Quantum Electrodynamics Contributions Research Articles

The study of atomic structure parameters for [formula omitted] = 4 - [formula omitted] = 3 transitions in Mg-like ions with [formula omitted

Multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction (RCI) methods are utilized for theoretical calculations of energy levels, wavelengths, line strengths, absorption oscillator strengths, and transition probabilities in 3s2-3s4p, 3s3p-3s4s, 3s3p-3s4d, 3s3p-3p4p, 3s3d-3s4f, 3p2-3p4s, 3p2-3p4d, and 3p3d-3p4f transitions in Mg-like ions with 15≤Z≤30. The calculations of energies account for Breit interaction and quantum electrodynamics (QED) contributions. The active space approximation is employed for the calculations, and energy converges when the active orbital set is increased to n = 7. The calculated lowest 78 levels including valence and core–valence correlations in Mg-like ions with Z=15–30 show good agreement with experiment results and other calculations. The relative difference between calculated wavelengths and experiment values exhibits a decreasing trend as the atomic numbers increase, and the difference is better than 1% for a majority of transitions. To evaluate the accuracy of the wave functions and transition parameters, the quantity dT is analyzed within 0.1 for most strong transitions. The transition probabilities are compared with other theoretical values to analyze with line strength S. Such calculations may provide valuable data for the experimental study of plasma diagnostics and modeling as there are others.

Toward XUV frequency comb spectroscopy of the 1 S–2 S transition in hbox {He}^+

The energy levels of hydrogen-like atoms and ions are accurately described by bound-state quantum electrodynamics (QED). hbox {He}^{+} ions have a doubly charged nucleus, which enhances the higher-order QED contributions and makes them interesting for precise tests of QED. Systematic effects that currently dominate the uncertainty in hydrogen spectroscopy, such as the second-order Doppler shift and time-of-flight broadening, are largely suppressed by performing spectroscopy on trapped and cooled hbox {He}^{+} ions. Measuring a transition in hbox {He}^{+} will extend the test of QED beyond the long-studied hydrogen. In this article, we describe our progress toward precision spectroscopy of the 1 S–2 S two-photon transition in hbox {He}^{+}. The transition can be excited by radiation at a wavelength of 60.8 nm generated by a high-power infrared frequency comb using high-order harmonic generation (HHG). The hbox {He}^{+} ions are trapped in a Paul trap and sympathetically cooled with laser-cooled hbox {Be}^{+} ions. Our recently developed signal detection scheme based on secular-scan spectrometry is capable of detecting hbox {He}^{+} excitation with single-event sensitivity.Graphic abstract

QED radiative corrections to electric dipole amplitudes in heavy atoms

We use the radiative potential method to perform a detailed study of quantum electrodynamics (QED) radiative corrections to electric dipole (E1) transition amplitudes in heavy alkali-metal atoms Rb, Cs, Fr, and alkali-metal-like ions Sr+, Ba+, and Ra+. The validity of the method is checked by comparing with the results of rigorous QED in simple atomic potentials. We study the effects of core relaxation, polarization of the core by the E1 field, and valence-core correlations on QED, which are shown to be important in some cases. We identify several transitions for which the QED contribution exceeds the deviation between atomic theory and experiment.

Generalized relativistic small‐core pseudopotentials accounting for quantum electrodynamic effects: Construction and pilot applications

AbstractA simple procedure to incorporate one‐loop quantum electrodynamic (QED) corrections into the generalized (Gatchina) nonlocal shape‐consistent relativistic pseudopotential model is described. The pseudopotentials for Lu, Tl, and Ra replacing only inner core shells (with principal quantum numbersn ≤ 3 for the two former elements andn ≤ 4 for the latter one) are derived from the solutions of reference atomic SCF problems with the Dirac–Coulomb–Breit Hamiltonian to which the model Lamb shift operator added. QED contributions to atomic valence excitation energies evaluated at the SCF level are demonstrated to exceed the errors introduced by the pseudopotential approximation itself by an order of magnitude. Pilot applications of the new model to calculations of excitation energies of two‐valence‐electron atomic systems using the intermediate‐Hamiltonian relativistic Fock space coupled cluster method reformulated here for incomplete main model spaces are reported. Implications for high‐accuracy molecular excited state calculations are discussed.

