We present a derivation of the last unknown part of the $α^7m$ contribution to the Lamb shift of a two-electron atom, induced by the radiative QED effects beyond the Bethe logarithm. This derivation is performed in the framework of nonrelativistic quantum electrodynamics and is valid for the triplet (spin $S = 1$) atomic states. The obtained formulas are free from any divergences and are suitable for a numerical evaluation. This opens a way for a complete numerical calculation of the $α^7m$ QED effects in helium, which will allow an accurate determination of the nuclear charge radius from measurements of helium transition frequencies.
Quantum electrodynamical (QED) calculations of ionization energies of the $1snd\,D$ states are performed for the helium atom. We reproduce the previously known relativistic and QED effects up to order $mα^5$ and extend the theory by calculating the complete $mα^6$ correction. The total contribution of the $mα^6$ effects is shown to be much smaller than previously estimated, due to a large cancelation between the radiative and non-radiative parts of this correction. As a result of our calculations, we confirm the previously reported deviations between measured transition energies and theoretica…
Theoretical predictions for the Lamb shift in helium are limited by unknown quantum electrodynamic effects of the order $α^7m$, where $α$ is the fine-structure constant and $m$ is the electron mass. We make an important step towards the complete calculation of these effects by deriving the most challenging part, which is induced by the virtual photon exchange between all three helium particles, the two electrons, and the nucleus. The complete calculation of the $α^7m$ effect including the radiative corrections will allow comparing of the nuclear charge radii determined from the electronic and …
We have derived the complete formula for the $α^7\,m$ contribution to energy levels of an arbitrary triplet state of the helium atom, performed numerical calculations for the $2^3S$ and $2^3P$ states, and thus improved the theoretical accuracy of ionization energies of these states by more than an order of magnitude. Using the nuclear charge radius extracted from the muonic helium Lamb shift, we obtain the theoretical prediction in excellent agreement with the measured $2^3S - 2^3P$ transition energy [X.~Zheng et al., Phys. Rev. Lett. {\bf 199}, 263002 (2017)]. At the same time we observe sign…
Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific kind of non-relativistic contributions. Here we consider such corrections to the Lamb shift in order $α^5m_μ$. These contributions are due to free vacuum polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows a non-relativistic consideration of the muon. Both kin…
A calculation of two-electron QED effects to all orders in the nuclear binding strength parameter $Zα$ is presented for the ground and $n = 2$ excited states of helium-like ions. After subtracting the first terms of the $Zα$ expansion from the all-order results, we identify the higher-order QED effects of order $mα^7$ and higher. Combining the higher-order remainder with the results complete through order $mα^6$ from [V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81, 022507 (2010)], we obtain the most accurate theoretical predictions for the ground and non-mixing $n=2$ states of helium-like ion…
A recent 15 parts-per-million (ppm) experiment on muonic hydrogen found a major discrepancy with QED and independent nuclear size determinations. Here we find a significant discrepancy in a different type of exotic atom, a medium-Z nucleus with two electrons. Investigation of the data collected is able to discriminate between available QED formulations and reveals a pattern of discrepancy of almost 6 standard errors of experimental results from the most recent theoretical predictions with a functional dependence proportional to Z^n where n=4. In both the muonic and highly charged systems, the …
The largest contributions to the $n=2$ Lamb-shift, fine structure interval and $2s$ hyperfine structure of muonic hydrogen are calculated by exact numerical evaluations of the Dirac equation, rather than by a perturbation expansion in powers of $1/c$, in the framework of non-relativistic quantum electrodynamics. Previous calculations and the validity of the perturbation expansion for light elements are confirmed. The dependence of the various effects on the nuclear size and model are studied
Fully relativistic approach to evaluate the correlation effects in highly charged ions is presented. The interelectronic-interaction contributions of first and second orders in $1/Z$ are treated rigorously within the framework of bound-state quantum electrodynamics, whereas the calculations of the third- and higher-order contributions are based on the Dirac-Coulomb-Breit Hamiltonian. The developed approach allows one to deal with single as well as degenerate or quasi-degenerate states. We apply this approach to the calculations of the correlation contributions to the $n=1$ and $n=2$ energy lev…
We perform ab initio calculations of the QED effects of order $mα^7$ for the $2^3S$ and $2^3P$ states of He-like ions. The computed effects are combined with previously calculated energies from [V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81, 022507 (2010)], thus improving the theoretical accuracy by an order of magnitude. The obtained theoretical values for the $2^3S$-$2^3P_{0,2}$ transition energies are in good agreement with available experimental results and with previous calculations performed to all orders in the nuclear binding strength parameter $Zα$. For the ionization energies, howe…
