The electron Lamb shift $(2P-2S)$ and the energy interval $(2S-1S)$ in muon-electron ions of lithium, beryllium, and boron have been calculated within the framework of the perturbation theory method for the fine structure constant and the electron-muon mass ratio. The corrections of the first and second orders of the perturbation theory, which include the effects of vacuum polarization, nuclear structure and recoil, are taken into account. The obtained analytical results in perturbation theory are compared with the results of calculations within the variational approach. The obtained values for the Lamb shift $(2P-2S)$ and the interval $(2S-1S)$ can be used for comparison with future experimental data and verification of quantum electrodynamics.
The results of highly accurate calculations are presented for all twenty-two known bound $S(L = 0)-, P(L = 1)-, D(L = 2)-$ and $F(L = 3)-$states in the six three-body muonic molecular ions $ppμ, pdμ, ptμ, ddμ, dtμ$ and $ttμ$. A number of bound state properties of these muonic molecular ions have been determined numerically to high accuracy. The dependence of the total energies of these muonic molecules upon particle masses is considered. We also discuss the current status of muon-catalysis of nuclear fusion reactions.
The ground state hyperfine splitting of light muon-electron ions of lithium, beryllium, boron and helium is calculated on the basis of analytical perturbation theory in terms of small parameters of the fine structure constant and electron-muon mass ratio. The corrections of vacuum polarization, nuclear structure and recoil effects and electron vertex corrections are taken into account. The dependence of the corrections on the nucleus charge Z is studied. The obtained total values of hyperfine splitting intervals can be used for comparison with future experimental data.
We make precise calculation of hyperfine structure of $S$-states in muonic ions of lithium, beryllium and boron in quantum electrodynamics. Corrections of orders $α^5$ and $α^6$ due to the vacuum polarization, nuclear structure and recoil in first and second orders of perturbation theory are taken into account. We obtain estimates of the total values of hyperfine splittings which can be used for a comparison with future experimental data.
We calculate hyperfine structure intervals $ΔE^{hfs}(2P_{1/2})$ and $ΔE^{hfs}(2P_{3/2})$ for P-states in muonic ions of lithium, beryllium and boron. To construct the particle interaction operator in momentum space we use the tensor method of projection operators on states with definite quantum numbers of total atomic momentum F and total muon momentum j. We take into account vacuum polarization, relativistic, quadruple and structure corrections of orders $α^4$, $α^5$ and $α^6$. The obtained numerical values of hyperfine splittings can be used for a comparison with future experimental data of …
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.
Rigorous quantum electrodynamical calculation is presented for energy levels of the 1^1S, 2^1S, 2^3S, 2^1P_1, and 2^3P_{0,1,2} states of helium-like ions with the nuclear charge Z=3...12. The calculational approach accounts for all relativistic, quantum electrodynamical, and recoil effects up to orders mα^6 and m^2/Mα^5, thus advancing the previously reported theory of light helium-like ions by one order in α.
The hyperfine structure splittings are determined for all five bound $S(L = 0)-$states in the three symmetric muonic molecular ions: $ppμ, ddμ$ and $ttμ$. The expectation values of all interparticle delta-functions used in our calculations have been determined in recent highly accurate computations.