4-component relativistic Hamiltonian with effective QED potentials for molecular calculations.

We report the implementation of effective quantum electrodynamics (QED) potentials for all-electron four-component relativistic molecular calculations using the DIRAC code. The potentials are also available for two-component calculations, being properly picture-change transformed. The latter point is important; we demonstrate through atomic calculations that picture-change errors are sizable. Specifically, we have implemented the Uehling potential [E. A. Uehling, Phys. Rev. 48, 55 (1935)] for vacuum polarization and two effective potentials [P. Pyykkö and L.-B. Zhao, J. Phys. B: At., Mol. Opt. Phys. 36, 1469 (2003) and V. V. Flambaum and J. S. M. Ginges, Phys. Rev. A 72, 052115 (2005)] for electron self-energy. We provide extensive theoretical background for these potentials, hopefully reaching an audience beyond QED specialists. We report the following sample applications: (i) We first confirm the conjecture of P. Pyykkö that QED effects are observable for the AuCN molecule by directly calculating ground-state rotational constants B0 of the three isotopomers studied by microwave spectroscopy; QED brings the corresponding substitution Au-C bond length rs from 0.23 to 0.04pm agreement with experiment. (ii) In regard to spectroscopic constants of van der Waals dimers M2 (M = Hg, Rn, Cn, Og), QED induces bond length expansions on the order of 0.15(0.30) pm for row 6(7) dimers. (iii) We confirm that there is a significant change of valence s population of Pb in the reaction PbH4 → PbH2 + H2, which is thereby a good candidate for observing QED effects in chemical reactions, as proposed in [K. G. Dyall et al., Chem. Phys. Lett. 348, 497 (2001)]. We also find that whereas in PbH4 the valence 6s1/2 population resides in bonding orbitals, it is mainly found in nonbonding orbitals in PbH2. QED contributes 0.32 kcal/mol to the reaction energy, thereby reducing its magnitude by -1.27%. For corresponding hydrides of superheavy flerovium, the electronic structures are quite similar. Interestingly, the QED contribution to the reaction energy is of quite similar magnitude (0.35 kcal/mol), whereas the relative change is significantly smaller (-0.50%). This curious observation can be explained by the faster increase in negative vacuum polarization over positive electron self-energy contributions as a function of nuclear charge.

Wavelengths, transition rates, weight oscillator strengths and line strengths for He-like to Ne-like cobalt ions

The multi-configuration Dirac–Hartree–Fock (MCDHF) and Relativistic Configuration Interaction (RCI) methods have been used to calculate the wavelengths, transition rates, absorption oscillator strengths, and line strengths for the 2s-3p, 2p-3s, and 2p-3d electric dipole (E1) transitions in He-like to Ne-like cobalt ions. Breit interaction and quantum-electrodynamical (QED) contributions were estimated in extensive RCI calculations. The results are in good agreement with previous experimental and theoretical results. Our calculated wavelengths differ from wavelengths reported in other work mostly by less than 0.78%, with two exceptions, where our results differ by 1.63% and 1.91%, showing good overall confirmation of the accuracy of our results. Most importantly, we have predicted some theoretical data that, will be helpful in some plasma experiments.

Energy Levels and Radiative Lifetimes for Four Times Ionized Xenon (Xe V)

By employing the general-purpose relativistic atomic structure package (GRASP) based on the fully relativistic multiconfiguration Dirac-Fock (MCDF) method, energies and radiative lifetimes of levels have been reported for four times ionized xenon (Xe V). In calculations, the Breit interactions for relativistic effects, quantum electrodynamical (QED) contributions have been included as a perturbation, as well as electron correlation effects (VV, CV, and CC). We have compared the results obtained from this work with other available experimental and theoretical values in available literature for Xe V. We have obtained good agreement with previous works.

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.

Measurement of a helium tune-out frequency: an independent test of quantum electrodynamics.

