field, second-order component Frequency ratio ν L /ν c v/c Speed of light Boltzmann constant + The theoretical value of the g-factor of 1 C 5 Various physical effects contribute to a theoretical value of the g-factor. Apart from the leading relativistic term there are the one- and two-loop QED terms as well as effects originating from the nucleus, namely, the recoil contribution and the nuclear finite size effect. Further small terms from nuclear structure may arise such as the nuclear polarisability and deformation correction. A recent review of the theoretical formalism might be found in Refs. [6, 7]. In Table SII we list individual contributions for hydrogenlike C and Si (Z=6,14, respectively). These terms have been reasonably well tested before in Ref. [8] by comparing the theoretical and experimental values at Z=14 where all bound-state effects are magnified as compared to the case of Z=6 due to power laws. Figure S3 shows the scaling properties of various contributions. 7

8 RESEARCH SUPPLEMENTARY INFORMATION FIG. S: Modulus of various contributions to the absolute g-factor as functions of the nuclear charge Z. As it has been already mentioned, the main challenge for the theory is related to the so far unknown two-loop QED correction at orders higher than ( Z ) 4, ho α which we denote by g ( Z ). L One may obtain an estimation of the effect for ho carbon by means of extraction of g ( Z 14) L = from comparison of the theory and the experimental result for silicon and subsequent rescaling of it from Z = 14 to Z = 6. In analogy to the Lamb shift, the higher-order two-loop QED effect is assumed to be described by the formula b 6 g ho L α π { [ ] 5 3 ( Z ) = ( Zα ) b +b ( Zα ) ln ( Zα ) ( Zα ) ln ( Zα ) [ ] +b ( Zα ) ln[ ( Zα ) ] +b ( Zα ) + }, where terms of higher order with respect to powers of ( Zα ) are not taken into account. 61 In the notation for the b nl coefficients, n stands for the power of ( Zα) and l is a power of the logarithmic term. The (S15) expansion coefficients with n 5 have not been calculated thus far. Formally, the leading contribution to the righthand side of Eq. (S15) is related to b 50 be significant., but, in principle, the logarithmically enhanced terms of the next order may also 8

9 SUPPLEMENTARY INFORMATION RESEARCH a The result was obtained in this work. The details are discussed at length in the text. Theory 1 C 5+ 8 Si 13+ Ref. <r > 1/ [fm].4703() 3.13 (4) [9] Dirac value (1) (5) Finite nuclear size (44) TW One-loop QED (Zα) (1) (1) [6] (Zα) [10] (Zα) [7] h.o. SE (4) (60) TW+ [11] h.o. SE-FS () (0) TW+ [11] h.o. VP-EL (1) [1] h.o. VP-ML (10) [13] Two-loop QED (Zα) [6] (Zα) [10] (Zα) [7] h.o (3) a (1650) following [7] Recoil m e /m nucl [14] rad-rec [1] h.o [15] Nuclear polarisability (0) TW+ [16] Nuclear deformation [17] Total (6) (165) TABLE SII: Values of individual contributions to g( 1 C 5+ ) and g( 8 Si 13+ ). The abbreviations stand for: h.o." - a higherorder contribution, SE" - self-energy correction, SE-FS" - mixed self-energy and nuclear finite size correction, VP- EL" - electric-loop vacuum polarisation correction, VP-ML" - magnetic-loop vacuum polarisation correction, rad-rec" - the leading term of the mixed radiative-recoil correction. TW indicates results obtained in this work. See the text for details. 9

10 RESEARCH SUPPLEMENTARY INFORMATION We determine b50 as follows: Firstly, we restrict ourselves to the leading term in Eq. (S15) that only includes the b50 parameter. Then a comparison of the experimental and theoretical value reads * where g th ( Z ) α π * 5 ( Z ) = g ( Z ) + ( Zα ) b, gexp th 50 (S16) denotes the theoretical prediction for the g factor including only the known corrections, i.e., those without the higher-order two-loop effect. Let us now recall the relation between g exp ( Z ) and the frequency ratio Γ that is determined in an experiment: me gexp( Z ) = ( Z 1) Γ( ion), (S17) m and employ it along with Eq. (S16) to obtain a set of equations for carbon and silicon, namely, ion α π 5 ( 6α) b = ( 6 1) 50 m ion m 1 5+ ( C ) 1 5+ * ( C ) g ( ), e Γ th 6 (S18) * ( Si ) g ( ), α 5 me ( 14α) b50 = ( 14 1) Γ th 14 (S19) π m ( Si ) with the ions' masses depending on the electron mass through the formula AA where dd BB ( XX) ion AA mm( XX qq+ AA AA ) = mm( XX) qqmm ee + dd BB ( XX) dd BB ( XX qq+ ), (S0) AA is the binding energy of electrons in atom AA AA XX, expressed in atomic mass units, and dd BB ( XX qq+ ) is the binding energy of the electrons in an ion AA XX qq+, expressed again in atomic mass units. Specifically, binding energies for carbon ions can be found in Ref. [18], whereas for silicon ions in Ref. [19]. For the purpose of our calculation, in the above formula it is sufficient to substitute an old value of the electron mass since it is small compared to the nuclear mass. Therefore, we can treat the ions' masses on the right-hand sides of Eqn. (S18-S19) as known parameters. Those equations can then readily be solved for the variables m e and b 50, mm ee = namely, 43 gg tth(14) 16807ggth (6) mmiiiiii ( 1 C 5+ )mm iiiiii ( 8 Si 13+ ) [3159mm iiiiii ( 1 C 5+ )ΓΓ( 8 Si 13+ ) 84035mm iiiiii ( 8 Si 13+ )ΓΓ( C 5+ bb 50 = ππ 13 gg tth(6)mmiiiiii 1 C 5+ ΓΓ 8 Si gg th(14)mmiiiiii ( 8 Si 13+ )ΓΓ C αα 7 [84035 mm iiiiii ( 8 Si 13+ )ΓΓ( 1 C 5+ ) 3159 mm iiiiii ( 1 C 5+ )ΓΓ( 8 Si 13+ )] 1 )], (S1). (S) One obvious source of uncertainty of our value of m e comes from the error bars of the quantities present in Eqn. (S1) and (S). That contribution can be obtained according to the standard error propagation formula. The theory values occurring in Eqn. (S1-S) together with uncertainties read: 10

