Abstract
The magnetic moment μ of a bound electron, generally expressed by the gfactor μ=−g μ_{B} s ħ^{−1} with μ_{B} the Bohr magneton and s the electron’s spin, can be calculated by boundstate quantum electrodynamics (BSQED) to very high precision. The recent ultraprecise experiment on hydrogenlike silicon determined this value to eleven significant digits, and thus allowed to rigorously probe the validity of BSQED. Yet, the investigation of one of the most interesting contribution to the gfactor, the relativistic interaction between electron and nucleus, is limited by our knowledge of BSQED effects. By comparing the gfactors 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 lithiumlike calcium ions. The good agreement between the theoretical predicted recoil contribution and the highprecision gfactor measurements paves the way for a new generation of BSQED tests.
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Introduction
Besides 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 boronlike argon^{4} and lithiumlike 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 gfactors of lithiumlike calcium isotopes ^{40}Ca^{17+} and ^{48}Ca^{17+}. 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 gfactors of highly charged ions, for example, the relativistic, radiative, nuclear size or interelectronicinteraction corrections, are calculated using boundstate quantum electrodynamics (QED) in the infinitenuclearmass 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, boundstate 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 gfactor difference Δg≡g(^{40}Ca^{17+})−g(^{48}Ca^{17+}), which is in the order of 1 × 10^{−8}, requires four independent highprecision measurements: the Larmortocyclotron 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 × 10^{−11}. For this purpose, we studied single ions confined in a dedicated Penningtrap setup^{9,10}. Aiming for atomic masses with relative uncertainties of about 4 × 10^{−10}, we also improved the atomic mass of ^{48}Ca by a factor of seven. Here, we used the offline configuration of the Penningtrap mass spectrometer SHIPTRAP (ref. 11) in combination with the novel phaseimaging ioncyclotron resonance technique (PIICR)^{12,13}. The finally obtained 1.0σ agreement between the predicted and measured gfactor 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 α, and nuclear properties via gfactor measurements in heavy atomic systems^{14}.
Results
Calculation of the gfactor difference
The theoretical value of the isotope shift in the atomic gfactors is mainly given by a sum of the nuclear recoil and nuclear size contributions. Considering sstates of highly charged ions, the leading order terms scale with and (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 gfactor difference. For Z=20 the isotope shift is essentially determined by the mass shift, which in the case of sstates is of pure relativistic origin. Considering the two double magic isotopes ^{40}Ca and ^{48}Ca, the nuclear charge radii r_{nucl}(^{40}Ca)=3.4776 (19) fm and r_{nucl}(^{48}Ca)=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 gfactor difference of the lithiumlike electron configuration to 99.96%.
The lowest order recoil correction, which is nonQED but relativistic, can be derived from Breit equation^{18,19,20,21}. The full relativistic theory of the nuclear recoil effect on the atomic gfactor 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}·m_{nucl}^{−1} and to all orders in Zα (ref. 22). As a result, the complete Zαdependence formula for the recoil effect on the gfactor of a hydrogenlike ion has been derived. To zeroth order in Z^{−1}, this formula describes also the recoil effect in a fewelectron 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 twoelectron nuclear recoil contributions. However, for the (1s)^{2}2sstate of a lithiumlike ion, the twoelectron contributions vanish, and, therefore, to zeroth order in Z^{−1}, one has to evaluate the oneelectron contribution only. In the present paper, we evaluate this contribution to all orders in Zα for the 2sstate at Z=20 using the corresponding formula^{22}. This result is combined with the radiative and second order in m_{e}·m_{nucl}^{−1} recoil corrections^{19,21,23,24} to get the total oneelectron contribution. To evaluate the interelectronicinteraction contribution to the recoil effect of the first and higher orders in Z^{−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 oneelectron approximation by solving the Dirac equation numerically. Moreover, it can be evaluated using an analytical formula^{26}. The rootmeansquare 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 Δg=g(^{40}Ca^{17+})−g(^{48}Ca^{17+}) 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 gfactor difference, we measured successively the Zeeman splitting of the respective lithiumlike ion in a homogeneous magnetic field B using single ions confined in a Penning trap. The Larmor frequency ν_{L}, which quantifies the energy difference between the spinup and the spindown state of the valence electron, is given by: . We determine the magnetic field by measuring the cyclotron frequency of the ion with electric charge q_{ion} and mass m_{ion}. In the concluding equation for the gfactor:
the magnetic field cancels, if in the ratio Γ≡ν_{L}·ν_{c}^{−1} both frequencies are probed simultaneously. To obtain the gfactor from the measured frequency ratios Γ, used in equation (1), the atomic masses of the ions are required. While the masses of ^{40}Ca m(^{40}Ca^{17+})=39.953272233 (22) u with a relative mass uncertainty of δm_{ion}·m_{ion}^{−1}=0.6 parts per billion (p.p.b.; refs 27, 28) and also of the electron with δm_{e}·m_{e}^{−1}=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 highprecision measurements of (i) the ^{48}Ca mass and (ii) the frequency ratios Γ(^{40}Ca^{17+}) and Γ(^{48}Ca^{17+}).
