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arXiv:2106.09607

Two-photon polarizability of Ba$^+$ ion: Control of spin-mixing process in an ultracold $^{137}$Ba$^+$--$^{87}$Rb mixture

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

Published Jun 17, 2021• 0 citations
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Abstract

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, 6$^2$S$_{\frac{1}{2}}$ and 5$^2$D$_{\frac{3}{2}, \frac{5}{2}}$, 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, $|0,0\rangle$ $\leftrightarrow$ $|+1,-1\rangle$ and $|0,0\rangle$ $\leftrightarrow$ $|-1,+1\rangle$, of an ultra-cold spin-1 mixture of the $^{137}$Ba$^+$ and $^{87}$Rb 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.

DOI
10.48550/arxiv.2106.09607

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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.
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.
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Higher-order perturbation theory is used to study frequency shifts of the spin-forbidden 3 P J - 1 S 0 transitions for Mg, Sr and Yb atoms in an external laser field. Calculations of the higher-order dynamic Stark effect have been performed in connection with precise measurements on cooled and trapped atoms and the design of ultra stable optical clock in an engineered light shift trap. The form of the model potential of the valence electrons suggested by Simons is used for the calculation of atomic dynamic characteristics given by second- and fourth-order perturbation theory for the interaction of the atoms with the electromagnetic field. Dispersion properties of the polarizability and hyperpolarizability components and their behavior in the vicinity of both resonances and so-called magic wavelengths for the upper and lower atomic states, are analyzed. Calculations of ac multipole polarizabilities and dipole hyperpolarizabilities for the 3 P 0 - 1 S 0 clock transition in 87 Sr indicate that the contribution of the higher- order light shifts can be reduced to less than 1 mHz, allowing for a projected accuracy of better than 10 -17 .
Using the relativistic coupled-cluster method, we calculate the static dipole polarizabilities of the 6p6p6p, 7s7s7s, 7p7p7p, 8s8s8s, 8p8p8p, and 6d6d6d states and the dynamic dipole polarizabilities of the 6p6p6p and 7s7s7s states of the thallium atom. The trivalent thallium atom is computationally treated as a monovalent system, together with all linear and nonlinear terms of single- and double-cluster operators included in the correlation calculations. We observe that the dominating contributions to the static scalar polarizabilities of the 7s1/27{s}_{1/2}7s1/2​, 7p1/27{p}_{1/2}7p1/2​, 8p1/28{p}_{1/2}8p1/2​, and 8p3/28{p}_{3/2}8p3/2​ states are from one or two specific transitions. The matrix elements of these specific transitions can be determined by combining the experimental values of relevant static scalar polarizabilities. A number of magic wavelengths for the 6p1/26{p}_{1/2}6p1/2​--7s7s7s and 6p3/26{p}_{3/2}6p3/2​--7s7s7s transitions in the range of 488--1300 nm and the longest tune-out wavelength of the ground state are determined. These magic wavelengths and tune-out wavelength may be useful for further thallium experiments. Experimental measurements of the magic wavelengths near 1245 nm would give estimates of the 7s1/27{s}_{1/2}7s1/2​--7p1/27{p}_{1/2}7p1/2​ and 7s1/27{s}_{1/2}7s1/2​--7p3/27{p}_{3/2}7p3/2​ transition matrix elements and their ratio. Furthermore, lifetimes of many excited states, as well as the Stark-induced scalar and vector dipole polarizabilities for the 6p1/26{p}_{1/2}6p1/2​--7p1/27{p}_{1/2}7p1/2​ transition, are also evaluated and compared with available theoretical and experimental values.
We find a simple solution to the problem of probe laser light shifts in two-photon optical atomic clocks. We show that there exists a magic polarization at which the light shifts of the two atomic states involved in the clock transition are identical. We calculate the differential polarizability as a function of laser polarization for two-photon optical clocks based on neutral calcium and strontium, estimate the magic polarization angle for these clocks, and determine the extent to which probe laser light shifts can be suppressed. We show that the light shift and the two-photon excitation rate can be independently controlled using the probe laser polarization.
Magic wavelengths for trapping of Sr+^++ with focused optical vortex
A theory of dynamic polarizability for trapping relevant states of Sr+^++ is presented here when the ions interact with a focused optical vortex. The coupling between the orbital and spin angular momentum of the optical vortex varies with focusing angle of the beam and is studied in the calculation of the magic wavelengths for 5s1/2→4d3/2,5/25s_{{1}/{2}}\rightarrow 4d_{{3}/{2}, {5}/{2}}5s1/2​→4d3/2,5/2​ transitions of Sr+^++. The initial state of our interest here is 5s1/25s_{{1}/{2}}5s1/2​ with mJ=−1/2m_J = -1/2mJ​=−1/2 of which is different possible trapping state compare to our recent work on Sr+^++ [Phys. Rev. A \textbf{97}, 022511 (2018)]. We find variation in magic wavelengths and the corresponding polarizabilities with different combinations of orbital and spin angular momentum of the vortex beam. The variation is very significant when the wavelengths of the beam are in the infrared region of electromagnetic spectrum. The calculated magic wavelengths will help the experimentalists to trap the ion for performing the high precision spectroscopic measurements.