In [Li ${\it et \, al.}$, Phys. Rev. A ${\bf 98}$, 020502(R) (2018)] it was claimed that the model-potential computations of the Lamb shift on the $^2P_{1/2}-{}^2P_{3/2}$ fine structure in fluorinelike uranium lead to a discrepancy between theory and experiment. Later, it was reported by [Volotka ${\it et \, al.}$, Phys. Rev. A ${\bf 100}$, 010502(R) (2019)] that {\it ab initio} QED calculation, including the first-order one-electron QED contributions and the related effects of two-electron screening, yields the result which restores the agreement between theory and experiment and strongly dis…
The technique of quantum electrodynamics (QED) calculations of energy levels in the helium atom is reviewed. The calculations start with the solution of the Schrödinger equation and account for relativistic and QED effects by perturbation expansion in the fine-structure constant $α$. The nonrelativistic wave function is represented as a linear combination of basis functions depending on all three interparticle radial distances, $r_1$, $r_2$ and $r = |\vec{r}_1-\vec{r}_2|$. The choice of the exponential basis functions of the form $\exp(-αr_1 -βr_2 -γr)$ allows us to construct an accurate and c…
A model operator approach to calculations of the QED corrections to energy levels in relativistic many-electron atomic systems is developed. The model Lamb shift operator is represented by a sum of local and nonlocal potentials which are defined using the results of ab initio calculations of the diagonal and nondiagonal matrix elements of the one-loop QED operator with H-like wave functions. The model operator can be easily included in any calculations based on the Dirac-Coulomb-Breit Hamiltonian. Efficiency of the method is demonstrated by comparison of the model QED operator results for the …
B-spline functions have been widely used in computational atomic physics. Different from the traditional B-spline basis (a simple product of two B-splines), the recently developed correlated B-spline basis functions(C-BSBF), in which the interelectronic coordinate $r_{12}$ is included explicitly, have greatly improved the computational accuracy of polarizability [S. J. Yang \textit{et al}., Phys. Rev. A \textbf{95}, 062505 (2017)] and bethe logarithm [ S. J. Yang \textit{et al}., Phys. Rev. A \textbf{100}, 042509 (2019)] for singlet states of helium. Here, we report the extension of the C-BSBF…
We calculate the one- and two-loop corrections of order alpha(Zalpha)^6 and alpha^2(Zalpha)^6 respectively, to the Lamb shift in hydrogen-like systems using the formalism of nonrelativistic quantum electrodynamics. We obtain general results valid for all hydrogenic states with nonvanishing orbital angular momentum and for the normalized difference of S-states. These results involve the expectation value of local effective operators and relativistic corrections to Bethe logarithms. The one-loop correction is in agreement with previous calculations for the particular cases of S, P, and D states.…
Comparison of high accuracy calculations with precision measurement of the 413 nm tune-out wavelength of the He($2\,^3S_1$) state provides a unique test of quantum electro-dynamic (QED). We perform large-scale relativistic-configuration-interaction (RCI) calculations of the tune-out wavelength, that include the mass-shift operator, and fully account for leading relativistic nuclear recoil terms in the Dirac-Coulomb-Breit (DCB) Hamiltonian. We obtain the QED correction to the tune-out wavelength using perturbation theory, and the effect of finite nuclear size is also evaluated. The resulting tu…
The QED contribution to the dipole polarizability of the $^4$He atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to $0.048\,557\,2(14…
The velocity renormalization group is used to determine ln(alpha) contributions to QED bound state energies. The leading order anomalous dimension for the potential gives the alpha^5 ln(alpha) Bethe logarithm in the Lamb shift. The next-to-leading order anomalous dimension determines the alpha^6 ln(alpha), alpha^7 ln^2(alpha), and alpha^8 ln^3 (alpha) corrections to the energy. These are used to obtain the alpha^8 ln^3(alpha) Lamb shift and alpha^7 ln^2(alpha) hyperfine splitting for Hydrogen, muonium and positronium, as well as the alpha^2 ln(alpha) and alpha^3 ln^2(alpha) corrections to the …
We present the first measurement for helium atoms of the tune-out wavelength at which the atomic polarizability vanishes. We utilise a novel, highly sensitive technique for precisely measuring the effect of variations in the trapping potential of confined metastable ($2^{3}S_{1}$) helium atoms illuminated by a perturbing laser light field. The measured tune-out wavelength of 413.0938($9_{Stat.}$)($20_{Syst.}$) nm compares well with the value predicted by a theoretical calculation (413.02(9) nm) which is sensitive to finite nuclear mass, relativistic, and quantum electro-dynamic (QED) effects. …
Improvements in both theory and frequency metrology of few-electron systems such as hydrogen and helium have enabled increasingly sensitive tests of quantum electrodynamics (QED), as well as ever more accurate determinations of fundamental constants and the size of the nucleus. At the same time advances in cooling and trapping of neutral atoms have revolutionized the development of increasingly accurate atomic clocks. Here, we combine these fields to reach the highest precision on an optical tranistion in the helium atom to date by employing a Bose-Einstein condensate confined in a magic wavel…
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 $2^{3\!}S_1$ state of helium between transitions to t…
Calculation of higher-order two-loop corrections is now a limiting factor in development of the bound state QED theory of the Lamb shift in the hydrogen atom and in precision determination of the Rydberg constant. Progress in the study of light hydrogen-like ions of helium and nitrogen can be helpful to investigate these uncalculated terms experimentally. To do that it is necessary to develop a theory of such ions. We present here a theoretical calculation for low energy levels of helium and nitrogen ions.
