Reply to referee

We are glad the Referee appreciate our work, and we appreciate the constructive comments. In the following we address the points raised by the Referee.


- Additional details should be given for the implementation. I did not
find for instance the temperature of the atoms or, more importantly,
the diameter of the beams used for the FWM. Also, the authors should
precise if the residual magnetic fields are canceled or not.

    We have extended the description of the experimental setup, including a
    detailed description of the magnetic field of the magneto-optical trap and
    the pump and collection modes. We have no explicit measurement of the
    temperature of the atoms in this set of experiments, but we now mention a
    conservative estimate of a temperature at or below 87Rb Doppler temperature of 146 uK from experiments on with similar configuration.


- In section IV, only coincidence histograms are given. It would be
interesting to provide the correlation function of the heralded single
photons.

    We modified figure 2, it now includes a secondary axis showing the rescaling of the coincidence histogram to show the correlation function. We have also modified the main text to highlight the derivation of the correlation function.


- Concerning the heralding efficiency, i.e., a major figure of merit,
it seems to saturate in the best case around 20% when the OD is
increased. The authors should detail this saturation and discuss the
fundamental limitation in this number. Also, as various experiments
are developing with very large OD, above 500, what will be the
behavior for such systems?

    The referee correctly highlights the relevance of this parameter and its
    correlation with the optical density. As indicated in the main text we do
    not have a satisfactory model to explain the saturation behavior. We
    mention this in two places explicitly: (1) Sec.V: "We do not have a
    complete explanation for the non-linear increase of the pair rate with the
    optical density", and (2) at the end of the conclusion: "The dynamics of
    the combined effect of collective.... rate, heralding efficiency, or
    bandwidth.").
    Instead, we used an heuristic approach to describe the behavior and
    speculated on the extension of this behavior to higher optical density
    (see after Eq. 11 in Section V: "This heuristic expression suggests that
    (a) a higher optical density of the atomic cloud leads to an increase of
    the pair rate at the expense of a larger photon bandwidth, and (b) for
    large enough OD there is no improvement of heralding efficiency.")
 
    To further detail the saturation behavior, in the last paragraph of
    section V we added a detailed description of the heuristic parameters that describe the system.


- In figure 8, the model is in good agreement for large powers but it
seems not adequate anymore for small values. Can the authors comment
on this discrepancy? Also, still on figure 8, the powers should be
labeled by the color of the corresponding curve.

    We agree with the referee on the importance of highlighting the limits of
    our theoretical description. To reinforce what already included in the
    main text in Section VI, paragraph 4 ("The intuition of a higher heralding
    efficiency at low pump powers due to a smaller contribution... or their
    respective overlap."), we have added the following to the caption of
    fig.8: 
    "The model fails do describe the experimental behavior for low pump
    powers. As discussed in the main text, in this region the power broadening
    is comparable with the pump lasers linewidths, a regime beside the model
    assumptions".
    We changed the color of the curve labels accordingly in figures 6 and 8.


List of major changes to the manuscript:

1 - Sec.III, 1st paragraph. from: "trapped and cooled with a Magneto-Optical trap (MOT) formed by six laser beams"
to: "trapped and cooled with a Magneto-Optical trap (MOT) formed by a pair of circular coils connected in an anti-Helmholtz configuration generating a magnetic field gradient of 24.8 G/cm in the radial direction and 49.6 G/cm in the axial direction and six laser beams"

2 - Sec.III, 1st paragraph. Added: "No compensation was used for any residual magnetic field."

3 - Sec.III, 1st paragraph. Added: "(estimated from similar experimental setups [31] to be equal to or smaller than the Doppler temperature of 87Rb of 146 uK)"

4 - Sec.III, 2nd paragraph. Added: "The pump and collection modes are focused  in the cloud. Both pumps have a beam waist of ~0.45 mm, while the collection modes are ~0.4 mm and ~0.5 mm for signal and idler respectively."

5 - Caption fig. 2. from "Histogram of coincidence events G (2)(Dt) as a function [...]"
to: "Histogram of coincidence events G(2)(Dt) (left vertical axis) and the normalized second order correlation g(2)(Dt) (right vertical axis) as a function [...]"

6 - Sec. IV, after eq. (6). Added: "This can be used to estimate of the second order cross-correlation function g(2) from Eq (6):" followed by new eq. (7)

7 - Sec. VI, last paragraph. replaced "Such considerations are discussed in section IX."
with "These considerations are particularly relevant considering the recent development of cold atomic systems with optical densities in excess of 500 [35].  By fitting Eq. (12) to the experimental data, we obtain~$\eta_{0s} = 0.190 \pm 0.001$ and~OD$_{0s} =9.7 \pm 0.1$ for the signal
  and~$\eta_{0i} = 0.150 \pm 0.001$ and~OD$_{0i} =11.3 \pm 0.2$ for the idler."

8 - Caption fig. 8. Added: "The model fails do describe the experimental behavior for low pump powers. As discussed in the main text, in this the power broadening is comparable with the pump lasers linewidths, a regime beside the model assumptions."

9 - Added new reference [31] and [35]

10 - moved the explanation of Gamma 1 and 2 to the beginning of the appendix
to avoid a formatting problem with the long expressions.

11 - Changed the color of curve labels in figures 6 and 8.

12 - Added second vertical axis to Figure 2

13 - Removed the symbol "*" between units  in fig 14-16.