J-coupling NMR spectroscopy with nitrogen vacancy centers at high fields

PHYSICAL REVIEW RESEARCH 6, 043017 (2024) J-coupling NMR spectroscopy with nitrogen vacancy centers at high fields Pol Alsina-Bolívar ,1,2,*A. Biteri-Uribarren,1,2,*C. Munuera-Javaloy,1,2and J. Casanova1,2,3,† 1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 2EHU Quantum Center, University of the Basque Country UPV/EHU, Leioa, Spain 3IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain (Received 5 December 2023; accepted 16 September 2024; published 4 October 2024) A diamond-based sensor utilizing nitrogen-vacancy (NV) center ensembles permits the analysis of micronsized samples through nuclear magnetic resonance (NMR) techniques at room temperature. Current efforts are directed towards extending the operating range of NV centers into high magnetic fields, driven by the potential for larger nuclear spin polarization of the target sample and the presence of enhanced chemical shifts. Especially interesting is the access to J-couplings as they carry information of chemical connectivity inside molecules. In this work, we present a protocol to access J-couplings in both homonuclear and heteronuclear cases with NV centers at high magnetic fields. Our protocol leads to a clear spectrum exclusively containing J-coupling features with high resolution. This resolution is limited primarily by the decoherence of the target sample, which is mitigated by the noise-filtering capacities of our method. DOI: 10.1103/PhysRevResearch.6.043017 I. INTRODUCTION Nuclear magnetic resonance (NMR) spectroscopy has established itself as a fundamental tool for exploring materials since its inception in the 1950s [1–3]. This technique is essential for analyzing molecular structure and function. It finds extensive applications in the examination of macroscopic biological tissues and functional studies in medical imaging [4–6]. However, the inherent low sensitivity of standard NMR spectroscopy restricts its application to bulky samples, typically requiring microliter volumes with conventional macroscopic coils or nanoliter volumes with microcoils [7]. In this scenario, newly developed solid-state quantum sensors open up new avenues for spectroscopic measurements with unprecedented sensitivity and spatial resolution, enabling studies at the cellular [8] and single-molecular level [9–11]. In particular, quantum sensors consisting on NV ensembles have demonstrated their utility in conducting NMR spectroscopy on microscopic samples at room temperature, reaching sample volumes of picoliters [7,12–14]. The pursuit of extending the operating range of NV-based sensors into the high magnetic field regime is driven by the advantages offered by elevated fields. Namely, a larger thermal nuclear spin polarization and enhanced energy shifts that simplify data extraction. The ability to access J-couplings—also known as scalar or indirect couplings—is of particular interest in the actual *These authors contributed equally to this work. †Contact author: jcasanov[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. spectroscopic framework. J-couplings are indirect intramolecular dipole-dipole interactions mediated through chemical bonds. They are responsible for complex splittings in NMR spectral lines, and the examination of J-coupling patterns allows scientists to obtain information regarding nuclear connectivity within a molecule. Hence, accessing J-couplings at the microscale regime would open new avenues in masslimited scenarios for applications in, e.g., combinatorial chemistry, as well as in cell and membrane biology [15]. Nevertheless, J-couplings are not trivial to detect since (i) they are independent of the externally applied field. Then, at high magnetic fields other field-dependent terms dominate the dynamics, and (ii) long coherence times are required to resolve the spectral peaks corresponding to J-couplings owing to their weak value (typically, on the order of a few Hertz). In this work, we present J-coupling induced nuclear signal with extended coherence time (J-INSECT). This protocol enables the detection of homonuclear and heteronuclear Jcouplings with NV ensembles (thus, accessing microscale samples) in the regime of high magnetic fields. J-INSECT integrates heterodyne techniques [16–19] and exerts noise filtering over the sample effectively enhancing its coherence. This results in narrower peaks in the spectra, thereby increasing frequency resolution and thus enabling to resolve quantities on the order of a few hertz (such as J-couplings). Additionally, our method provides the capability to target specific types of couplings. In complex molecules, where numerous peaks may arise due to extensive couplings between nuclei, this allows for the selective targeting of only the couplings of interest, simplifying the spectra. We investigate the performance of J-INSECT in regimes including both a reduced and a large number of measurements, and analyze the impact of distinct noise sources (including the presence of always-on J-couplings) demonstrating the feasibility of the method in common experimental conditions. 