Dear Dr. Belsole,

first, we would like to thank the Referees again for their constructive remarks again for further improvement of the manuscript. We amended the text and some figures, following the recommendations.

Our detailed replies to the specific points raised by Referee 1:

(1) Proper referencing to the "struggle for strong coupling": We now included two references to what we believe are the most appropriate early demonstrations, namely by Thompson/Rempe/Kimble in PRL 68, 1132 (1992), and Rempe/Thompson/Kimble/Lalezari in Optics Letters 17, 363 (1992) as our new references 3 and 4 to reflect both the atomic physics and the optical cavity aspect. We hope that this gives proper credit to these heroic experiments.

(2) Normal mode splitting in the previous figure 4 shown too early: We moved this part of the figure to figure 5 (now forming part e of it), together with the cavity transmission, and the transverse mode structures. We adjusted the figure captions of both figures 4 and 5 accordingly.

(3) "Small" difference in discussion of figure 5: We agree with the comment, and removed the word "small".

(4) Discussion of figure 5d in the text: We agree that this should be explained better. We rewrote this section as described in change (E) below.

(5) Addition of a simulated normal mode spectrum for figure 7. We added this simulation as suggested, but we restricted this to the contribution from the fundamental cavity mode. Indeed, the change is significant. We also changed the figure caption accordingly, as well as the corresponding text in the main part on page 10 of the manuscript, as summarized under (G) below.

(6) Change in calculated line width from previous version: As the error bars were quite large in the first version, we repeated the experiment with  better  mechanical stability of the cavity. The mirrors that have been used in the new measurement have a slightly different reflectivity. In this case the R_m^2 in equation (2) has changed accordingly, which changes the finesse (equation (3)) and, consequently the linewidth. The now presented measurements are all consistent.

(7)  The expression for the cooperativity formula is corrected, so is the formula in figure 9. There  are different nomenclature for the cavity cooperativity parameter; since we followed reference [2] it is now cited in the related statement to make it easier for a reader to identify the convention we use.


The changes we made to the manuscript are as follows:
(A)  Inclusion of new references 3,4 for the single atom cavity QED experiments, as suggested by comment (1).

(B) Figure 4 changed, the mode splitting is now shown as part of figure 5. We removed the corresponding part in figure caption 4 as well.

(C) Change of figure 5 to now include the simulated mode splitting. The figure caption was amended accordingly.

(D)  Removed the word "small" in the discussion of figure 5d as suggested in comment (3).

(E) We rewrote the following discussion of figure 5  (formerly "In Fig. 5b the higher order modes.... (circles around the center of the fundamental mode)."). It now reads:
"The snapshot shown in Fig. 5c is taken at the frequency that corresponds to the peak apex, where many higher order modes are present, and the fundamental mode is excited at ~50\% of its maximum as the frequency is half linewidth away from the resonance frequency of the fundamental transverse mode of the cavity. However, the higher order modes are also visible (circles around the central fundamental mode). For a detuning
above the main resonance structure, the transverse profile (see Figure~\ref{aberrations}d ) is dominated by higher order transverse modes."

(F) We changed figure 7 to now include a simulated normal mode splitting, and changed the figure caption accordingly.

(G) Inclusion of a description of the simulated normal mode splitting:
On page 10 of the previous revision, we stated "The cavity transmission spectrum corresponds to a focusing parameter of 0.36 that gives maximum cavity cooperativity value of 150 (see Fig. 9). Some residual excitation of higher order modes is visible, which we attribute to possibly non-ideal quality of the aspheric surface, as well as to mismatch in input beam waist and non-perfect beam alignment in experiment. Mode-hop-free tuning of an external cavity diode laser over this range was accomplished by synchronizing the rotation of
the grating with adjustment of the diode current, resulting in continuous tuning over more than 30 GHz." 
This is now replaced by the following:
"Mode-hop-free tuning of an external cavity diode laser over this range was accomplished by synchronizing the rotation of the grating with adjustment of the diode current, resulting in continuous tuning over more than 30 GHz. The transmission spectrum corresponds to a focusing parameter of $0.36$ that gives maximum cavity cooperativity value of 150 (see Figure 9). Figure 5b shows that the spatial intensity profile of the cavity output for the peak
transmission resembles a clean Gaussian profile. Figure 5c shows some residual excitation of
higher transverse modes, which we could be attributed to both a non-ideal quality of the aspheric surface and an imperfect alignment. Figure 5d shows the expected normal-mode splitting with a single atom in the center of the cavity, with a coupling coefficient $g_{0}(u=0.36)=163$\,MHz for this cavity geometry. Even though this model assumes no excitation of higher transverse modes,  the change in the cavity transmission due to the presence of an atom in the cavity should clearly be visible."

(H) We included the factor 1/2 in the cooperativity expression near equation (10), and in figure 9.


With this, we hope to have adequately addressed the additional points raised by referee 1 to be considered for publication in the New Journal of Physics.

With Best Regards on behalf of all authors,

Christian Kurtsiefer