We consider one-loop self-energy and vacuum polarization radiative corrections to the shift of atomic energy level due to finite nuclear size. Analytic expressions for vacuum polarization corrections are derived. For the self-energy of p1/2 and p3/2 states in addition to already known terms we derive next-to-leading nonlogarithmic Zα-terms. Together with contributions obtained earlier the terms derived in the present work give explicit analytic expressions for s1/2 and p1/2 corrections which agree with results of previous numerical calculations up to Z=100 (Z is the nuclear charge number). We also show that the finite nuclear size radiative correction for a p3/2 state is not small compared to the similar correction for a p1/2 state at least for small Z.
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…
The extraction of nuclear charge radii from spectroscopy experiments in muonic atoms is currently limited by the large uncertainties associated with the theoretical evaluation of the nuclear polarizability effects. To facilitate calculations, these polarizability corrections are conventionally expressed as an expansion in a dimensionless parameter $η$, which has been argued in previous literature to hold an approximate value of 0.33 in light-nuclear systems. In this work, we check this claim by doing a Bayesian analysis of the nuclear-polarizability corrections to the Lamb shift in $μ^2$H and …
The nuclear size effect on the one-loop self energy and vacuum polarization is evaluated for the 1s, 2s, 3s, 2p_{1/2}, and 2p_{3/2} states of hydrogen-like ions. The calculation is performed to all orders in the binding nuclear strength parameter Zα. Detailed comparison is made with previous all-order calculations and calculations based on the expansion in the parameter Zα. Extrapolation of the all-order numerical results obtained towards Z=1 provides results for the radiative nuclear size effect on the hydrogen Lamb shift.
Stimulated by the proton radius conundrum, measurements of the Lamb shift in various light muonic atoms are planned at PSI. The aim is to extract the rms charge radius with high precision, limited by the uncertainty in the nuclear polarization corrections. We present an ab-initio calculation of the nuclear polarization for mu-4He+ leading to an energy correction in the 2S-2P transitions of $δ_{pol}=-2.47$ meV $\pm$ 6%. We use two different state-of-the-art nuclear Hamiltonians and utilize the Lorentz integral transform with hyperspherical harmonics expansion as few-body methods. We take into a…
We report high-precision calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter $Zα$. The results are in excellent agreement with the known terms of the $Zα$ expansion and allow an accurate identification of the nonperturbative higher-order remainder. For hydrogen, the higher-order remainder was found to be much larger than anticipated. This result resolves the long-standing disagreement between the numerical all-order and the analytical $Zα$-expansion a…
We present an experimental measurement of the cooperative Lamb shift and the Lorentz shift using an atomic nanolayer with tunable thickness and atomic density. The cooperative Lamb shift arises due to the exchange of virtual photons between identical atoms. The interference between the forward and backward propagating virtual fields is confirmed by the thickness dependence of the shift which has a spatial frequency equal to $2k$, i.e. twice that of the optical field. The demonstration of cooperative interactions in an easily scalable system opens the door to a new domain for non-linear optics.
Accurate calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms are presented. Numerical results are reported for the $ns$ states with $n \leq 5$ and for the $2p_{1/2}$ and $2p_{3/2}$ states. The calculations are performed to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter $Zα$ (where $Z$ is the nuclear charge number and $α$ is the fine structure constant). The obtained results provide accurate predictions for the higher-order remainder beyond the known $Zα$-expansion terms. In the case of hydrogen, the …
We evaluate a number of systematic effects that are important for an experimental microwave measurement of the $n$$=$$2$ S-to-P intervals in atomic hydrogen. The analysis is important for both re-evaluating the best existing measurement [Lundeen and Pipkin, PRL 46, 232 (1981)] of the 2S$_{1/2}$-to-2P$_{1/2}$ Lamb shift, and for a new measurement that is ongoing in our laboratory. This work is part of a larger program to understand the several-standard-deviation discrepancies between various methods for determining the proton charge radius.
We present calculations of nuclear structure effects to the Lamb shift in light muonic atoms. We adopt a modern ab-initio approach by combining state-of-the-art nuclear potentials with the hyperspherical harmonics method. Our calculations are instrumental to the determination of nuclear charge radii in the Lamb shift measurements, which will shed light on the proton radius puzzle.