2643-1564/2024/6(4)/043017(9) 043017-1 Published by the American Physical Society POL ALSINA-BOLÍVAR et al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024) II. RESULTS AND DISCUSSION A. Molecular target Hamiltonian To exemplify the method we consider an isotropic liquid sample [20] containing a target molecular ensemble with two types of atoms, i.e., each molecule comprises NHhydrogens (H from now on) and a number NAof nuclei of the “A” species. The resulting Hamiltonian is H/¯h=ωH NH  i=1 Sz i+ NH  i=1 δH iSz i+ NH  i<j JH i,j Si· Sj +ωA NA  i Iz i+ NA  i=1 δA iIz i+ NA  i<j JA i,j Ii· Ij + NH  i NA  j JH−A i,j Si· Ij + NH  i H i(t)Sx icos (ω1t)+ NA  i A i(t)Ix icos (ω2t), (1) where we denote the spin operators of H nuclei as Sx,y,z k, and those of the Aspecies as Ix,y,z k. On the first line we have Larmor terms of the NHH (with ωH=|γH|Bext), their chemical shifts, and the homonuclear J-couplings of magnitude JH i,j.The terms on the second line are the equivalent interactions for the Aspecies in the target molecule. The third line comprises heteronuclear J-couplings between H and Anuclei (JH−A i,j), while the last one contains the RF controls over H and A nuclear spins. Due to significant differences in gyromagnetic ratios among distinct spin species, crosstalk effects are safely neglected using the rotating-wave approximation (RWA), especially in high-field scenarios. By setting the RF fields such that ω1≡ωHand ω2≡ωA, in a rotating frame with respect to (w.r.t) Larmor terms, Eq. (1) simplifies to H/¯h=Hshift + NH  i<j neq JH i,jSz iSz j+ NA  i<j JA i,jIz iIz j + NH  i NA  j JH−A i,jSz iIz j+Hc,(2) where Hshift =NH i=1δH iSz i+NA i=1δA iIz iand Hc= NH iH iSx i+NA iA iIx i. For the sake of simplicity in the presentation of the method, Eq. (2) assumes the elimination of the transversal spin operators (Ix,y i,Ix,y j,Sx,y i, and Sx,y j). This applies in a scenario such that |ωH−ωi|JH−A i,j, |δH i−δH j| JH i,j, and |δA i−δA j|JA i,j(note the validity of these conditions is stronger at high magnetic fields). In addition, the label “neq” in the first summation indicates that the homonuclear J-couplings among magnetically equivalent nuclei (i.e., those with identical chemical shifts) are not included in Eq. (2) since they do not have an impact in the dynamics as confirmed via numerical simulations. As a summary, Eq. (2) features only ZZ-type interactions and serves as a model to explain the basics of our protocol. We remark that our numerical simulations consider deviations w.r.t. Eq. (2) due to shortcomings in the application of prior approximations and the consideration of all J-couplings throughout the entire protocol. This encompasses instances during the application of finite-width RF pulses, where deviations in the Rabi frequency are also taken into account. B. Protocol J-INSECT is illustrated in Fig. 1. This starts with an initial π 2pulse over the H nuclei that transfer their thermal polarizations to the x-axis and continues by alternating encoding and detection stages. During free evolutions (of length τ/2) at the encoding stages, Eq. (2) simplifies to H/¯h=Hshift + NH  i<j neq JH i,jSz iSz j+ NH  i NA  j JH−A i,jSz iIz j,(3) as NA i<jJA i,jIz iIz jcommutes with every other term in Eq. (3) and does not impact the dynamics of H nuclei—the emitters in J-INSECT—(check the readout stage). πpulses are delivered in the middle of the encoding stage with a twofold objective. On the one hand, they cancel Hshift mitigating the broadening caused by spatial inhomogeneities in the external magnetic field. In other words, the refocusing πpulses lengthen the nuclear coherence time from T∗ 2to T2. This enables a longer scanning duration of the nuclear sample, which proves especially advantageous for estimating J-couplings due to their relatively weak values. On the other hand, πpulses over H (and A) enable the selective encoding of JH i,j(and JH−A i,j)in the system dynamics. This becomes evident upon calculating the effective propagator at the end of the encoding stage after delivering πpulses over H nuclei, resulting in U1(t)=exp ⎡ ⎢ ⎢ ⎣ −it NH  i<j neq JH i,jSz iSz j⎤ ⎥ ⎥ ⎦ .