The response of matter to fields underlies the physical sciences, from particle physics to astrophysics, and from chemistry to biophysics. We observe an atom's response to an electric quadrupole field to second- and higher orders; this arises from the atom's electric quadrupole polarizability and hyperpolarizabilities. We probe a single atomic ion which is excited to Rydberg states and confined in the electric fields of a Paul trap. The quadrupolar trapping fields cause atomic energy level shifts and give rise to spectral sidebands. The observed effects are described well by theory calculations.
Three different measurements have been reported for the ground state quadrupole polarizability in the singly ionized barium (Ba$^+$) which disagree with each other. Our calculation of this quantity using the relativistic coupled-cluster method disagrees with two of the experimental values and is within the error bars of the other. We discuss the issues related to the accuracy of our calculations and emphasize the need for further experiments to measure the quadrupole polarizability for this state and/or the 5D states.
Selection of "magic" trapping conditions with ultracold atoms or molecules, where pairs of internal states experience identical trapping potentials, brings substantial benefits to precision measurements and quantum computing schemes. Working at such conditions could ensure that detrimental effects of inevitable inhomogeneities across an ultracold sample are significantly reduced. However, this aspect of confinement remains unexplored for ultracold polar molecules. Here, we present means to control the AC Stark shift of rotational states of ultracold polar molecules, when subjected to both trap…
Highly excited Rydberg states are usually extremely polarizable and exceedingly sensitive to electric fields. Because of this Rydberg ions confined in electric fields have state-dependent trapping potentials. We engineer a Rydberg state that is insensitive to electric fields by coupling two Rydberg states with static polarizabilities of opposite sign, in this way we achieve state-independent magic trapping. We show that the magically-trapped ion can be coherently excited to the Rydberg state without the need for control of the ion's motion.
We present precise values of electric polarizabilities for the ground state of Zn due to second-order dipole and quadrupole interactions, and due to third-order dipole-quadrupole interactions. These quantities are evaluated in the linear response theory framework by employing a relativistic version of the normal coupled-cluster (NCC) method. The calculated dipole polarizability value is compared with available experimental and other theoretical results including those are obtained using the ordinary coupled-cluster (CC) methods in both finite-field and expectation value evaluation approaches. …
$^{129}$Xe is an important candidate for the searches of electric dipole moments due to violations of time-reversal and parity violations in diamagnetic atoms. In view of the similarities between the electric dipole moment and the polarizability from the point of view of many-body theory, we have performed rigorous calculations of the ground state polarizability of $^{129}$Xe using a self consistent relativistic coupled-cluster method and the relativistic normal coupled-cluster method. The discrepancy between the results from these two methods is two percent, but each of them differs from the …
The dynamic electric dipole polarizability function for the magnesium atom is formed by assembling the atomic electric dipole oscillator strength distribution from combinations of theoretical and experimental data for resonance oscillator strengths and for photoionization cross sections of valence and inner shell electrons. Consistency with the oscillator strength (Thomas-Reiche-Kuhn) sum rule requires the adopted principal resonance line oscillator strength to be several percent lower than the values given in two critical tabulations, though the value adopted is consistent with a number of th…
We develop a method based on the relativistic coupled-cluster theory to incorporate a perturbative interaction to the no-pair Dirac-Coulomb atomic Hamiltonian. The method is general and suitable to incorporate any perturbation Hamiltonian in a many electron atom or ion. Using this perturbed relativistic coupled-cluster (PRCC) theory we calculate the electric dipole polarizability, $α$, of Neon. The linearized PRCC results are in very good agreement with the experimental value. However, the results of the nonlinear PRCC shows larger uncertainty but it is consistent with the observations from ea…
