Zero- to ultralow-field nuclear magnetic resonance

Danila Barskiy, John Blanchard, Dmitry Budker, James Eills, Szymon Pustelny, Kirill Sheberstov, Michael C.D. Tayler, Andreas Trabesinger

► To cite this version:

Danila Barskiy, John Blanchard, Dmitry Budker, James Eills, Szymon Pustelny, et al. Zero- to ultralow-field nuclear magnetic resonance. Progress in Nuclear Magnetic Resonance Spectroscopy, 2025, 148-149, pp.101558. 10.1016/j.pnmrs.2025.101558. hal-05480259

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Submitted on 27 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Dedication: Dedicated to Prof. Alexander Pines (1945–2024), pioneer of zero-field NMR and extraordinary mentor who continues to inspire our work.
Zero- to ultralow-field nuclear magnetic resonance

Danila A. Barskiy<sup>a,b</sup>, John W. Blanchard<sup>c</sup>, Dmitry Budker<sup>a,b,d</sup>, James Eills<sup>e</sup>, Szymon Pustelny<sup>f</sup>, Kirill F. Sheberstov<sup>g</sup>, Michael C. D. Tayler<sup>h</sup>, Andreas H. Trabesinger<sup>i</sup>

<sup>a</sup>Helmholtz-Institut Mainz, 55099 Mainz, Germany; GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany

<sup>b</sup>Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

<sup>c</sup>Quantum Technology Center, University of Maryland, College Park, MD 20742, USA

<sup>d</sup>Department of Physics, University of California, Berkeley, CA 94720-7300 USA

<sup>e</sup>Institute for Bioengineering of Catalonia, 08028 Barcelona, Spain

<sup>f</sup>Jagiellonian University in Kraków, 30-348 Kraków, Poland

<sup>g</sup>Laboratoire des Biomolécules, LBM, Département de Chimie, École Normale Supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France

<sup>h</sup>ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

<sup>i</sup>Brown Boveri Platz 4, 5400 Baden, Switzerland

Abstract

Zero and ultralow-field nuclear magnetic resonance (ZULF NMR) is an NMR modality where experiments are performed in fields at which spin-spin interactions within molecules and materials are stronger than Zeeman interactions. This typically occurs at external fields of microtesla strength or below, considerably smaller than Earth's field. In ZULF NMR, the measurement of spin-spin couplings and spin relaxation rates provides a nondestructive means for identifying chemicals and chemical fragments, and for conducting sample or process analyses. The absence of the symmetry imposed by a strong external magnetic field enables experiments that exploit terms in the nuclear spin Hamiltonian that are suppressed in high-field NMR, which in turn opens up new capabilities in a broad range of fields, from the search for dark matter to the preparation of hyperpolarized contrast agents for clinical imaging. Furthermore, as in ZULF NMR the Larmor frequencies are typically in the audio band, the nuclear spins can be manipulated with d.c. magnetic field pulses, and highly sensitive magnetometers are used for detection. In contrast to high-field NMR, the low-frequency signals readily pass through conductive materials such as metals, and heterogeneous samples do not lead to resonance line broadening, meaning that high-resolution spectroscopy is possible. Notable practical advantages of ZULF NMR spectroscopy are the low cost and relative simplicity and portability of the spectrometer system. In recent years ZULF NMR has become more accessible, thanks to improvements in magnetometer sensitivity and commercial availability, and the development of hyperpolarization methods that provide a simple means to boost signal strengths by several orders of magnitude. These topics are reviewed and a perspective on potential future avenues of ZULF-NMR research is presented.

