The current theoretical status of the Lamb shift in He+ is discussed. Recent calculations of two-loop binding corrections to the Lamb shift significantly shift the theoretical value of the "classic" Lamb shift in He+, i.e. of the 2S_1/2-2P_1/2-interval. In this brief research note, we present a new (theoretical) value for this interval which reads 14041.474(42) Mhz. The theoretical uncertainty is reduced as well as the discrepancy between theory and experiment. Planned measurements should be of help in further elucidating the situation.
We report on recent progress in the treatment of two-loop binding corrections to the Lamb shift, with a special emphasis on S and P states. We use these and other results in order to infer an updated theoretical value of the Lamb shift in 4He+.
My previous calculations of the Lamb shift in muonic hydrogen are reviewed and compared with other work. In addition, numerical results for muonic deuterium are presented.
The Lamb shift in muonic hydrogen continues to be a subject of experimental and theoretical investigation. Here my older work on the subject is updated to provide a complementary calculation of the energies of the 2p-2s transitions in muonic hydrogen.
Higher order $(α/π)^2 (Z α)^6$ logarithmic corrections to the hydrogen Lamb shift are calculated. The results obtained show the two-loop contribution has a very peculiar behavior, and significantly alter the theoretical predictions for low lying S-states.
In the comment we discuss an additional term to the $α^3$ correction to the Lamb shift, which was missed in the calculation by Kinoshita and Nio (Phys. Rev. Lett. 82, 3240 (1999)).
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 …
General expressions for quantum electrodynamic corrections to the one-loop self-energy [of order alpha(Zalpha)^6] and for the two-loop Lamb shift [of order alpha^2(Zα)^] are derived. The latter includes all diagrams with closed fermion loops. The general results are valid for arbitrary excited non-S hydrogenic states and for the normalized Lamb shift difference of S states, defined as Delta_n = n^3 DeltaE(nS) - DeltaE(1S). We present numerical results for one-loop and two-loop corrections for excited S, P and D states. In particular, the normalized Lamb shift difference of S states is calculat…
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.
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.
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
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.
Quantum electrodynamics in very strong Coulomb fields is one scope which has not yet been tested experimentally with suffcient accuracy to really determine whether the perturbative approach is valid. One sensitive test is the determination of the 1s Lamb Shift in highly-charged very heavy ions. The 1s Lamb Shift of hydrogen-like lead (Pb81+) and gold (Au78+) has been determined using the novel detector concept of silicon microcalorimeters for the detection of hard X-rays. The results of (260 +- 22) eV for lead and (208 +- 13) eV for gold are within error bars in good agreement with theoretical…
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 …
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.
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…
The deuteron charge radius is calculated from the measurement of the Lamb shift in muonic deuterium, taking into account the electron vacuum polarization correction to the nuclear-structure effects. This correction is unexpectedly large and gives a mean-square charge-radii difference $ r_d^2-r_p^2 = 3.81747(346)\, {\rm fm}^2 $, which is now consistent with that obtained from the ordinary H-D isotope shift in the 1$S$-2$S$ transition. This suggests that the long-standing discrepancy in the proton charge radius obtained from electronic and muonic systems is due to an underestimated uncertainty i…
In muonic atoms the Uehling potential (an effect of a free electronic vacuum polarization loop) is responsible for the leading contribution to the Lamb shift causing the splitting of states with Delta n = 0 and Delta l \neq 0. Here we consider the Lamb shift in the leading nonrelativistic approximation, i.e., within an approach based on a certain Schrodinger equation. That is valid for low and medium $Z$ as long as (Z alpha)^2 >> 1. The result is a function of a few parameters, including kappa = Z alpha m_ mu/m_e, n and l. We present various asymptotics and in particular we study a region of v…
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 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.
The 7 standard deviations between the proton rms charge radius from muonic hydrogen and the CODATA-10 value from hydrogen spectroscopy and electron-scattering has caused considerable discussions. Here, we review the theory of the 2S-2P Lamb shift and 2S hyperfine splitting in muonic hydrogen combining the published contributions and theoretical approaches. The prediction of these quantities is necessary for the determination of both proton charge and Zemach radii from the two 2S-2P transition frequencies measured in muonic hydrogen.
