Dear Editor,

first of all, we like to thank the referees for their positive assessment and
their constructive suggestions on how to improve on our manuscript.

Let us first address the reservations of referee B regarding a
publication in PRL.

The main question was perhaps the novelty of the work at hand in comparison to
the work presented in reference [26]. The main aim of this work was to
demonstrate that such a photon source can indeed generate Fourier-limited
photons which is necessary to implement a transfer process from photons
into atoms. In this paper, we provide clear experimental evidence that this is
the case with independent measurements of the spectral properties of the
photons emerging from the process, as well as the correlation time for the
photon pairs. We are not aware of such a direct demonstration so far.
To make this clearer as suggested by the referees, we mentioned
that now explicitly in the modified title, a comment in the abstract and a
when we compare the measured bandwidth with the inverse correlation time.
This, together with the significant advancement on the purity of
the photon pair source with an increase of the degree of violation of a
Cauchy-Schwartz inequality by three orders of magnitude, and a direct
experimental demonstration of the thermal character of the emitted light makes
us believe that this is a significant advance in the field compatible with the
guidelines for publication in Physical Review Letters.

As for the other comments raised, we like to address them individually:

1. Referee A, remark on the smaller bandwidth of unheralded idler photons in
comparison with heralded photons.
   We expanded the discussion on this topic via an attempt to separate the
   collectively enhanced part of the light from the "incoherent" two-step
   decay by subtracting a loss-corrected fraction of the heralded spectrum
   from the unheralded spectrum, and showed this result in a new partial
   figure 3c. The uncertainties in the losses seemed low enough to allow for
   this procedure to be meaningful. What we find is a residual width that is
   narrower than both the heralded and unheralded spectrum, but does not lead
   yet to the perhaps expected natural line width on that transition. A very
   probable explanation of this could be self-absorption on this line in the
   atomic cloud, which would widen the spectrum significantly. In an attempt
   to model that, we found qualitative agreement, but had to make too many
   assumptions on the exact trap geometry that would make such a discussion
   not very meaningful. We thus limited ourselves with a presentation of this
   subtraction process.

2. Referee A, a discussion on the quantitative connection between slope of
bandwidth and the measured optical density.
   We tried to clarify how we present the model prediction in the paper by
   making explicit reference to the atom number N rather than an optical
   density, since we measure this density on the D2 line, but for the idler
   bandwidth we would need to use the OD on the D1 transition. We tried to
   make this clearer in the manuscript whenever we refer to measured optical
   densities. Again, a quantitatively meaningful discussion about the
   proportionality factor between OD_m and bandwidth would require a more
   quantitative knowledge of the atomic cloud geometry during the emission
   process  which we don't have. However, we tried to make the dependency on a
   geometric factor and the atom number N more clear in the discussion.

3. Referee B, reference to polarization correlations
   While we agree that observing polarization correlations is an important
   aspect of such a photon pair source, we felt that a focus on spectral
   aspects and temporal correlations would be more appropriate for this
   manuscript. We do see the polarization correlations, and are
   quantitatively characterizing the polarization-correlated entangled states
   for various parameters, but feel that this should be part of a more
   detailed report.

4. Referee B, improvement of discussion of correlation time and bandwidth.
   This comment is very similar to comment 1 of referee A; we tried to address
   this both by a better highlighting of the transform-limited photons
   emerging from the source, as well as an expanded discussion on the
   collective enhancement/two-step decay process, as highlighted under comment
   1 above.

5. Referee B, discussion of polarization aspects
   We now explicitly state that the polarization combination we chose for the
   measurements here were guided in an attempt of maximizing the product of
   the Clebsch-Gordan coefficients, and thereby the effective nonlinearity.

6. Referee B, Reference to conditioned homodyne analysis by the Innsbruck group
   We now include this reference, together with the reference to the recent
   Calgary experiments.

7. Referee B, spelling mistake in the introduction
   Thanks for pointing this out, this is fixed now.

8. We also updated references 9 and 23 which have been published by now, and
   corrected the date of reference 39 (Steck et al.)

With this, especially the inclusion of a corrected spectrum for photons
arising from a two-step decay, we hope to have addressed the concerns of the
referees, and hope that the amended manuscript can be considered for
publication in Physical Review Letters.

With Best Regards on behalf of all authors,

Christian Kurtsiefer