Accurate QED evaluations of the one- and two-photon interelectron interaction for low lying two- and three-electron configurations for ions with nuclear charge numbers $60\le Z \le 93$ are performed. The three-photon interaction is also partly taken into account. The Coulomb gauge is employed. The results are compared with available experimental data and with different calculations. A detailed investigation of the behaviour of the energy levels of the configurations $1s_{1/2}2s_{1/2} {}^1 S_0$, $1s_{1/2}2p_{1/2} {}^3 P_0$ near the crossing points Z=64 and Z=92 is carried out. The crossing poin…
We have studied the radiative decay of the 2s state of one-electron and one-muon ions, where the two-photon mechanism plays an important role. Due to the nuclear size corrections the radiative decay of the 2s state in the electron and muon ions is qualitatively different. Based on the accurate relativistic calculation, we introduced a two-parameter approximation, which makes it possible to describe the two-photon angular-differential transition probability for the polarized emitted photons with high accuracy. The emission of photons with linear and circular polarizations was studied separately…
It is shown that one can stimulate, by using the present-day laboratory magnetic fields, transitions between the $lm$ sub-levels of fast $μHe^+$ ions formating in muon catalyzed fusion. Strong fields also cause the self-ionization from highly excited states of such muonic ions. Both effects are the consequence of the interaction of the bound muon with the oscillating field of the Stark term coupling the center-of-mass and muon motions of the $μHe^+$ ion due to the non-separability of the collective and internal variables in this system. The performed calculations show a possibility to drive th…
The both total and binding energies of the (1,1)-states in the weakly-bound three-body muonic $d d μ$ and $d t μ$ ions are determined to high numerical accuracy. The binding energy of the (1,1)-state in the muonic $d t μ$ ion is evaluated as $\varepsilon(d t μ)$ = -0.66033003831(30) $eV$, while for the same state in the muonic $d d μ$ ion we have found that $\varepsilon(d d μ)$ = -1.9749806166970(30) $eV$. These energies are the most accurate numerical values obtained for these systems and they are sufficient for all current and future experimental needs.
The ground bound states in the five-body muonic ions $a b μe_2$ (or $(a b μe_2)^{-}$), where $(a, b) = (p, d, t)$, are considered for the first time. As follows from accurate numerical computations of these five-body ions they are similar to the negatively charged hydrogen ion H$^{-}$ with a three-particle central quasi-nucleus $a b μ$ (or $(a b μ)^{+}$). These five-body ions play some role in the muon-catalyzed fusion of nuclear reaction in liquid deuterium and/or deuterium-tritium mixtures. We also discuss the bound states in the six-body $a b c μe_2$ quasi-molecules, where $(a, b, c) = (p, …
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…
Energies of the low-lying bound S-states (L=0) of exotic three-body systems, consisting a nuclear core of charge +Ze (Z being atomic number of the core) and two negatively charged valence muons, have been calculated by hyperspherical harmonics expansion method (HHEM). The three-body Schrődinger equation is solved assuming purely Coulomb interaction among the binary pairs of the three-body systems X$^{Z+}μ^-μ^-$ for Z=1 to 54. Convergence pattern of the energies have been checked with respect to the increasing number of partial waves $Λ_{max}$. For available computer facilities, calculations ar…
Laser spectroscopy of the Lamb shift (2S-2P energy difference) in light muonic atoms or ions, in which one negative muon $μ^-$ is bound to a nucleus, has been performed. The measurements yield significantly improved values of the root-mean-square charge radii of the nuclei, owing to the large muon mass, which results in a vastly increased muon wave function overlap with the nucleus. The values of the proton and deuteron radii are 10 and 3 times more accurate than the respective CODATA values, but 7 standard deviations smaller. Data on muonic helium-3 and -4 ions is being analyzed and will give…
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…
Based on the results of accurate variational calculations we demonstrate stability of the five-body negatively charged ions $a^{+} b^{+} μ^{-} e^{-} e^{-}$. Each of these five-body ions contains two electrons $e^{-}$, one negatively charged muon $μ^{-}$ and two nuclei of the hydrogen isotopes $a, b = (p, d, t)$. The bound state properties of these five-body ions, including their hyperfine structure, are briefly discussed. We also investigate the hyperfine structure of the ground states of the four-body muonic quasi-atoms $a^{+} b^{+} μ^{-} e^{-}$. In particular, we determine the hyperfine stru…
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 …
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.