**Keywords:** Zero- to ultralow-field (ZULF) NMR, J-coupling, single-spin NMR, axions and axion-like particles, nitrogen-vacancy (NV) color center in diamond, quantum control, biomedical diagnostics, battery research, hyperpolarization

Contents

1 Introduction 3

1.1 What is ZULF NMR and what it is good for 3

1.2 Magnetic field regimes 4

1.3 A short, incomplete history 4

1.4 A generic ZULF-NMR experiment 6

1.5 Notations for spin systems in ZULF NMR 7

2 Theory 7

2.1 Nuclear spin Hamiltonians 8

2.2 Density-operator formalism 9

2.2.1 Spin density operator 9

2.2.2 Observable quantity 9

2.3 Vector-model description of XA spin systems 9

2.4 XA<sub>n</sub> spin systems at zero field 11

2.4.1 Energy-level structure 11

2.4.2 Selection rules 12

2.4.3 Zero-field-NMR signal for a thermally polarized system 12

2.4.4 Experimental spectra 15

2.5 (XA<sub>n</sub>)B<sub>m</sub> spin systems at zero field 16

2.5.1 Energy-level structure 17

2.5.2 Selection rules 18

2.5.3 Experimental spectra 18

2.6 XA<sub>n</sub> spin systems in ultralow magnetic field 18

2.6.1 Energy-level structure 18

2.6.2 Selection rules 20

2.6.3 Experimental spectra 20

2.7 Untruncated couplings 21

3 Spin polarization for ZULF NMR 22

3.1 Thermal polarization 22

3.2 From thermal polarization to hyperpolarization 22

3.3 Spin-exchange optical pumping 23

3.4 Parahydrogen-induced polarization 25

3.5 Dynamic nuclear polarization 28
3.5.1 Overhauser Dynamic Nuclear Polarization 28

3.5.2 Dissolution Dynamic Nuclear Polarization 29

3.6 Chemically induced nuclear polarization 29

3.7 Statistical polarization 33

3.8 Spin polarization induced by rotating magnetic fields 34

4 Spin evolution in ZULF NMR 34

4.1 Initiating free evolution 34

4.2 Coherent control 34

4.2.1 Pulsed-field excitation 34

4.2.2 Selective excitation 35

4.2.3 Scaling and suppression of spin interactions 36

4.2.4 Magnetic field sweeps in the ZULF regime 36

5 Detection in ZULF NMR 38

5.1 Detection modalities 38

5.1.1 Inductive detection 38

5.1.2 Superconducting interferometers 38

5.1.3 Atomic magnetometers 39

5.1.4 Color centers in diamond 40

5.1.5 Anisotropic magnetoresistive sensors 40

5.1.6 Radioactively detected ZULF NMR 41

5.1.7 Nuclear spin optical rotation 41

5.1.8 Remote detection 41

5.2 Practical considerations 41

5.3 Effects of geometry in detection 42

5.3.1 Detector arrangement 42

5.3.2 Sample geometry 43

5.3.3 Applied magnetic field 45

6 Extensions of ZULF NMR 45

6.1 Two-dimensional spectroscopy 45

6.2 Imaging 45

6.3 Relaxometry 47

6.3.1 $T_1$ relaxometry of uncoupled nuclear spins 47

6.3.2 Relaxometry of coupled nuclear spins 48

6.4 Partial molecular alignment and orientation 48

6.5 ZULF NMR of solids 50

7 Applications of ZULF NMR 50

7.1 Physics 50

7.1.1 Polarized nuclear targets 50

7.1.2 Dark matter searches 51

7.1.3 Parity violation 51

7.1.4 Search for spin-gravity coupling 51

7.1.5 Quantum control 51

7.1.6 Quantum simulation 52

7.2 Chemistry 52

7.2.1 Study of chemical exchange 53

7.2.2 Chemical reaction monitoring 53

7.2.3 Battery diagnostics 53

7.2.4 Detection of molecular chirality 53

7.3 Precision metrology 55

7.4 ZULF NMR to prepare specific hyperpolarized spin states 55

7.5 Biomedicine 56

7.5.1 Studies with biomolecules 56

7.5.2 Dynamic nuclear polarization for in vivo studies 57

7.6 Portability and fieldable devices 57

8 Conclusions and outlook 58

1. Introduction

1.1. What is ZULF NMR and what it is good for

