g factor of hydrogenlike ²⁸Si¹³⁺

Testing quantum electrodynamics in extreme fields using helium-like uranium

Quantum electrodynamics (QED), the quantum field theory that describes the interaction between light and matter, is commonly regarded as the best-tested quantum theory in modern physics. However, this claim is mostly based on extremely precise studies performed in the domain of relatively low field strengths and light atoms and ions1-6. In the realm of very strong electromagnetic fields such as in the heaviest highly charged ions (with nuclear charge Z ≫ 1), QED calculations enter a qualitatively different, non-perturbative regime. Yet, the corresponding experimental studies are very challenging, and theoretical predictions are only partially tested. Here we present an experiment sensitive to higher-order QED effects and electron-electron interactions in the high-Z regime. This is achieved by using a multi-reference method based on Doppler-tuned X-ray emission from stored relativistic uranium ions with different charge states. The energy of the 1s1/22p3/2 J = 2 → 1s1/22s1/2 J = 1 intrashell transition in the heaviest two-electron ion (U90+) is obtained with an accuracy of 37 ppm. Furthermore, a comparison of uranium ions with different numbers of bound electrons enables us to disentangle and to test separately the one-electron higher-order QED effects and the bound electron-electron interaction terms without the uncertainty related to the nuclear radius. Moreover, our experimental result can discriminate between several state-of-the-art theoretical approaches and provides an important benchmark for calculations in the strong-field domain.

High-Precision Determination of g Factors and Masses of ^{20}Ne^{9+} and ^{22}Ne^{9+}

We present the measurements of individual bound electron g factors of ^{20}Ne^{9+} and ^{22}Ne^{9+} on the relative level of 0.1 parts per billion. The comparison with theory represents the most stringent test of bound-state QED in strong electric fields. A dedicated mass measurement results in m(^{20}Ne)=19.992 440 168 77(9)  u, which improves the current literature value by a factor of 18, disagrees by 4 standard deviations, and represents the most precisely measured mass value in atomic mass units. Together, these measurements yield an electron mass on the relative level of 0.1 ppb with m_{e}=5.485 799 090 99(59)×10^{-4}  u as well as a factor of seven improved m(^{22}Ne)=21.991 385 098 2(26)  u.

Stringent test of QED with hydrogen-like tin

Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 1015 V cm-1 for the innermost electrons1. Especially in few-electron, highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. Consequently, their Lamb shifts were intensively studied in the past several decades2,3. Another approach is the measurement of gyromagnetic factors (g factors) in highly charged ions4-7. However, so far, either experimental accuracy or small field strength in low-Z ions5,6 limited the stringency of these QED tests. Here we report on our high-precision, high-field test of QED in hydrogen-like 118Sn49+. The highly charged ions were produced with the Heidelberg electron beam ion trap (EBIT)8 and injected into the ALPHATRAP Penning-trap setup9, in which the bound-electron g factor was measured with a precision of 0.5 parts per billion (ppb). For comparison, we present state-of-the-art theory calculations, which together test the underlying QED to about 0.012%, yielding a stringent test in the strong-field regime. With this measurement, we challenge the best tests by means of the Lamb shift and, with anticipated advances in the g-factor theory, surpass them by more than an order of magnitude.

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

Quantum electrodynamics (QED) is one of the most fundamental theories of physics and has been shown to be in excellent agreement with experimental results1-5. In particular, measurements of the electron's magnetic moment (or g factor) of highly charged ions in Penning traps provide a stringent probe for QED, which allows testing of the standard model in the strongest electromagnetic fields6. When studying the differences between isotopes, many common QED contributions cancel owing to the identical electron configuration, making it possible to resolve the intricate effects stemming from the nuclear differences. Experimentally, however, this quickly becomes limited, particularly by the precision of the ion masses or the magnetic field stability7. Here we report on a measurement technique that overcomes these limitations by co-trapping two highly charged ions and measuring the difference in their g factors directly. We apply a dual Ramsey-type measurement scheme with the ions locked on a common magnetron orbit8, separated by only a few hundred micrometres, to coherently extract the spin precession frequency difference. We have measured the isotopic shift of the bound-electron g factor of the isotopes 20Ne9+ and 22Ne9+ to 0.56-parts-per-trillion (5.6 × 10-13) precision relative to their g factors, an improvement of about two orders of magnitude compared with state-of-the-art techniques7. This resolves the QED contribution to the nuclear recoil, accurately validates the corresponding theory and offers an alternative approach to set constraints on new physics.

