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 electric fields generated inside these molecules are approximately of 73 GV/cm, -17 GV/cm, and -18 GV/cm respectively. As a byproduct of this discussion, we also explain how to make estimates of the size of the effective electric field acting in a molecule, using commercially available, nonrelativistic molecular structure software.
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.
In this Letter, we investigate the non-dipole effects in time delay of photoelectrons emitted by multi-electron atoms, negative ions, and respective endohedrals. We present the necessary general formulas in the frame of the random phase approximation with exchange (RPAE) applied to atoms, negative ions, and properly adjusted to endohedrals. We concentrate on low photon energy region, where non-dipole effects are very small in the cross-sections but become observable in angular distributions. We not only derive the formulas for non-dipole effects in time delay, but perform corresponding numeric…
The one-electron quantum-electrodynamic corrections to the magnetic-dipole transition amplitude between the fine-structure levels (1s^2 2s^2 2p) ^2P_3/2 - ^2P_1/2 in boronlike ions are calculated to all orders in (αZ). The results obtained serve for improving the theoretical accuracy of the lifetime of the (1s^2 2s^2 2p) ^2P_3/2 level in boronlike argon.
Apropos to the growing interest in the study of long-range interactions which for their applications in cold atom physics, we have performed theoretical calculation for the two-dipole $C_6$ and three-dipole $C_9$ dispersion coefficients involving alkaline-earth atoms with alkaline-earth atoms and alkaline-earth ions. The $C_6$ and $C_9$ coefficients are expressed in terms of the dynamic dipole polarizabilities, which are calculated using relativistic methods. Thereafter, the calculated $C_6$ coefficients for the considered alkaline-earth atoms among themselves are compared with the previously …
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.
We use the perturbed relativistic coupled-cluster theory to compute the static electric dipole polarizabilities of the singly ionized alkali atoms, namely, Na$^+$, $^+$, Rb$^+$, Cs$^+$ and Fr$^+$. The computations use the Dirac-Coulomb-Breit Hamiltonian with the no-virual-pair approximation and we also estimate the correction to the static electric dipole polarizability arising from the Breit interaction.
The electron beam ion trap (EBIT) at the National Institute of Standards and Technology was used to produce highly-charged ions of hafnium, tantalum and gold with an open 3d shell. The extreme-ultraviolet (EUV) spectra from these ions were recorded with a flat-field grazing-incidence spectrometer in the wavelength range of 4.5 nm to 25 nm. A total of 133 new spectral lines, primarily due to magnetic-dipole transitions within the ground-state $3d^n$ configurations of the Co-like to K-like ions, were identified by comparing energy-dependent experimental spectra with a detailed collisional-radiat…
Theoretical studies are presented how the electric dipole moment (EDM) of the electron in H-like ions in electrostatic storage rings can sensitively be determined. With the proposed experiments a new constraint of about $10^{-29}$ e cm for the electron EDM can be established what is by an order of magnitude more restrictive than the existing bounds. Experiments with H-like ions may provide a possibility to distinguish between the electron EDM effect and the effect of P,T violating interaction between the atomic electron and the nucleus.