In the context of nuclear magnetic resonance (NMR) spectroscopy, we define the zero- to ultralow-field (ZULF) regime as the magnetic field range in which ‘internal’ spin interactions dominate ‘external’ ones. That is, in ZULF NMR couplings to magnetic fields originating from the sample itself are stronger than couplings to magnetic fields generated by the experimental apparatus [1, 2, 3]. As a direct consequence, and in stark contrast to ‘conventional’ NMR experiments, in the ZULF regime nuclear spin dynamics are not dominated by the imposed symmetry of a large magnetic field.

Operation under the ZULF conditions means in particular that all information encoded in the interaction tensors is preserved. The access to terms of the interaction tensors that are suppressed—or truncated—in high-field spectra opens up unique ways to exploring spin physics that remains largely inaccessible in high-field NMR (see Sec. 2.7). We consider this feature to be the main motivation for research in this traditionally neglected corner of the NMR landscape, and the driver of the renewed interest in it since the turn of the millennium.

Progress in ZULF NMR has been fuelled by partially independent developments that have substantially changed how ZULF NMR experiments are performed. Advances have been made in two areas in particular. First, versatile external sources of nuclear spin polarization have become available (see Sec. 3), removing the dependence on a strong magnetic field to polarize the spin ensembles. Second, there have been major developments in magnetometry, providing detection modalities that are not limited by a direct dependence of signal strength on precession frequency, and therefore external field strength, as traditional inductive detection does (see Sec. 5).

Beyond access to untruncated interaction Hamiltonians, further motivation for ZULF NMR arises from advantages traditionally associated with NMR at ‘low field’. These include decreased complexity and increased portability of spectrometers that do not require a strong magnet at their core. In addition, many applications benefit from the absence of line broadening induced by magnetic susceptibility, which is especially advantageous for studying inhomogeneous samples. Moreover, in the typical frequency range of ZULF NMR experiments—hundreds of hertz and below—the skin effect is negligible for most materials, so that for example probing through metal enclosures becomes possible.
The elephant-in-the-room question is, however, what spectroscopic insight can be gained in a regime where chemical-shift effects are negligible, by definition (see Sec. 1.2). Chemical shifts dominate the spectroscopic content in high-field liquid-state NMR, but have no role in ZULF NMR. However, the indirect spin-spin (or, *J*-coupling) interaction is independent of external field, and therefore is present even at zero field. This interaction is exquisitely sensitive to changes in local molecular geometry, conformation, and electronic structure [1, 2, 3]. Measuring *J*-coupling values with high spectral resolution provides therefore valuable physicochemical information in the ZULF regime, and the spectra will depend most sensitively on the topology of the *J*-coupling network (see also Sec. 1.5) and changes thereof, for example due to chemical reactions (see Sec. 7.2).

In the present review, when we discuss spectroscopic capabilities, we mainly refer to *J*-spectroscopy [4] in the liquid state, whereas we largely ignore solid-state NMR and nuclear quadrupole resonance (NQR). The latter is sometimes referred to as the original zero-field NMR and is a separate well-developed field that we do not cover here.

