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Two-Photon Polarizability of Ba+ Ion: Control of Spin-Mixing Processes in an Ultracold 137Ba+ − 87Rb Mixture

Arghya Das, Anal Bhowmik, Narendra Nath Dutta, Sonjoy Majumder

Published Oct 3, 2022• 1 citation
Publisher

Abstract

In this work, we present a scheme of a two-photon interaction to calculate magic wavelengths for the 62S12 − 52D32,52 clock transitions of Ba+ ion employing the relativistic coupled-cluster method. These magic wavelengths can be essential inputs to achieve better accuracy in the future ionic clock experiments. In this paper, we further show an application of a two-photon interaction to the spin-mixing processes, |0,0⟩↔|+1,−1⟩ and |0,0⟩↔|−1,+1⟩, of an ultra-cold spin-1 mixture of 137Ba+ ions and 87Rb atoms. We determine the protocols for selecting these spin-mixing oscillations by changing the strength and frequencies of the externally applied magnetic field and laser beams, respectively.

DOI
10.3390/atoms10040109

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Physica Scripta•Aug 24, 2026•0 citations
In this work, we present a scheme of a two-photon interaction to calculate magic wavelengths for the 62S12 − 52D32,52 clock transitions of Ba+ ion employing the relativistic coupled-cluster method. These magic wavelengths can be essential inputs to achieve better accuracy in the future ionic clock experiments. In this paper, we further show an application of a two-photon interaction to the spin-mixing processes, |0,0⟩↔|+1,−1⟩ and |0,0⟩↔|−1,+1⟩, of an ultra-cold spin-1 mixture of 137Ba+ ions and 87Rb atoms. We determine the protocols for selecting these spin-mixing oscillations by changing the strength and frequencies of the externally applied magnetic field and laser beams, respectively.
Two-photon polarizability of Ba+^++ ion: Control of spin-mixing process in an ultracold 137^{137}137Ba+^++--87^{87}87Rb mixture
Ionic clocks exhibit as the most promising candidates for the frequency standards. Recent investigations show the profound advantages of interrogating two laser beams with different frequencies in developing the frequency standards. Here we present a scheme of a two-photon mechanism to calculate the dynamic polarizabilities for the clock states, 62^22S12_{\frac{1}{2}}21​​ and 52^22D32,52_{\frac{3}{2}, \frac{5}{2}}23​,25​​, of Ba+^++ by employing relativistic coupled-cluster method. We illustrate the Stark-shift cancellation between these clock states at the two-photon magic wavelengths. These magic wavelengths can be essential inputs to achieve better accuracy in the ionic clock experiments. We also calculate the magic wavelengths under the single-photon interaction to serve as the reference and for a comparative study. The calculated single- and two-photon magic wavelengths lie in the optical region and thus are significant for future state-of-the-art experiments. Moreover, as an application of the two-photon polarizabilities, we investigate the impact of these polarizabilities on the spin-mixing processes, ∣0,0⟩|0,0\rangle∣0,0⟩ ↔\leftrightarrow↔ ∣+1,−1⟩|+1,-1\rangle∣+1,−1⟩ and ∣0,0⟩|0,0\rangle∣0,0⟩ ↔\leftrightarrow↔ ∣−1,+1⟩|-1,+1\rangle∣−1,+1⟩, of an ultra-cold spin-1 mixture of the 137^{137}137Ba+^++ and 87^{87}87Rb atoms. We determine the protocols of selecting these spin-mixing oscillations by changing the strength of an externally applied magnetic field and the frequencies of the interrogating laser beams.
Experiments with vortex beams have shown a surge of interest in controlling cold atoms. Most of the controlling protocols are dominated by circularly polarized light due to its ability to induce vector polarization at atoms, which is impossible for paraxial linearly polarized light. Here we develop a theory for frequency dependent polarizability of an atomic state interacting with a focused linearly polarized vortex beam. The naturally induced spin-orbit coupling in this type of linearly polarized beam produces vector component of the valence polarizability to an atomic state, obeying the total angular-momentum conservation of the beam. The theory is employed on 87Sr+^{87}\mathrm{Sr}^{+}87Sr+ to accurately calculate the magic wavelengths for the clock transitions and tune-out wavelengths for the clock states using the relativistic coupled-cluster method. The induced vector component in the dynamic polarizability due to the linearly polarized focused vortex beam promotes a fictitious magnetic field to the atomic state. We demonstrate that this fictitious magnetic field, depending on the focusing angle and orbital angular momentum of the beam, improves the flexibility of the coherent heteronuclear spin oscillations in a spin-1 mixture of 87Rb^{87}\mathrm{Rb}87Rb and 23Na^{23}\mathrm{Na}23Na atoms.
In the mixture of ultracold spin-1 atoms of two different species A and B (e.g., 23Na^{23}\mathrm{Na}23Na (A) and 87Rb^{87}\mathrm{Rb}87Rb (B)), the interspecies singlet-pairing process ${\mathrm{A}}_{+1}+{\mathrm{B}}_{\ensuremath{-}1}\ensuremath{\rightleftharpoons}{\mathrm{A}}_{\ensuremath{-}1}+{\mathrm{B}}_{+1}$ can be induced by the spin-dependent interatomic interaction, where subscript $\ifmmode\pm\else\textpm\fi{}1$ denotes the magnetic quantum number. Nevertheless, one cannot isolate this process from other spin-changing processes, which are usually much stronger, by tuning the bias real magnetic field. As a result, it is difficult to clearly observe the singlet-pairing process and precisely measure the corresponding interaction strength. In this work we propose to control the singlet-pairing process via combining the real magnetic field and a laser-induced species-dependent synthetic magnetic field. With our approach one can significantly enhance this process and simultaneously suppress all other spin-changing processes. We illustrate our approach for both a confined two-atom system and a binary mixture of spinor Bose-Einstein condensates. Our control scheme is helpful for the precise measurement of the weak singlet-pairing interaction strength and the entanglement generation of two different atoms.