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 compute the ground-state electric dipole polarizability of group-IIIA ions using the perturbed relativistic coupled-cluster (PRCC) theory. To account for the relativistic effects and QED corrections, we use the Dirac-Coulomb-Breit Hamiltonian with the corrections from the Uehling potential and the self-energy. The effects of triple excitations are considered perturbatively in the PRCC. Our PRCC results for $α$ are good in agreement with the previous theoretical results for all the ions. From our computations we find that the nonlinear terms in PRCC have significant contributions and must be…
The magnetic-dipole transition probabilities between the fine-structure levels (1s^2 2s^2 2p) ^2P_1/2 - ^2P_3/2 for B-like ions and (1s^2 2s 2p) ^3P_1 - ^3P_2 for Be-like ions are calculated. The configuration-interaction method in the Dirac-Fock-Sturm basis is employed for the evaluation of the interelectronic-interaction correction with negative-continuum spectrum being taken into account. The 1/Z interelectronic-interaction contribution is derived within a rigorous QED approach employing the two-time Green function method. The one-electron QED correction is evaluated within framework of the…
An electron beam ion trap (EBIT) is used to measure extreme ultraviolet spectra between 10 nm and 25 nm from highly-charged ions of tungsten with an open 3d shell (W XLVIII through W LVI). We found that almost all strong lines are due to the forbidden magnetic-dipole (M1) transitions within $3d^n$ ground configurations. A total of 37 spectral lines are identified for the first time using detailed collisional-radiative (CR) modeling of the EBIT spectra. A new level-merging scheme for compactification of rate equations is described. The CR simulations for Maxwellian plasmas show that several lin…
Using perturbed relativistic coupled-cluster (PRCC) theory we compute the ground state electric dipole polarizability, $α$, of doubly ionized alkaline earth metal ions $\rm{Mg}^{2+}$, $\rm{Ca}^{2+}$, $\rm{Sr}^{2+}$, $\rm{Ba}^{2+}$ and $\rm{Ra}^{2+}$. In the present work we use the Dirac-Coulomb-Breit atomic Hamiltonian and we also include the Uehling potential, which is the leading order term in the vacuum polarization corrections. We examine the correction to the orbital energies arising from the Uehling potential in the self-consistent field calculations as well as perturbatively. Our result…
In this paper we present a technique for the precise measurement of electric dipole allowed transitions in trapped ions. By applying a probe and a cooling laser in quick succession, the full transition can be probed without causing distortion from heating the ion. In addition, two probes can be utilized to measure a dispersion-like signal, which is well suited to stabilizing the laser to the transition. We have fully characterized the parameters for the measurement and find that it is possible to measure the transition frequency to better than 100kHz with an interrogation time of 30s. The long…
We employ the closed-shell perturbed relativistic coupled-cluster (RCC) theory developed by us earlier [Phys. Rev. A {\bf 77}, 062516 (2008)] to evaluate the ground state static electric dipole polarizabilities (αs) of several atomic systems. In this work, we have incorporated a class of higher order many-body effects in our calculations that had not been taken into account in the above paper. We highlight their importance in improving the accuracy of $α$. We also calculate the ground state αs of the inert gas atoms and several iso-electronic singly and doubly charged ions in order to make a c…
Ion-atom interactions are a comparatively recent field of research that has drawn considerable attention due to its applications in areas including quantum chemistry and quantum simulations. In first experiments, atomic ions and neutral atoms have been successfully overlapped by devising hybrid apparatuses combining established trapping methods, Paul traps for ions and optical or magneto-optical traps for neutral atoms, respectively. Since then, the field has seen considerable progress, but the inherent presence of radiofrequency (rf) fields in such hybrid traps was found to have a limiting im…
The interaction between the electric dipole moment of a trapped molecular ion and the configuration of the confined Coulomb crystal couples the orientation of the molecule to its motion. We consider the practical feasibility of harnessing this interaction to initialize, process, and read out quantum information encoded in molecular ion qubits without optically illuminating the molecules. We present two schemes wherein a molecular ion can be entangled with a co-trapped atomic ion qubit, providing, among other things, a means for molecular state preparation and measurement. We also show that vir…
The angular distribution of the emitted electrons, following the two-photon ionization of the hydrogen-like ions, is studied within the framework of second order perturbation theory and the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron--photon interaction. For medium- and high-Z ions, in particular, the non-dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if compared with th…
We briefly introduce the formalism for describing dipolar molecules in an ultracold gaseous environment. This treatment includes the quantum mechanical states of the dipoles, the electric fields they produce, and their interactions with one another.
Signatures of multi-keV LMN dielectronic resonances in highly-charged 3d^n ions of tungsten were detected in the intensity ratios of extreme-ultraviolet magnetic-dipole lines within ground configurations. The measurements were performed with an electron beam ion trap at beam energies of about 6 keV. Large-scale collisional-radiative modeling incorporating magnetic sublevels of autoionizing levels showed the significance of anisotropy effects due to the monodirectional propagation of the electron beam. The observation method allows simultaneous resolved registration of dielectronic resonances f…
We study a hybrid quantum system composed of an ion and an electric dipole. We show how a trapped ion can be used to measure the small electric field generated by a classical dipole. We discuss the application of this scheme to measure the electric dipole moment of cold polar molecules, whose internal state can be controlled with ultrafast laser pulses, by trapping them in the vicinity of a trapped ion.
