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The quantum electrodynamic (QED) description of light-and-matter interaction is one of the most fundamental theories of physics and has been shown to be in excellent agreement with experimental results. Specifically, measurements of the electronic magnetic moment (or $g$ factor) of highly charged ions (HCI) in Penning traps can provide a stringent probe for QED, testing the Standard model in the strongest electromagnetic fields. When studying the difference of isotopes, even the intricate effects stemming from the nucleus can be resolved and tested as, due to the identical electron configuration, many common QED contributions do not have to be considered. Experimentally however, this becomes quickly limited, particularly by the precision of the ion masses or the achievable magnetic field stability. Here we report on a novel measurement technique that overcomes both of these limitations by co-trapping two HCIs in a Penning trap and measuring the difference of their $g$ factors directly. The resulting correlation of magnetic field fluctuations leads to drastically higher precision. We use a dual Ramsey-type measurement scheme with the ions locked on a common magnetron orbit, separated by only a few hundred micrometres, to extract the coherent spin precession frequency difference. We have measured the isotopic shift of the bound electron $g$ factor of the neon isotopes of $^{20}$Ne$^{9+}$ and $^{22}$Ne$^{9+}$ to 0.56 parts-per-trillion ($5.6 \cdot 10^{-13}$) precision relative to their $g$ factors, which is an improvement of more than two orders of magnitude compared to state-of-the-art techniques. This resolves the QED contribution to the nuclear recoil for the very first time and accurately validates the corresponding theory. Furthermore, the agreement with theory allows setting constraints for a fifth-force, resulting from Higgs-portal-type dark-matter interactions., Comment: 22 pages, 5 figures

We present a theoretical investigation of the resonant elastic scattering of laser photons by ultra-relativistic partially stripped ions, that is the core process of the Gamma Factory project. Special emphasis in our study is placed on the angular distribution and polarization of scattered photons as observed in the collider and ion-rest reference frames. In order to describe these (angular and polarization) properties for arbitrary relativistic many-electron ion, the general approach, based on the application of irreducible polarization tensors, is laid down. By making use of the polarization tensors we explore in detail the scattering of both, conventional plane-wave- and twisted (or vortex) photons. For the former case we show how the propagation directions and polarization states of incident and outgoing photons are related to each other for the $n S_{0} \to n' P_{1} \to n S_{0}$, $n S_{1/2} \to n' P_{1/2} \to n S_{1/2}$ and $n S_{1/2} \to n' P_{3/2} \to n S_{1/2}$ resonant transitions. For the scattering of initially twisted light, that carries non-zero orbital angular momentum, we explore the angular distribution of secondary photons and discuss the conditions under which they are also twisted., Comment: 16 pages, 4 figures

Large-scale relativistic calculations are performed for the transition energy and line strength of the $ 1s^{2} 2s 2p$ $^1P_{1} \,-\ 1s^{2} 2s^{2}$ $^1S_{0} $ transition in Be-like carbon. Based on the multiconfiguration Dirac-Hartree-Fock~(MCDHF) approach, different correlation models are developed to account for all major electron-electron correlation contributions. These correlation models are tested with various sets of the initial and the final state wave functions. The uncertainty of the predicted line strength due to missing correlation effects is estimated from the differences between the results obtained with those models. The finite nuclear mass effect is accurately calculated taking into account the energy, wave functions as well as operator contributions. As a result, a reliable theoretical benchmark of the $E1$ line strength is provided to support high precision lifetime measurement of the $ 1s^{2} 2s 2p$ $^1P_{1} $ state in Be-like carbon.

We investigate the process of nuclear excitation via a two-photon electron transition (NETP) for the case of the doubly charged thorium. The theory of the NETP process has been devised originally for heavy helium-like ions. In this work, we study this process in the nuclear clock isotope $^{229}$Th in the $2+$ charge state. For this purpose, we employ a combination of configuration interaction and many-body perturbation theory to calculate the probability of NETP in resonance approximation. The experimental scenario we propose for the excitation of the low lying isomeric state in $^{229}$Th is a circular process starting with a two-step pumping stage followed by NETP. The ideal intermediate steps in this process depend on the supposed energy $\hbar\omega_N$ of the nuclear isomeric state. For each of these energies, the best initial state for NETP is calculated. Special focus is put on the most recent experimental results for $\hbar\omega_N$.

