The discussion by Stern and Sheberstov introduces to readers some basic principles by which to simulate NMR spectra of simplistic molecules during free evolution of their nuclear magnetization in a zero or ultralow magnetic field (ZULF). Specifically, the compounds must contain two sets of magnetically equivalent spins-1/2, each with a different gyromagnetic ratio, for example a carbon-13 and hydrogen-1.

Technically speaking, the work is correct, but readers should note that the content is not so original, and in my opinion a narrow viewpoint on the topic of near-zero-field NMR. Butler et al. wrote a key paper back in 2013 describing the theory of zero-field NMR in not only AX_n systems as presented in this work, but more complex spin systems A_mX_n and AX_mB_n as well. Many of the results were derived analytically without simulations, for example, using perturbation theory.

Simulations of zero-and-ultralow-field NMR spectra in AX_n, A_mX_n and AX_mB_n are also presented in detail in several PhD theses from the early 2010s: see for example Dr Thomas Theis (2012, UC Berkeley, https://escholarship.org/uc/item/01d528kh ), Dr John Blanchard (2014, UC Berkeley, https://escholarship.org/uc/item/2mp738zn ) and Dr Tobias Sjolander (2017, UC Berkeley, https://escholarship.org/uc/item/2kj4v04n ). Readers are encouraged to consult these original sources.

I encourage the authors to revise the paper by including more original content.  For example, simulating the zero-field NMR spectra of compound that has not yet been studied experimentally, or magnetic fields bordering the zero-field condition where the perturbation theory starts to break down. Alternatively, by going beyond summarizing the main results of Butler-2013 and reviewing other works where zero-field NMR spectra are calculated. I believe this would be very useful to readers interested in simulation, not only to do justice to the works listed above. Perhaps a quick way is to use a table: compound or spin system, simulation approach (e.g. exact, perturbation theory), software, literature reference .

Additional comments:

- Authors and Readers should be aware of review article on zero-and-ultralow-field NMR, plus applications, published by Jiang M, Peng X et al. in 2021. This deserves to be mentioned: https://doi.org/10.1016/j.fmre.2020.12.007

- Some of the equations can provide valuable physical insight into how a ZULF NMR experiment works, but it is not explained. Let us take Equation 57 as an example. The total magnetization operator M_z = g_I I_z + g_S S_z is applied to an eigenstate of the zero field, that is |F, m_F> in the case of AX_n. The result is immediately quoted in terms of Clebsch-Gordan coefficients.  There is a slightly different way to write the result where the operator is given as Mz = g_I (I_z + S_z) + (g_S - g_I) S_z. Here the first term on the right-hand side of the (=) sign leads to the eigenvalue m_F g_I, while the second term leads to a sum over other operators, and overall proportional to (g_S - g_I) times the C-G coefficient. The authors may want to mention that this second term leads to ZULF signals where the amplitude of peaks scale with (g_S - g_I).

General comment:

I really enjoyed reading this paper. It reminded me of my time as a PhD student, when I tried to understand the density matrix formalism and program the transverse magnetization response to some odd excitation using assembler code. In those days long gone I found the then recent paper by P.D. Buckley, K.W. Jolley, D.N. Pinder “Application of density matrix theory to NMR line-shape calculations”, PNMRS 10 (1975) 1-26 most helpful as it gave hands-on examples which I could adapt to my own case of interest. Compared to that old paper, the manuscript at stake is written even more in a tutorial style, working out the details of the A₃X system as an example. In my view this is a valuable guide to beginning PhD students interested in ZULF NMR. Clearly this manuscript is not a review, nor does it cover all common cases encountered with ZULF NMR, but I find it to be quite useful as a starting point for one’s own simulations. I recommend it to be published following revision. In particular, the corrections and comments voiced in the discussion so far should be considered and implemented so long as the length of the manuscript can largely be maintained.