We present a general approach within the relativistic coupled-cluster theory framework to calculate exactly the first order wave functions due to any rank perturbation operators. Using this method, we calculate the static dipole and quadrupole polarizabilities in some alkali atoms and alkaline earth-metal ions. This may be a good test of the present theory for different rank and parity interaction operators. This shows a wide range of applications including precise calculations of both parity and CP violating amplitudes due to rank zero and rank one weak interaction Hamiltonians. We also give …
Dipole-dipole interactions between helium atoms in Rydberg-Stark states with principal quantum number $n=53$ and approximately linear Stark energy shifts, resulting from induced electric dipole moments of approximately 7900 D, have been investigated experimentally. The experiments were performed in pulsed supersonic metastable helium beams, with particle number densities of up to $\sim10^9$ cm$^{-3}$. In the presence of amplitude-modulated, radio-frequency electric fields, changes in the spectral intensity distributions associated with the transitions to these states that are attributed to dip…
The angular distribution of the emitted electrons, following the two--photon ionization of the hydrogen--like ions, is studied within the framework of second order perturbation theory \textit{and} the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron--photon interaction. For medium-- and high--Z ions, in particular, the non--dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if co…
It has been shown in an earlier publication that magnetic field gradients applied to particles in traps can induce Larmor frequency shifts that may falsely be interpreted as electric-dipole moment (EDM) signals. This study has now been extended to include nonuniform magnetic field gradients due to the presence of a local magnetic dipole. It is found that, in the high orbit-frequency regime, the magnitude of the shifts can be enhanced beyond the simple expectation of proportionality to the volume-averaged magnetic-field gradient.
Resonant electric dipole-dipole interactions between cold Rydberg atoms were observed using microwave spectroscopy. Laser-cooled Rb atoms in a magneto-optical trap were optically excited to 45d Rydberg states using a pulsed laser. A microwave pulse transferred a fraction of these Rydberg atoms to the 46p state. A second microwave pulse then drove atoms in the 45d state to the 46d state, and was used as a probe of interatomic interactions. The spectral width of this two-photon probe transition was found to depend on the presence of the 46p atoms, and is due to the resonant electric dipole-dipol…
In this article, we have employed relativistic many-body theory to theoretically assess the suitability of TlF+ molecular ion in its ground state for electron electric dipole moment searches. To that end, we have computed values of the effective electric field as well as the molecular permanent electric dipole moment using both configuration interaction and coupled cluster methods with high quality basis sets, followed by an analysis on the role of electron correlation in the considered properties. We find that TlF+ has a large value of effective electric field of about 163 GV/cm, which is abo…
We present our progress towards a new measurement of the electron electric dipole moment using a beam of YbF molecules. Data are currently being taken with a sensitivity of 10^{-27} e.cm / \sqrt{day}.
Heavy polar diatomic molecules are the primary tools for searching for the T-violating permanent electric dipole moment of the electron (eEDM). Valence electrons in some molecules experience extremely large effective electric fields due to relativistic interactions. These large effective electric fields are crucial to the success of polar-molecule-based eEDM search experiments. Here we report on the results of relativistic ab initio calculations of the effective electric fields in a series of molecules that are highly sensitive to an eEDM, the mercury monohalides (HgF, HgCl, HgBr,and HgI). We …
We propose an experiment to measure the electric dipole moment of the electron using ultracold YbF molecules. The molecules are produced as a thermal beam by a cryogenic buffer gas source, and brought to rest in an optical molasses that cools them to the Doppler limit or below. The molecular cloud is then thrown upward to form a fountain in which the electric dipole moment of the electron is measured.