The magnetic dipole (M1) line strength between the fine-structure levels of the ground configurations in B-, F-, Al- and Cl-like ions are calculated for the four elements argon, iron, molybdenum and tungsten. Systematically enlarged multiconfiguration Dirac-Hartree-Fock~(MCDHF) wave functions are employed to account for the interelectronic interaction with the Breit interaction included in first-order perturbation theory. The QED corrections are evaluated to all orders in $\alpha Z$ utilizing an effective potential approach. The calculated line strengths are compared with the results of other theories. The M1 transition rates are reported using accurate energies from the literature. Moreover, the lifetimes in the range of millisecond to picoseconds are predicted including the contributions also from the transition rate due to the E2 transition channel. The discrepancies of the predicted rates from those available from the literature are discussed and a benchmark dataset of theoretical lifetimes is provided to support future experiments., Comment: 9 pages, 4 tables, 1 figure

A recent measurement of the hyperfine splitting in the ground state of Li-like $^{209}$Bi$^{80+}$ has established a "hyperfine puzzle" -- the experimental result exhibits a 7$\sigma$ deviation from the theoretical prediction [J. Ullmann et al., Nat. Commun. 8, 15484 (2017); J. P. Karr, Nat. Phys. 13, 533 (2017)]. We provide evidence that the discrepancy is caused by an inaccurate value of the tabulated nuclear magnetic moment ($\mu_I$) of $^{209}$Bi. We perform relativistic density functional theory and relativistic coupled cluster calculations of the shielding constant that should be used to extract the value of $\mu_I(^{209}{\rm Bi})$ and combine it with nuclear magnetic resonance measurements of Bi(NO$_3$)$_3$ in nitric acid solutions and of the hexafluoridobismuthate(V) BiF$_6^-$ ion in acetonitrile. The result clearly reveals that $\mu_I(^{209}{\rm Bi})$ is much smaller than the tabulated value used previously. Applying the new magnetic moment shifts the theoretical prediction into agreement with experiment and resolves the hyperfine puzzle.

The thorium nucleus with mass number $A=229$ has attracted much interest because its extremely low lying first excited isomeric state at about $8$eV opens the possibility for the development of a nuclear clock. However, neither the exact energy of this nuclear isomer nor properties, such as nuclear magnetic dipole and electric quadrupole moment are known to a high precision so far. The latter can be determined by investigating the hyperfine structure of thorium atoms or ions. Due to its electronic structure and the long lifetime of the nuclear isomeric state, Th$^{2+}$ is especially suitable for such kind of studies. In this letter we present a combined experimental and theoretical investigation of the hyperfine structure of the $^{229}$Th$^{2+}$ ion in the nuclear ground and isomeric state. A very good agreement between theory and experiment is found for the nuclear ground state. Moreover, we use our calculations to confirm the recently presented experimental value for the nuclear magnetic dipole moment of the thorium nuclear isomer, which was in contradiction to previous theoretical studies.

Relativistic effects in the non-resonant two-photon K-shell ionization of neutral atoms are studied theoretically within the framework of second-order perturbation theory. The non-relativistic results are compared with the relativistic calculations in the dipole and no-pair approximations as well as with the complete relativistic approach. The calculations are performed in both velocity and length gauges. Our results show a significant decrease of the total cross section for heavy atoms as compared to the non-relativistic treatment, which is mainly due to the relativistic wavefunction contraction. The effects of higher multipoles and negative continuum energy states counteract the relativistic contraction contribution, but are generally much weaker. While the effects beyond the dipole approximation are equally important in both gauges, the inclusion of negative continuum energy states visibly contributes to the total cross section only in the velocity gauge.