Determination of the atomic mass of ^{48}Ca^{17+}
With the Penningtrap 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 cyclotronfrequency ratio R of the mass doublet of singly charged ^{48}Ca^{+} ions and ^{12}C_{4}^{+} carbon cluster ions: R≡ν_{c}(^{48}Ca^{+})/ν_{c}(^{12}C_{4}^{+})=m(^{12}C_{4}^{+})/m(^{48}Ca^{+}). Instead of using the Brown–Gabrielse invariance theorem ν_{c}^{2}=ν_{+}^{2}+ν_{z}^{2}+ν_{−}^{2} (ref. 29), both cyclotron frequencies have been determined as the sum of the ion’s two radial eigenfrequencies ν_{c}=ν_{+}+ν_{−}, where ν_{+} is the modified cyclotron frequency and ν_{−} the magnetron frequency. Considering a mass difference of Δm=m(^{12}C_{4}^{+})−m(^{48}Ca^{+})≈4.8 × 10^{−2} u we derive a systematic shift of the mass ratio ΔR <1 × 10^{−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 (≤5 ions) of ^{48}Ca^{+} and ^{12}C_{4}^{+} with a laserablation ion source^{30} and separately transfer them into a preparation trap for cooling and centring via massselective buffergas cooling^{31} (Fig. 1). Then, the particular cyclotron frequency is measured in the measurement trap with the novel PIICR (refs 12, 13; Methods section). Combining the measured cyclotronfrequency ratio R=1.00099010175 (35)_{stat} (17)_{syst} (δR·R^{−1}=0.39 p.p.b.) with the known carbon cluster mass m(^{12}C_{4}^{+}) and correcting for the missing electrons and their corresponding binding energies, we obtain the following value for the mass of lithiumlike ^{48}Ca (Methods section):
The resulting atomic mass agrees within its uncertainty with the previous less accurate measurements^{32,33}.
Measurement of the Larmortocyclotron frequency ratios
Using a triple Penning trap setup located at the University of Mainz, and described in detail in refs 34, 35, we measured the Larmortocyclotron frequency ratio Γ of both calcium isotopes. Within a cryogenic (T=4.2 K) ultrahigh vacuum chamber (P<10^{−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=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}∼5 K) as a minimum (‘dipsignal’) 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 rfsideband coupling to the axial resonator generating double dipsignals 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 highprecision phasesensitive measurement of the modified cyclotron frequency^{37}, lasting about 5 s, we inject microwaves (MW) at the assumed Zeeman transition frequency (ν_{MW}≈105 GHz) into the apparatus to induce spinflips. To assess the success of a spinflip attempt in our Precision trap (PT), we analyse the electron spinstate before and after the probing in a spatially separated Penning trap, the Analysis trap (AT). Here, a large magnetic bottle (B_{2,z}=10(1)·10^{3} T m^{−2}) couples the magnetic moment to the axial motion, resulting in frequency jumps of the axial oscillation , which are caused by changes of the electron’s spin direction. This socalled continuous Stern–Gerlach effect^{38} enables the spinstate detection. In case of the ^{48}Ca^{17+} ion amounts to only 140 mHz at an absolute frequency of ν_{z}=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 ν_{MW}. Combining the corresponding measured frequency ratios Γ*=ν_{MW}·ν_{c}^{−1} with the binary information of the spinflip, we obtain a Γresonance (Fig. 2c), which depicts the spinflip probability in the PT versus the measured frequency ratios. With a weighted Gaussian maximumlikelihood fit, we extract the mean value Γ_{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 Γ(^{40}Ca^{17+})=4,282.42953545 (30) and Γ(^{48}Ca^{17+})=5,138.83795612 (42), we derive the most precise gfactor values for lithiumlike ions from equation (1):
The statistical, systematic and ion mass uncertainties are given separately. The absolute values for the gfactors (Table 2) provide a stringent test of manyelectron QED calculations in a magnetic field^{40,41}. The gfactor difference finally yields the soughtafter 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 gfactor difference. Consequently, the denoted systematic uncertainty of the frequency ratios is smaller than the quadratically summed statistical uncertainties of the Γratios given in equations (3) and (4). The comparison of the measured value of the gfactor 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 gfactor difference in combination with the measured frequency ratios and the mass of ^{48}Ca^{17+} to determine the isotopic mass difference: Δm=m(^{48}Ca)−m(^{40}Ca)=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 highprecision measurements of Larmortocyclotron frequency ratios, atomic masses of the lithiumlike isotopes ^{40}Ca^{17+} and ^{48}Ca^{17+} and corresponding gfactor calculations, presented in this paper, enables a variety of fundamental studies. Besides the test of manyelectron QED calculations in a magnetic field by considering the absolute values of the gfactors or the indirect determination of the isotopic mass difference, the analysis of the measured and predicted gfactor 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 α via gfactor measurements of heavy, highly charged ions^{14}.