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.
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.
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 …
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.
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
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 correction to the hydrogen Lamb shift due to the proton electric and magnetic polarizabilities is expressed analytically through their static values, which are known from experiment. The numerical value of the correction is $ - 71 \pm 11 \pm 7$ Hz.
The correction to the hydrogen Lamb shift due to the proton electric and magnetic polarizabilities is expressed analytically through their static values, which are known from experiment. The numerical value of the correction to the hydrogen 1S state is $ - 71 \pm 11 \pm 7$ Hz. Correction to the H-D 1S-2S -- isotope shift due to the proton and neutron polarizabilities is estimated as $ 53 \pm 9 \pm 11$ Hz.
Corrections of order $α^3(Zα)^5m$ to the Lamb shift and corrections of order $α^3(Zα)E_F$ to hyperfine splitting generated by the insertions of the three-loop one-particle reducible diagrams with radiative photons in the electron line are calculated. The calculations are performed in the Yennie gauge.
We consider three-loop radiative corrections to the Lamb shift and hyperfine splitting. Corrections of order $α^3(Zα)^5m$ are the largest still unknown contributions to the Lamb shift in hydrogen. We calculated radiative corrections to the Lamb shift and hyperfine splitting generated by the diagrams with insertions of one radiative photon and electron polarization loops in the graphs with two external photons. We also obtained corrections generated by the gauge invariant sets of diagrams with two reducible radiative photon insertions in the electron line and polarization operator insertion in …
Recent improvements in the determination of the running of the fine-structure constant also allow an update of the hadronic vacuum-polarization contribution to the Lamb shift. We find a shift of -3.40(7) kHz to the 1S level of hydrogen. We also comment on the contribution of this effect to the determination by elastic electron scattering of the r.m.s. radii of nuclei.
We report direct, time-resolved observations of single-photon superradiance in a highly extended, elliptical sample of cold 87Rb atoms. The observed rapid decay rate is accompanied by its counterpart, the cooperative Lamb shift. The rate of the strongly directional decay, and the associated shift, scale linearly with the number of atoms and show a strong dependence on the sample shape, demonstrating the collective nature of the observed quantities.
We calculate radiative corrections to the Lamb shift of order $α^3(Zα)^5m$ and radiative corrections to hyperfine splitting of order $α^3(Zα)E_F$ generated by the diagrams with insertions of radiative photons and electron polarization loops in the graphs with two external photons. We also obtain the radiative-recoil correction to hyperfine splitting in muonium generated by the diagrams with the $τ$ polarization loop.
The results and main steps of an analytic calculation of radiative-recoil corrections of order $α(Zα)^5(m/M)m$ to the Lamb shift in hydrogen are presented. The calculations are performed in the infrared safe Yennie gauge. The discrepancy between two previous numerical calculations of these corrections existing in the literature is resolved. Our new result eliminates the largest source of the theoretical uncertainty in the magnitude of the deuterium-hydrogen isotope shift.
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.
From the comparison between the experimental and theoretical results for the energy difference between 2s1/2 and 2p1/2 states in hydrogen and hydrogen-like ions with different values of Z, we estimate bounds for the presence of an extra fractional dimension. These bounds are higher when we work with muonic hydrogen and muonic deuterium, which open the possibility that future measurements of the Lamb shift in muonic atoms with one electron and higher values of Z may show evidence of the extra dimension.
The Lamb shift measurement and theory are now both a dynamically developing field and we give a review of the current data. Critical comparison of theory and experiment can be done using a value of the proton charge radius and we pay attention to several results of its determination.
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…
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.