We measure the near-resonant transmission of light through a dense medium of potassium vapor confined in a cell with nanometer thickness in order to investigate the origin and validity of the collective Lamb-shift. A complete model including the multiple reflections in the nano-cell accurately reproduces the observed strong asymmetry of the line shape and allows extraction of a density dependent shift of the atomic resonance. We observe an additional, unexpected dependence of this shift with the thickness of the medium. This extra dependence demands further experimental and theoretical investi…
Atomic electrons are sensitive to the properties of the nucleus they are bound to, such as nuclear mass, charge distribution, spin, magnetization distribution, or even excited level scheme. These nuclear parameters are reflected in the atomic transition energies. A very precise determination of atomic spectra may thus reveal information about the nucleus, otherwise hardly accessible via nuclear physics experiments. This work reviews theoretical and experimental aspects of the nuclear effects that can be identified in atomic structure data. An introduction to the theory of isotope shifts and hy…
The two-loop self-energy correction to the Lamb shift of hydrogen-like ions is calculated for the $1s$, $2s$, and $2p_{1/2}$ states and nuclear charge numbers $Z = 30$-$100$. The calculation is performed to all orders in the nuclear binding strength parameter $Zα$. As compared to previous calculations of this correction, numerical accuracy is improved by an order of magnitude and the region of the nuclear charges is extended. An analysis of the $Z$-dependence of the obtained results demonstrates their consistency with the known $Zα$-expansion coefficients.
We present a precise calculation of the Lamb shift $(2P_{1/2}-2S_{1/2})$ in muonic ions $(μ^6_3Li)^{2+},~(μ^7_3Li)^{2+}$, $(μ^9_4Be)^{3+},~(μ^{10}_4Be)^{3+}$, $(μ^{10}_5B)^{4+},~(μ^{11}_5B)^{4+}$. The contributions of orders $α^3\divα^6$ to the vacuum polarization, nuclear structure and recoil, relativistic effects are taken into account. Our numerical results are consistent with previous calculations and improve them due to account of new corrections. The obtained results can be used for the comparison with future experimental data, and extraction more accurate values of nuclear charge radii.
We report on the direct observation of resonant electric dipole-dipole interactions in a cubic array of atoms in the many-excitation limit. The interactions, mediated by single-atom couplings to the shared electromagnetic vacuum, are shown to produce spatially-dependent cooperative Lamb shifts when spectroscopically interrogating the mHz-wide optical clock transition in strontium-87. We show that the ensemble-averaged shifts can be suppressed below the level of evaluated systematic uncertainties for state-of-the-art optical atomic clocks. Additionally, we demonstrate that excitation of the ato…
We calculate the longitudinal structure function of the deuteron up through next-to-next-to-leading order in the framework of pionless effective field theory. We use these results to compute the two-photon polarizability contribution to Lamb shift in muonic deuterium, which can be utilized to extract the nuclear charge radius of the deuteron. We present analytical expressions order-by-order for the relevant transition matrix elements and the longitudinal structure function, and we give numerical results for the corresponding contributions to the Lamb shift. We also discuss the impact of relati…
Theoretical energy levels of the $n = 1$ and $n = 2$ states of hydrogenlike atoms with the nuclear charge numbers $1 \le Z \le 110$ are tabulated. The tabulation is based on ab initio quantum electrodynamical calculations performed to all orders in the nuclear binding strength parameter $Zα$, where $α$ is the fine structure constant. Theoretical errors due to various effects are critically examined and estimated.
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 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…
We provide a systematic assessment of the order-$α^5$ nuclear contributions to the Lamb shift of muonic deuterium, including the accompanying radiative corrections due to vacuum polarization, up to next-to-next-to-next-to-leading order (N3LO) within the pionless effective field theory (EFT). We also evaluate higher-order corrections due to the single-nucleon structure, which are expected to be the most important corrections beyond N3LO. We find a correlation between the deuteron charge and Friar radii, which can be useful to judge the quality of charge form factor parametrisations. We refine t…
Probabilities of various final states are determined numerically for a number of ß- decaying light atoms. In our evaluations of the final state probabilities we have used the highly accurate atomic wave functions constructed for each few-electron atom/ion. We also discuss an experimental possibility to observe negatively charged ions which form during the nuclear ß+ decays. High order corrections to the results obtained for ß+/- decays in few-electron atoms with the use of sudden approximation are considered.
The resonant process of nuclear excitation by electron transition (NEET) in highly charged ions is investigated. In NEET, a bound electronic decay transition occurs with the simultaneous excitation of the nucleus, provided that the energies of the atomic and nuclear transition match. By varying the atomic charge state, the atomic transition energy can be tuned to a better match of the nuclear transition energy. We propose a new way to create the atomic hole in highly charged ions by dielectronic capture of a free electron. In order to derive the cross section for the three-step process compose…
We compute an $\mathcal{O}\left(α^2(Zα)^6\right)$ contribution to the hydrogen-atom Lamb shift arising from the light-by-light scattering. Analogous diagrams, with one atomic electric field insertion replaced by an external magnetic field, contribute to the gyromagnetic factor of the bound electron at $\mathcal{O}\left(α^2(Zα)^4\right)$. We also calculate the contribution to the gyromagnetic factor from the muon magnetic loop.