(4) Alternatively, in the scenario where synchronized πpulses are applied to H and A, resembling a double nucleus-nucleus resonance, one gets U2(t)=exp ⎡ ⎢ ⎢ ⎣ −it⎛ ⎜ ⎜ ⎝ NH  i<j neq JH i,jSz iSz j+ NH  i NA  j JH−A i,jSz iIz j⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ . (5) In addition, the influence of additional nuclear species (such as B) is negated within the dynamics due to the refocusing introduced by J-INSECT. That is, the unavoidable presence of extra nuclear spin isotopes in the target sample would not alter the system dynamics unless a πpulse that synchronously flips the Bspecies is introduced on purpose (see later). Then, the initial π 2pulse along the y-axis takes the thermal state of hydrogen nuclei to the x-axis (see Fig. 1) while ZZ interactions that govern the encoding stage—i.e., those in Eqs. (4)or(5)—encode J-couplings in the dynamics of Sx i and Sy iof H nuclei. Importantly, owing to the initialization of the thermal polarization along the x-axis, it can be demonstrated that following the encoding stage, Sx iexhibits 043017-2 J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024) FIG. 1. Scheme of J-INSECT with RF and MW controls. Each channel (black bold horizontal line) denotes the radiation pattern over each system constituent. The upper RF channel corresponds to the pulses over nonhydrogen nuclear species. In this respect: to detect heteronuclear J-couplings, simultaneous πpulses over H and another isotope are delivered. The central channel pictures the RF that must be applied to H nuclei, i.e., an initial (π/2)ypulse, followed by an alternation of (π)xpulses and two full Rabi oscillations along Yfor detection, with winding-unwinding mechanism for robustness as originally proposed in [17] and further developed in [21]. The resultant magnetization of hydrogen nuclei,  M∝NH i[Sx i,Sy i,Sz i] (see Appendix B), at different instants is represented with red arrows in the spheres: dashed arrows indicate the magnetization at the beginning of each protocol step (pulses or free evolutions), while the solid arrow indicates the magnetization at the end of the step. The evolution of the magnetization outside the xy-plane (induced by RF drivings) is depicted with blue arrows indicating the magnetization’s trajectory. During the detection stage, the Y¯ Ypulse induces a magnetic field (Bz) on the NV ensemble along the NV axis that precess at a controllable speed depending on the RF Rabi frequency. The amplitude of Bzat the nth detection period is proportional to the amplitude of the red solid arrow (i.e., to the magnetization at the end of the nth encoding period). The Y¯ Ypulses bring the magnetization back to the state at the start of the detection stage as they induce two 2πrotations. Bzis captured by the NV ensemble via a MW control sequence (e.g., the XY4 in the lower channel), while green pulses indicate the standard initialization and readout of the NVs. significantly greater amplitude than Sy i(approximately 105 times larger), as detailed in Appendix A1. In this manner, J-INSECT is tuned to target Sx iin the detection stages. At a generic nth detection stage, owing to a rotation of the H nuclei along the y-axis (see Fig. 1), the magnetic field Bn z induced by the sample over the NV ensemble is along the NV axis and reads [22] (see Appendix B) Bn z(t)=B0(nτ)sin(Ht),(6) where His the Rabi frequency of the Yand ¯ Ypulses (see Fig. 1) while, importantly, B0(nτ)∝2 NHSx it=nτ.(7) In conclusion, the subsequent magnetic field Bn z(t) has an amplitude proportional to Sx iof H nuclei, and oscillates with a controllable frequency H. This is, irrespective of the involved Larmor frequencies, what enables to operate at high magnetic fields. Note that, at 2 T hydrogen Larmors reach ≈(2π)×85 MHz, posing a technically challenging task for their tracking. On the contrary, Hcan be tuned, for instance, to tens of kHz [23,24], such that the oscillations of Bn z(t) are easily followed by the NV ensemble by applying, e.g., a standard XY4 sequence. Recording the resultant phase acquisition of the NV ensemble after every detection period leads to a direct reconstruction of