When the hydrogen atom moves, the proton current generates a magnetic field which interacts with the hydrogen electron. A simple analyze shows that this interaction between the hydrogen momentum and the electron is of order of $α^3\frac{v}{c}m$, where $α$ is the fine structure constant, $v$ is the atom velocity, $c$ is the speed of light, and $m$ is the electron mass. Using the Bethe-Salpeter equation, the two velocity-dependent operators of this order are derived for the hydrogen velocity $\frac{v}{c}<<α$. As well known, the degeneracy of the energy levels with the same principal quantum numb…
The quantum vacuum fundamentally alters the properties of embedded particles. In contrast to classical empty space, it allows for creation and annihilation of excitations. For trapped particles this leads to a change in the energy spectrum, known as Lamb shift. Here, we engineer a synthetic vacuum building on the unique properties of ultracold atomic gas mixtures. This system makes it possible to combine high-precision spectroscopy with the ability of switching between empty space and quantum vacuum. We observe the phononic Lamb shift, an intruiguing many-body effect orginally conjectured in t…
We calculate hard spin-independent three-loop radiative corrections to energy levels in muonium and positronium which are due to radiative insertions in two-photon exchange diagrams. These corrections could be relevant for the new generation of precise $1S-2S$ and $2S-2P$ measurements in muonium and positronium.
Hard spin-independent three-loop radiative contributions to energy levels in positronium generated by the two-photon-exchange diagrams with one-loop radiative insertions in the fermion lines and exchanged photons are calculated. This is a next step in calculations of all corrections of order $mα^7$ inspired by the new generation of precise $1S-2S$ and $2S-2P$ measurements in positronium.
Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meantime, the involved models are not consistent eit…
By enhancing electron position fluctuations, equilibrium electromagnetic radiation modifies the potential for an electron in a Hydrogen atom. This can have significant effects for weakly bound states and especially at finite temperature. This implies a 2% correction to Bethe's value for 2S-2P Lamb shift for weak fluctuations, but the effect is an order of magnitude larger for strong fluctuations where it provides direct measure of the proton diameter.
We report a new measurement of the $n=2$ Lamb shift in Muonium using microwave spectroscopy. Our result of $1047.2(2.3)_\textrm{stat}(1.1)_\textrm{syst}$ MHz comprises an order of magnitude improvement upon the previous best measurement. This value matches the theoretical calculation within one standard deviation allowing us to set limits on CPT violation in the muonic sector, as well as on new physics coupled to muons and electrons which could provide an explanation of the muon $g-2$ anomaly.
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…
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 meas…
Whenever several quantum light emitters are brought in proximity with one another, their interaction with common electromagnetic fields couples them, giving rise to cooperative shifts in their resonance frequency. Such collective line shifts are central to modern atomic physics, being closely related to superradiance on one hand and the Lamb shift on the other. Although collective shifts have been theoretically predicted more than fifty years ago, the effect has not been observed yet in a controllable system of a few isolated emitters. Here, we report a direct spectroscopic observation of the …
We consider one-loop light-by-light-scattering contributions to the Lamb shift of the $1s, 2s, 2p$ states in light muonic hydrogen like atoms at $Z\leq10$. The contributions are of the order $α^5m_μ$ (with diverse dependence on the nuclear charge $Z$). Those include the contributions of the so-called Wichmann-Kroll potential ($α(Zα)^4m_μ$), the virtual Delbrück scattering ($α^2(Zα)^3m_μ$), etc. The results are obtained in a nonrelativistic approximation. For the calculation of the virtual-Delbrück-scattering contribution, we have constructed an effective potential in the coordinate space which…
We review the theory of the Lamb shift for muonium, provide an updated numerical value and present the prospects of the Mu-MASS collaboration at PSI to improve upon their recent measurement. Due to its smaller nuclear mass, the contributions of the higher-order recoil corrections (160kHz level) and nucleus self-energy 40kHz level) are enhanced for muonium compared to hydrogen where those are below the level of the latest measurement performed by Hessels et al. and thus could not be tested yet. The ongoing upgrades to the Mu-MASS setup will open up the possibility to probe these contributions a…
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.
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…
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…
We compute the spin-independent structure functions of the forward virtual-photon Compton tensor of the proton at one loop using heavy baryon chiral perturbation theory and dispersion relations. We study the relation between both approaches. We use these results to generalize some sum rules to virtual photon transfer momentum and relate them with sum rules in deep inelastic scattering. We then compute the leading chiral term of the polarizability correction to the Lamb shift of the hydrogen and muonic hydrogen. We obtain -87.05/n^3 Hz and -0.148/n^3 meV for the correction to the hydrogen and m…