Despite quantum electrodynamics (QED) being one of the most stringently tested theories underpinning modern physics, recent precision atomic spectroscopy measurements have uncovered several small discrepancies between experiment and theory. One particularly powerful experimental observable that tests QED independently of traditional energy level measurements is the "tune-out" frequency, where the dynamic polarizability vanishes and the atom does not interact with applied laser light. In this work, we measure the tune-out frequency for the 23S1 state of helium between transitions to the 23P and 33P manifolds and compare it with new theoretical QED calculations. The experimentally determined value of 725,736,700(260) megahertz differs from theory [725,736,252(9) megahertz] by 1.7 times the measurement uncertainty and resolves both the QED contributions and retardation corrections.

Abinitio determination of the polarizability of neon

The static electric-dipole polarizability $\ensuremath{\alpha}$ of the neon atom was determined with a relative uncertainty of only about 0.003% using state-of-the-art ab initio approaches. The new value, $\ensuremath{\alpha}=2.661\phantom{\rule{0.16em}{0ex}}067(77)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, is almost five times more accurate than the previous ab initio estimate, $\ensuremath{\alpha}=2.660\phantom{\rule{0.16em}{0ex}}80(36)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, by Lesiuk et al. [Phys. Rev. A 102, 052816 (2020)]. Similar to their work, we calculated $\ensuremath{\alpha}$ using ab initio methods up to full configuration interaction and added corrections for finite nuclear mass and size, relativistic, and quantum electrodynamics (QED) effects. The uncertainty reduction of this work was achieved in particular by employing extremely large basis sets, including newly developed ones of 11Z, 12Z, and 13Z quality. Moreover, the finite nuclear mass effects and most of the relativistic contributions were calculated at much higher levels of theory than in the work of Lesiuk et al. However, we adopted their values for the orbit-orbit part of the relativistic correction and for the Bethe logarithm needed to compute the QED correction. The uncertainty of our final value is still an order of magnitude larger than that of the experimental value recently measured by Gaiser and Fellmuth [Phys. Rev. Lett. 120, 123203 (2018)] with an uncertainty of only a few parts per million using dielectric-constant gas thermometry. Yet, our ab initio value agrees with their value, ${\ensuremath{\alpha}}_{\mathrm{exp}}=2.661\phantom{\rule{0.16em}{0ex}}057(7)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, almost within the experimental uncertainty. This could indicate that the higher-order relativistic corrections and QED contributions, which dominate our uncertainty budget, are more accurate than expected considering the uncontrolled approximations involved in their calculation.

Precision measurement of quasi-bound resonances in H2 and the H + H scattering length

Quasi-bound resonances of are produced via two-photon photolysis of S molecules as reactive intermediates or transition states and detected before decay of the parent molecule into three separate atoms. As was previously reported [K. F. Lai et al., Phys. Rev. Lett. 127, 183001 (2021)] four centrifugally bound quantum resonances with lifetimes of multiple μs, lying energetically above the dissociation limit of the electronic ground state of , were observed as , , , and , while also the short-lived ( ns) quasi-bound resonance was probed. The present paper gives a detailed account on the identification of the quasi-bound or shape resonances, based on laser detection via - two-photon transitions, and their strongly enhanced Franck-Condon factors due to the shifting of the wave function density to large internuclear separation. In addition, the assignment of the rotational quantum number is verified by subsequent multi-step laser excitation into autoionisation continuum resonances. Existing frameworks of full-fledged ab initio computations for the bound region in , including Born-Oppenheimer, adiabatic, non-adiabatic, relativistic and quantum-electrodynamic contributions, are extended into the energetic range above the dissociation energy. These comprehensive calculations are compared to the accurate measurements of energies of quasi-bound resonances, finding excellent agreement. They show that the quasi-bound states are in particular sensitive to non-adiabatic contributions to the potential energy. From the potential energy curve and the correction terms, now tested at high accuracy over a wide range of energies and internuclear separations, the s-wave scattering length for singlet H+H scattering is determined at . It is for the first time that such an accurate value for a scattering length is determined based on fully ab initio methods including effects of adiabatic, non-adiabatic, relativistic and QED with contributions up to .

The role of QED effects in transition energies of heavy-atom alkaline earth monofluoride molecules: A theoretical study of Ba+, BaF, RaF, and E120F.