Experiments on one- and two-photon laser excitation of 232Th+ ions in a radiofrequency ion trap are reported. As the first excitation step, the strongest resonance line at 402 nm from the 6d^2 7s J=3/2 ground state to the 6d7s7p J=5/2 state at 24874 cm^{-1} is driven by radiation from an extended cavity diode laser. Spontaneous decay of the intermediate state populates a number of low-lying metastable states, thus limiting the excited state population and fluorescence signal obtainable with continuous laser excitation. We study the collisional quenching efficiency of helium, argon, and nitroge…
Accurate QED calculation of transition probabilities for the low-lying two-electron configurations of multicharged ions is presented. The calculation is performed for the nondegenerate states $(1s2s) 3S1$, $(1s2p_{3/2}) 3P2$ ($M 1$ and $M 2$ transitions, respectively) and for the quasidegenerate states $(1s2p) 1P1$, $(1s2p) 3P1$ ($E 1$ transitions) decaying to the ground state $(1s1s) 1S0$. Two-electron ions with nuclear charge numbers $Z=10-92$ are considered. The line profile approach is employed for the description of the process in multicharged ions within the framework of QED.
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.
We demonstrate a Doppler cooling and detection scheme for ions with low-lying D levels which almost entirely suppresses scattered laser light background, while retaining a high fluorescence signal and efficient cooling. We cool a single ion with a laser on the 2S1/2 to 2P1/2 transition as usual, but repump via the 2P3/2 level. By filtering out light on the cooling transition and detecting only the fluorescence from the 2P_3/2 to 2S1/2 decays, we suppress the scattered laser light background count rate to 1 per second while maintaining a signal of 29000 per second with moderate saturation of th…
The energy levels of the muonium ($μ^+ e^-$) atom, which consists of two ''point-like'' leptonic particles, can be calculated to very high accuracy in the framework of bound state Quantum Electrodynamics (QED), since there are no complications due to internal nuclear structure and size which is the case for all atoms and ions of natural isotopes including atomic hydrogen. The ground state hyperfine splitting and the 1s-2s energy interval can provide both tests of QED theory and determinations of fundamental constants like the muon mass $m_μ$ and the fine structure constant $α$. The excellent u…
Energy levels, wavelengths, magnetic-dipole and electric-quadrupole transition rates between the low-lying states are evaluated for W$^{51+}$ to W$^{54+}$ ions with $3d^n$ (n = 2 to 5) electronic configurations using an approach combining configuration interaction with linearized coupled-cluster single-double method. The QED corrections are directly incorporated into the calculations and their effect is studied in detail. Uncertainties of the calculations are discussed. This first study of such highly charged ions with the present method opens the way for future applications allowing an accura…
We have developed one electron detachment theory from a closed-shell atomic configuration in the relativistic Fock-space coupled-cluster ansatz. Using this method, we determine sensitivity coefficients to the variation of the fine structure constant in the first three important low-lying transitions of the astrophysically interesting highly charged Mn IX, Fe X, Co XI and Ni XII ions. The potential of this method has been assessed by evaluating the detachment energies of the removed electrons and determining lifetimes of the atomic states in the above ions. To account the sensitivity of the hig…
The low-lying n(=3,4,5)d $^2D_{5/2}$ states alkaline earth ions are of vital importance in a number of different physical applications. The hyperfine structure constants of these states are characterized by unusually strong electron correlation effects. Relativistic coupled-cluster theory has been employed to carry out {\it ab initio} calculations of these constants. The role of the all order core-polarization effects was found to be decisive in obtaining good agreement of the results of our calculations with accurate measurements. The present work is an apt demonstration of the power of the c…
