Dear Editor, first, we like to thank both referee for the careful reading and their constructive comments. We tried to address the issues raised with a revised manuscript. In particular, we believe we can address the concerns regarding suitability for PRL. Our responses in detail: Referee A comment: 1) Near concentric cavities have the advantage to have a very small waist at the center of the mode, allowing the authors to get a coupling g0 on the order of 1.7 times the natural decay rate of the atom. Although this is impressive, the cavity itself is very lossy (with a linewidth on the order of 100 MHz), and the final cooperativity is C~0.08, which has already been achieved in the context of near concentric cavity. As a result, I do not believe the paper in this present form should be published in PRL. Reply: A cooperativity of one has been observed in a near-concentric cavity by the Blatt group Ref.[9,10], however, there the cavity was operated relatively far from the concentric point, about 84um. As a consequence the demonstrated coupling to linewidth ratio was low (<0.2); a high cooperativity was nevertheless achieved by using a high finesse cavity F=70000. In this sense, we believe that our work represents a very different approach, and shows a significant advancement in the coupling to linewidth ratio by working much closer to the concentric length (less than three wavelengths). ---------------------------------------------------------------------- Referee A comment: 2) Formulae (3): it is not clear how the value of \Delta n is measured. Is it a guess to fit l_cav? It seems to me that 1044 or 1042 would work as well. How did you determine \nu_780 and \nu_810? With which precision do you know their values? Reply: We completely agree with the referee comment, we just carried out the double resonance measurement as a consistency check. In the revised manuscript, we omitted the reference to this length measurement to avoid the confusion this apparently created, as this method is sensitive to the exact value of R_c and considering the machining tolerances we cannot reliable extract the distance to the concentric length by this method. However, our conclusion from the analysis of the transverse mode spacing remains, and gives a precise measurement of the distance to the concentric length, 1.7(1)um. ---------------------------------------------------------------------- Referee A comment: 3) Figure 3: How is determined the threshold? To what correspond the event at 1 s? Reply: The increase of fluorescence at 1s corresponds to an atom loaded into a side of the intra-cavity optical lattice which does not couple strongly to the cavity mode. We choose a high threshold to ensure that we only consider strongly coupled atoms. To make this clear, we included corresponding statements at two locations in the revised manuscript: - Caption,Fig.3 "At 1\,s an atom is loaded into a side of the intra-cavity optical lattice which does not couple strongly to the cavity mode. We choose a high threshold value to select only strongly coupled atoms." - Page 3, 1st paragraph:"By choosing a high threshold value, we select atoms which couple strongly to the cavity mode." ---------------------------------------------------------------------- Referee A comment: 4) Finally, even after the averaging over the CB coefficients, the expected value of g0 is still larger than what is measured. Can you comment on that? Reply: On one hand, the ac-Stark shift induced by the dipole trap leads spin-state dependent energy shifts in the excited state which are on the order of the natural linewidth. Without state preparation as in our experiment, this leads to a reduction in the observed atom-light interaction. On the other hand, our threshold detection technique for looking at atoms may not only select atoms in the strongest coupling lattice site of the trap. There may even be some finite temperature effects that we can not fully characterize at this point. We acknowledge the disagreement between our observation and a simple theoretical model by about a factor of two, but wanted to avoid speculation on these different reasons in the manuscript because we really don't have good evidence yet what the main contributor to this discrepancy is. ---------------------------------------------------------------------- Referee B comment: 1) I find the experimental results presented in the manuscript interesting, but it's also clear from the manuscript that there are serious costs to this implementation. The authors found that the cavity mirrors needed to be re- aligned in situ every 15 minutes, a substantial experimental constraint. Reply: The cavity alignment is automatically checked every 15min but the re- alignment procedure takes only 1-10 seconds, and thus this does not affect the experimental duty cycle significantly. The fact that this is done automatically makes it probably a very stable experiment in comparison. In addition, the need for transverse re- alignment is mainly caused by temperature changes and could be strongly reduced by using different materials for the mechanical mount of the cavity (currently the holder is made out of aluminium). We modified the manuscript to clarify the stabilization procedure: "Every 15\,minutes an automatized alignment algorithm optimizes the transversal mirror position using the transmission of the 780\,nm and 810\,nm light as feedback signals; this procedure takes between 1-10 seconds and thus does not significantly reduce the experimental duty cycle." ---------------------------------------------------------------------- Referee B comment: 2) Furthermore, the mirror losses in their current system are quite high, rendering the system not suitable for many cavity-QED experiments, but with a higher finesse (as suggested at the end of the manuscript), wouldn't the system be even more sensitive to misalignment? Reply: We do not expect any significant increase in sensitivity to misalignment for modestly increased finesse compared to non-concentric cavities. Experimentally we verified this for cavities up to F=600. Currently our linewidth is dominated by intra-cavity absorption losses rather than transmission through the mirrors or diffraction losses. These losses are caused by contamination of the mirrors during the baking out the vacuum chamber. We are positive that this can be prevented and does not pose a fundamental limit to the concentric cavity approach. We included a statement in the manuscript (page 4, last paragraph) to clarify that the intracavity loss is not dominated by diffraction losses: "We note that although we operate the cavity only $1.7(1)\,\mu$m shorter than "the critical length, the expected intracavity diffraction losses are "negligibly low as the mode radius on the mirror is an order of magnitude "smaller than the aperture of the mirror~\cite{Durak2014}. ---------------------------------------------------------------------- Referee B comment: 3) The authors also suggest the a factor of 10 improvement in the distance from the concentric threshold might be possible, but they don't elaborate on how