Helicity Transfer in Strong Laser Fields via the Electron Anomalous Magnetic Moment

Electron beam longitudinal polarization during the interaction with counterpropagating circularly polarized ultraintense laser pulses is investigated, while accounting for the anomalous magnetic moment of the electron. Although it is known that the helicity transfer from the laser photons to the electron beam is suppressed in linear and nonlinear Compton scattering processes, we show that the helicity transfer nevertheless can happen via an intermediate step of the electron radiative transverse polarization, phase matched with the driving field, followed up by spin rotation into the longitudinal direction as induced by the anomalous magnetic moment of the electron. With spin-resolved QED Monte Carlo simulations, we demonstrate the consequent helicity transfer from laser photons to the electron beam with a degree up to 10%, along with an electron radial polarization up to 65% after multiple photon emissions in a femtosecond timescale. This effect is detectable with currently achievable laser facilities, evidencing the role of the leading QED vertex correction to the electron anomalous magnetic moment in the polarization dynamics in ultrastrong laser fields.

g Factor of Lithiumlike Silicon and Calcium: Resolving the Disagreement between Theory and Experiment

The bound-electron g factor is a stringent tool for tests of the standard model and the search for new physics. The comparison between an experiment on the g factor of lithiumlike silicon and the two recent theoretical values revealed the discrepancies of 1.7σ [Glazov et al. Phys. Rev. Lett. 123, 173001 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.173001] and 5.2σ [Yerokhin et al. Phys. Rev. A 102, 022815 (2020)PLRAAN2469-992610.1103/PhysRevA.102.022815]. To identify the reason for this disagreement, we accomplish large-scale high-precision computation of the interelectronic-interaction and many-electron QED corrections. The calculations are performed within the extended Furry picture of QED, and the dependence of the final values on the choice of the binding potential is carefully analyzed. As a result, we significantly improve the agreement between the theory and experiment for the g factor of lithiumlike silicon. We also report the most accurate theoretical prediction to date for lithiumlike calcium, which perfectly agrees with the experimental value.

Ground-State g Factor of Highly Charged ^{229}Th Ions: An Access to the M1 Transition Probability between the Isomeric and Ground Nuclear States

A method is proposed to determine the M1 nuclear transition amplitude and hence the lifetime of the "nuclear clock transition" between the low-lying (∼8  eV) first isomeric state and the ground state of ^{229}Th from a measurement of the ground-state g factor of few-electron ^{229}Th ions. As a tool, the effect of nuclear hyperfine mixing in highly charged ^{229}Th ions such as ^{229}Th^{89+} or ^{229}Th^{87+} is used. The ground-state-only g-factor measurement would also provide first experimental evidence of nuclear hyperfine mixing in atomic ions. Combining the measurements for H-, Li-, and B-like ^{229}Th ions has a potential to improve the initial result for a single charge state and to determine the nuclear magnetic moment to a higher accuracy than that of the currently accepted value. The calculations include relativistic, interelectronic-interaction, QED, and nuclear effects.

Measuring Ion Oscillations at the Quantum Level with Fluorescence Light

We demonstrate an optical method for detecting the mechanical oscillations of an atom with single-phonon sensitivity. The measurement signal results from the interference between the light scattered by a trapped atomic ion and that of its mirror image. We detect the oscillations of the atom in the Doppler cooling limit and reconstruct average trajectories in phase space. We demonstrate single-phonon sensitivity near the ground state of motion after electronically induced transparency cooling. These results could be applied for motion detection of other light scatterers of fundamental interest, such as trapped nanoparticles.

g Factor of Lithiumlike Silicon: New Challenge to Bound-State QED

The recently established agreement between experiment and theory for the g factors of lithiumlike silicon and calcium ions manifests the most stringent test of the many-electron bound-state quantum electrodynamics (QED) effects in the presence of a magnetic field. In this Letter, we present a significant simultaneous improvement of both theoretical g_{th}=2.000 889 894 4 (34) and experimental g_{exp}=2.000 889 888 45 (14) values of the g factor of lithiumlike silicon ^{28}Si^{11+}. The theoretical precision now is limited by the many-electron two-loop contributions of the bound-state QED. The experimental value is accurate enough to test these contributions on a few percent level.