We present an introduction to the principles behind atomic energy level calculations with Quantum Electrodynamics (QED) and the two-time Green's function method; this method allows one to calculate an effective Hamiltonian that contains all QED effects and that can be used to predict QED Lamb shifts of degenerate, quasidegenerate and isolated atomic levels.
Energy levels of the $1sns$ and $1snp$ states of ions along the helium isoelectronic sequence from carbon to uranium are calculated, with $n=3$-$7$. The computation is performed within the relativistic configuration-interaction method, including the relativistic nuclear recoil effect, the leading QED effects, and the frequency dependence of the Breit interaction. All theoretical energies are supplied with uncertainty estimates.
High-precision laser spectroscopy of atomic hydrogen has led to an impressive accuracy in tests of bound-state quantum electrodynamics (QED). At the current level of accuracy many systematics have to be studied very carefully and only independent measurements provide the ultimate cross-check. This has been proven recently by measurements in muonic hydrogen, eventually leading to a significant shift of the CODATA recommended values of the proton charge radius and the Rydberg constant. We aim to contribute to tests of fundamental physics by measuring the 1S-2S transition in the He$^+$ ion for th…
Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for $Z \leq 20$ the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with $Z \leq 20$ the finite nuclear mass effects do not change 4-5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less…
Recently the muonic hydrogen lamb shift has been measured with unprecedented accuracy, allowing for a precise determination of the proton radius. This determination is 5 sigma away from the previous CODATA value obtained from (mainly) the hydrogen lamb shift and the electron-proton scattering. Within an effective field theory formalism, I will define the proton radius and briefly review some aspects of the theoretical prediction for the muonic hydrogen lamb shift, studying both the pure QED-like computation and the hadronic effects.
On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order $α(Z α)^5$ to the Lamb shift in muonic hydrogen and helium. To construct the interaction potential of particles, which gives the necessary contributions to the energy spectrum, we use the method of projection operators to states with a definite spin. Separate analytic expressions for the contributions of the muon self-energy, the muon vertex operator and the amplitude with spanning photon are obtained. We present also numerical results for these contributions using m…
We consider the $1s$ Lamb shift in hydrogen and helium ions, a quantity, required for an accurate determination of the Rydberg constant and the proton charge radius by means of hydrogen spectroscopy, as well as for precision tests of the bound-state QED. The dominant QED contribution to the uncertainty originates from $α^8m$ external-field contributions (i.e., the contributions at the non-recoil limit). We discuss the two- and three-loop cases and in particular, we revisit calculations of the coefficients $B_{61}, B_{60}, C_{50}$ in standard notation. We have found a missing logarithmic contri…
The derivation of leading quantum electrodynamic corrections to the nuclear magnetic shielding in light hydrogen- and helium-like atomic systems is described in detail. The presented theoretical approach applies to any light atomic and molecular systems, enabling the determination of the magnetic moment of light nuclei with much higher precision than known presently.