## 1.2. Magnetic field regimes

The hierarchy of interactions provides the basis for a practical definition of typically used denominations for regimes of the magnetic field strength, $|B|$, in NMR: 'high field', 'low field', 'ultralow field' and 'zero field'. Let us consider here a solution-state experiment with a molecule containing two different nuclear spin species, *I* and *S*, characterized by following parameters (see Fig. 1): homo- and heteronuclear *J*-couplings ($J_{II}$ and $J_{IS}$), differences in homonuclear chemical shifts ($\delta_{I1} - \delta_{I2}$) and heteronuclear gyromagnetic ratios ($\gamma_I - \gamma_S$), and characteristic spin relaxation times ($T_{relax}^{(I,S)}$). Here, the high-field regime is where the Zeeman interaction dominates, $|\gamma_I(\delta_{I1} - \delta_{I2})B| \gg 2\pi|J_{II}|$ and the boundary to the low-field regime is where the two terms are comparable. Once the difference in Larmor frequency between spins of different species becomes comparable to their mutual *J*-coupling $J_{IS}$, we are entering the ultralow-field regime, with the boundary at $|\gamma_I - \gamma_S|B| \approx |2\pi J_{IS}|$. On further reduction of the magnetic field strength, we reach the regime where the Larmor precession period $1/|\gamma_{IS}B|$ becomes comparable for both nuclear species to their respective relaxation time $T_{relax}^{(I,S)}$. We define the zero-field regime as the magnetic field range in which spin polarization relaxes to thermal equilibrium faster than it precesses about any residual magnetic field. In Fig. 1, we give a typical example with numerical values for the different regimes.

For a sample without measurable *J*-couplings (e.g., a bulk sample of water), the only regimes are high- and zero-field; it would be meaningless to talk about low or ultralow fields in this case. Note that in other areas of research, different naming conventions and definitions are used, and our definitions would also need modification if dipolar or quadrupolar couplings were considered.

Our definition of field regimes is guided exclusively by the hierarchy of spin interactions and ignores technological considerations. For example, field ranges can be categorized

<figure><img src="image_2.png" /><figcaption>Figure 1: Practical definitions of the different field regimes. The boundaries between regimes vary significantly depending on the system under study.</figcaption></figure>

also according to the most suitable detection modality at the corresponding Larmor frequency [5]. Historically, however, whether a magnetic field is considered 'high' or 'low' typically depended on how that field was generated, that is, by a superconducting, resistive or permanent magnet—or by the Earth's core. By contrast, operation in the ZULF regime typically requires shielding of the Earth's and laboratory fields.

## 1.3. A short, incomplete history

The history of NMR is, in large part, a history of pushing towards ever higher field strengths, and for good reasons. In a traditional NMR experiment the sample is polarized by the magnetic field in which it is immersed, spectral dispersion is dominated by chemical shift differences, and the signal is detected by electromagnetic induction. All these interactions typically take place in one and the same magnet, and they all scale favorably with increasing field strength.

In the shadow of the technical development and scientific exploitation of spectrometers built around increasingly powerful magnets, a small but intriguing body of work emerged over the decades dedicated to experiments in the Earth-field range (~50 µT). Many of these experiments were motivated by sensing the magnetic field of the Earth. In 1954, Martin Packard and Russell Varian first demonstrated free nuclear induction as a tool for measuring the geomagnetic field [6]. In their experiment, they polarized a 500 cm³ water sample in a field of 10 mT, which was applied perpendicular to the Earth's field. Subsequent nonadibatic reduction of the polarizing field to zero set off precession of the nuclear magnetization, which they detected inductively. The approach was soon adopted for geophysical surveying [7, 8]. As a notable spin-off, Erwin Hahn proposed
in 1960 that sea-water motion could be detected by recording, with his eponymous echo sequences, phase shifts due to movement of spins in a gradient field [9]. This method later inspired high-field experiments for quantifying blood flow and elastic deformations.

The instrumental simplicity of Earth-field NMR means that it can be employed for the study of unusually large samples. Spectacular applications include studies of diffusion in Antarc- tic ice [10] and experiments with a helicopter-borne NMR coil system—six meters in diameter and weighing some 1000 kg— with the aim of searching for oil spills under ice [11]. Moreover, the homogeneity of the Earth’s magnetic field inspired a series of magnetic resonance imaging (MRI) experiments [12, 13], some of which involved the combination with the measurement of dispersion due to slow molecular motions [14, 15].