Methods
Calculation of the isotope shift
The main contribution to the isotope shift Δg=g(^{40}Ca^{17+})–g(^{48}Ca^{17+}) results from the nuclear recoil effect that must be calculated including the relativistic, QED and interelectronicinteraction contributions. As the nuclear size effect is rather small, it can be evaluated in oneelectron approximation by solving the Dirac equation.
Consider first the nuclear recoil effect on the atomic gfactor to zeroth order in 1/Z. In this approximation, the m_{e}·m_{nucl}^{−1} nuclear recoil contribution to the g factor of an ion with one electron over closed shells is given in refs 22, 23.
Here, ℏ=c=1, e<0, μ_{B} is the Bohr magneton, m_{a} is the angular momentum projection of the state a, p^{k}=−i∇^{k} is the momentum operator, A_{cl}=[B × r]/2 is the vector potential of the homogeneous magnetic field B directed along the z axis, D^{k}(ω)=−4παZα^{l}D^{lk}(ω),
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 CoulombGreen function ) 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 CoulombGreen function is defined as , where is the Fermi energy and η→0. In equation (7), the summation over the repeated indices (k=1,2,3), which enumerate components of the threedimensional vectors, is implicit. Formula (7) incorporates both one and twoelectron nuclear recoil contributions to zeroth order in 1/Z. For the (1s)^{2}2sstate of a lithiumlike ion, the (1/Z)^{0} twoelectron contribution is zero and, therefore, we restrict our consideration to the oneelectron contribution only. For the practical calculations, the oneelectron contribution is conveniently represented by a sum of loworder (‘nonQED’) and higher order (‘QED’) term, Δg=Δg_{L}+Δg_{H}:
where V(r)=−(αZ)/(r) is the Coulomb potential induced by the nucleus and n=r ·r^{−1}. The loworder term can be derived from the relativistic Breit equation, while the derivation of the higherorder term requires using QED beyond the Breit approximation. For this reason, we call them the nonQED and QED contributions, respectively.
The loworder term Δg_{L} can be evaluated analytically^{42}:
where E is the Dirac energy and . To the two lowest orders in αZ, we have
As follows from this formula, for an sstate (κ=−1) the nonrelativistic contribution to Δg_{L} vanishes and the loworder term comes from pure relativistic (∼(αZ)^{2}) origin.
The calculation of the higher order term, Δg_{H}, is a much more difficult task. For the 1sstate it is calculated in ref. 42. In the present paper we performed the corresponding calculation for the 2sstate. Details of this calculation and the corresponding results for other ions will be published elsewhere.
In addition to the main oneelectron nuclear recoil contribution, we have to consider the radiative (∼α) nuclear recoil correction and the (m_{e}/m_{nucl})^{2} nuclear recoil correction. To the lowest order in αZ, these corrections were evaluated in refs 19, 21, 23, 24. We need also to account for the interelectronicinteraction 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 interelectronicinteraction contribution is mainly due to uncalculated higherorder relativistic and QED corrections.
To get the total value of the isotope shift, we also evaluate the nuclear size correction. The rootmeansquare 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 gfactor for ^{40}Ca^{17+} and ^{48}Ca^{17+} are presented in Table 1.