the evolution of Sx iof H nuclei, thus of the J-couplings encoded in Sx i.Inthis regard, it is worth noting that, unlike standard heterodyne measurement schemes, J-INSECT does not impose restrictions on the separation between consecutive measurements. This means they can be placed arbitrarily in time, provided that the elapsed time between measurements is recorded, as this is crucial for extracting chemical information. For simplicity in the protocol, they are placed equidistant, separated by τ. Notice that since hydrogens are left in the same state as before the measurement (due to the rotation being a multiple of 2π, the next encoding period can be applied directly, without the need to repolarize the sample). Finally, it is important to note that the NV centers are active only during the detection periods; this implies a high spectral resolution constrained not by the sensor, but by the T2(>T∗ 2) of H nuclei. C. Numerical results We consider a picoliter sized liquid sample on top of a diamond hosting an NV ensemble at micrometer depth [12] and a static magnetic field of Bext =2 T aligned with the NV axis. To exemplify J-INSECT we address a sample consisting on fluoromethanol molecules (CH2FOH) that contains five spin-1/2 nuclei of three different species: H, carbon, and fluorine (13Cand F from now on, note oxygen has no spin) in a thermal state, see the scheme of the molecule in 043017-3 POL ALSINA-BOLÍVAR et al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024) FIG. 2. Simulated spectra for the fluoromethanol molecule. Obtained from 1000 repetitions of the protocol with 600 encoding +detection stages. Due to the dephasing time T2=0.6 s, the FWHM is approximately 1 Hz. Within each figure, there is an illustrative depiction of the molecule, highlighting the targeted J-couplings. Colored nuclei signify the species upon which πpulses are applied. Additionally, each figure includes a boxed scheme detailing the RF pulse train applied on each case. (a) Targeting only H and 13C, five peaks arise, which encode J,J1and J2. (b) Targeting the H, the 13Cand the F, ten peaks arise, encoding: J,J1,J2,JF 1,andJF 2. inset of Fig. 2. Our protocol enables accessing heteronuclear J-couplings among H and the 13Cnucleus, as well as with the F. Moreover, it gives access to the homonuclear coupling between nonequivalent H (the two H nuclei in Fig. 2labeled as Haare magnetically equivalent, whereas Hbis magnetically unequivalent with respect to the former). We choose values falling within the typical range found in the literature [2], namely: JHaHb≡J=(2π)×8Hz,JHaC≡ J1=(2π)×130 Hz, JHbC≡J2=(2π)×6 Hz, while interactions with the F nuclei are JHaF≡JF 1=(2π)×80 Hz and JHbF≡JF 2=(2π)×4 Hz, whereas JCF =(2π)×160. Furthermore, for a magnetic field Bext =2 T, the chemical shifts are δHa=(2π)×512 Hz, δHb=(2π)×236Hz, δC= (2π)×85Hz,and δF=(2π)×450 Hz. Our numerical simulations start with Hamiltonian (1) in an interaction picture w.r.t the Larmor terms ωHNH i=1Sz i+ωANA iIz i. In this manner, the only approximation made is simplifying the heteronuclear coupling term NH iNA jJH−A i,j Si· Ijin Eq. (1) to only ZZ interactions [i.e., to NH iNA jJH−A i,jSz iIz j,likeinEq.(2)]. This can be done since |ωH−ωi|∼(2π)×10 MHz JH−A i,j∼ (2π)×100 Hz, thus, the rotating wave approximation safely applies. The reader can find the full simulated Hamiltonian in Appendix B. To maintain the Hamiltonian close to Eq. (2) the condition τ>1/(|δHa−δHa|) must hold, such that the homonuclear term of H spins approaches to a ZZ interaction. The previous condition imposes a lower bound on τ, limiting the amount of measurements that can be carried out for a given experimental time. In our case, we set τ=1.2/(|δHa−δHa|)= 4.3 ms (note later we explore the J-INSECT performance with a larger number of measurements, i.e., departing from this condition). In addition, the amplitude of the RF targeting HisH=(2π)×50 kHz while, for robustness purposes, πpulses over H are implemented via a CORPSE architecture [25] which reduces the impact of detunings during pulse execution. We observe no discernible impact on the system dynamics when standard top-hat pulses are applied to 13Cand F, thus we exclusively deploy CORPSE pulses over Hs. This results in tCORPSE π=43.3µs,tC π=39.7 µs and tF π=10.6µs. Our simulations first consider 600 measurements (that is, 600 encoding and detection stages in a single experimental run) hence the duration of the J-INSECT per experimental run is Ttotal =600 (τ+tCORPSE π+2/H)≈2.66 s, where the term 2/Hcorresponds to the duration of two full RF Rabi oscillations during the readout stage. Regarding error and imperfections, our numerical simulations comprise the following: (1) always on chemical shifts and J-couplings; (2) RF cross-talk effects during RF irradiation; (3) errors on the amplitude of the RF drivings (these are simulated through an Ornstein-Uhlenbeck process [26–28] with a 1% amplitude shift and a noise correlation time of 1 ms); and (4) a nuclear dephasing time T2=0.6s[12], incorporated via an exponential decay in the field emitted by the sample. We consider two cases. When the J-INSECT targets H and 13Cnuclei (case 1) we get the spectrum in Fig. 2(a). Note that in this case, F does not actively participate since we do not apply πpulses over it, although it is included in our numerical model. Case 1 exhibits five resonances at different frequencies fiencoding heteronuclear J-couplings between H and 13Cand homonuclear J-coupling between H nuclei. The assignment of J-couplings based on the obtained fican be accomplished either by analytically solving the evolution in Eq. (5)(see Appendix A2) or by employing standard spin multiplicity techniques [29]. Conversely, simultaneous targeting of the F nuclei with an additional πpulse (case 2) results in the emergence of ten resonances, as illustrated in Fig. 2(b).Note in this second scenario the heteronuclear coupling between the F and H nuclei is also encoded in B0(nτ). The resulting estimations for the J-couplings in cases 1 and 2 are collected in Table I. In summary, J-INSECT effectively discloses J-couplings with large spectral resolution resulting in a full width at half maximum (FWHM) of, at most, 1 Hz, corresponding to the chosen value of T2=0.6 s. Regarding sensitivities, the use of elevated magnetic fields allows achieving a signal-to-noise ratio (SNR) of 30 with approximately 18 000 repetitions of the protocol. (The estimation carried out for TABLE I. J-coupling values obtained from the spectra shown in Fig. 2(a) (first row) and Fig. 2(b) (second row). Values and their uncertainty are given in Hertz. JJ1J2JF 1JF 1 Case 1 8.1±0.5 130.1±0.86.1±0.9– – Case 2 8.1±0.5 130.1±0.96±180.0±0.94±1 043017-4 J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024) case 1 with photon statistics of NV experiments, assuming a 7% of luminescence contrast. Additional details concerning case 1 and the development of case 2, which yields similar results, are available in Appendix C.) This implies an experimental time of ∼13 h. Hence, J-INSECT facilitates a high-fidelity retrieval of J-couplings within a reasonable signal-collecting time, with the potential to further reduce the experimental duration by incorporating complementary methods, such as prior hyperpolarization of the sample reported in Refs. [13,14] or advanced readout techniques such as repetitive readout [30,31]. In terms of sensitivity, previous works using quantum heterodyne measurements at moderate magnetic fields (88 mT) [12] report sensitivities in the range of ∼25–75pT ·Hz−1/2. Since our work operates at high magnetic fields, which enhance polarization by a factor of ∼30, we anticipate a sensitivity of approximately ∼1–5pT ·Hz−1/2. Additionally, our approach extends the coherence time of the sample from T∗ 2to T2, which will further reduce the sensitivity. An interesting final aside to consider is enhancing the rate of measurements, as a strategy to extra reduce experimental time. To exemplify the procedure we inspect the simpler case 1. Enhancing the measurement rate implies reducing τin the encoding stages, hence departing from the condition τ>1/(|δHa−δHa|). In this scenario the spectrum is no longer given by the usual multiplicity rules [i.e., the homonuclear terms, H-H, the interaction Hamiltonian Eq. (2)areno longer ZZ interactions], thus we estimate the values of the J-couplings by performing Bayesian inference. In particular, we choose to extend the number of measurements on each run (i.e., the number of detection stages) from 600 to 6553 and reduce the overall experimental time four times (from ∼13 h to ∼3h30 ). In this case, the inference gives J=8.2±0.5, J1=130.1±0.6,and J2=5.7±0.4 (see Appendix D). III. CONCLUSION We introduce J-INSECT, a protocol that enables the detection of homonuclear and heteronuclear J-couplings with NV ensembles at high magnetic fields. J-INSECT incorporates heterodyne techniques and exerts noise filtering, enhancing