As argued in Sec. 1.2, for spins without measurable cou- plings to each other, the only field regimes are high- and zero- field. The majority of Earth’s field experiments, including those described above, therefore belong in the former category. How- ever, there are several works in which heteronuclear J-coupling interactions between ¹H-²⁹Si [16], ¹H-¹⁹F [17, 18] and ¹H-³¹P [19] species were resolved. These fall into the low-field regime.

Independently of the developments in Earth-field NMR, a few groups tackled the challenge of performing NMR at field strengths in which intramolecular interactions dominate. With NQR and electron paramagnetic resonance (EPR), such experiments are readily possible even at zero magnetic field, as nuclear quadrupolar interactions and zero-field splittings aris- ing from dipole–dipole interactions between unpaired electrons are often sufficiently large to make inductive detection feasible. However, we are aware of only one case where magnetic reso- nance signals of non-quadrupolar nuclei have been detected in the ZULF regime using ‘conventional NMR’: in experiments with solid hydrogen, as first performed by Frederick Reif and Edward Purcell in 1953 [20] using the resonance absorption technique [21]. The unusually large line splittings due to dipo- lar interactions, combined with high polarization at cryogenic temperatures, made it possible to obtain resolved spectra at zero magnetic field.

It would take 30 years until the conceptual ideas and experimental tools were in place to enable zero-field NMR with less extreme samples. Seeking methods for removing orientational broadening from so-called powder spectra while retaining resolved dipole–dipole and quadrupole couplings, Daniel Weitekamp and co-workers in the laboratory of Alexan- der Pines at Berkeley started to develop time-domain zero-field NMR [22]. The motivation was as follows: The broad spec- tra of powder samples are a consequence of the truncation of local interactions due to the preferred axis imposed by a strong magnetic field. Magic-angle spinning and multiple- pulse techniques commonly employed in solid-state NMR can be used to remove such anisotropy-induced broadening. However, eliminating the anisotropic contributions of the local spin interactions necessarily also removes potentially valuable information. An elegant solution is to remove the external field, so that there is no privileged direction and all crystallites of a polycrystalline sample or all sites of an amorphous solid

become equivalent. In this way resolution can be improved, comparable to the case of an oriented single crystal in high-field NMR, without sacrificing the information encoded in local spin couplings.

The challenges related to polarizing the samples and detect- ing a signal were addressed by adopting field-cycling methods (see Fig. 2a). In this approach the initial spin state is prepared in a high-field spectrometer, where also the final spin state is read out [23, 24]. In between, the sample is shuttled out of the superconducting magnet into an intermediate field. Spin evo- lution is then initiated by a sudden transition to zero magnetic field, mirroring the strategy employed by Packard and Varian in their Earth’s field experiments [6]. After a time t₁, the magnetic field is turned back on, preserving the components of the nu- clear spin magnetization that are projected along the magnetic field. The zero-field spectrum is then obtained by performing the experiment at several values of t₁ and Fourier transforming the resulting time-domain data. As an example, panels b and d of Fig. 2 show the high-field and zero-field spectra of powdered barium chlorate hydrate [Ba(ClO₃)₂ · H₂O], respectively. As expected, the zero-field linewidth is comparable to that of the high-field single-crystal spectrum (Fig. 2c).

Exploring this approach further, Pines and co-workers in- troduced a series of isotropic averaging pulse sequences that selectively average nuclear spin Hamiltonians while preserv- ing invariance with respect to sample orientation [25, 26, 27, 28] (see also Sec. 4.2.3). In parallel, the group started to de- velop alternative ways for detecting NMR at low fields. Pines teamed up with the fellow Berkeley group of John Clarke, a pioneer in the field of superconducting quantum interference devices (SQUIDs), and in particular in their use as magnetome- ters (see Sec. 5.1.2). The availability of a magnetometer sensi- tive enough to detect nuclear magnetization offered the intrig- ing possibility of recording NMR signals with a sensitivity that does not depend on precession frequency, and therefore on the strengths of the field present during detection. The potential of using SQUIDs (or, more generally, Josephson junctions) for the detection of NMR signals had been demonstrated already in seminal experiments starting from the late 1960s [29, 30, 31]. But the Pines–Clarke collaboration pushed the approach con- siderably further, for example by combining SQUID detection with laser-polarized noble gases—another area in which the Pines group made substantial advances starting from the 1990s [32, 33]—and generating images of ³He and ¹²⁹Xe samples in millitesla fields [34]. In these fields, xenon chemical shifts in different physico-chemical environments were resolved as well [35], benefiting from the exceptionally large chemical shift range of xenon due to the highly polarizable electron cloud of the atom.