g Factor of Boronlike Argon ^{40}Ar^{13+}

We have measured the ground-state g factor of boronlike argon ^{40}Ar^{13+} with a fractional uncertainty of 1.4×10^{-9} with a single ion in the newly developed Alphatrap double Penning-trap setup. The value of g=0.663 648 455 32(93) obtained here is in agreement with our theoretical prediction of 0.663 648 12(58). The latter is obtained accounting for quantum electrodynamics, electron correlation, and nuclear effects within the state-of-the-art theoretical methods. Our experimental result distinguishes between existing predictions that are in disagreement, and lays the foundations for an independent determination of the fine-structure constant.

The Heidelberg compact electron beam ion traps

Electron beam ion traps (EBITs) are ideal tools for both production and study of highly charged ions (HCIs). In order to reduce their construction, maintenance, and operation costs, we have developed a novel, compact, room-temperature design, the Heidelberg Compact EBIT (HC-EBIT). Four already commissioned devices operate at the strongest fields (up to 0.86 T) reported for such EBITs using permanent magnets, run electron beam currents up to 80 mA, and energies up to 10 keV. They demonstrate HCI production, trapping, and extraction of pulsed Ar¹⁶⁺ bunches and continuous 100 pA ion beams of highly charged Xe up to charge state 29+, already with a 4 mA, 2 keV electron beam. Moreover, HC-EBITs offer large solid-angle ports and thus high photon count rates, e.g., in x-ray spectroscopy of dielectronic recombination in HCIs up to Fe²⁴⁺, achieving an electron-energy resolving power of E/ΔE > 1500 at 5 keV. Besides traditional on-axis electron guns, we have also implemented a novel off-axis gun for laser, synchrotron, and free-electron laser applications, offering clear optical access along the trap axis. We report on its first operation at a synchrotron radiation facility demonstrating the resonant photoexcitation of highly charged oxygen.

Recoil Effect on the g Factor of Li-Like Ions

The nuclear recoil effect on the g factor of Li-like ions is evaluated. The one-electron recoil contribution is treated within the framework of the rigorous QED approach to the first order in the electron-to-nucleus mass ratio m/M and to all orders in the parameter αZ. These calculations are performed in a range Z=3-92. The two-electron recoil term is calculated for low- and middle-Z ions within the Breit approximation using a four-component approach. The results for the two-electron recoil part obtained in the Letter strongly disagree with the previous calculations performed using an effective two-component Hamiltonian. The obtained value for the recoil effect is used to calculate the isotope shift of the g factor of Li-like ^{A}Ca^{17+} with A=40 and A=48 which was recently measured. It is found that the new theoretical value for the isotope shift is closer to the experimental one than the previously obtained value.

High-Precision Measurement of the Proton's Atomic Mass

We report on the precise measurement of the atomic mass of a single proton with a purpose-built Penning-trap system. With a precision of 32 parts per trillion our result not only improves on the current CODATA literature value by a factor of 3, but also disagrees with it at a level of about 3 standard deviations.

g Factor of Light Ions for an Improved Determination of the Fine-Structure Constant

A weighted difference of the g factors of the H- and Li-like ions of the same element is theoretically studied and optimized in order to maximize the cancellation of nuclear effects between the two charge states. We show that this weighted difference and its combination for two different elements can be used to extract a value for the fine-structure constant from near-future bound-electron g factor experiments with an accuracy competitive with or better than the present literature value.

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 μ_Bs ħ⁻¹ with μ_B 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.

In addition to hyperfine splitting effects, isotope shifts of atomic electronic energy levels allow the investigation nuclear properties. Here, the authors study the isotope dependence of the Zeeman effect in litihium-like calcium isotopes in a Penning-trap setup and find good agreement with QED calculations.

Ultrarelativistic electron states in a general background electromagnetic field

The feasibility of obtaining exact analytical results in the realm of QED in the presence of a background electromagnetic field is almost exclusively limited to a few tractable cases, where the Dirac equation in the corresponding background field can be solved analytically. This circumstance has restricted, in particular, the theoretical analysis of QED processes in intense laser fields to within the plane wave approximation even at those high intensities, achievable experimentally only by tightly focusing the laser energy in space. Here, within the Wentzel-Kramers-Brillouin approximation, we construct analytically single-particle electron states in the presence of a background electromagnetic field of general space-time structure in the realistic assumption that the initial energy of the electron is the largest dynamical energy scale in the problem. The relatively compact expression of these states opens, in particular, the possibility of investigating analytically strong-field QED processes in the presence of spatially focused laser beams, which is of particular relevance in view of the upcoming experimental campaigns in this field.