the sample’s coherence thus enabling highly precise estimations of J-couplings limited by T2>T∗ 2. We explore J-INSECT’s performance in two scenarios, considering both reduced and enhanced data harvesting. We assess the influence of different noise sources, including the persistent presence of always-on J-couplings, showcasing the method’s feasibility in experimental conditions. The complete dataset supporting the findings of this work is available upon request from the corresponding author. On the one hand, secondary data reused and/or analyzed from other studies is specified along the text. On the other hand, original data generated in this study (via numerical simulations) are also available upon request. The source code and associated materials used in this research are available upon request. ACKNOWLEDGMENTS P.A.B. and A.B.U. acknowledge the financial support of the IKUR STRATEGY (IKUR-IKA-23/22) and (IKUR-IKA23/04), respectively. C.M.J. acknowledges the predoctoral MICINN Grant No. PRE2019-088519. J.C. acknowledges the Ramón y Cajal (RYC2018-025197-I) research fellowship. The authors acknowledge the Quench project that received funding from the European Union’s Horizon Europe–The EU Research and Innovation Programme under Grant Agreement No. 101135742, the financial support from the Spanish Government via the Nanoscale NMR and complex systems (PID2021-126694NB-C21) project, the ELKARTEK project Dispositivos en Tecnologías Cuánticas (KK-2022/00062), and the Basque Government Grant No. IT1470-22. APPENDIX A: SPIN DYNAMICS OF THE SAMPLE In this section, we provide further details regarding the nuclear spin dynamics in the sample. 1. Amplitudes of the σx iand σy icomponents We consider an evolution governed by the next propagator [note this is similar to those in Eqs. (4) and (5)inthemain text] Ut=i,je−iφi,j 2σz iσz j,(A1) over an initial state such that ρx 1... ρx kρz k+1... ρz N,(A2) where ρx,z jare thermal states polarized along xand zdirections. This is, ρz j=1 2I+Bn 4σzwhile ρx j=eiπ 4σy jρz je−iπ 4σy j= 1 2I−BH 4σx, with BH=¯hγhBext KbTand Bn=¯hγnBext KbTbeing small quantities at room temperature (on the order of 10−5for a magnetic field of 2 T). In the previous expressions, Bext =2T is the external magnetic field, ¯h=1.054×10−34 J·sisthe Planck constant divided by 2π,γH=(2π)×42.57 MHz/Tis the gyromagnetic ratio of the H and γnthe corresponding one of other nuclei, KB=1.38×10−23 J/K the Boltzmann constant, and the temperature T=300 K. Here we differentiate among thermal states corresponding to the H nuclei, i.e., the nuclear spin which is rotated to the Xaxis at the start of the protocol (denoted by ρx j), and the other generic nuclear spin, which is represented with ρz j. On the one hand, one can demonstrate that σx 1=TrUtρx 1... ρx kρz 1... ρz kU† tσx 1 =Tr=1e|ρx 1|g... ρx kρz k+1... ρz Nj>1e−iφ1,jσz j +g|ρx 1|e... ρx kρz k+1...ρz Nj>1eiφ1,jσz j,(A3) where Tr=1[...] represents the trace over all spins excepting the first one. Using e|ρx 1|g=g|ρx 1|e=− BH 4one reaches σx 1=− BH 4k j=1cos φ1,jN j=k+1cos φ1,j−iBn 2sin φ1,j −BH 4k j=1cos φ1,jN j=k+1cos φ1,j+iBn 2sin φ1,j. (A4) 043017-5 POL ALSINA-BOLÍVAR et al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024) On the other hand, one can demonstrate that σy 1leads to σy 1=− iBH 4k j=1cos φ1,jN j=k+1cos φ1,j−iBn 2sin φ1,j +iBH 4k j=1cos φ1,jN j=k+1cos φ1,j+iBn 2sin φ1,j. (A5) Note the above expressions hold for any H, thus for any quantity σx iand σy ithat involves a H spin in the target sample. Notice how the leading order for each expectation value differs by five orders of magnitude: σx 1∝O(BH) and σy 1∝ O(BHBn). Thus, σx iis ∼105times greater than σy i. 2. Fluoromethanol resonances We compute the resonance positions for the specific case of the fluoromethanol molecule. For the sake of simplicity we consider a polarized H, and the other nuclei in the maximally mixed state, that is, ρx i=1 2(I+σx i) for H and ρm i=1 2Ifor other nuclei. Note these considerations only alter the amplitude of the signal, not the resonance position. However, the results in the main text consider thermal states for all nuclei in the fluoromethanol molecule. Thus, following Eq. (A3), we have Sx i=σx i 2=1 2N j=1cos φi,j, where φ1,icorresponds to the phase accumulated due to the interaction between the ith and jth spins during a time t, that is, φi,j(t) =Ji,j 2t. When H nuclei and 13Care targeted in the fluoromethanol molecule (corresponding to case 1 in the