The finite field calculations are performed for two heavy frequency-standard candidates In$^+$ and Sr. The progressive hierarchy of electron correlations is implemented by the relativistic coupled-cluster and configuration interaction methods combined with basis set of increasing size. The dipole polarizabilities, dipole hyperpolarizabilities, quadrupole moments, and quadrupole polarizabilities are recommended for the ground state 5s$^2$ $^1S_0$ and low-lying states 5s5p $^3P^{\rm o}_{0,1,2}$ of In$^+$ and Sr. Comparative study of the fully and scalar relativistic electron correlation calculat…
The AC Stark shift induced by blackbody radiation is a major source of systematic uncertainty in present-day cesium microwave frequency standards. The shift is parametrized in terms of a third-order electric polarizability $α_0^{(3)}$ that can be inferred from the static electric field displacement of the clock transition resonance. In this paper, we report on an all-optical CPT pump-probe experiment measuring the differential polarizability $Δα_0^{(3)} = α_0^{(3)}(F{=}4) - α_0^{(3)}(F{=}3)$ on a thermal Cs atomic beam, from which we infer $ α_0^{(3)}(F{=}4) = 2.023(6)_{stat}(9)_{syst}~Hz/(kV/…
We have developed a Fock-space relativistic coupled-cluster theory based method for the calculation of electric dipole polarizability of one-valence atoms and ions. We employ this method to compute the ground-state and spin-orbit coupled excited state electric dipole polarizability of Al and In. To check the quality of many-electron wavefunctions, we also compute the excitation energies of some low-lying states of Al and In. The effects of Breit interaction and QED corrections from the Uehling potential and the self-energy are included to improve the accuracy of $α$ further. Our recommended va…
Light guided by an optical nanofibre can have a very steep evanescent field gradient extending from the fibre surface. This can be exploited to drive electric quadrupole transitions in nearby quantum emitters. In this paper, we report on the observation of the $5S_{1/2}$ $\rightarrow$ $4D_{3/2}$ electric quadrupole transition at 516.6 nm (in vacuum) in laser-cooled $^{87}$Rb atoms using only a few $μ$W of laser power propagating through an optical nanofibre embedded in the atom cloud. This work extends the range of applications for optical nanofibres in atomic physics to include more fundament…
Interactions between atoms and light fields are usually described in the electric-dipole approximation. We show that electric-quadrupole terms are important in the Kapitza-Dirac arrangement for light gratings on resonance with a quadrupole atomic transition. We derive the diffraction patterns, which in some cases are experimentally verifiable with the same techniques used with dipole transitions.
We present a theoretical analysis of the Stark effect in the hyperfine structure of the cesium ground-state. We have used third order perturbation theory, including diagonal and off-diagonal hyperfine interactions, and have identified terms which were not considered in earlier treatments. A numerical evaluation using perturbing levels up to n=18 yields new values for the tensor polarizability $α_2(6S_{1/2})$ and for the Stark shift of the clock transition frequency in cesium. The polarizabilities are in good agreement with experimental values, thereby removing a 40-year-old discrepancy. The cl…
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…
We employ a fully relativistic coupled-cluster theory to calculate the ground-state electric dipole polarizability and electron correlation energy of superheavy elements Cn, Nh$^+$ and Og. To assess the trend of electron correlation as function of $Z$, we also calculate the correlation energies for three lighter homologs--Zn, Cd and Hg; Ga$^+$, In$^+$ and Tl$^+$; Kr, Xe and Rn--for each superheavy elements. The relativistic effects and quantum electrodynamical corrections are included using the Dirac-Coulomb-Breit Hamiltonian with the corrections from the Uehling potential and the self-energy.…
The perturbed relativistic coupled-cluster (PRCC) theory is applied to calculate the electric dipole polarizabilities of alkaline Earth metal atoms. The Dirac-Coulomb-Breit atomic Hamiltonian is used and we include the triple excitations in the relativistic coupled-cluster (RCC) theory. The theoretical issues related to the triple excitation cluster operators are described in detail and we also provide details on the computational implementation. The PRCC theory results are in good agreement with the experimental and previous theoretical results. We, then, highlight the importance of consideri…