By the early 2000s, steady advances made it possible to detect with SQUIDs NMR signals in microtesla fields from milliliter amounts of liquid samples [36, 37]. In these exper- iments the samples were thermally prepolarized in fields of a few millitesla, so that a wide range of samples and nuclear species could be studied. Importantly, in the microtesla field range the *absolute* homogeneity of even the simplest solenoids proved to be good enough to obtain NMR linewidths on the or-
<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Schematic diagram of the field-cycling apparatus and the time-dependent field profile for early zero-field NMR experiments. (b-d) $^1$H NMR spectra of barium chlorate hydrate [Ba(ClO$_3$)$_2$ · H$_2$O]. Figure reprinted with permission from Ref. [23] by the American Physical Society.</figcaption></figure>

der of one hertz (despite the *relative* homogeneity being in the percent range). This, in turn, enabled the extension of SQUID-detected NMR to the detection of heteronuclear *J*-couplings in neat liquids [36] and $^{13}$C-enriched solutes [37], including some venturing towards the ultralow-field regime as defined in Sec. 1.2 [37]. The ‘microtesla platform’ was also exploited for MRI [38], demonstrating several distinct advantages of operating at such low fields, including a large dispersion of longitudinal relaxation times [39] and the imaging of samples next to or inside metallic objects [40].

The Clarke group, however, was not the only one developing sensitive magnetometers in Berkeley. The group led by one of us (D. Budker) was working to push the sensitivity of atomic magnetometers [41] (see Sec. 5.1.3). Such optically pumped magnetometers (OPMs) had been used already in the late 1960s by Claude Cohen-Tannoudji and co-workers for the detection of

nuclear magnetization, in their case, from a gas of nuclear polarized $^3$He atoms [42]. Here too, the ambition of the newly formed Pines-Budker collaboration was to push the approach much further and to develop the basis for using atomic magnetometers as versatile detectors for NMR in the ZULF regime. Starting from the mid-2000s, the team did just that, from the initial demonstration of detecting hyperpolarized gaseous xenon with an optical atomic magnetometer [43] to the detection of intramolecular $^{13}$C-$^1$H *J*-couplings in the ZULF regime [44].

Nowadays, when we talk about ‘ZULF-NMR experiments’, we typically refer to direct detection of NMR using magnetometers, rather than indirect detection using field-cycling. In the following subsection we describe a generic ‘modern’ ZULF-NMR experiment.

### 1.4. A generic ZULF-NMR experiment

A typical NMR experiment is composed of three stages: spin polarization, encoding, and detection. In conventional NMR, each of these stages depends on the presence of a magnetic field on the order of hundreds of millitesla to tens of tesla. In ZULF NMR, by contrast, the need for strong persistent magnets is eliminated at every stage. This is possible owing to, first, the development of hyperpolarization techniques that do not require strong magnetic fields (Sec. 3); second, the realization that scalar spin-spin or *J*-coupling provides for spectroscopic encoding at ZULF, without having to rely on chemical shifts (Sec. 4); and, third, the availability of sensitive detection modalities that enable non-inductive sensing (Sec. 5).

The experimental setup for ZULF NMR can be relatively simple, see Fig. 3. Layers of mu-metal and/or ferrite shielding can be used to attenuate the ambient magnetic field by several orders of magnitude to create the ZULF region. The residual field can be further reduced by ‘shimming’, that is, by applying compensating magnetic fields through coils inside the magnetic shield.