main text), the resulting signal is 3  i=1 Sx i(t) =1 6cos J 2t2 cos J1 2t +cos J 2tcos J2 2t, which leads to the resonance peaks in Fig. 2(a) of the main text, f1=±J2/2, f2=±(J2−2J)/2, f3=±(J2+2J)/2, f4=±(J1−J)/2, and f5=±(J1+J). In the case where the F is also targeted with πpulses (corresponding to case 2 in the main text), one finds 3  i=1 Sx i(t) =1 6cos J 2t2 cos J1 2tcos JF 1 2t +cos J 2tcos J2 2tcos JF 2 2t, with the corresponding resonances depicted in Fig. 2(b) of the main text. These are f1=±(J2−JF 2)/2, f2=±(+2J−J2− JF 2)/2, f3=±(J2+JF 2)/2, f4=±(2J+JF 2−J2)/2, f5= ±(2J+J2−JF 2)/2, f6=±(2J+J1+JF 1)/2, f7=±(J1− JF 1−J)/2, f8=±(J+J1−JF 1)/2, f9=±(J1+JF 1−J)/2, and f10 =±(J+J1+JF 1)/2. Alternatively, the resonant frequencies can be obtained via standard NMR multiplicity rules [29]. That is, when a nuclei A is coupled with nxspin-1 2nuclei of an isotope X,theAsignal splits into nx+1 lines. If Ais also coupled to an element different to X(lets sayY) this induces additional ny+1 splittings in each previous peak. In our case, since the broadening is mitigated, all Jcoupling peaks tend to be centered at 0 Hz. Thus, when H and 13Care targeted, the signal of Ha(recall the Haand Hbare magnetically inequivalent hydrogen groups in the molecule, refer to Fig. 2of the main text) is split into 2×2=4 peaks due to couplings with Hband the 13C, while the signal of Hb is split into 3×2=6 peaks, three due to the pair of equivalent Ha, and two due to the 13C. Then, we are left with 6 +4=10 peaks around 0 Hz. If J coupling interactions with F are also excited, the signal of Hais split into 2×2×2=8(two per every other element), while the signal of Hbsplits into 3×2×2=12 peaks, leading to a total of 20 peaks, see, e.g., Ref. [6] for more information. APPENDIX B: FROM THE SAMPLE TO THE NV CENTERS The oscillation of hydrogens in the detection stage of JINSECT impacts on the NV ensemble (which is considered at a micrometer depth from diamond surface) as a classical magnetic field. In this respect, see, e.g., Refs. [12,22]). In the high field regime, the magnetic signal induced by the sample in the orthogonal directions (i.e., xand y) with respect to the external magnetic field (oriented with the NV axis, i.e., z) can be neglected as it rapidly oscillate at a speed γNBzwith Bzbeing several Tesla. The resultant field emanated from the sample during an nth detection period (n∈N) can be well approximated by [12,22] Bz(t)=Bn 0sin(Ht),(B1) where H=γHBRF is the amplitude of the RF driving. On the other hand, the amplitude of the magnetic field is Bn 0=(2π)2(¯hγH)2μ0ρHBzF3 16πKBTMx,(B2) where Mxis the xcomponent of the normalized magnetization, defined as  M=2 NHi Si(nτ). Note that the quantity i Si(nτ) is computed via numerically evolving the next molecular Hamiltonian H/¯h=δHASz 1+Sz 2+δHBSz 3+δCIz+δFPz +JHaHa S1· S2+JHaHb( S1· S3+ S2· S3) +JHaCSz 1+Sz 2Iz+JHbCSz 3Iz +JHaFSz 1+Sz 2Pz+JHbFSz 3Pz+JCFIzPz.(B3) In addition, ¯h=1.054×10−34 J·s is the Planck’s constant divided by 2π,γH=(2π)×42.57 MHz/T is the gyromagnetic ratio of the H, μ0=4π×10−7H/m the magnetic permeability of free space, ρH≈6.6×1028 m−3the H density of the sample, Bz=2 T the external magnetic field, KB=1.38×10−23 J/K the Boltzmann constant, T=300 K the temperature and F3=4.1 characterizes the geometry of the sample [12]. 043017-6 J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024) FIG. 3. Simulating an experiment. Panel (a) corresponds to case 1, while (b) holds for case 2. Insets: In blue are the signals without considering luminescence issues, while red squares are the outcome that results from averaging 18000 repetitions of the respective protocol with 2.5×108NV centers assuming a contrast of 7%. Main figures: In blue are the spectra that correspond to the signals without luminescence issues, while the red squares are the FT of averaged signal in the inset (thus, carrying readout noise). Both spectra are normalized by a common factor, ensuring that the maximum peak in case 1 has amplitude 1. In case 1, referencing the amplitude of the smallest peak, the corresponding SNR is just above 30. In case 2 under the same reference criterion, it is slightly below 15. The Hamiltonian of a generic NV in the ensemble under the influence of Bz(t)is HNV =σzγe 2Bz(t)+Hc,(B4) where Hcaccounts for the MW driving on the NV ensemble. In particular, applying a XY4 control sequence during the detection stage over the NV ensemble with tRF =21 Hand the spacing between the MW