This paper is a theoretical work in support of a newly proposed experiment (R. Stutz and E. Cornell, Bull. Am. Soc. Phys. 89, 76 2004) that promises greater sensitivity to measurements of the electron's electric dipole moment (EDM) based on the trapping of molecular ions. Such an experiment requires the choice of a suitable molecule that is both experimentally feasible and possesses an expectation of a reasonable EDM signal. We find that the molecular ions PtH+, HfH+, and HfF+ are suitable candidates in their low-lying triplet Delta states. In particular, we anticipate that the effective elect…
Anomalously enhanced lifetimes of low angular momentum Rydberg states in singly charged alkaline-earth metal ions
Institut für Theoretische Physik and Center for Integrated Quantum Science and Technology, Universität Tübingen（蒂宾根大学理论物理研究所与集成量子科学与技术中心）；School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, The University of Nottingham（诺丁汉大学物理与天文学学院及量子非平衡系统数学与理论物理中心）
High precision spectroscopy of the low-lying dielectronic resonances in fluorine-like nickel ions were determined by employing the merged electron-ion beam at the heavy-ion storage ring CSRm. The measured dielectronic resonances are identified by comparing with the most recent relativistic calculation utilizing the FAC code. The first resonance at about 86 meV due to the dielectronic recombination via (2s2p6[2S1/2]6s)J=1 intermediate state was recognized. The experimental determination of the resonance position at 86 meV reaches an uncertainty of 4 meV, which allows precise determination of th…
An effective optimization strategy has been developed to construct highly accurate bound state wave functions in various three-body systems. Our procedure appears to be very effective for computations of weakly bound states and various excited states, including rotationally excited states, i.e. states with $L \ge 1$. The efficiency of our procedure is illustrated by computations of the excited $P^{*}(L = 1)-$states in the $ddμ, dtμ$ and $ttμ$ muonic molecular ions, $P(L = 1)-$states in the non-symmetric $pdμ, ptμ$ and $dtμ$ ions and $2^{1}P(L = 1)-$ and $2^{3}P(L = 1)-$states in He atom(s).
Relativistic many-body perturbation theory is applied to study properties of ions of the francium isoelectronic sequence. Specifically, energies of the 7s, 7p, 6d, and 5f states of Fr-like ions with nuclear charges Z = 87 - 100 are calculated through third order; reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for 7s - 7p, 7p - 6d, and 6d - 5f electric-dipole transitions; and 7s - 6d, 7s - 5f, and 5f_5/2 - 5f_7/2 multipole matrix elements are evaluated to obtain the lifetimes of low-lying excited states. Moreover, for the ions Z = 87 - 92 calculati…
Large-scale self-consistent multiconfiguration Dirac--Hartree--Fock and relativistic configuration interaction calculations are reported for the $n \leq 6$ levels in Be-like ions from B II to Ne VII. Effects from electron correlation are taken into account by means of large expansions in terms of a basis of configuration state functions, and a complete and accurate data set of excitation energies, lifetimes, wavelengths, and electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole line strengths, transition rates, and oscillator strengths for these levels is provided for …
Ionization potentials, excitation energies, transition properties, and hyperfine structure constants of the low-lying $3p^6 3d^{9} \ ^2D_{5/2}$, $3p^6 3d^{9} \ ^2D_{3/2}$, $3p^5 3d^{10} \ ^2P_{3/2}$ and $3p^5 3d^{10} \ ^2P_{1/2}$ atomic states of the Co-like highly-charged ions such as Y$^{12+}$, Zr$^{13+}$, Nb$^{14+}$, Mo$^{15+}$, Tc$^{16+}$, Ru$^{17+}$, Rh$^{18+}$, Pd$^{19+}$, Ag$^{20+}$ and Cd$^{21+}$ are investigated. The singles and doubles approximated relativistic coupled-cluster theory in the framework of one electron removal Fock-space formalism is employed over the Dirac-Hartree-Fock…
We investigate black-body induced energy shifts for low-lying levels of atomic systems, with a special emphasis on transitions used in current and planned high-precision experiments on atomic hydrogen and ionized helium. Fine-structure and Lamb-shift induced black-body shifts are found to increase with the square of the nuclear charge number, whereas black-body shifts due to virtual transitions decrease with increasing nuclear charge as the fourth power of the nuclear charge. We also investigate the decay width acquired by the ground state of atomic hydrogen, due to interaction with black-body…
The total energies of twenty eight bound S-, P-, D-, F-, G-, H-, and I-states in the three-electron systems Li atom and Be+ ion, respectively, are determined with the use of the Configuration Interaction (CI) with Slater orbitals and L-S eigenfunctions, and the Hylleraas-configuration-interaction (Hy-CI) methods. We discuss the construction and selection of the configurations in the wave functions, optimization of the orbital exponents and advanced computational techniques. Finally, we have developed a very effective procedure which allows one to determine the energies of the excited states in…