We investigate the deexcitation of the $^{229}$Th nucleus via the excitation of an electron. Detailed calculations are performed for the enhancement of the nuclear decay width due to this so called electron bridge (EB) compared to the direct photoemission from the nucleus. The results are obtianed for triply ionized thorium by using a B-spline pseudo basis approach to solve the Dirac equation for a local $x_\alpha$ potential. This approach allows for an approximation of the full electron propagator including the positive and negative continuum. We show that the contribution of continua slightly increases the enhancement compared to a propagator calculated by a direct summation over bound states. Moreover we put special emphasis on the interference between the direct and exchange Feynman diagrams that can have a strong influence on the enhancement.

The non-resonant two-photon one-electron ionization of neutral atoms is studied theoretically in the framework of relativistic second-order perturbation theory and independent particle approximation. In particular, the importance of relativistic and screening effects in the total two-photon ionization cross section is investigated. Detailed computations have been carried out for the K-shell ionization of neutral Ne, Ge, Xe, and U atoms. The relativistic effects significantly decrease the total cross section, for the case of U, for example, they reduce the total cross section by a factor of two. Moreover, we have found that the account for the screening effects of the remaining electrons leads to occurrence of an unexpected minimum in the total cross section at the total photon energies equal to the ionization threshold, for the case of Ne, for example, the cross section drops there by a factor of three.

A systematic investigation of the nuclear-polarization effects in one- and few-electron heavy ions is presented. The nuclear-polarization corrections in the zeroth and first orders in $1/Z$ are evaluated to the binding energies, the hyperfine splitting, and the bound-electron g factor. It is shown, that the nuclear-polarization contributions can be substantially canceled simultaneously with the rigid nuclear corrections. This allows for new prospects for probing the QED effects in strong electromagnetic field and the determination of fundamental constants., Comment: 5 pages, 4 figures

The nuclear recoil effect to the g factor of boron-like ions is investigated. The one-photon-exchange correction to the nuclear recoil effect is calculated in the non-relativistic approximation for the nuclear recoil operator and in the Breit approximation for the interelectronic-interaction operator. The screening potential is employed to estimate the higher-order contributions. The updated g-factor values are presented for the ground 2P_1/2 and first excited 2P_3/2 states of B-like argon 40^Ar^13+, which are presently being measured by the ARTEMIS group at GSI., Comment: 5 pages, 2 figures, submitted to Journal of Physics

The g factor of lithiumlike 28Si11+ has been measured in a triple-Penning trap with a relative uncertainty of 1.1x10^{-9} to be g_exp=2.0008898899(21). The theoretical prediction for this value was calculated to be g_th=2.000889909(51) improving the accuracy to 2.5x10^{-8} due to the first rigorous evaluation of the two-photon exchange correction. The measured value is in excellent agreement with the state-of-the-art theoretical prediction and yields the most stringent test of bound-state QED for the g factor of the 1s^22s state and the relativistic many-electron calculations in a magnetic field.

We performed a laser spectroscopic determination of the $2s$ hyperfine splitting (HFS) of Li-like $^{209}\text{Bi}^{80+}$ and repeated the measurement of the $1s$ HFS of H-like $^{209}\text{Bi}^{82+}$. Both ion species were subsequently stored in the Experimental Storage Ring at the GSI Helmholtzzentrum f\"ur Schwerionenforschung Darmstadt and cooled with an electron cooler at a velocity of $\approx 0.71\,c$. Pulsed laser excitation of the $M1$ hyperfine-transition was performed in anticollinear and collinear geometry for $\text{Bi}^{82+}$ and $\text{Bi}^{80+}$, respectively, and observed by fluorescence detection. We obtain $\Delta E^{(1s)}= 5086.3(11)\,\textrm{meV}$ for $\text{Bi}^{82+}$, different from the literature value, and $\Delta E^{(2s)}= 797.50(18)\,\textrm{meV}$ for $\text{Bi}^{80+}$. These values provide experimental evidence that a specific difference between the two splitting energies can be used to test QED calculations in the strongest static magnetic fields available in the laboratory independent of nuclear structure effects. The experimental result is in excellent agreement with the theoretical prediction and confirms the sum of the Dirac term and the relativistic interelectronic-interaction correction at a level of 0.5% confirming the importance of accounting for the Breit interaction., Comment: 5 pages, 2 figures