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

Round 1

Reviewer 1 Report

Manuscript contains both a review and original results.

Author Response

We thankfully acknowledge the respected reviewer’s appreciation of our work and the manuscript. Minor corrections on spelling suggested by him are taken care of in the modified manuscript.

Reviewer 2 Report

This article two-photon polarizability of Ba ions. It is very well written. I only have two minor comments.

1. About “The number of GTO basis functions used in our calculations for the s-, p-, d-, f-, g-, and h-type symmetries are 33, 30, 28, 25, 21, and 20 respectively” in line 104, how those numbers were chosen?
2. About “The magnetic sublevels of the 87Rb atom or the 137Ba+ ion are degenerate in the absence of an external magnetic field or a circularly polarized light.” in line 241, please explain why linearly polarized light is excluded in this statement.

Author Response

We thankfully acknowledge the respected reviewer’s appreciation of our work and the manuscript. We have made English corrections to our manuscript.

Reply to 1^st comment: The number of GTO basis functions of Ba ions are chosen based on the agreement of energy, r-, 1/r-, and r^2 matrix elements of all the bound DF orbitals considered in the RCC calculations with the same obtained from the highly precise numerical program “GRASP92”. In this optimization technique, we also optimize the exponents of the GTO basis together. This approach has been discussed many times in our old papers cited in the manuscript. Therefore, we did not mention them in the manuscript but cited our old work.

Reply to 2^nd comment:  We know that the magnetic sublevels splitting vector term in the polarizability calculation vanishes or contributes zero for linearly polarized light. Please see Equation (2) in our manuscript. The only other magnetic sublevel-depending term is tensor polarizability, but the parameter appears here quadratic in power. Therefore, linearly polarized light does not contribute to magnetic sublevel splitting.

Reviewer 3 Report

The authors presented their research on the two photon interaction to calculate the magic wavelength of the clock transition of Ba+ ions. The two photon interaction could also be extended to spin-mixing processes. Experimentally, they successfully determined the applied magnetic field and laser beam as the significant factors for spin mixing oscillation. The scientific content is sound and well supported. I was wondering how representative the system could be and are there any thoughts when applying into other molecules. How accurate this method could be? I recommend it to be accepted after minor revisions.

Author Response

We thankfully acknowledge the respected reviewer’s appreciation of our work and the manuscript. We have done English corrections in our manuscript to our best.

Reply to 1^st comment: In our manuscript, we have demonstrated spin-mixing in the atom-ion mixture. Our endeavor is on the atomic mix-system as one of the important applications of the two-photon polarizability apart from the two-photon clock transition. We used a ¹³⁷Ba⁺-⁸⁷Rb ion-atom mixture. It can be shown in other ion-atom mixtures also. For instance, a recent experiment (Nature Comm. Vol-9, article no. 920 (2018)) showed spin-exchange in the ⁸⁸Sr⁺-⁸⁷Rb mixture or other diatomic molecule ⁴⁰K⁸⁷Rb (Nature Physics, vol-17, 1144 (2021), Science, vol-327, page  853 (2010), ...), etc. We have also mentioned a few more examples in the manuscript. The theory's accuracy depends on how accurately the hyperfine states are defined and the calculation of polarizability. As shown in our manuscript, the precession polarizability calculations rely on the accuracy of the electric dipole transition matrix, which is excellent in our work. Discussions on the other molecules are beyond the scope of the submitted manuscript. Therefore, we are not augmenting any additional statements on the other molecules in our manuscript.