Relativistic atomic structure calculations are carried out in alkaline-earth-metal ions using a semiempirical-core-potential approach. The systems are partitioned into frozen-core electrons and an active valence electron. The core orbitals are defined by a Dirac-Hartree-Fock calculation using the grasp2k package. The valence electron is described by a Dirac-like Hamiltonian involving a core-polarization potential to simulate the core-valence electron correlation. The associated equation is solved with the Lagrange-mesh method, which is an approximate variational approach having the form of a m…
We present tabulated data for the nuclear magnetic shielding constants ($σ$) of the Dirac one-electron atoms with a pointlike, motionless and spinless nucleus of charge $Ze$. Utilizing the exact general analytical formula for $σ$ derived by us \mbox{[P. Stefa{ń}ska, Phys. Rev. A. 94 (2016) 012508/1-15],} valid for an arbitrary discrete energy eigenstate, we have computed the numerical values of the magnetic shielding factors for the ground state and for the first and the second set of excited states, i.e.: 2s$_{1/2}$, 2p$_{1/2}$, 2p$_{3/2}$, 3s$_{1/2}$, 3p$_{1/2}$, 3p$_{3/2}$, 3d$_{3/2}$, and …
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…
In order to study ultracold charge-transfer processes in hybrid atom-ion traps, we have mapped out the potential energy curves and molecular parameters for several low lying states of the Rb, Yb$^+$ system. We employ both a multi-reference configuration interaction (MRCI) and a full configuration interaction (FCI) approach. Turning points, crossing points, potential minima and spectroscopic molecular constants are obtained for the lowest five molecular states. Long-range parameters, including the dispersion coefficients are estimated from our {\it ab initio} data. The separated-atom ionization…
Trapped Rydberg ions represent a flexible platform for quantum simulation and information processing which combines a high degree of control over electronic and vibrational degrees of freedom. The possibility to individually excite ions to high-lying Rydberg levels provides a system where strong and long-range interactions between pairs of excited ions can be engineered and tuned via external laser fields. We show that the coupling between Rydberg pair interactions and collective motional modes gives rise to effective long-range multi-body interactions, consisting of two, three, and four-body …
The thorium nucleus with mass number $A=229$ has attracted much interest because its extremely low lying first excited isomeric state at about $8$eV opens the possibility for the development of a nuclear clock. However, neither the exact energy of this nuclear isomer nor properties, such as nuclear magnetic dipole and electric quadrupole moment are known to a high precision so far. The latter can be determined by investigating the hyperfine structure of thorium atoms or ions. Due to its electronic structure and the long lifetime of the nuclear isomeric state, Th$^{2+}$ is especially suitable f…
A relativistic description of the structure of heavy alkali atoms and alkali-like ions using S-spinors and L-spinors has been developed. The core wavefunction is defined by a Dirac-Fock calculation using an S-spinors basis. The S-spinor basis is then supplemented by a large set of L-spinors for the calculation of the valence wavefunction in a frozen-core model. The numerical stability of the L-spinor approach is demonstrated by computing the energies and decay rates of several low-lying hydrogen eigenstates, along with the polarizabilities of a $Z=60$ hydrogenic ion. The approach is then appli…
In this paper we present tabulated data for magnetic-dipole-to-electric-quadrupole cross-susceptibilities ($χ_{\textrm{M}1 \to \textrm{E}2}$) for Dirac one-electron atoms with a pointlike, spinless and motionless nucleus of charge $Ze$. Numerical values of this susceptibility for the hydrogen atom ($Z=1$) and for hydrogenic ions with $2 \leqslant Z \leqslant 137$ are computed from the general analytical formula, recently derived by us [P. Stefa{ń}ska, Phys. Rev. A 93 (2016) 022504], valid for an arbitrary discrete energy eigenstate. In this work we provide 30 tables with the values of $χ_{\tex…
