Dear Editor,

we thank the reviwers for the careful second reading of our manuscript. We try to address the remaining concerns of Referee 1 more quantitatively, but essentially remain with our position. To address the concerns point by point:


1. COMMENTS TO THE AUTHOR(S)
The re-submitted paper is considerably improved, but there are two issues raised in the initial review that the authors did not sufficient address.  

First, there is extensive discussion in the literature about how vector light shifts in a tightly-focused trap are very important for coherence (refs [13] and [14]).  This was brought up in the first review, yet the authors chose to essentially not address it by saying that their waist is large so the effect is negligible.  This is not a quantitative statement, and is even qualitatively dubious – the difference in waist divided by wavelength is only about a factor of 1.5 compared to reference [14].  The authors should either include this effect by scaling the calculations from [14], or quantitatively show that it is indeed negligible.  Simply claiming that the vector light shift “vanishes” is not sufficient for a paper that centers around light shifts and coherences in single-atom tweezers.

Reply:
 We followed the suggestion of the referee and clarified the strength of the vector light shift caused by the non-paraxial focus. In the manuscript we included an estimate of this effect based on the analysis in Ref.14. For our focusing strength, magnetic field amplitude, FORT detuning and beam intensity the polarization gradients induced by the focused trap beam are very weak and our qubit coherence is currently not limited by this effect. The observed dephasing rate is about 100 times faster than what we expect from the polarization gradients. Our observed coherence is quite similar what other groups measured [13,35] and is most likely dominated by technical noise such as magnetic field fluctuation.

In Section II we added:
"Notably for very tightly focused beams, even a linearly polarized FORT
results in a spatially varying vector light shift and rapid qubit dephasing (as observed in Ref. [14]).
For our focusing strength, detuning and beam intensity, however, this
nonparaxial focus effect is negligible (see Section IV)."


In Section IV we added:
"We note that the dephasing time is not limited by the polarization gradients
induced by the tightly focused FORT but other noise sources – following Ref. [14], we estimate
that the polarization-gradient dephasing rate is two orders of magnitude lower than the observed value."


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2. Second, the authors claim that [14] does not contain coherence measurements (despite the title of the paper…) and therefore does not need to be discussed in that context.  Not only does the paper have coherence measurements, but extensive analysis and interpretation of the coherence, which is a main theme.  The most relevant findings are summarized in figure 1d of [14], which includes analysis of the role of a bias magnetic field in light of the virtual magnetic fields from the trap polarization gradients.  The authors of the present manuscript claimed in their response that no such measurements were made.  I therefore reiterate that the authors should discuss their results in context of existing publications, especially the highly-relevant reference [14].

Reply:
As mentioned in our reply to the first comment, the nonparaxial effects of the trap beam are quite negligible for our experimental parameters. To the best of our knowledge, only Ref [14] and [35] report on the dephasing time of non-clock states in single atom dipole traps. Our dephasing time is slightly shorter than in Ref [14,35] (40us compared to 80us), and we added a statement in the manuscript. However, dephasing times can be easily dominated by technical noise such as slowly varying magnetic fields, and are not a good figure of merit of the coherence of a system. A better measure is the coherence time obtained by spin echoes. However, [14,35] report only dephasing times T2*, which makes a fair comparison of the coherence properties difficult.  

We hope to have addressed the concerns in a quantitative enough way. For ease of referencing, we added a pdf file highlighting the differences, and are looking forward for your reply.

With Best Regards on behalf of all authors,

Christian Kurtsiefer