pulses TMW =tRF/4, yields σyNV ≈2γetRF πBn 0,(B5) provided that 2γetRF πBn 01. APPENDIX C: READOUT CONSIDERATIONS To evaluate the impact of the measurement rate of the protocol under the high external magnetic-field paradigm, we make an estimation of the required signal averaging time leading to a signal to noise ration (SNR) of 30. In Ref. [30] the authors recorded the mean number of photon-count per measurement when a single NV is in |0, that is, n0=0.016, and a photon count difference between the ground and excited spin states of n0−n1=0.005, yielding a 30% increase of luminescence contrast between both states. In this case, since we are working with NV ensembles, we consider a fluorescence contrast of 7%, which was achieved by lowering the photon emission n0. Employing the latter photon statistics, and taking as a basis the signal with no readout error [in particular, the inverse Fourier transform of the spectra in Fig. 2(a), note that this signal carries the rest of errors described in the main text], we reproduce an experimental outcome. In particular, we assume a configuration similar to that in Ref. [12]. This is an NV center density of 0.8×1023 m3,a beam diameter of 20 µm, and a 10 µm thick layer of NV centers sensitive to the thermal spin signal. This leads to a sensor containing ≈2.5×108active NV centers. The protocol is repeated 18 000 times. That is, we average the emitted signal across 18 000 repetitions of the protocol, which leads to a duration of 18 000 ×2.66s ≈13h20. Specifically, in case 1, the noise in the resulting averaged signal has a standard deviation of approximately 6 ×10−5 while the signal’s amplitude is σz NV∼1×10−3,seeinsetof Fig. 3(a). This leads to a SNR of about 30 in the spectrum by taking the the height of the smallest peak. For case 2, the noise standard deviation is similar. However, the SNR is approximately 15. This is because even though both signals (from case 1 and case 2) have similar amplitudes, the spectrum of case 2 exhibits more peaks which results in peaks of smaller amplitude as it can be seen in Fig. 3for clarification. In conclusion, the thermal polarization induced by the high external magnetic field is enough to balance out the relatively 043017-7 POL ALSINA-BOLÍVAR et al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024) FIG. 4. (a) Top, scheme of the RF sequence employed for the high measurement-rate protocol [with (π)xand (π)−xpulses altogether with CORPSE structures over Hs during encoding for enhanced robustness] and the corresponding spectrum (bottom). This shows the result obtained with 6553 measurements (solid blue) while, for comparison, we include the spectrum obtained with 600 measurements (dashed red), which corresponds to the same spectrum shown in Fig. (a) of the main text. (b) Posterior distributions for each scanned parameter via Bayesian inference. I: Posterior distribution corresponding to J; II: to J1; and III: to J2. The exact value of each parameter is marked by a vertical dotted line in I, II, and III. The uncertainties are the standard deviation σθ,givenbyσ2 θ=(θ−θ)2=(θ−θ)2P(θ|D) dθ,whereθstands for J,J1,andJ2. low measurement rate, yielding a detectable signal (thus of clean J-coupling spectra) under reasonable experimental time in both cases. APPENDIX D: BEYOND τ>1.2/(|δHa−δHa|) To increase the number of measurements on each experimental run, we reduce τin encoding stages from τ=4.3ms to τ=75 µs and perform 6553 measurements instead of 600. We consider imperfect pulses (with 1% error on the amplitude driving) as well as always-on J-couplings and readout errors (see previous section). This increase in the number of measurements leads to a reduction on the total experimental repetitions to be performed. In this case the protocol includes an extra (π) pulse during the encoding period to further reduce the impact of imperfect drivings (see the scheme of the pulse sequence in Fig. 4). On the other hand, the reduction on τleads to an intricate spectrum without clear resonance peaks, see Fig. 4(a). Thus, to infer the values of J-couplings we perform Bayesian analysis. As an input to the Bayesian inference we employ the previously described noisy signal with imperfect pulses and readout errors. Regarding the parameter set, as the prior knowledge we assume a uniform distribution. To sum up, via Bayes inference, simplicity of the spectra can be traded for shorter experimental times. 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