Detailed additional comments:

Line 20 ff: “Increasing magnetic field strength boosts the sensitivity thanks to higher Boltzmann nuclear polarization and higher Larmor frequency”. Although stated many times in the literature, this only applies to high-resolution spectroscopy at constant linewidth in frequency units. It is the peak amplitude and not the peak integral in the spectrum that defines the sensitivity. (This raises the question of the “homogeneity” of the zero-field and its impact on spectral resolution and sensitivity at a given polarization.) Perhaps one can write “increasing the field strength while maintaining the linewidth …”.

Line 133: Figure 1B

Line 170: What defines the directions of the axes at zero field?

Lines 224 and 226: Are “density operator” and “density matrix” used synonymously? That is confusing to the beginner.

Equation (19) and throughout the entire manuscript: The format of constants, variables, functions is inconsistent. This poses extra barriers for a student struggling to understand the math. The formatting rules apply independently to the quantity under consideration and its superscripts and subscripts. For example, the subscript “eq” in (19) is not a variable and should not be written italic.

Lines 335: What is a “Fourier transform function”? The Fourier transformation is an operation, the result of which is a transform. Both, input and output of the Fourier transformation are functions. See also line 345.

Lines 399, 416 “… multiply the frequency domain signal ….” Should probably read “… multiply the abscissa of the frequency-domain signal …”.

Line 435, “Hermitian”: Charles Hermite was a French mathematician. Consequently, the attribute referring to his name is correctly spelled “Hermitean”. Admittedly, contrary to the older literature, one often finds “Hermitian” in the modern literature. You may want to pay tribute to the correct spelling of his name in the manuscript.

Caption to Fig. 5 and elsewhere in the text: “with a zero-filling of 65’536 points” should be replaced by “with zero filling to 65,536 points”, because you did not fill in 65,536 zeroes.

Overall the manuscript is well written, with just a few language issues , which I am sure, the Copernicus editors will pick up. 

This is a well written and clearly presented tutorial paper. I think that a pedagogical paper supported by a well-documented and accessible simulation code to bridge the gap between a basic NMR understanding using the vector picture and advanced product operator simulation packages like SPINACH and Spin Dynamica will be of benefit to the NMR community. This paper goes a long way to providing this link; however, I think that focus on ZULF NMR simulations, particularly in the earlier sections of the paper, may be confusing for many non-experts and may therefore limit the audience for this very nice contribution. The final section of the paper is quite theoretically challenging for the non-expert and will likely only be of interest to those in the ZULF community. Overall I think that this paper is publishable with minor revisions; however, the addition of some clearer links to high-field NMR simulations (see specific suggestions below) along with a slightly modified version of the MATLAB code to allow the user to perform a more familiar HF NMR simulation to compare with the ZULF simulations would make this a much more broadly useful contribution.

Specific Comments:

