The mass corrections to the bound state energies of the three-body muonic molecular ions $ppμ, pdμ, ptμ, ddμ, dtμ$ and $ttμ$ are determined numerically from the results of highly accurate computations. The total energies and some other bound state properties of these ions are evaluated to very high accuracy for the bound $S(L = 0)-, P(L = 1)-$ and $D(L = 2)-$states. In these highly accurate calculations we used the most recent and accurate masses of particles $m_p, m_d, m_t$ and $m_μ$ known from high energy experiments.
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.
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…
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 A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-d…
The reduced-mass dependence of relativistic and radiative effects in simple muonic bound systems is investigated. The spin-dependent nuclear recoil correction of order (Zalpha)^4 mu^3/m_N^2 is evaluated for muonic hydrogen and deuterium, and muonic helium ions (mu is the reduced mass and m_N is the nuclear mass). Relativistic corrections to vacuum polarization of order alpha (Zalpha)^4 mu are calculated, with a full account of the reduced-mass dependence. The results shift theoretical predictions. The radiative-recoil correction to vacuum polarization of order alpha (Zα)^5 ln^2(Zalpha) mu^2/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 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 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 hyperfine splitting in hydrogenlike $^{209}$Bi, $^{203}$Tl, and $^{205}$Tl is calculated with the nuclear magnetization determined from experimental data on the hyperfine splitting in the corresponding muonic atoms. The single-particle and configuration-mixing nuclear models are considered. The QED corrections are taken into account for both electronic and muonic atoms. The obtained results are compared with other calculations and with experiment.
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.
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…
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.
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 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, …
Resonant scattering of muonic hydrogen atoms via back decay of molecular complex, a key process in the understanding of epithermal muonic molecular formation, is analyzed. The limitations of the effective rate approximation are discussed and the importance of the explicit treatment of the back decay is stressed. An expression of the energy distribution for the back-decayed atoms is given.
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…
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.
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 muonic vacuum polarization contribution to the $g$-factor of the electron bound in a nuclear potential is investigated theoretically. The electric as well as the magnetic loop contributions are evaluated. We found these muonic effects to be observable in planned trapped-ion experiments with light and medium-heavy highly charged ions. The enhancement due to the strong Coulomb field boosts these contributions much above the corresponding terms in the free-electron $g$-factor. Due to their magnitude, muonic vacuum polarization terms are also significant in planned determinations of the fine-s…
The relativistic recoil contributions to the Uehling corrections are revisited. We consider a controversy in recent calculations based on different approaches including Breit-type and Grotch-type calculations. We have found that calculations of those authors were in fact done in different gauges and in some of those gauges contributions the retardation and two-photon-exchange effects were missed. We have evaluated such effects and obtained a consistent result from those approaches. We present a correct expression for the Grotch-type approach which produces a correct gauge-invariant result. We …
Muonic molecular ions $^6$Li$^{3+}dμ$ and $^7$Li$^{3+}dμ$ are studied numerically. Using the complex coordinate rotation method we found six rotational states (three for each isotope), which are resonant states with the life-time of an order of picoseconds. These molecular systems may be of interest for studying low-energy fusion reactions. A key quantity, $|Ψ(0)|^2$, the wave function squared at the coalescence point of the nuclei is calculated for $S$ states for both isotopes with a precision better than 3\%.
Total energies and various bound state properties are determined for the ground states in all six four-body muonic $a^{+} b^{+} μ^{-} e^{-}$ quasi-atoms. These quasi-atoms contain two nuclei of the hydrogen isotopes $p^{+}, d^{+}, t^{+}$, one negatively charged muon $μ^{-}$ and one electron $e^{-}$. In general, each of the four-body muonic $a^{+} b^{+} μ^{-} e^{-}$ quasi-atoms, where $(a, b) = (p, d, t)$, can be considered as the regular one-electron (hydrogen) atom with the complex nucleus $a^{+} b^{+} μ^{-}$ which has a finite number of bound states. Furthermore, all properties of such quasi…
In this theoretical work, we calculate potential energy curves, spectroscopic pa- rameters and transition dipole moments of molecular ions BeX^+ (X=Na, K, Rb) composed of alkaline ion Be and alkali atom X with a quantum chemistry approach based on the pseudopotential model, Gaussian basis sets, effective core polarization potentials, and full configuration interaction (CI). We study in detail collisions of the alkaline ion and alkali atom in quantum regime. Besides, we study the possibility of the formation of molecular ions from the ion-atom colliding systems by stimu- lated Raman adiabatic p…
The potential energy curves, permanent and transition dipole moments, and the static dipolar polarizability, of molecular ions composed of one alkali-metal atom and a Strontium ion are determined with a quantum chemistry approach. The molecular ions are treated as effective two-electron systems and are treated using effective core potentials including core polarization, large gaussian basis sets, and full configuration interaction. In the perspective of upcoming experiments aiming at merging cold atom and cold ion traps, possible paths for radiative charge exchange, photoassociation of a cold …
The static dipole polarizability for the hydrogen molecular ions H$_2^+$, HD$^+$, and D$_2^+$ are calculated. These new data for polarizability takes into account the leading order relativistic corrections to the wave function of the three-body system resulted from the Breit-Pauli Hamiltonian of $mα^4$ order. Our study covers a wide range of rotational ($L=0-5$) and vibrational ($v=0-10$) states, which are of practical interest for precision spectroscopy of the hydrogen molecular ions.