We use perturbed relativistic coupled-cluster (PRCC) theory to compute the electric dipole polarizabilities $α$ of Zn, Cd and Hg. The computations are done using the Dirac-Coulomb-Breit Hamiltonian with Uehling potential to incorporate vacuum polarization corrections. The triple excitations are included perturbatively in the PRCC theory, and in the unperturbed sector, it is included non-perturbatively. Our results of $α$, for all the three elements, are in excellent agreement with the experimental data. The other highlight of the results is the orbital energy corrections from Breit interaction…
We measured the ground state electric dipole polarizability of sodium, potassium, and rubidium using a Mach-Zehnder atom interferometer with an electric field gradient. We find alpha_Na=24.11(2)_stat(18)_sys x 10^-24 cm^3, alpha_K=43.06(14)(33), and alpha_Rb=47.24(12)(42). Since these measurements were all performed in the same apparatus and subject to the same systematic errors we can present polarizability ratios with 0.3% precision. We find alpha_Rb/alpha_Na=1.959(5), alpha_K/alpha_Na=1.786(6), and alpha_Rb/alpha_K=1.097(5). We combine our ratio measurements with the higher precision measur…
To operate an optical lattice clock at a fractional uncertainty below $10^{-17}$, one must typically consider not only electric-dipole (E1) interaction between an atom and the lattice light field when characterizing the resulting lattice light shift of the clock transition but also higher-order multipole contributions, such as electric-quadrupole (E2) and magnetic-dipole (M1) interactions. However, strongly incompatible values have been reported for the E2-M1 polarizability difference of the clock states $(5s5p)\,{}^{3}\mathrm{P}_{0}$ and $(5s^2)\,{}^{1}\mathrm{S}_{0}$ of strontium [Ushijima e…
We propose a nuclear mechanism which can induce an atomic electric dipole moment (EDM). Parity and time reversal violating (P,T-odd) nuclear forces generate a mixed P,T-odd nuclear polarizability beta_{ik} (defined by an energy shift U = -beta_{ik} E_{i}H_{k}, E is electric field and H magnetic field). The interaction of atomic electrons with beta_{ik} produces an atomic EDM. We performed an analytical calculation of the P,T-odd nuclear polarizability and estimated the value for the induced atomic EDM. The measurements of atomic EDMs can provide information about P,T-odd nuclear forces and tes…
We apply relativistic many-body methods to compute static differential polarizabilities for transitions inside the ground-state hyperfine manifolds of monovalent atoms and ions. Knowing this transition polarizability is required in a number of high-precision experiments, such as microwave atomic clocks and searches for CP-violating permanent electric dipole moments. While the traditional polarizability arises in the second-order of interaction with the externally-applied electric field, the differential polarizability involves additional contribution from the hyperfine interaction of atomic el…
Relativistic formulas for the electric quadrupole moment of a hydrogenlike atom, induced by the hyperfine interaction, are derived for $ns_{1/2}$ and $np_{1/2}$ states. Both the magnetic dipole and electric quadrupole hyperfine interactions are taken into account. The formulas are valid for ions with arbitrary nuclear charge and spin. The induced quadrupole moment is compared with the nuclear quadrupole moment for a wide range of hydrogenlike ions.
We present revised measurements of the static electric dipole polarizabilities of K, Rb, and Cs based on atom interferometer experiments presented in [Phys. Rev. A 2015, 92, 052513] but now re-analyzed with new calibrations for the magnitude and geometry of the applied electric field gradient. The resulting polarizability values did not change, but the uncertainties were significantly reduced. Then we interpret several measurements of alkali metal atomic polarizabilities in terms of atomic oscillator strengths $f_{ik}$, Einstein coefficients $A_{ik}$, state lifetimes $τ_{k}$, transition dipole…
Effect of the electric quadrupole moment, $Q$, is studied for positron-atom bound systems. It is demonstrated that for $Q >50$ a.u. the electric quadrupole potential is sufficiently strong to bind positron (or electron) even in the absence of the dipole polarization potential. Such large values of $Q$ are not known for atomic ground states, however, they exist in molecules and excited atoms. In the state $2s2p~^3P^o_2$ of beryllium, the quadrupole contribution makes difference between stable bound state and decay to Be$^+$ ion and positronium. In a majority of atoms the quadrupole contribution…
Using the operator representation of the Dirac Coulomb Green function the analytical method in perturbation theory is employed in obtaining solutions of the Dirac equation for a hydrogen-like atom in a time-dependent electric field. The relativistic dynamical polarizability of hydrogen-like atoms is calculated and analysed.