The sample is polarized using one of the various methods available (see Sec. 3), for example by thermal polarization in a magnet outside the ZULF region (Fig. 3a and Sec. 3.1) or a hyperpolarization technique such as parahydrogen-induced polarization (PHIP; Fig. 3b and Sec. 3.4). A signal can be initiated by applying brief magnetic field pulses to the sample in the ZULF region, for instance with a Helmholtz pair. The signal is then measured, typically with an atomic magnetometer (see Sec. 5.1.3). Whereas traditionally home-built atomic magnetometer have been used, recently commercial magnetometers have become available that are suitable for this task, both in terms of sensitivity and functionality [45], thereby removing one of the main hurdles for new groups entering the field of ZULF NMR.

In order to illustrate the basic principles underlying ZULF *J*-spectroscopy, let us consider the evolution of the thermally polarized state of a coupled $^1$H-$^{13}$C spin pair. This is illustrated in Fig. 4. The thermal equilibrium state at high field corresponds to spin polarization along the external field, where the polarization of $^1$H spins is approximately four times higher than that of $^{13}$C spins, due to the difference in gyromagnetic ratio. This state does not evolve with time. However, if the field is
a

<figure><img src="image_4.png" /></figure>

Figure 3: Typical examples of setups for liquid-state ZULF NMR. Some of the key components are shown: a magnetic shield and shim coils to create a near-zero-field region; a magnetometer for signal detection; a sample to be measured; a source of polarization to create detectable magnetization in the sample. (a) An apparatus in which the sample is thermally polarized in a permanent magnet, and is pneumatically shuttled to the zero-field region for detection. (b) An apparatus in which the sample is polarized by chemical reaction with parahydrogen (pH₂) gas that is bubbled through the sample. Reused from Ref. [46], with the permission of AIP Publishing.

suddenly removed, the state starts to evolve (see Fig. 4). The original ¹H polarization coherently transfers to the ¹³C spins, while the original ¹³C polarization transfers to the ¹H spins. The oscillation occurs at the *J*-coupling frequency (for instance, a typical value for the one-bond ¹H-¹³C *J*-coupling in a formyl group, ¹H¹³COO-, is 210 Hz [47, 48, 49]). The result is an observable oscillation of the sample magnetization, from which the *J*-spectrum is obtained.

The origin of the oscillation can be understood as follows. While the field is on, the Zeeman interaction dominates the Hamiltonian and the spin system is in an eigenstate; it does not evolve. When the field is suddenly switched off, the state is initially unchanged, but it is no longer an eigenstate of the Hamiltonian, which is now dominated by the *J*-coupling interaction. In fact, the initial state corresponds to a superposition of the two eigenstates of the 'new' Hamiltonian, whose energies in our example are separated by *J* (see Sec. 2.3). The temporal evolution of the superposition of eigenstates corresponds to quantum beats, and these are exactly the oscillations sketched in Fig. 4.

### 1.5. Notations for spin systems in ZULF NMR

In high-field NMR experiments, spin systems are typically classified according to the Pople notation [50]. Therein, each nucleus is assigned a capital letter of the Roman alphabet. Pairs of weakly coupled nuclei of the same species (i.e.,

<figure><img src="image_5.png" /></figure>

Figure 4: A minimal illustration of ZULF NMR *J*-spectroscopy. A ¹H-¹³C spin pair with spin-spin scalar coupling *J*<sub>CH</sub> is thermally polarized in a high-field regime, leading to ¹H and ¹³C spin ordering along the field axis in the ratio 1:0.25. The field is then suddenly switched off, and the polarization difference proceeds to oscillate about the mean at the frequency *J*<sub>CH</sub>. While the sum of polarizations remains constant in the ZULF regime (ignoring relaxation effects) the total magnetization does not, due to the different gyromagnetic ratios of the spin species — see Sec. 2. The along-axis magnetization oscillation is typically sampled directly using a magnetometer — see Sec. 5.