We present the preparation and deterministic delivery of a selectable number of externally cold molecular ions. A laser cooled ensemble of Mg^+ ions subsequently confined in several linear Paul traps inter-connected via a quadrupole guide serves as a cold bath for a single or up to a few hundred molecular ions. Sympathetic cooling embeds the molecular ions in the crystalline structure. MgH^+ ions, that serve as a model system for a large variety of other possible molecular ions, are cooled down close to the Doppler limit and are positioned with an accuracy of one micrometer. After the producti…
We discuss the design and optimisation of two types of junctions between surface-electrode radiofrequency ion-trap arrays that enable the integration of experiments with sympathetically cooled molecular ions on a monolithic chip device. A detailed description of a multi-objective optimisation procedure applicable to an arbitrary planar junction is presented, and the results for a cross junction between four quadrupoles as well as a quadrupole-to-octupole junction are discussed. Based on these optimised functional elements, we propose a multi-functional ion-trap chip for experiments with transl…
The direct photodissociation of trapped $^{85}$Rb$_2^+$ (rubidium) molecular ions by the cooling light for the $^{85}$Rb magneto-optical trap (MOT) is studied, both experimentally and theoretically. Vibrationally excited Rb$_{2}^{+}$ ions are created by photoionization of Rb$_{2}$ molecules formed photoassociatively in the Rb MOT and are trapped in a modified spherical Paul trap. The decay rate of the trapped Rb$_{2}^{+}$ ion signal in the presence of the MOT cooling light is measured and agreement with our calculated rates for molecular ion photodissociation is observed. The photodissociation…
Multipole radiofrequency ion traps are a highly versatile tool to study molecular ions and their interactions in a well-controllable environment. In particular the cryogenic 22-pole ion trap configuration is used to study ion-molecule reactions and complex molecular spectroscopy at temperatures between few Kelvin and room temperatures. This article presents a tutorial on radiofrequency ion trapping in multipole electrode configurations. Stable trapping conditions and buffer gas cooling, as well as important heating mechanisms, are discussed. In addition, selected experimental studies on cation…
The direct laser cooling of neutral diatomic molecules in molecular beams suggests that trapped molecular ions can also be laser cooled. The long storage time and spatial localization of trapped molecular ions provides the opportunity for multi-step cooling strategies, but also requires a careful consideration of rare molecular transitions. We briefly summarize the requirements that a diatomic molecule must meet for laser cooling, and we identify a few potential molecular ion candidates. We then perform a detailed computational study of the candidates BH+ and AlH+, including improved ab initio…
A general method for rotational microwave spectroscopy and control of polar molecular ions via direct microwave addressing is considered. Our method makes use of spatially varying AC Stark shifts, induced by far off-resonant, focused laser beams to achieve an effective coupling between the rotational state of a molecular ion and the electronic state of an atomic ion. In this setting, the atomic ion is used for read-out of the molecular ion state, in a manner analogous to quantum logic spectroscopy based on Raman transitions. In addition to high-precision spectroscopy, this setting allows for r…
We propose a new method for the production of ultracold molecular ions. This method utilizes sympathetic cooling due to the strong collisions between appropriately chosen molecular ions and laser-cooled neutral atoms to realize ultracold, internal ground-state molecular ions. In contrast to other experiments producing cold molecular ions, our proposed method efficiently cools both the internal and external molecular ion degrees of freedom. The availability of an ultracold, absolute ground-state sample of molecular ions would have broad impact to fields as diverse as quantum chemistry, astrophy…
Radiative emission during cold collisions between trapped laser-cooled Rb atoms and alkaline-earth ions (Ca$^+$, Sr$^+$, Ba$^+$) and Yb$^+$, and between Li and Yb$^+$, are studied theoretically, using accurate effective-core-potential based quantum chemistry calculations of potential energy curves and transition dipole moments of the related molecular ions. Radiative association of molecular ions is predicted to occur for all systems with a cross section two to ten times larger than the radiative charge transfer one. Partial and total rate constants are also calculated and compared to availabl…
We report the sympathetic cooling and Coulomb crystallization of molecular ions above the surface of an ion-trap chip. N$_2^+$ and CaH$^+$ ions were confined in a surface-electrode radiofrequency ion trap and cooled by the interaction with laser-cooled Ca$^{+}$ ions to secular translational temperatures in the millikelvin range. The configuration of trapping potentials generated by the surface electrodes enabled the formation of planar bicomponent Coulomb crystals and the spatial separation of the molecular from the atomic ions on the chip. The structural and thermal properties of the Coulomb …
Trapped diatomic molecular ions could prove to be a sensitive probe for a permanent electron electric dipole moment (eEDM). We propose to use a ground or metastable $^3Δ_1$ level, due to its high polarizability and large EDM enhancement factor. Ions allow for simple trapping and long interrogation times, but require a time-varying electric bias field in order to probe the eEDM. We discuss experimental design as well as challenges in performing a precision spectroscopic measurement in rapidly time-varying electric fields.