We present electric dipole polarizabilities ($α_d$) of the alkali-metal negative ions, from H$^-$ to Fr$^-$, by employing four-component relativistic many-body methods. Differences in the results are shown by considering Dirac-Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DC Hamiltonian with the lower-order quantum electrodynamics interactions. At first, these interactions are included self-consistently in the Dirac-Hartree-Fock (DHF) method, and then electron correlation effects are incorporated over the DHF wave functions in the second-order many-body perturbation …
We describe the first precision measurement of the electron's electric dipole moment (eEDM, $d_e$) using trapped molecular ions, demonstrating the application of spin interrogation times over 700 ms to achieve high sensitivity and stringent rejection of systematic errors. Through electron spin resonance spectroscopy on $^{180}{\rm Hf}^{19}{\rm F}^{+}$ in its metastable $^{3}Δ_{1}$ electronic state, we obtain $d_e = (0.9 \pm 7.7_{\rm stat} \pm 1.7_{\rm syst}) \times 10^{-29}\,e\,{\rm cm}$, resulting in an upper bound of $|d_e| < 1.3 \times 10^{-28}\,e\,{\rm cm}$ (90% confidence). Our result pro…
The quadrupole interaction between the Rydberg electronic states of a Rydberg ion and the radio frequency electric field of the ion trap is analyzed. Such a coupling is negligible for the lowest energy levels of a trapped ion but it is important for a trapped Rydberg ion due to its large electric dipole moment. This coupling cannot be neglected by the standard rotating-wave approximation because it is comparable to the frequency of the trapping electric field. We investigate the effect of the quadrupole coupling by performing a suitable effective representation of the Hamiltonian. For a single…
Dipole-phonon quantum logic (DPQL) leverages the interaction between polar molecular ions and the motional modes of a trapped-ion Coulomb crystal to provide a potentially scalable route to quantum information science. Here, we study a class of candidate molecular ions for DPQL, the cationic alkaline-earth monoxides and monosulfides, which possess suitable structure for DPQL and can be produced in existing atomic ion experiments with little additional complexity. We present calculations of DPQL operations for one of these molecules, CaO$^+$, and discuss progress towards experimental realization…
We evaluate the feasibility of using magnetic-dipole (M1) transitions in highly-charged ions as a basis of an optical atomic clockwork of exceptional accuracy. We consider a range of possibilities, including M1 transitions between clock levels of the same fine-structure and hyperfine-structure manifolds. In highly charged ions these transitions lie in the optical part of the spectra and can be probed with lasers. The most direct advantage of our proposal comes from the low degeneracy of clock levels and the simplicity of atomic structure in combination with negligible quadrupolar shift. We dem…
The relativistic theory for the electric dipole moment (EDM) of paramagnetic atoms arising from the electric dipole moment of the electron is presented. A novel approach using the relativistic coupled-cluster method that incorporates the residual Coulomb interaction to all orders and a weak parity and time-reversal violating interaction to one order has been employed in Fr to obtain the enhancement of the EDM of that atom compared to the EDM of the electron. Trends of the different correlation effects and leading contributions from different physical states are discussed. Our result in combina…
The time evolution of a two-level quantum mechanical system can be geometrically described using the Bloch sphere. By mapping the Bloch sphere evolution onto the dynamics of oscillating electric dipoles, we provide a physically intuitive link between classical electromagnetism and the electric dipole transitions of atomic & molecular physics.
Using trapped ions in an entangled state we propose detecting a magnetic dipole of a single atom at distance of a few $μ$m. This requires a measurement of the magnetic field gradient at a level of about 10$^{-13}$ Tesla/$μ$m. We discuss applications e.g. in determining a wide variation of ionic magnetic moments, for investigating the magnetic substructure of ions with a level structure not accessible for optical cooling and detection,and for studying exotic or rare ions, and molecular ions. The scheme may also be used for measureing spin imbalances of neutral atoms or atomic ensembles trapped …
Gauge transformation leaves the electric and the magnetic fields unchanged as long as the gauge function is treated classically. In this paper we consider the gauge transformation commonly used to obtain the electric dipole interaction Hamiltonian in a system of dipoles and the electromagnetic field (Goeppert-Mayer transformation) and treat the vector potential that appear in the gauge function as an operator. While it modifies the electric field, the static interaction between the dipoles is derived.
The rovibrational electric dipole transitions in the ground electronic state of the HD molecule are studied. A simple, yet rigorous formula is derived for the transition rates in terms of the electric dipole moment function $D(R)$, which is calculated in a wide range of $R$. Our numerical results for transition rates are in moderate agreement with experiments and previous calculations, but are at least an order of magnitude more accurate.
We report the magnetic dipole and electric quadrupole hyperfine coupling constants of 6 P_{3/2} state of ^{133} Cs(I= {7/2}) obtained from the relativistic coupled cluster (RCC) method. To our knowledge, no prior electric quadrupole hyperfine CC calculation is available for ^{133} Cs(6 P_{3/2}) state. Our computed magnetic dipole (A) and electric quadrupole (B) coupling constants are in excellent agreement with the experiment.