Collisions between ultracold neutral atoms have for some time been the subject of investigation, initially with hydrogen and more recently with laser cooled alkali atoms. Advances in laser cooling and trapping of neutral atoms in a Magneto-Optic Trap (MOT) have made cold atoms available as the starting point for many laser cooled atomic physics investigations. The most spectacularly successful of these, the observation of Bose-Einstein Condensation (BEC) in a dilute ultra-cold spin polarized atomic vapor, has accelerated the study of cold collisions. Experimental and theoretical studies of BEC and the long range interaction between cold alkali atoms is at the boundary of atomic and low temperature physics. Such studies have been difficult and would not have been possible without the development and advancement of laser cooling and trapping of neutral atoms. By contrast, ion-atom interactions at low temperature, also very difficult to study prior to modern day laser cooling, have remained largely unexplored. But now, many laboratories worldwide have almost routine access to cold neutral atoms. The combined technologies of ion trapping, together with laser cooling of neutrals has made these studies experimentally feasible and several very important, novel applications might come out of such investigations . This paper is an investigation of ion-atom interactions in the cold and ultra-cold temperature regime. Some of the collisional ion-atom interactions present at room temperature are very much reduced in the low temperature regime. Reaction rates for charge transfer between unlike atoms, A + B(+) approaches A(+) + B, are expected to fall rapidly with temperature, approximately as T(sup 5/2). Thus, cold mixtures of atoms and ions are expected to coexist for very long times, unlike room temperature mixtures of the same ion-atom combination. Thus, it seems feasible to cool ions via collisions with laser cooled atoms. Many of the conventional collisional interactions, exploited as a useful tool at room temperature and higher, are greatly enhanced at low energy. For example, collisional spin transfer from one species of polarized atoms to another has long been a useful method for polarizing a sample of atoms where no other means was available. Because optical pumping cannot be used to polarize the nuclear spin of Xe-129 or He-3 (for use in nmr imaging of the lungs), the nuclear spins are polarized via collisions with an optically pumped Rb vapor in a cell containing both gases. In another case, a spin polarized thermal Cs beam was used to polarize the hyperfine states of trapped He(+)-3 ions in order to measure their hyperfine clock transition frequency. The absence of an x-ray light source to optically pump the ground state of the He(+)-3 ion necessitated this alternative state preparation. Similarly, Cd(+) and Sr(+) ions were spin-oriented via collisions in a cell with optically pumped Rb vapor. Resonant RF spin changing transitions in the ground state of the ions were detected by changes in the Rb resonance light absorption. Because cold collision spin exchange rates scale with temperature as T(sup -1/2) this technique is expected to be a far more powerful tool than the room temperature counterpart. This factor of 100 or more enhancement in spin exchange reaction rates at low temperatures is the basis for a novel trapped ion clock where laser cooled neutrals will cool, state select and monitor the ion clock transition. The advantage over conventional direct laser cooling of trapped ions is that the very expensive and cumbersome UV laser light sources, required to excite the ionic cooling transition, are effectively replaced by simple diode lasers.
Extensive calculations of the electric-dipole matrix elements in alkali-metal atoms are conducted using the relativistic all-order method. This approach is a linearized version of the coupled-cluster method, which sums infinite sets of many-body perturbation theory terms. All allowed transitions between the lowest nsnsns, np1∕2n{p}_{1∕2}np1∕2​, np3∕2n{p}_{3∕2}np3∕2​ states and a large number of excited states are considered in these calculations and their accuracy is evaluated. The resulting electric-dipole matrix elements are used for the high-precision calculation of frequency-dependent polarizabilities of the excited states of alkali-metal atoms. We find ``magic'' wavelengths in alkali-metal atoms for which the nsnsns and np1∕2n{p}_{1∕2}np1∕2​ and np3∕2n{p}_{3∕2}np3∕2​ atomic levels have the same ac Stark shifts, which facilitates state-insensitive optical cooling and trapping.
Abstract In present work, we have investigated the roles of dynamic quadrupole polarizabilities and the effect of electric field gradient in the accurate determination of magic wavelengths of the clock transitions for Ca + , Sr + and Ba + ions for linearly polarized light. We have employed relativistic all-order method for the evaluation of matrix elements for accurate determination of polarizability. The effect of gradient of electric field on magic wavelengths is analyzed with reference to small and large values of applied electric field-gradients in context to the experiments. Additional magic wavelengths have been revealed when the ratio of the electric field-gradient to electric field strength ( η = ‖ ( ∇ ε ∗ ⊗ ∇ ε ) ( k ) ‖ ‖ ( ε ∗ ⊗ ε ) ( k ) ‖ ) exceeds 0.1 a.u.. It is also observed that the characteristics of the trap can be switched with alteration in η . We have also estimated additional parameters such as trap depths, photon scattering rates and sensitivity to laser linewidths for these newly emerged magic wavelengths. Therefore, we believe that it is crucial to consider the spatial variation of any electric field while determining magic wavelengths for better understanding and implementation of laser trapping techniques.