Compact bulk-machined electromagnets for quantum gas experiments

1) The authors present an impressively compact design with excellent heat management that will be of interest to other experimentalists.
2) The description of the manufacturing process is comprehensive and easy-to-follow.
3) The authors have made an effort to explain their design considerations, which means the design will be more easily adapted by other scientists.

1) Phrasing and grammar impedes comprehension of the manuscript
2) Explanation of design considerations is somewhat confusing in terms of heat management concerns. Equations 1 and 2 have little accompanying justification or explanation, so their accuracy and relevance are hard to understand.
3) Discussions of the magnetic field itself are lacking in regards to the actual deployment of a pair of coils. For instance, the manuscript lacks a discussion of the axial field homogeneity for a pair of coils in the configuration that they will use in their experiment.

In this manuscript, the authors present a new design for a pair of electromagnets for quantum gas studies. Their design minimizes the amount of space occupied by the magnets and maximizes the amount of heat removed by water cooling. Since the maximum magnetic field that can be achieved is proportional to the square root of the dissipated power, this means that they can achieve large magnetic fields ($\sim$1000 G) at a distance of 43 mm from the centres of their coils. The authors describe the manufacture of their electromagnet and the water cooling considerations that they undertook to optimize their design. In their results, they show the magnetic field (as G/A) for a single coil, and they present measurements of the heat management of the electromagnet. These measurements are comparable to other modern designs of electromagnets for ultracold atomic physics with the added advantage of a compact design.

In general, I think that this manuscript will be of the most interest to experimentalists who are either setting up a new experiment or investigating improvements to an existing apparatus. In particular, the authors' descriptions of how the electromagnet was manufactured and assembled are detailed enough to allow someone else to build a similar device and to avoid pitfalls that the authors clearly encountered. The authors' measurements make a clear case that their design is effective at managing heat for reasonable pressure heads. Although their design was necessitated by the limited space into which they could put their magnets, a similar design methodology could be used for ultracold atom systems that use small, rectangular 'science cells' for their experiments. In such a situation, the improved heat management would be an asset, although other systems (such as in Ref 7) might be a better option. One measurement which is not reported but would be of interest to other experimentalists is the frequency response of the coils. As the authors do not report this, it seems that they are unconcerned with the dynamical response, but others who wish to implement this design may want an estimate of how fast the field can be changed.

I have two main concerns with the manuscript. The first is that their explanation of their design considerations is unclear. Equation 1, which is used to calculate the top-to-bottom temperature gradient on the coil, is presented without a reference or derivation. Equation 2 is similarly presented with little explanation, and I find its purpose vague and unclear. On a similar note, the authors include a photograph of the top surface of their coil to showcase its surface quality. It is, however, unclear why this is important. Would a roughly cut top surface significantly degrade the heat management?

My second concern is that the discussion of the magnetic field generated by the coil is limited. For instance, the authors state that a pair of coils is to be used for generating homogeneous fields at the location of their sample of atoms; however, the arrangement of this pair of coils is not stated. Is it supposed to be in a Helmholtz configuration? What is the homogeneity of the axial field at the location of the atoms? How fast can the magnetic field be changed: i.e., what is the dynamical response of the coils? How much does the magnetic field change with changes in the temperature of the coil?

As a last comment: this manuscript would benefit from further proofreading, as some of the phrasing and grammar negatively impacts the clarity of the article.

Overall, I think that this manuscript is of sufficient interest to those in the field of ultracold atomic physics, and to those outside, to be published in SciPost Physics.