11 SUPPLEMENTARY INFORMATION RESEARCH g tth ( 1 C 5+ ) = (47), g tth ( 8 Si 13+ ) = (81). The uncertainties of relevant contributing corrections can be found in Table SII. The values of the one-loop higher-order self-energy and mixed self-energy-finite-size corrections were extrapolated from results for higher Z values in Ref. [11]. The masses and the fine-structure constant occurring in Eqn. (S1-S) read: and the experimental frequency ratio values are: mm iiiiii ( 1 C 5+ ) = (11)uu, mm iiiiii ( 8 Si 13+ ) = (5)uu, α = (4) Г( 1 C 5+ ) = (11)(7) and Г( 8 Si 13+ ) = (13)(13) (from Ref. []). Another source of uncertainty is the presence of unknown b6k parameters in Eq. S15. Clearly, one cannot rigorously fit more than one b parameter since one has only two equations at hand. Therefore, we tested various configurations of the b's to assess the sensitivity of our results subjected to changes of these parameters. Our estimation obtained in this way 16 is Δ m = This uncertainty was linearly added to the theory's uncertainty. Our final value for the electron's mass b e and the b 50 parameter read: mm ee = (14)(9)()uu, bb 50 = 4.0(5.1). [1] H. Häffner, T. Beier, N. Herrmanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú and G. Werth, High- Accuracy Measurement of the Magnetic Moment Anomaly of the Electron, Phys. Rev. Lett., vol. 85, no. 5, pp , 000. [] S. Sturm, A. Wagner, M. Kretzschmar, W. Quint, G. Werth and K. Blaum, g-factor measurement of hydrogenlike 8 Si 13+ as a challenge to QED calculations, Phys. Rev. A, vol. 87, no. 3, p (R), 013. [3] L. S. Brown and G. Gabrielse, Geonium theory: Physics of a single electron or ion in a Penning trap, Rev. Mod. Phys., vol. 58, no. 1, p , [4] S. Sturm, A. Wagner, B. Schabinger and K. Blaum, Phase-Sensitive Cyclotron Frequency Measurements at Ultralow Energies, Phys. Rev. Lett., vol. 107, no. 14, p , 011. [5] J. Verdú, S. Djekić, T. Valenzuela, M. Vogel, G. Werth, T. Beier, H.-J. Kluge and W. Quint, Electronic g Factor of Hydrogenlike Oxygen 16 O 7+, Phys. Rev. Lett., vol. 9, no. 9, p ,

12 RESEARCH SUPPLEMENTARY INFORMATION [6] P. J. Mohr, B. N. Taylor and D. B. Newell, CODATA recommended values of the fundamental physical constants: 010, Rev. Mod. Phys., vol. 84, no. 4, p , 01. [7] K. Pachucki, A. Czarnecki, U. D. Jentschura and V. A. Yerokhin, Complete two-loop correction to the bound-electron g factor, Phys. Rev. A, vol. 7, no., p. 0108, 005. [8] S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. Keitel and K. Blaum, g Factor of Hydrogenlike 8 Si 13+, Phys. Rev. Lett., vol. 107, no., p. 0300, 011. [9] I. Angeli, A consistent set of nuclear rms charge radii: properties of the radius surface R(N,Z), At. Data Nucl. Data Tables, vol. 87, no., p , 004. [10] H. Grotch, Electron g Factor in Hydrogenic Atoms, Phys. Rev. Lett., vol. 4, no., pp. 39-4, [11] V. Yerokhin, P. Indelicato and V. Shabaev, Evaluation of the self-energy correction to the g factor of S states in H-like ions, Phys. Rev. A, vol. 69, no. 5, p , 004. [1] T. Beier, The g j factor of a bound electron and the hyperfine structure, Phys. Rep., vol. 339, no., pp , 000. [13] R. N. Lee, A. I. Milstein and I. S. Terekhov, Virtual light-by-light scattering and the g factor of a bound electron, Phys. Rev. A, vol. 71, no. 5, p , 005. [14] V. M. Shabaev and V. A. Yerokhin, Recoil Correction to the Bound-Electron g Factor in H-Like Atoms to All Orders in αz, Phys. Rev. Lett, vol. 88, no. 9, p , 00. [15] K. Pachucki, Nuclear mass correction to the magnetic interaction of atomic systems, Phys. Rev. Lett., vol. 78, no. 1, p , 008. [16] A. V. Nefiodov, G. Plunien and G. Soff, Nuclear-Polarization Correction to the Bound-Electron g Factor in Heavy Hydrogenlike Ions, Phys. Rev. Lett., vol. 89, no. 8, p , 00. [17] J. Zatorski, N. S. Oreshkina, C. H. Keitel and Z. Harman, Nuclear Shape Effect on the g Factor of Hydrogenlike Ions, Phys. Rev. Lett., vol. 108, no. 6, p , 01. [18] P. J. Mohr and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 00, Rev. Mod. Phys., vol. 77, no. 1, pp , 005. [19] W. C. Martin and R. Zalubas, Energy Levels of Silicon, Si I through Si XIV, J. Phys. Chem. Ref. Data, vol. 1, no., p. 33,