this could be achieved. Reply: As mentioned in the manuscript we currently observe that the cavity finesse and transmission drop when we operate closer to the concentric length; We suspect deviations of the mirror from an ideal spherical surface as a possible cause but we do not have conclusive results on this topic yet. ---------------------------------------------------------------------- Referee B comment: 4) I have a concern about the measurement of the cavity length, as it is claimed in both the abstract and in the manuscript that this is determined with error bars of 10 nm (!). However, the radius of curvature of the mirrors is stated as 5.5 mm, with no error bars given. Certainly there are error bars inherent in the mirror polishing, which should then play a role in the determination of l_cav and its uncertainty? Reply: We included the machining tolerances of the mirrors in the manuscript, 5.500(6)mm and omitted the cavity length measurement via two simultaneously resonant light fields because this method is indeed sensitive to the exact value of R_c. Our conclusion from the analysis of the transverse mode spacing is valid and gives a precise measurement of the distance to the concentric length, 1.7(1)um. ---------------------------------------------------------------------- Referee B comment: 5) 1st page, 1st paragraph: "by coupling atoms (or other quantum emitter)" - emitter should be plural Reply: Corrected, thanks for the note. ---------------------------------------------------------------------- Referee B comment: 6) 1st page, 2nd paragraph: The mode volume is introduced but not defined. It is particularly important to define the mode volume clearly in this paper because in the case of near-concentric cavities (or any cavity where the length is not much shorter than the mirror radii of curvature), the relevant parameter in the coupling strength is not the actual mode volume but, as the authors mention in the same paragraph, the effective mode volume. Reply: We completely agree that it is important to give a clear definition of the mode volume - indeed, we like to refer to it as an effective mode volume, as it differs from the physical volume of the cavity. The difference is explained in our reference [13], but we included the definition of the effective mode volume in this manuscript for better reference on the 1st page, 2nd paragraph: "The mode function u(x) (normalized to one at the field maximum) is tightly focused in the center of the cavity, leading to a small effective mode volume V =\int dx |u(x)|^2 while the physical size of the cavity is large." To obtain the projected coupling strength g_0, we derive the mode volume in paraxial approximation which is valid for our cavity waist of 4um. We added the corresponding expression for the effective mode volume, page 4 2nd paragraph. "The experimentally obtained value for g_0 is lower than expected for a two-level atom from the cavity geometry g_0 =3λ2cγ/(4πV) = 2π × 12.1 MHz where V = π/4 w_0^2 l_cav is the effective mode volume in paraxial approximation." ---------------------------------------------------------------------- Referee B comment: 7) 1st page, 2nd paragraph: "A cavity is concentric when the separation of the two mirrors l_cav..." Strictly speaking, the relevant length is not just the mirror separation but also takes into account penetration of the cavity field into the mirror coatings; see, e.g., the discussion in Ref. 20. Reply: We agree. We corrected the statement and refer to "cavity length" instead of mirror separation. ---------------------------------------------------------------------- Referee B comment: 8) 1st page, 4th paragraph: A concentric cavity is described as being "marginally stable." It might be also worth mentioning that the spot size on the mirrors would have to be infinite. Reply: We included the spot size in the manuscript. "Thus, a concentric cavity with l_cav = 2R_C is a limiting case at which the cavity is only marginally stable; the mode diameter at the position of the mirrors becomes infinite and the cavity highly susceptible to misalignment." ---------------------------------------------------------------------- Referee B comment: 9) 2nd page, 2nd paragraph: I'm confused about why the reflection measurement is disregarded in determining the cavity linewidth. I understand that using the transmission value is a worst case, but shouldn't the error bar at least reflect the two measurements? Reply: The results of the two linewidth measurements are very close. We use the linewidth obtained from the transmission spectrum to further deduce cavity parameters, because the reflection spectrum is sensitive to the spatial filtering by the single mode fiber connected to the detector. ---------------------------------------------------------------------- Referee B comment: 10) 3rd page, 1st paragraph: "From the low frequency of loading events we infer that the probability of simultaneously loading two atoms in the center region of the trap to be negligible." This statement ought to be made more quantitative. What is the loading frequency? What is the estimated two-atom rate? Also, why is the inferred survival probability in Fig. 3b 75% at time t=0? What role is the motion (temperature) of the atoms understood to play in these measurements? What about the registration of the FORT with respect to the cavity? Reply: We included a typical value for the loading rate. "The average duration between loadings events is typically 3-4 seconds. Thus, the low loading rate makes the simultaneous loading of two atoms in the center region of the cavity negligible." The reason for the 75% survival probability at t=0 is the use of a very high threshold value. We use a high threshold value to make sure that we only consider atoms strongly coupled to the cavity. However, in the lifetime measurement this leads to a reduced recapture probability as minor fluctuations in the number of detected photons can appear as atom loss. However, the inferred trap lifetime is not affected by the high threshold. ---------------------------------------------------------------------- Referee B comment: 11) 4th page, 2nd paragraph: "for a clean two-level atom" - perhaps the authors could be more precise about what they mean by "clean" Reply: We omit the word 'clean' as "two-level atom" describes exactly what we mean (i.e., a closed two-level system). ---------------------------------------------------------------------- Referee B comment: 12) There seems to be some mix-up among citations 27-29. Presumably 27 and 28 belong to the Rydberg atom example and 27 to ions. Reply: We agree - the citations are corrected. With this, we hope to have addressed the concerns of the referees, and believe that the limitations we faced in the particular experiment are not inherent of this approach, which may permit exploring cavity QED experiments where it is hard with the conventional approaches with very high finesse mirrors. We therefore would ask you for reconsidering this manuscript, and are looking forward for your reply. With Best Regards on behalf of all authors,