Nuclear-spin-dependent (NSD) parity violating (PV) effects can be strongly enhanced in diatomic molecules containing heavy atoms. Future measurements are anticipated to provide nuclear anapole moments and strength constants for PV nuclear forces. In light molecules, the NSD electroweak electron-nucleus interaction may also be detected. Here we calculate NSD PV effects for molecular ions. Our calculations are motivated by rapid developments in trapping techniques for such systems at low temperatures.
We describe the current status of high-precision ab initio calculations of the spectra of molecular hydrogen ions (H_2^+ and HD^+) and of two experiments for vibrational spectroscopy. The perspectives for a comparison between theory and experiment at a level of 1 ppb are considered.
A complete effective Hamiltonian for relativistic corrections at orders $mα^6$ and $mα^6(m/M)$ in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calcu…
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).
We consider the practical feasibility of using the direct, electric dipole-dipole interaction between co-trapped molecular ions for robust quantum logic without the need for static polarizing fields. The use of oscillating dipole moments, as opposed to static electric dipoles, dynamically decouples the dipoles from laboratory fields, including the electric fields of the trap itself. Further, this implementation does not require quantum control of motion, potentially removing a major roadblock to ion trap quantum computing scalability. Since the polarizing field is electromagnetic radiation, ev…
We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spin-spin tensor interactions in hydrogen molecular ions. The $mα^7 \ln(α)$-order radiative correction is derived in the NRQED framework. We present complete numerical calculations, including as well the $mα^6$-order relativistic correction, for the case of H$_2^+$. The theoretical uncertainty is reduced by more than one order of magnitude with respect to the Breit-Pauli level, down to a few ppm. We also compare our results with available rf spectroscopy data.
We present a new method for the generation of rotationally and vibrationally state-selected, translationally cold molecular ions in ion traps. Our technique is based on the state-selective threshold photoionization of neutral molecules followed by sympathetic cooling of the resulting ions with laser-cooled calcium ions. Using N$_2^+$ ions as a test system, we achieve > 90 % selectivity in the preparation of the ground rovibrational level and state lifetimes on the order of 15 minutes limited by collisions with background-gas molecules. The technique can be employed to produce a wide range of a…
We review recent experiments on HD$^+$ spectroscopy. The reduced proton-deuteron mass to electron mass ratio, $μ_{pd}/m_e$, is inferred from the comparison of theory and experiment. Theoretical issues related to the calculations of the spin structure, which are currently the main limiting factor of theoretical accuracy, are discussed.
Time-dependent close-coupling calculations of the ionization and excitation cross section for antiproton collisions with molecular hydrogen ions are performed in an impact-energy range from 0.5 keV to 10 MeV. The Born-Oppenheimer and Franck-Condon approximations as well as the impact parameter method are applied in order to describe the target molecule and the collision process. It is shown that three perpendicular orientations of the molecular axis with respect to the trajectory are sufficient to accurately reproduce the ionization cross section calculated by [Sakimoto, Phys. Rev. A 71, 06270…
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…
High-precision variational calculations of the operators for the relativistic corrections in the leading $mα^4$ order are presented. The ro-vibrational states in the range of the total orbital angular momentum $L\!=\!0\!-\!4$ and vibrational quantum number $v\!=\!0\!-\!10$ for the $\mbox{H}_2^+$ and $\mbox{HD}^+$ molecular ions are considered. We estimate that about ten significant digits are obtained. This high precision is required for making theoretical predictions for transition frequencies at the level of $10^{-12}$ relative uncertainty.
The major goal of the high-precision studies of ro-vibrational states in the hydrogen molecular ions is to provide an alternative way for improving the electron-to-proton mass ratio, or the atomic mass of electron. By now the complete set of relativistic and radiative corrections have been obtained for a wide range of ro-vibrational states of H_2^+ and HD^+ up to order R_\inftyα^4. In this work we complete calculations of various contributions to the R_\inftyα^4 order by computing the relativistic corrections to the binding energy of electron.