We provide an up to date summary of the theory contributions to the 2S-2P Lamb shift and the fine structure of the 2P state in the muonic helium ion $(\mathrm{μ^4He})^+$. This summary serves as the basis for the extraction of the alpha particle charge radius from the muonic helium Lamb shift measurements at the Paul Scherrer Institute, Switzerland. Individual theory contributions needed for a charge radius extraction are compared and compiled into a consistent summary. The influence of the alpha particle charge distribution on the elastic two-photon exchange is studied to take into account pos…
A calculation valid to all orders in the nuclear-strength parameter is presented for the two-loop Lamb shift, notably for the two-loop self-energy correction, to the 2p-2s transition energies in heavy Li-like ions. The calculation removes the largest theoretical uncertainty for these transitions and yields the first experimental identification of two-loop QED effects in the region of the strong binding field.
Precision spectroscopy of the hydrogen molecule is a test ground of quantum electrodynamics (QED), and may serve for determination of fundamental constants. Using a comb-locked cavity ring-down spectrometer, for the first time, we observed the Lamb-dip spectrum of the R(1) line in the overtone of HD. The line position was determined to be 217 105 182.79 $\pm0.03_{stat}\pm0.08_{syst}$ MHz ($δν/ν=4\times 10^{-10}$), which is the most accurate transition ever measured for the hydrogen molecule. Moreover, from calculations including QED effects up to the order $m_eα^6$, we obtained predictions for…
The present status of tests of QED with highly charged ions is reviewed. The theoretical predictions for the Lamb shift and the transition energies are compared with available experimental data. Recent achievements in studies of the hyperfine splitting and the $g$-factor isotope shift with highly charged ions are reported. Special attention is paid to tests of QED within and beyond the Furry picture at strong-coupling regime. Prospects for tests of QED at supercritical fields that can be created in low-energy heavy-ion collisions are discussed as well.
In connection with recent and proposed experiments, and new theoretical results, my previous calculations of the Lamb shift in muonic hydrogen will be reviewed and compared with other work. In addition, numerical results for muonic deuterium and helium will be presented. Some previously neglected (but very small) effects are included.
We present a detailed study of the Flambaum-Ginges radiative potential method which enables the accurate inclusion of quantum electrodynamics (QED) radiative corrections in a simple manner in atoms, ions, and molecules over the range 10<=Z<=120, where Z is the nuclear charge. Calculations are performed for binding energy shifts to the lowest valence s, p, and d waves over the series of alkali atoms Na to E119. The high accuracy of the radiative potential method is demonstrated by comparison with rigorous QED calculations in frozen atomic potentials, with deviations on the level of 1%. The many…
The fine structure interval of P states in hydrogenlike systems can be determined theoretically with high precision, because the energy levels of P states are only slightly influenced by the structure of the nucleus. Therefore a measurement of the fine structure may serve as an excellent test of QED in bound systems or alternatively as a means of determining the fine structure constant $α$ with very high precision. In this paper an improved analytic calculation of higher-order binding corrections to the one-loop self energy of 3P and 4P states in hydrogen-like systems with low nuclear charge n…
Cavity-modified static polarizabilities of atoms and small molecules within QED full configuration interaction
St. Petersburg State University（圣彼得堡国立大学）；Petersburg Nuclear Physics Institute named by B. P. Konstantinov of National Research Centre “Kurchatov Institute”（国家研究中心“库尔恰托夫研究所”下属的以B.P.康斯坦丁诺夫命名的圣彼得堡核物理研究所）
Expressions for the effective Quantum Electrodynamics (QED) Hamiltonian due to self-energy screening (self-energy correction to the electron-electron interaction) are presented. We use the method of the two-time Green's function, which handles quasidegenerate atomic states. From these expression one can evaluate energy corrections to, e.g., 1s2p 3P1 and 1s2p 1P1 levels in helium and two-electron ions, to all orders in Zα
The present knowledge of Lamb shift, fine-, and hyperfine structure of the 2S and 2P states in muonic helium-3 ions is reviewed in anticipation of the results of a first measurement of several $\mathrm{2S\rightarrow2P}$ transition frequencies in the muonic helium-3 ion, $\mathrm{μ^3He^+}$. This ion is the bound state of a single negative muon $μ^-$ and a bare helium-3 nucleus (helion), $\mathrm{^3He^{++}}$. A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-d…
Theoretical calculations of the Lamb shift provide the basis required for the determination of the Rydberg constant from spectroscopic measurements in hydrogen. The recent high-precision determination of the proton charge radius drastically reduced the uncertainty in the hydrogen Lamb shift originating from the proton size. As a result, the dominant theoretical uncertainty now comes from the two- and three-loop QED effects, which calls for further advances in their calculations. We review the present status of theoretical calculations of the Lamb shift in hydrogen and light hydrogen-like ions …