Heavy-atom alkaline earth monofluoride molecules are considered as prospective systems to study spatial parity or spatial parity and time-reversal symmetry violating effects such as the nuclear anapole moment or the electron electric dipole moment. A comprehensive and highly accurate theoretical study of the electronic structure properties and transition energies in such systems can simplify the preparation and interpretation of the experiments. However, almost no attempts to calculate quantum electrodynamics (QED) effects' contribution into characteristics of these neutral heavy-atom molecules have been performed. Recently, we have formulated and implemented such an approach to calculate QED contributions to transition energies of molecules [L. V. Skripnikov, J. Chem. Phys. 154, 201101 (2021)]. In this paper, we perform a benchmark theoretical study of the transition energies in the Ba+ cation and BaF molecule. The deviation of the calculated values from the experimental ones is of the order 10 cm-1 and is more than an order of magnitude better than the "chemical accuracy," 350 cm-1. The achievement of such an agreement has been provided, in particular, by the inclusion of the QED effects. The latter appeared to be not less important than the high-order correlation effects beyond the coupled cluster with single, double, and perturbative triple cluster amplitude level. We compare the role of QED effects for transition energies with heavier molecules-RaF and E120F, where E120 is the superheavy Z = 120 homologof Ra.

Relativistic multiconfiguration Dirac-Hartree-Fock calculations for energies, lifetimes, and E1, E2, M1, and M2 line strengths of Y XXXII

Energies, wavelengths, line strengths, transition probabilities, and weighted oscillator strengths for electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions among the levels arising from the n ≤ 3 configurations of oxygen-like yttrium, Y XXXII, are calculated using the multiconfiguration Dirac-Hartree-Fock (MCDHF) method. In the calculations, the valence-valence (VV) and core-valence (CV) correlations, as well as the Breit interaction (BI) and the leading quantum electrodynamics (QED) contributions are considered. Comparisons with the available results are used to determine the uncertainty for energies, wavelengths, and oscillator strengths. For lifetimes, E1, E2, M1, and M2 line strengths, the comparisons are also made between the two sets of the present MCDHF calculations to determine the accuracy of the results.

Ground-state gj factors of the Cd+, Yb+, and Hg+ ions

Accurate calculations of ${g}_{j}$ factors of the ground state of the ${\mathrm{Cd}}^{+},$ ${\mathrm{Yb}}^{+},$ and ${\mathrm{Hg}}^{+}$ ions are presented by employing the normal order unrelaxed $\mathrm{\ensuremath{\Lambda}}$-approach relativistic coupled-cluster $(\mathrm{\ensuremath{\Lambda}}\text{\ensuremath{-}}\mathrm{RCC})$ theory. Contributions from the quantum electrodynamics (QED) are estimated from the free electron QED contributions and the roles of electron correlation effects are analyzed with different cutoff of occupied and virtual active orbitals in the $\mathrm{\ensuremath{\Lambda}}$-RCC method. Our final ${g}_{j}$ factors come out to be 2.002291(4), 2.002798(113), and 2.003128(41) for the ${\mathrm{Cd}}^{+},$ ${\mathrm{Yb}}^{+},$ and ${\mathrm{Hg}}^{+}$ ions, respectively. Our result for ${\mathrm{Hg}}^{+}$ agrees up to the fourth decimal places with its experimental value 2.0031745(74) indicating that our calculations of the other ions are of similar accuracies. The understanding of roles of electron correlation effects from this study will be useful in improving accuracies of the ${g}_{j}$ factors in the considered ions further by adding more physical effects in the future and performing calculations with similar accuracies in other heavier atomic systems. The reported ${g}_{j}$ factors can also be used to scrutinize the background noise related to the stray magnetic fields in the laser-cooled microwave ion clocks using the above ions.

Energies and E1, M1, E2, and M2 line strengths for Nb XXXIV

The multiconfiguration Dirac-Hartree-Fock (MCDHF) method is used to calculate energies, lifetimes, wavelengths, transition probabilities, weighted oscillator strengths, and line strengths for electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), and magnetic quadrupole (M2) transitions among the lowest 226 fine-structure levels arising from the 2s22p4, 2s2p5, 2p6, 2s22p33l, 2s2p43l, and 2p53l (l = s, p, d) configurations in O-like Nb XXXIV. The valence and core-valence correlation effects are included in the calculations. The Breit-interaction and the leading quantum electrodynamics contributions are taken into account in the relativistic configuration interaction calculations. The present energies, wavelengths, line strengths, and oscillator strengths are compared with the available experimental and other theoretical results. The accuracy of the present results is estimated using the comparison between the results from the two largest layers in the present MCDHF calculations.