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.
Fock-space perturbed relativistic coupled-cluster calculations of electric dipole polarizability and nuclear spin-dependent parity non-conservation in Cs
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.
We evaluate the electric quadrupole moments of the ${^1}\!S_0$ and ${^3}\!P_0$ clock states of $^{27}$Al$^+$ and $^{115}$In$^+$. To capture all dominant contributions, our analysis extends through third order of perturbation theory and includes hyperfine coupling of the electrons to both the magnetic dipole and electric quadrupole moments of the nucleus. For $^{27}$Al$^+$, a fortuitous cancellation leads to a suppressed frequency shift. This should allow for continued improvement of the clock without special techniques to control or cancel the shift, such as the averaging schemes that are crit…
We have measured the strongly suppressed electric tensor polarizability of the Cs ground state using an optical pump-probe technique in a thermal atomic beam. The result agrees with a previous measurement and confirms the long-standing discrepancy with a theoretical value. The anticipated future reduction of the total uncertainty to the 1% level makes this quantity a valuable test for atomic structure calculations involving short-range inner-atomic interactions.
Second-order [O(k^2), k=omega/c] nondipole effects in soft-x-ray photoemission are demonstrated via an experimental and theoretical study of angular distributions of neon valence photoelectrons in the 100--1200 eV photon-energy range. A newly derived theoretical expression for nondipolar angular distributions characterizes the second-order effects using four new parameters with primary contributions from pure-quadrupole and octupole-dipole interference terms. Independent-particle calculations of these parameters account for a significant portion of the existing discrepancy between experiment a…
We report the first driving of the $7s\ ^2S_{1/2}\rightarrow 6d\ ^2D_{3/2}$ and $7s\ ^2S_{1/2}\rightarrow 6d\ ^2D_{5/2}$ electric quadrupole (E2) transitions in Ra$^{+}$. We measure the frequencies of both E2 transitions, and two other low-lying transitions in $^{226}$Ra$^{+}$ that are important for controlling the radium ion's motional and internal states: $6d\ ^2D_{3/2}\rightarrow 7p\ ^2P_{3/2}^{0}$ and $6d\ ^2D_{5/2}\rightarrow 7p\ ^2P_{3/2}^{0}$.
Hyperfine-induced electric dipole contributions may significantly increase probabilities of otherwise very weak electric octupole and magnetic quadrupole atomic clock transitions (e.g. transitions between $s$ and $f$ electron orbitals). These transitions can be used for exceptionally accurate atomic clocks, quantum information processing and search for dark matter. They are very sensitive to new physics beyond the Standard Model, such as temporal variation of the fine structure constant, the Lorentz invariance and Einstein equivalence principle violation. We formulate conditions under which th…
A method for highly accurate calculations of atomic electric quadrupole moments (EQM) is presented, using relativistic general-excitation-rank configuration interaction wavefunctions based on Dirac spinors. Application to the clock transition states of the thulium atom employing up to full Quadruple excitations for the atomic wavefunction yields a final value of $Q_{zz}({^2F}_{7/2}) = 0.07 \pm 0.07$ a.u., establishing that the thulium electronic ground state has an exceptionally small EQM. A detailed analysis of this result is presented which has implications for EQMs of other atoms with unpai…
The electric quadrupole moment for the $5d^{2}D_{3/2}$ state of $\mathrm{^{171}Yb^{+}}$, has been calculated using the relativistic coupled-cluster method. Earlier a similar calculation was performed for the 4d $^2D_{5/2}$ state of $\mathrm{^{88}Sr^{+}}$ which is the most accurate determination to date {[}PRL, \textbf{96}, 193001 (2006)]. The present calculation of the electric quadrupole moment of $\mathrm{^{171}Yb^{+}}$ yielded a value $2.157 ea_{0}^{2}$ where the experimental value is $2.08(11)ea_{0}^{2}$; $a_{0}$ is the Bohr radius and $e$ the elementary charge. We discuss in this paper ou…