1. There are a few concepts that are mentioned by not defined clearly and so may confuse a less knowledgeable reader
   1. Hilbert space (p6 line 154) is mentioned but not explicitly defined
   2. Bra-ket notion is used in eq. 17 and 18 without being explained/introduced
      1. On a related note it would be helpful to define the alpha and beta kets with the matrix notation when they are first introduced. I don’t think it is made quite clear the relationship between these states and the columns/rows of the matrices.
   3. P8 line 201 is the first explicit mention of eigenstates. I think it would be useful when introducing the Zeeman Hamiltonian (eq 5) to explicitly define the alpha and beta kets as the eigenstates and to show the relationship between these states and the matrix representation. This would also be an opportunity to introduced the time-independent Schrodinger equation for 1 spin before the introduction of the Liouville von-Neuman equation for the density matrix (see point 3 below).
2. Eq. 11 – the use of I_x, I_y and I_z to denote the sum of the anguluar momentum operators over a number of spins is confusing as this same notation is used in eqs. 3 and 13 to denote just the 2x2 angular momentum matrices. I suggest using a different letter such as L to denote a sum over multiple spins. I had a similar issue with the definitions in eqs 48-51, where I was unsure of the definitions of various terms and found the use of bold and italics unclear/inconsistent.
3. Eq. 23 for the equilibrium matrix includes implicitly within it a two-spin-order term that emerges from the product. It is common in NMR textbooks, when analysing pulse sequences using product operators to express the equilibrium starting state as just the sum of the Iz operators for the various spins of interest and to omit any higher spin order terms. I think it would be useful to explicitly show the expansion and explain why, and under what circumstances, the single spin operators are a reasonable approximation of the equilibrium density matrix at thermal equilibrium.
4. I think it would be useful prior to Eq. 26 to explicitly give the Liouville-von Newumann equation to which it is a solution.
5. On p14 line 350 T2 is defined as the “coherence time constant”, which is a true definition but will be unfamiliar to most readers who will better know this as the spin-spin relaxation or transverse relaxation time constant.
6. In section 2.7, the focus is put on the acquisition parameters for ZULF spectra without any discussion of how these relate to the acquisition parameters for standard HF spectra, where the spectrum is acquired in the rotating frame and Larmor frequencies are defined as chemical shift offsets relative to a reference frequency. As the ppm scale and HF NMR is the most natural reference point for most readers, I think it is important to describe the two regimes and the relationship between them. Indeed, that is what I was expecting in the “comparison with high-field NMR” section.
7. Due to the focus on ZULF NMR the non-standard case of static field pulses is introduced but the standard representation of ideal RF pulses in the high field regime using rotation operators is not described. The first mention of a rotation operator is on line 599 (p26). This is potentially confusing as the role of rotation operators in NMR simulations has not been described previously. I think it would be helpful in the pedagogical spirit of this paper to include a brief description of this in the theory section.
8. In the introduction to ZULF NMR the authors choose to define ULF NMR as where there is a Zeeman contribution but this is not dominant. This definition excludes NMR in the tens of uT regime, notably Earth’s field NMR. By excluding EFNMR a key step in the development of ZULF NMR is omitted, from the initial EFNMR experiments by Packard and Varian in 1954 (Phys Rev), where pre-polarisation and non-adiabatic field switching was first used to the first pulsed EFNMR work of Callaghan and LeGros (Americal Journal of Physics, 1982) and up to the work by Appelt et al (Nature Physics 2006), who was the first to introduce the idea of using a Halbach for prepolarisation at a few Tesla before detection in uT fields.

Technical corrections

I noted a couple equation references that appear to be incorrect:

37. I think on p17 line 411 it should be eq 36 while line 412 should be eq 37.
38. I think on p18 line 456 it should be eq 35.

The work from Q. Stern and K. Sheberstov is a tutorial paper addressed to PhD students approaching NMR experiments simulations at ultra-low or zero field.

Despite the manuscript is well written and over all correct, I struggle to see a real pedagogical aim in this work. In my honest opinion, I doubt that first year PhD students will be able to open their MATLAB and start to run NMR simulation thanks to this manuscript. I must agree with Reviewer 1 about the fact that the authors assume a lot more of the readers’ pre-knowledge than what they think. Through the text, apart some NMR theory that can be found in any topic specific textbook (e.g. Spin Dynamics from Prof Levitt), I could not see any real trick that would “carry by hand” the newbie from analytical formulas to computation. Moreover, especially at the beginning, jumping straight to ZULF simulation might be confusing for the non-experts. The paper could be useful for a broader audience if a “standard/high-field” NMR section was added.

The following is just a suggestion, but, why not starting with a very simple system (non-interacting spins ½ at thermal equilibrium with Zeeman interaction only, no propagation of the density matrix yet just CW style) and show step-by-step how to go from the Hamiltonian to the spectrum in MATLAB, reporting even chunks of code into the main text. Then, we can add the J-coupling and see how the spectrum changes, propagate the density matrix etc.

As a first year PhD student I would love to find in the literature something like this!

Below some minor details:

Line 27: the second “interaction” is redundant

Line 41: add a reference after “coils” about OPM detection for ZULF

Line 42: I would remove “This simple idealization of”

Line 427: “positioned” instead of “position”