Nuclear-structure effects often provide an irreducible theory error that prevents using precision atomic measurements to test fundamental theory. We apply newly developed effective field theory tools to Hydrogen atoms, and use them to show that (to the accuracy of present measurements) all nuclear finite-size effects (e.g. the charge radius, Friar moments, nuclear polarizabilities, recoil corrections, Zemach moments {\it etc.}) only enter into atomic energies through exactly two parameters, independent of any nuclear-modelling uncertainties. Since precise measurements are available for more th…
The accuracy of atomic theory calculations limits the extraction of nuclear charge radii from isotope shift measurements of odd-proton nuclei. For Na isotopes, though precise spectroscopic measurements have existed since more than half a century, calculations by different methods offer a wide range of values. Here, we present accurate atomic calculations to reliably extract the Na charge radii. By combining experimental matter radii with nuclear coupled-cluster calculations based on nucleon-nucleon and three-nucleon forces, we constrain the parameters obtained from the atomic calculations. The…
A state-of-the-art approach for calculating the finite nuclear size correction to atomic energy levels and the bound-electron $g$ factor is introduced and demonstrated for a series of highly charged hydrogen-like ions. Firstly, self-consistent mean-field calculations based on the Skyrme-type nuclear interaction are employed in order to produce a realistic nuclear proton distribution. In the second step, the obtained nuclear charge density is used to construct the potential of an extended nucleus, and the Dirac equation is solved numerically. The ambiguity in the choice of a Skyrme parametrizat…
A Lamb-shift polarimeter, which has been built for a fast determination of the polarization of protons and deuterons of an atomic-beam source and which is frequently used in the ANKE experiment at COSY-Jülich, is shown to be an excellent device for atomic-spectroscopy measurements of metastable hydrogen isotopes. It is demonstrated that magnetic and electric dipole transitions in hydrogen can be measured as a function of the external magnetic field, giving access to the full Breit-Rabi diagram for the $2^2S_{1/2}$ and the $2^2P_{1/2}$ states. This will allow the study of hyperfine structure, $…
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
Calculations of the magnetic hyperfine structure rely on the input of nuclear properties -- nuclear magnetic moments and nuclear magnetization distributions -- as well as quantum electrodynamic (QED) radiative corrections for high-accuracy evaluation in heavy atoms. The uncertainties associated with assumed values of these properties limit the accuracy of hyperfine calculations. For example, for the heavy alkali-metal atoms Cs and Fr, these uncertainties may amount collectively to almost 1\% or 2\%, respectively. In this paper we propose a method for removing the dependence of hyperfine struct…
Approximate probabilities of formation of various atoms and ions in different bound states are determined for the exothermic nuclear $(n,{}^{6}$Li$;t,{}^{4}$He)-reaction of atomic lithium-6 with slow neutrons. In our calculations of the final state probabilities we have assumed that the incident lithium atom is in its ground (doublet) atomic ${}^2S(L = 0)-$state. It is straightforward to generalize our analysis to other bound states of the three-electron Li atom.
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 have considered an effect of atomic electrons due to the electronic bridge process on the nuclear $^{229m}$Th -- $^{229g}$Th transition in ^{229}$Th$^{3+}$. Based on a recent experimental result we assumed the energy difference between the isomeric and the ground nuclear states to be equal to 7.6 eV. We have calculated the ratios of the electronic bridge process probability Gamma_EB to the probability of the nuclear radiative transition (Gamma_N) for the electronic 5f_5/2 --> 6d_3/2,6d_5/2,7s and the 7s --> 7p_1/2,7p_3/2 transitions and found Gamma_EB/Gamma_N ~ 0.01-0.1 for the former and G…
We continue the analysis of quantum two-particle bound systems we have started in (Kholmetskii, A.L., Missevitch, O.V. and Yarman, T. Phys. Scr., 82 (2010), 045301), where we re-postulated the Dirac equation for the bound electron in an external EM field based on the requirement of total momentum conservation, when its EM radiation is prohibited. It has been shown that the modified expression for the energy levels of hydrogenic atoms within such a pure bound field theory (PBFT) provides the same gross and fine structure of energy levels like the standard theory. Now we apply the PBFT to the an…
We study the collective response of a dense atomic sample to light essentially exactly using classical-electrodynamics simulations. In a homogeneously broadened atomic sample there is no overt Lorentz-Lorenz local field shift of the resonance, nor a collective Lamb shift. However, addition of inhomogeneous broadening restores the usual mean-field phenomenology.