In Table 2 we list the various contributions to the gfactor of ^{40}Ca^{17+} and ^{48}Ca^{17+}. The Dirac value, as well as the QED, interelectronicinteraction, 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 PIICR 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^{+} and ions due to the anharmonicity of the trap potential and inhomogeneity of the magnetic field to a level well below 10^{−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 ‘magnetronmotion phase’ and ‘cyclotronmotion phase’, are applied alternately to measure the ion cyclotron frequency ν_{c}. In the ‘magnetronmotion 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 ‘cyclotronmotion 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 positionsensitive detector. The angle between the ionposition images corresponding to two patterns with respect to the trap centre image is proportional to the ion cyclotron frequency ν_{c}. Pulse patterns are applied for a total measurement time of ∼5 min. On this measurement scale the ‘magnetronmotion phase’ and ‘cyclotronmotion phase’ can be considered to be measured simultaneously. Data with >5 detected ions per cycle are not considered in the analysis to reduce a possible shift in the ratio of the ^{48}Ca^{+} and ^{12}C_{4}^{+} ions due to ion–ion interaction. To eliminate a possible cyclotronfrequency 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^{+} and ^{12}C_{4}^{+} ion masses due to the possible distortion of the ionmotion projection onto the detector to a level well below 10^{−10} (ref. 13).
Data sets for the ion cyclotronfrequency ratio R
The cyclotron frequencies ν_{c} of the ^{48}Ca^{+} and ^{12}C_{4}^{+} ions are measured alternately for several days. The total measurement period is divided in 45 ∼1h periods. In addition, each 5 min measurement is divided in 10 30s periods. For each of the 45 1h periods the ratio R_{1 h} of the cyclotron frequencies ^{48}Ca^{+} and ^{12}C_{4}^{+} ions is obtained along with the inner and outer errors^{44} by fitting to the 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^{+} frequency points a polynomial P_{1}(t)=R_{1 h} × P_{2}(t). The final cyclotronfrequency 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}=1.00099010175 (35)_{stat} (17)_{syst}. The systematic uncertainty in the frequencyratio determination originates from the anharmonicity of the trap potential, the inhomogeneity of the magnetic field and the distortion of the ionmotion projection onto the detector^{13}.
The atomic mass of ^{48}Ca^{17+}
The mass of a C_{4}^{+} cluster is calculated by considering the dissociation energy: E_{diss}=18.0(17) eV (ref. 45), the ionization energy: E_{ion}=11.0(7) eV (ref. 46) and the missing electron: The mass differences between all three possible cluster structures—linear, rhombus and triangular pyramidal—are already covered by the uncertainties of the dissociation and ionization energies. For the determination of the mass of lithiumlike ^{48}Ca we have to correct the mass of singly charged ^{48}Ca, m(^{48}Ca^{1+})=m(C_{4}^{+})/R, by the 16 missing electron masses and the corresponding ionization energies: Δm(E_{bind})=7.2438 (43) × 10^{−6} u, where E_{bind}=6,747.5 (40) eV (ref. 47) and 1u=931,494,061 (21) eV c^{−2}:
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 binding energy E_{bind}=6.11315520 (25) eV (ref. 47) we obtain:
which is in good agreement with the literature value of m(^{48}Ca)=47.952522765 (129) (ref. 28) but a factor seven more precise.
Cycle weighting of the Γresonances
In the magnetic bottle of the AT the axial frequency jump caused by an induced spinflip scales with the inverse of the ion’s mass. In contrast to our previous measurements, where the axial frequency shifts have been: for ^{12}C^{5+} (ref. 10), for ^{28}Si^{13+} (ref. 9) and ^{28}Si^{11+} (ref. 40), it is a particular challenge to resolve the spinstates for the calcium isotopes, where for ^{40}Ca^{17+} and only for ^{48}Ca^{17+}. We measure axial phase differences of subsequent measurements by applying a coherent detection technique, which includes three steps: (i) The axial phase is imprinted by a 10 ms dipolar excitation. (ii) The axial phase evolves for a certain time T_{evol}. (iii) The phase is measured via the axial detection system. With a phaseevolution time of T_{evol}=1 s and a readouttime of 552 ms, a spinflip corresponds to an axial phaseshift of for ^{40}Ca^{17+} and for ^{48}Ca^{17+}. In Fig. 4 1,790 averaged axial frequency differences of ^{48}Ca^{17+} are histogrammed. Here, we determine each axial frequency by averaging over four successive phase measurements. Between these measurement sequences, we try to induce spinflips for 30 s at maximum MWpower and at a fixed MWfrequency. The plotted probability density ρ_{AT} is modelled by a superposition of three Gaussian distributions:
where G_{no sf} is the Gaussian distribution of the axial frequency differences without spinflips with an amplitude (1A), a mean value of zero and a s.d. of . G_{sf up} and G_{sf down} are the Gaussian distributions with spinflip up (mean value: ) and spinflip down (mean value: ). From a maximumlikelihood fit, the following three parameters are extracted: (I) the spinflip rate: 26.5%, (II) the frequency jitter: and (III) the axial frequency jump due to a spinflip: . For the different data sets of ^{48}Ca^{17+}, we determine a frequency jitter of . More precisely, we started with a tiny frequency jitter of 25 mHz and ended with a larger jitter of 35 mHz, although we optimized the trap harmonicity and checked the ion temperature. The reason of the declined frequency stability is unclear, but probably related to varying radiofrequency noise from external sources. As the largest measured jitter is only 2–2.9 times smaller than the cutfrequency difference of the probability of error of 0.5–4.5% is not negligible and has to be considered.