The electric quadrupole moment for the $4d ^2D_{5/2}$ state of $\mathrm{^{88}Sr^+}$, one of the most important candidates for an optical clock, has been calculated using the relativistic coupled-cluster theory. The result of the calculation is presented and the important many-body contributions are highlighted. The calculated electric quadrupole moment is $(2.94\pm0.07)ea_{0}^{2}$, where $a_{0}$ is the Bohr radius and $e$ the electronic charge while the measured value is $(2.6\pm0.3)ea_{0}^{2}$. This is so far the most accurate determination of the electric quadrupole moment for the above ment…
Molecular hydrogen ions are of metrological relevance due to the possibility of precise theoretical evaluation of their spectrum and of external-field-induced shifts. We report the results of the calculations of the rate of laser-induced electric quadrupole transitions between a large set of ro-vibrational states of ${\rm H_2^+}$. The hyperfine and Zeeman structure of the E2 transition spectrum and the effects of the laser polarization are treated in detail. We also present the nuclear spin-electron spin coupling constants, computed with a precision 10 times higher than previously.
We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earli…
We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. This solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. The charg…
We have employed the relativistic coupled cluster theory to calculate the magnetic dipole and electric quadrupole hyperfine constants for the ground and low lying excited states of singly ionized magnesium. Comparison with experimental and the other theoretical results are done and predictions are also made for a few low lying excited states which could be of interest. We have made comparative studies of the important many body effects contributing to the hyperfine constants for the different states of the ion.
We investigate electric quadrupole hyperfine structure constant ($B$) results in $^{135}$Ba$^+$ and $^{137}$Ba$^+$ and compare their ratios with the reported measurements of the ratio between the nuclear quadrupole moment ($Q$) values of these isotopes. We carry out confidence tests for the reported experimental $B$ values and calculations of $B/Q$ from the present work. Inconsistencies in the experimental $B$ values are observed in both the isotopes from different experiments performed using the same techniques. The present calculations are carried out using an all order relativistic many-bod…
The $6S_{1/2}-5D_{5/2}$ electric quadrupole transition is investigated in Cesium vapor at room temperature via nonlinear Doppler-free 6P-6S-5D three-level spectroscopy. Frequency-resolved studies of individual E2 hyperfine lines allow one to analyze optical pumping dynamics, polarization selection rules and line intensities. It opens the way to studies of transfer of light orbital angular momentum to atoms, and the influence of metamaterials on E2 line spectra.
Intensities of all fine-structure resolved rovibrational electric quadrupole absorption lines in $^{16}$O$_2$($X^{3}Σ^{-}_{g}$) calculated with a new $\textit{ab initio}$ quadrupole moment curve
The proton polarizability effect in the muonic-hydrogen Lamb shift comes out as a prediction of baryon chiral perturbation theory at leading order and our calculation yields for it: $ΔE^{(\mathrm{pol})} (2P-2S) = 8^{+3}_{-1}\, μ$eV. This result is consistent with most of evaluations based on dispersive sum rules, but is about a factor of two smaller than the recent result obtained in {\em heavy-baryon} chiral perturbation theory. We also find that the effect of $Δ(1232)$-resonance excitation on the Lamb-shift is suppressed, as is the entire contribution of the magnetic polarizability; the elec…
Recent advances in the high sensitivity spectroscopy have made it possible, in combination with accurate theoretical predictions, to observe for the first time very weak electric quadrupole transitions in a polar polyatomic molecule of water. Here we present accurate theoretical predictions of the complete quadrupole ro-vibrational spectrum of a non-polar molecule CO$_2$, important in atmospheric and astrophysical applications. Our predictions are validated by recent cavity enhanced absorption spectroscopy measurements and are used to assign few weak features in the recent ExoMars ACS MIR spec…