Many-electron QED corrections to the g factor of lithiumlike ions

A rigorous QED evaluation of the two-photon exchange corrections to the g factor of lithiumlike ions is presented. The screened self-energy corrections are calculated for the intermediate-Z region, and its accuracy for the high-Z region is essentially improved in comparison with that of previous calculations. As a result, the theoretical accuracy of the g factor of lithiumlike ions is significantly increased. The theoretical prediction obtained for the g factor of (28)Si(11+) g(th) = 2.000 889 892(8) is in an excellent agreement with the corresponding experimental value g(exp) = 2.000 889 889 9(21) [A. Wagner et al., Phys. Rev. Lett. 110, 033003 (2013).

Control of the conformations of ion Coulomb crystals in a Penning trap

Laser-cooled atomic ions form ordered structures in radiofrequency ion traps and in Penning traps. Here we demonstrate in a Penning trap the creation and manipulation of a wide variety of ion Coulomb crystals formed from small numbers of ions. The configuration can be changed from a linear string, through intermediate geometries, to a planar structure. The transition from a linear string to a zigzag geometry is observed for the first time in a Penning trap. The conformations of the crystals are set by the applied trap potential and the laser parameters, and agree with simulations. These simulations indicate that the rotation frequency of a small crystal is mainly determined by the laser parameters, independent of the number of ions and the axial confinement strength. This system has potential applications for quantum simulation, quantum information processing and tests of fundamental physics models from quantum field theory to cosmology.

Resolution of single spin flips of a single proton

The spin magnetic moment of a single proton in a cryogenic Penning trap was coupled to the particle's axial motion with a superimposed magnetic bottle. Jumps in the oscillation frequency indicate spin flips and were identified using a Bayesian analysis.

g Factor of lithiumlike silicon 28Si11+

The g factor of lithiumlike silicon (28)Si(11+) has been measured in a triple-Penning trap with a relative uncertainty of 1.1×10(-9) to be g(exp)=2.000 889 889 9(21). The theoretical prediction for this value was calculated to be g(th)=2.000 889 909(51) improving the accuracy to 2.5×10(-8) due to the first rigorous evaluation of the two-photon exchange correction. The measured value is in excellent agreement with the theoretical prediction and yields the most stringent test of bound-state QED for the g factor of the 1s(2)2s state and the relativistic many-electron calculations in a magnetic field.

Cryogenic linear Paul trap for cold highly charged ion experiments

Storage and cooling of highly charged ions require ultra-high vacuum levels obtainable by means of cryogenic methods. We have developed a linear Paul trap operating at 4 K capable of very long ion storage times of about 30 h. A conservative upper bound of the H(2) partial pressure of about 10(-15) mbar (at 4 K) is obtained from this. External ion injection is possible and optimized optical access for lasers is provided, while exposure to black body radiation is minimized. First results of its operation with atomic and molecular ions are presented. An all-solid state laser system at 313 nm has been set up to provide cold Be(+) ions for sympathetic cooling of highly charged ions.

Nuclear shape effect on the g factor of hydrogenlike ions

The nuclear shape correction to the g factor of a bound electron in the 1S state is calculated for a number of nuclei in the range of charge numbers from Z=6 up to Z=92. The leading relativistic deformation correction has been derived analytically, and also its influence on one-loop quantum electrodynamic terms has been evaluated. We show the leading corrections to become significant for mid-Z ions and for very heavy elements to even reach the 10(-6) level.

Phase-sensitive cyclotron frequency measurements at ultralow energies

A novel technique for a direct and coherent measurement of the modified cyclotron frequency of an ion in a Penning trap at energies close to the thermal cooling limit is presented. This allows a rapid and both precise and accurate determination of the free-space cyclotron frequency in real Penning traps despite the existence of electric and magnetic field imperfections and relativistic shifts. The demonstrated performance paves the way for considerably improved bound-state g-factor measurements on the 10 ppt level and mass measurements in the 1 ppt range and possibly below.