Instead of using a data analysis based on simple quality cuts, to decrease the probability of error and in that way losing statistics, we introduce the following ATweight w_{AT} for each spinflip, in a way that w_{AT}=0, if the electron is in spindown, w_{AT}=1, if the electron is in spinup and w_{AT}=0.5, if the spinstate is unknown:
where the spinflip cut is 70 mHz for ^{48}Ca^{17+}. In a normal measurement cycle, we try to induce a spinflip at least three times in the AT and then proceed with this measurement process, until the cutcriterion Δν_{z}> spinflip cut is fulfilled for the first time. For the first and the last frequency jump in the AT, which fulfils this criterion, the ATweight is calculated. The spinflip probability in the PT (w_{PT}) is calculated from the two ATweights: before entering the PT and directly after leaving the PT :
w_{PT}=1 corresponds to a spinflip in the PT, w_{PT}=0 corresponds to no spinflip in the PT and w_{PT}=0.5 corresponds to no spinflip information in the PT. The Gaussian lineshape of the Γresonance, which has been analysed in refs 9, 34, gets modified by adding a fourth fitparameter (off_{Γ}), which describes the wrong spinflip detection rate in the PT:
The PTweight finally has to be included in the maximumlikelihood function:
which is used, to extract the final mean value Γ_{mean}. In comparison to the common cutanalysis, we improve the relative uncertainty of Γ_{mean} by 20 p.p.t.
Data sets of the Γresonances
Various Γresonances are recorded at different modified cyclotron energies during the phaseevolution time of the modified cyclotron mode and the simultaneous probing of the Larmor frequency ν_{MW} in the PT. In Fig. 5 the mean values from the maximumlikelihood fit, see equation (19), are plotted for ^{40}Ca^{17+} (a) and ^{48}Ca^{17+} (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 . From linear extrapolations to zero modified cyclotron energy we derive our statistical Γvalues:
which have to be corrected by systematic shifts.
Systematic shifts and uncertainties of Γ(^{40}Ca^{17+}) and Γ(^{48}Ca^{17+})
The systematic shifts of the Larmortocyclotron 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 the Penning trap electrode surfaces generate an additional effective electric potential, which shifts the radial eigenfrequencies of the ion. In ref. 39 the shift of the cyclotron frequency is analytically calculated:
where r is the inner radius of the Penning trap. Due to the r^{−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.
Additional information
How to cite this article: Köhler, F. et al. Isotope dependence of the Zeeman effect in lithiumlike calcium. Nat. Commun. 7:10246 doi: 10.1038/ncomms10246 (2016).
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Acknowledgements
This work was supported by the Max Planck Society, the EU (ERC grant no 290870; MEFUCO), International Max Planck Research School for Quantum Dynamics in Physics, Chemistry and Biology (IMPRSQD), GSI Helmholtzzentrum für Schwerionenforschung, the Helmholtz Alliance HA216/EMMI, Russian Foundation for Basic Research (RFBR) (grants no. 130200630 and No. 140231316), the St Petersburg State University (grant nos. 11.38.269.2014 and 11.38.237.2015), Deutsche Forschungsgemeinschaft (DFG) (grant no. VO1707/12), Bundesministerium für Bildung und Forschung (BMBF) (grant no. 01DJ14002) and the FAIRRussia Research Center.
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F.K., S.S., A.K. and J.H. performed the measurements on the Larmortocyclotron frequency ratios. V.M.S., D.A.G. and A.V.V. carried out the QED calculations. S.E., M.G. and E.M.R. performed the measurement of the mass of ^{48}Ca. F.K., S.S., V.M.S., S.E., G.W. and K.B. prepared the manuscript. All authors discussed the results and contributed to the manuscript at all stages.
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Köhler, F., Blaum, K., Block, M. et al. Isotope dependence of the Zeeman effect in lithiumlike calcium. Nat Commun 7, 10246 (2016). https://doi.org/10.1038/ncomms10246
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DOI: https://doi.org/10.1038/ncomms10246
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