1) In the second and third paragraphs of the introduction the authors describe so-called parallel coolant flow as being problematic for hollow wire coils but as being a benefit for Bitter-type magnets. The authors need to clarify what is meant by 'parallel coolant flow' and revisit these descriptions to ensure that they are consistent.
2) In the second paragraph of the 'Concept and design' section, the authors state that they will use a pair of their coils to generate the large magnetic fields used for accessing the lithium Feshbach resonance. It would be useful to the reader to know what the configuration of this pair of coils will be.
3) Figure 1 could be improved by the use of additional colours to indicate where epoxy is used and where the water flows in the coils. On my first reading this was very confusing.
4) The last three paragraphs of the 'Concept and design' section need to be revisited. Equation 1 has no reference or derivation despite its clear importance to the manuscript. It may also be incorrect: my quick derivation indicates that their expression for the temperature difference is missing a factor of 3. Similarly, the importance of equation 2 is not clearly explained. In general, I found this section to be hard to follow.
5) The importance of the surface quality, as shown in Figure 2b, needs to be explained. Additionally, Figure 2b does not necessarily show well the surface finish of the coils.
6) In section 4.1, the authors should replace the word 'vertical' with 'axial' as being a more generally applicable description.
7) Also in section 4.1, the authors should discuss, measure, or calculate the field distribution resulting from a pair of coils in their desired configuration. In particular, the axial field homogeneity would be of interest to readers.
8) A discussion of the dynamical response of the coils should be included, as well as the response of the magnetic field to temperature changes.
9) In the conclusion, the authors say that making the coil took 40 hours on a wire-erosion machine, and in the abstract they say that their coil is a 'cost-effective' solution. Other institutions without a wire-erosion machine and trained machinist would be forced to have this made off-site, and it is not clear if making the part in this way would still be cost-effective. An estimate of the cost (I imagine their machinist would be able to provide this) would be very useful to readers.
10) The last sentence of the conclusion contains the confusing phrase '..a feature which revealed crucial in transport measurements...' and should be fixed.
11) References 3 and 8 have the full URL for the DOI in the DOI field, as opposed to just the DOI. This means the link does not function. Reference 21 does not have a DOI, and one should be included if it exists.

1-An interesting solution to a very common experimental problem in cold atom experiments.
2-Well written and laid out manuscript.

1-Some technical information is not provided with sufficient detail (e.g. the composition of some materials, the derivation of Equation 1 - see requested changes).
2-Perhaps some of the figures could be more informative.
3. There could be more of a discussion regarding how the coils will perform in their designated task - producing a stable and uniform magnetic field across a degenerate quantum gas of Li atoms. It is unclear what expected variation in scattering length - spatially and temporally - these coils are likely to produce under experimental conditions.

The paper by Roux et. al. reports on the design of an electromagnet machined from bulk copper for use in quantum gas experiments. Their design is capable of producing fields suitable for the 832G Feshbach Resonance in $^6$Li, without excessive heating or demanding current requirements and while fitting within the available space constraints of a small re-entrant window.

Overall, this is a well written paper reporting on an interesting approach to the technical design of a ubiquitous component (magnetic coils) in cold atoms experiments. While their approach may not be entirely novel and does not yield any new physics, it is a different technical solution that appears to fulfill the design constraints well. Given the number of magnetic field coils used in cold atoms (and other) experiments worldwide, the paper will thus be of interest to the experimental cold atoms community. I therefore think that it is worthy of publication, subject to the following suggested changes.

1- Fig. 1 should have a few more parts labeled for clarity (either in the caption or on the image), and the materials that components such as the black insulators, green dots, plastic caps are made from should be mentioned. Indicating the physical meaning of some of the variables (w, H etc), on this Fig might also be helpful.
2- It would be helpful to provide some more information of where equation (1) comes from. At the minimum a reference is needed, although some more information in the text might be helpful, as parts of it are based on assumptions (e.g. why does the distance which $\Delta T$ is over not come into the equation?).
3 - Similarly, I feel that the general reader would benefit if a comment and/or reference could be provided to expand on the estimate of a Nusselt number of ~70.
4 - What is the expected field variation across a typical atomic cloud size that will be achieved with these coils, and what variation in scattering length will this lead to at some typical FB resonance B field values? From some very rough estimates looking at Fig. 3, it seems like in the z direction it might be non-negligible. Maybe of order 1G across a 100um cloud size? A couple of sentences discussing this would be nice. What about temporal variations due to thermal heating of the coils?
5 - Why were the screws made of titanium? Was it for the low magnetisation? Is this use of a dissimilar metal likely to cause corrosion problems due to the relatively large electrode potential difference between Ti and Cu?
6 - What is the cooling water temperature, and is it actively maintained at a precise temperature?
7 - While the English throughout the paper is generally at a high standard, there are a couple of minor errors I noticed including:
- First sentence of last paragraph of section 2: missing "an" before "extreme".
- Last sentence section 3.1: should be "...a fiber-glass...".
- First sentence of section 3.2: "in" should be "of".
- Last paragraph of section 4.2: "Last" should be "Finally", plus the last sentence is extremely long and should ideally be broken up for readability.