Energy levels and transition probabilities for transitions in Zr XXXV

Excitation energies, wavelengths, transition probabilities, weighted oscillator strengths, and line strengths for electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), and magnetic quadrupole (M2) transitions between the states of the 2s22p2, 2s2p3, 2p4, 2s22p3l, 2s2p23l, and 2p33l(l = s, p, d) configurations are calculated using the multiconfiguration Dirac-Hartree-Fock (MCDHF) method for C-like Zr XXXV. The Breit interaction and quantum electrodynamics (QED) contributions are included in the relativistic configuration interaction calculations. The calculated energies, wavelengths, and oscillator strengths are compared with the other theoretical results.

Isotope Shifts of Energy Levels in Helium-Like Highly Charged Ions

Isotope shifts of $$n = 1$$ and $$n = 2$$ levels and corresponding transitions in helium-like highly charged ions are calculated with high precision. The total value of the isotope shift is determined mainly by the sum of the field and mass shifts. The field shift is calculated by using the Dirac–Fock–Sturm method taking into account interaction of configurations. Quantum-electrodynamic corrections to this contribution are taken into account approximately by using the corresponding one-electron expressions. The mass shift is calculated both within the framework of the Breit approximation and by taking into account quantum-electrodynamic contributions that become substantial for heavy ions. In the case of thorium and uranium ions, corrections due to nuclear polarization and deformation are taken into account additionally.

Accurate determination of energy levels, hyperfine structure constants, lifetimes and dipole polarizabilities of triply ionized tin isotopes

We have investigated the energies, magnetic dipole hyperfine structure constants (Ahyf) and electric dipole (E1) matrix elements of a number of low-lying states of triply ionized tin (Sn3+) by employing relativistic coupled-cluster theory. Contributions from the Breit interaction and lower-order quantum electrodynamic (QED) effects in the determination of the above quantities are also given explicitly. These higher-order relativistic effects are found to be important for the accurate evaluation of energies, while only QED contributions are seen to contribute significantly to the determination of Ahyf values. Our theoretical results for the energies are in agreement with one of the measurements but show significant differences for some states with another measurement. Reported Ahyf will be useful in guiding the measurements of hyperfine levels in the stable isotopes of Sn3+. The calculated E1 matrix elements are further used to estimate the oscillator strengths, transition probabilities and dipole polarizabilities (α) of many states. Large discrepancies between the present results and previous calculations of oscillator strengths and transition probabilities are observed among a number of states. The estimated α values will be useful for carrying out high precision measurements using the Sn3+ ion in future experiments.

Изотопические сдвиги уровней энергии в гелиеподобных многозарядных ионах

The precision calculations of the isotope shifts of the $n = 1$, $n = 2$ energy levels and the corresponding transition energies in helium-like highly charged ions are performed. The total isotope shift is mainly determined by a sum of the field and mass shifts. The field shift is calculated by the configuration-interaction Dirac-Fock-Sturm method. The quantum electrodynamics corrections to this contribution are taken into account approximately by using the corresponding one-electron formulas. The calculation of the mass shift is performed within the framework of the Breit approximation and includes the quantum electrodynamics contributions, which become very significant for heavy ions. For the thorium and uranium ions the nuclear polarization and deformation corrections are taken into account.

Energies and electric dipole transitions for triply ionized Kr, Xe and Rn

Energies and transition parameters such as wavelengths, transition probabilities (rates) and oscillator strengths for the electric dipole (E1) transitions have been reported for triply ionized krypton (Z = 36), xenon (Z = 54) and radon (Z = 86) using the general-purpose relativistic atomic structure package based on a fully relativistic multiconfiguration Dirac–Fock method. The Breit interactions for the relativistic effects, quantum electrodynamical contributions and correlation effects have been included in calculations. Obtained results have been compared with available theoretical and experimental values in the literature. Most of results, in particular for Rn IV, have been presented firstly.