Reviewer #1 (Remarks to the Author):

In this manuscript, the authors report the coupling of a few-photon
light field to a single atom in free-space making use of a "4pi" lens
configuration, as known from high-resolution imaging. The achieved
extinction of probe photon transmission due to scattering from the
single atom reaches 37%, a value comparable to state-of-the-art
waveguide systems.
The presented results are an interesting step forward in the domain of
freespace-QED. Both the experiment as well as the analysis and
comparison to theory are nicely presented and interesting, I only have a
few comments which are listed below. I think this work deserves
publication in Nat. Comm. after the authors have considered these comments.

* I think the very limited length of the manuscript stems from previous
restrictions. As I understand it in NatComm these restrictions are much
less severe. I would highly appreciate some more detail at various
places in the manuscript. Most importantly to me are the presentations
of eq. 1 and 2, which now are simply given together with references. A
short overview of the derivation and/or somewhat more detailed
discussion would be very useful to a reader not extremely familiar with
the topic.

* I find the repeatedly used phrase "weak probe beam" somewhat unclear.
What exactly is meant by weak here? That the effective coupling strength
(per time) is small compared to the excited state decay rate \Gamma?
This more quantitative definition should be particularly important for
the validity of eq. (2). In case of the coupling strength exceeding the
decay rate, the single TLS should undergo Rabi oscillations, as recenlty
observed for a Rydberg superatom in https://arxiv.org/abs/1705.04128.
The theory presented there gives the full expression for g2(t1,t2) for
any coupling strength. Eq. (2) should be valid in the "weak coupling
regime".

* The agreement between theory and experiment in Fig. 3 is not too
convincing (it is called "qualitative" in the text). The authors seem to
understand their system very well, as e.g. shown by the nice analysis of
the g2 data. Why does this (more basic) measurement of the transmission
not agree so well? A bit of discussion would be good here.

* I do not fully understand the discussion of the "single-photon
converter" at the end of the manuscript, reachable with a stronger
coupling. Do the authors mean that for stronger coupling a single photon
is reflected while all others are transmitted? Such a "single-photon
subtractor" has recently been demonstrated with a single atom couped to
a nanofiber (Nature Photonics 10, 19-22), although the physics of this
system is slightly different as it makes use of 3 atomic levels and the
polarization dependence of the light in the fiber. A somewhat more
detailed discussion how this scheme would work for the single TLS would
be nice.

* A question in connection to these nanofiber systems: with the tight
focussing of the light, to what extend is the field still transverse? Or
does spatially varying polarization already play a role here, as in the
(sub-wavelength) fiber-systems? Will that be more relevant for the
future higher NA system mentioned in the outlook?


Reviewer #2 (Remarks to the Author):

In their manuscript “Nonlinear photon-atom coupling with 4Pi
microscopy” the authors implement a technique known from
high-resolution imaging to increase the atom-light coupling. Principle
of this technique is to illuminate the atom from opposite directions
such that the driving field forms a standing wave. Then, the atom-photon
coupling is strongly increased if the atom is held in an antinode of the
driving field. The authors observe an extinction of the driving field by
about 36 %, a value that has not been reached before. Furthermore, they
demonstrate in a photon-correlation measurement that the nonlinear
interaction between atom and photon leads to a modified photon
statistics in the transmitted field.
 The experimental claims are convincing, the manuscript well written and
the representation of the experimental results clear. The results are
important for other groups and will certainly motivate them to increase
their emitter-photon coupling by using this technique. I recommend
publication in Nature Communications.

However, there are a few (minor) points the authors should address
before publication:

-       For easier comparison of Fig. 1 b/c with d/e, it would be nice if the
authors give the NA of their lenses in the figure caption. It is only
given in the Supplementary Information.

-       If space constrains allow it, it would be helpful for the general
reader to explain the terms of Eq. 1 intuitively.
-       There must be mistake in the optimal power splitting, it must read
P_{2,in}= P_{1,in}\Lambda_2/\Lambda_1

-       Can the authors comment on how the total coupling \Lambda_{total}
depends on the chosen post selection? Could they increase the measured
value by an even stricter post selection, or is the residual difference
due to atomic motion in the 1D lattice? Could the value reach the
theoretical limit if a 3D lattice would be used?

-       It does not become clear to me why larger values of \Lambda will lead
to photon bunching in the transmitted field. Could the authors explain
this a bit more?
-       Is a value of \Lambda=0.25 within experimental reach? What NA lenses
would be necessary?
-       The authors motivate their study with the prospect of deterministic
all optical quantum logic. Here, I see a weak point in the 4Pi
technique: one has to use beam splitters to separate the input mode from
the output mode. Therefore, there is always a trade of between the
fraction of input light that is sent towards the atom and the fraction
of light that can be detected. In an all-optical quantum processor, it
might be important to not lose input light as well as output light. Have
the authors thought about this problem? Have they ideas how to deal with
it? It would be great if they could include a short discussion on this
in the outlook.

-       Methods: The post selection process does not become clear only from
this paragraph, I could only understand it after reading the SI. It is
not mentioned here, that the transmission in the second interval must be
below a certain value, which signals good photon-atom coupling, in order
to take data in the first interval into account.

-       SI: on page 3, instead of referencing to Fig3, they have to reference
to Fig. 4. This mistake has been made twice.
-       SI: normalization of g^{(2)}(\tau). It seems to me that the
normalization of each interval makes the data quite noisy, since there
are only about 150 counts per interval with an error of +/-12 due to
counting statistics. I expect that the fluctuations due to movement of
the optical lattice are slow, so maybe it would make more sense to
normalize about 20 intervals together I order to reduce noise? Have the
authors considered this?


Reviewer #3 (Remarks to the Author):

The manuscript "Nonlinear photon-atom coupling with 4pi microscopy"
reports on experiments where light is transmitted through/by a single
atom held at the tight focus of the laser beam. The main findings are
that double-sided illumination decrease the transmission significantly,
and that the transmitted light show sub-Poissonian photon statistics. I
consider these important experimental demonstrations in the field, and
well supported by the data presented, indicating that the conclusions
are valid. The findings are clearly beyond statistical fluctuations and
further supported by comparison to simple theoretical predictions. In
the case of the sub-poisonian photon statistics the "unprocessed data"
is also shown in the supplementary information, which removes any doubt
that conclusions are drawn based on artifacts from the data-processing.
The topic is of high interest to the scientific community and will
definitely be of interest outside its specific field. The manuscript is w
ell
written in clear English. I enjoyed reading it. I am therefore happy to
recommend publication in Nature Communications. I have few minor comments:
1. Where the light-atom coupling efficiency \Lambda is introduced, I did
not understand its definition without looking in Ref 15. In particular
what the maximal possible amplitude referred to. As \Lambda plays a
central role in the paper it would be good to clarify this.
2. As far as I can see, it is only said in the supplementary material,
that for one-sided illumination, the postselection procedure does not
change the observed transmission. This is a crucial point for
interpretation of the main data, so I suggest that this is stated in the
main text or methods.
3. The caption of Fig. 2 is a bit misleading. I presume that red and
blue in b refers to the two detectors rather than illumination paths as
is indicated.