The multiconfiguration Dirac-Hartree-Fock (MCDHF) method has been employed to calculate atomic electric dipole moments (EDM) of 225^Ra, 199^Hg, and 171^Yb. For the calculations of the matrix elements we extended the relativistic atomic structure package GRASP2K. The extension includes programs to evaluate matrix elements of (P, T)-odd e-N tensor-pseudotensor and pseudoscalar-scalar interactions, the atomic electric dipole interaction, the nuclear Schiff moment, and the interaction of the electron electric dipole moment with nuclear magnetic moments. The interelectronic interactions were accoun…
The electric dipole moments of various neutral elementary particles, such as neutron, neutrinos, certain hypothetical dark matter particles and others, are predicted to exist by the standard model of high energy physics and various extensions of it. However, the predicted values are beyond the present experimental capabilities. We propose to simulate and emulate the electric dipole moment of neutral relativistic particles and the ensuing effects in the presence of electrostatic field by emulation of an extended Dirac equation in ion traps.
We report the first measurements and detailed analysis of extreme ultraviolet (EUV) spectra (4 nm to 20 nm) of highly-charged tungsten ions W$^{54+}$ to W$^{63+}$ obtained with an electron beam ion trap (EBIT). Collisional-radiative modelling is used to identify strong electric-dipole and magnetic-dipole transitions in all ionization stages. These lines can be used for impurity transport studies and temperature diagnostics in fusion reactors, such as ITER. Identifications of prominent lines from several W ions were confirmed by measurement of isoelectronic EUV spectra of Hf, Ta, and Au. We als…
We have considered a mechanism for inducing a time-reversal violating electric dipole moment (EDM) in atoms through the interaction of a nuclear EDM (d_N) with the hyperfine interaction, the "magnetic moment effect". We have derived the operator for this interaction and presented analytical formulas for the matrix elements between atomic states. Induced EDMs in the diamagnetic atoms 129Xe, 171Yb, 199Hg, 211Rn, and 225Ra have been calculated numerically. From the experimental limits on the atomic EDMs of 129Xe and 199Hg, we have placed the following constraints on the nuclear EDMs, |d_N(129Xe)|…
Recent advances in quantum technologies have enabled the precise control of single trapped molecules on the quantum level. Exploring the scope of these new technologies, we studied theoretically the implementation of qubits and clock transitions in the spin, rotational, and vibrational degrees of freedom of molecular nitrogen ions including the effects of magnetic fields. The relevant spectroscopic transitions span six orders of magnitude in frequency illustrating the versatility of the molecular spectrum for encoding quantum information. We identified two types of magnetically insensitive qub…
We demonstrate explicitly that some recent calculations of atomic electric dipole moments (EDM) are incomplete. A contribution overlooked therein is pointed out. When included, it cancels exactly the result of those calculations, and thus restores the standard conclusions for nuclear EDM in atoms.
In the recent work [Phys. Rev. D 91, 111702(R) (2015)], C. Hill concludes that the axion electromagnetic anomaly induces an oscillating electron electric dipole moment of frequency $m_a$ and strength $\sim 10^{-32}~e$ cm, in the limit $v/c \to 0$ for the axion field. Here, we demonstrate that a proper treatment of this problem in the lowest order yields $\textit{no}$ electric dipole moment of the electron in the same limit. Instead, oscillating electric dipole moments of atoms and molecules are produced by different mechanisms.
Electric dipole moments of diamagnetic atoms of experimental interest are calculated using the relativistic Hartree-Fock and random-phase approximation methods, the many-body perturbation theory and configuration interaction technique. We consider P,T-odd interactions which give rise to atomic electric dipole moment in the second order of the perturbation theory. These include nuclear Schiff moment, P,T-odd electron-nucleon interaction and electron electric dipole moment. Interpretation of a new experimental constraint of a permanent electric dipole moment of $^{199}$Hg [W. C. Griffith {\it et…
The sensitivity much better than $10^{-24} e$ cm may be expected in the searches for electric dipole moments (EDM) of $β$-active nuclei at ion storage rings. It would be a serious progress in studies of the CP-violation problem.