We investigate the precisely measured anomalous magnetic moment and Lamb shift as tests for the possible existence of the radiatively induced Lorentz and CPT violation effects in quantum electrodynamics. To this end we calculate the one-loop vacuum polarization tensor and vertex radiative correction in dimensional reduction and on-shell renormalization scheme. We explicitly show how the Lorentz and CPT violation sector affects the anomalous magnetic moment and Lamb shift. Remarkably, we find infrared divergences coming from Lorentz and CPT violating term that do not cancel in physical cross se…
We comprehensively analyse the theoretical prediction for the Lamb shift in muonic hydrogen, and the associated determination of the proton radius. We use effective field theories. This allows us to relate the proton radius with well-defined objects in quantum field theory, eliminating unnecessary model dependence. The use of effective field theories also helps us to organize the computation so that we can clearly state the parametric accuracy of the result. In this paper we review all (and check several of) the contributions to the energy shift of order $α^5$, as well as those that scale like…
In this review, we have investigated the asymmetry of the line profile in precision one- and two-photon spectroscopy of hydrogen and helium atoms within the framework of a rigorous QED approach. A detailed analysis of the angular correlations of the quantum interference effect has been carried out using various examples. Nonresonant effects are also considered in relation to some astrophysical problems. In particular, a rigorous QED derivation of the nonresonant extension for the Lorentz line profile is given using the Ly$_α$ transition as an example; such a QED derivation has been lacking in …
Corrections to energy levels in light muonic atoms are investigated in order $α^2(Zα)^4m$. We pay attention to corrections which are specific for muonic atoms and include the electron vacuum polarization loop. In particular, we calculate relativistic and relativistic-recoil two-loop electron vacuum polarization contributions. The results are obtained for the levels with $n=1,2$ and in particular for the Lamb shift ($2p_{1/2}-2s_{1/2}$) and fine-structure intervals ($2p_{3/2}-2p_{1/2}$) in muonic hydrogen, deuterium, and muonic helium ions.
The electron-electron interaction correction of first order in $1/Z$ to the one-electron part of the nuclear recoil effect on binding energies in atoms and ions is considered within the framework of the rigorous QED approach. The calculations to all orders in $αZ$ are performed for the $1s^2$ state in heliumlike ions and the $1s^2 2s$ and $1s^2 2p_{1/2}$ states in lithiumlike ions in the range $Z=5$--$100$. The results obtained are compared with the Breit-approximation values. The performed calculations complete a systematic treatment of the QED nuclear recoil effect up to the first order in $…
We investigate the finite nuclear mass corrections in the helium atom in order to resolve a significant disagreement between the $2^3S - 2^3P$ and $2^3S - 2^1S$ transition isotope shifts. These two transitions lead to discrepant results for the nuclear charge radii difference between $^4$He and $^3$He. The accurate treatment of the finite nuclear mass effects is quite complicated and requires the use of the quantum field theoretical approach. We derive $α^6 m^2/M$ correction with the help of nonrelativistic QED and dimensional regularization of the three body Coulombic system, and present accu…
The measurement of the Lamb shift in muonic hydrogen and the subsequent emergence of the proton-radius puzzle have motivated an experimental campaign devoted to measuring the Lamb shift in other light muonic atoms, such as muonic deuterium and helium. For these systems it has been shown that two-photon exchange nuclear structure corrections are the largest source of uncertainty and consequently the bottleneck for exploiting the experimental precision to extract the nuclear charge radius. Utilizing techniques and methods developed to study electromagnetic reactions in light nuclei, recent calcu…
The total energies and various bound state properties of the excited $2^1S(L = 0)-$states in two-electron helium atoms, including the ${}^{\infty}$He, ${}^4$He and ${}^3$He atoms, are determined to very high numerical accuracy. The convergence of the results obtained for some electron-nuclear and electron-electron expectation values and, in particular, for the electron-nuclear and electron-electron cusp values, is discussed. The field component of the isotope shift and lowest order QED correction are estimated for the $2^1S(L = 0)-$states in the ${}^4$He and ${}^3$He atoms. We also apply our h…
An electrodynamical coupled cluster (CC) methodology starting from a covariant formalism and an equal time approximation, and finally based on the Dirac-Fock picture of the electron and positron fields and Coulomb gauge, is given here. The formalism first leads to different physical interactions from the use of an exponential cluster operator for radiative effects. Lamb, Breit and hyperfine interactions are obtained. Next, relativistic many-body effects are determined using the matter cluster in a way familiar from the nonrelativistic CC. This step can be nontrivial. By allowing the matter clu…