$|\gamma_I(\delta_{I_1} - \delta_{I_2})B| \gg 2\pi|J_{II}|$ are assigned letters with a large distance between them in the alphabet, such as A and X. If the coupling between the spins is on the order of or larger than the absolute value of the chemical-shift difference, then adjacent letters of the alphabet are used, for example A and B. Nuclei that are chemically equivalent but magnetically nonequivalent are given the same letter but with a prime added to one of them (e.g., AA'XX' or AA'BB').

In ZULF NMR, chemical shift differences between like spins vanish, and for heteronuclear pairs $|(\gamma_I - \gamma_S)B| \lesssim |2\pi J_{IS}|$, by definition (see Sec. 1.2). Therefore, typically a modified Pople notation is used, with letters that are well separated in the alphabet denoting exclusively heteronuclear pairs. Adjacent letters stand for chemically nonequivalent nuclei of the same species and parentheses group together the spins with the largest coupling. For instance, $(XA_n)B_m$ denotes a spin system with two spin species, where X represents for example a ¹³C nucleus, $A_n$ a set of *n* equivalent ¹H nuclei coupled to X, and $B_m$ a chemically distinct set of *m* equivalent ¹H nuclei that are less strongly coupled to X and $A_n$.

An alternative way to classify spin systems is according to the topology of the *J*-coupling network [51, 52, 53]. Whereas to our knowledge this approach has not been explored systematically in the context of ZULF NMR, we expect that this classification of spin systems might be useful across a broad range of ZULF-NMR scenarios.

## 2. Theory

Whereas ZULF NMR naturally shares many commonalities with its conventional counterpart [54], the fact that spin-spin interactions originating from the sample itself—rather than the Zeeman interaction—dominate the dynamics of the nuclear spin system leads to a number of unique properties and behaviors. A general description of these features, basic and more advanced, is explained in detail in the following section, together
with experimental examples.

## 2.1. Nuclear spin Hamiltonians

Free evolution of nuclear spins is governed by a Hamiltonian $\hat{H}$ that, in the case of NMR of isotropic liquids, can be split into two terms. The first term, $\hat{H}_Z$, describes the Zeeman interaction with the external magnetic field $\mathbf{B}_0$, and the second term, $\hat{H}_J$, the indirect spin-spin interaction (J-coupling):

$$ 
\begin{aligned}
\hat{H}_{iso} &= \hat{H}_Z + \hat{H}_J, \\
\hat{H}_Z &= -\hbar \sum_i \gamma_i (1 - \delta_i) \mathbf{B}_0 \cdot \hat{\mathbf{I}}_i, \\
\hat{H}_J &= 2\pi\hbar \sum_{i>j} J_{ij} \hat{\mathbf{I}}_i \cdot \hat{\mathbf{I}}_j.
\end{aligned}
\quad (1) $$

Here the summation is taken over the entire system of coupled spins; $\hbar = h/(2\pi)$ denotes the reduced Planck constant; $\gamma_i$ is the gyromagnetic ratio of the $i^{th}$ spin; $\delta_i$ is the isotropic chemical shift for the $i^{th}$ spin; $\hat{\mathbf{I}}_i$ denotes a spin operator of the $i^{th}$ spin; and $J_{ij}$ is the isotropic spin-spin coupling constant between spins $i$ and $j$. We note that while J-coupling is written as a scalar interaction in Eq. (1), it is generally a tensor interaction [55, 56] (see also Sec. 2.7).

If the sample under study is anisotropic, then other internal interactions may manifest themselves in ZULF NMR. For example, direct spin-spin interactions have been observed [23, 57]. These dipole-dipole interactions are described with the Hamiltonian

$$ 
\begin{aligned}
\hat{H}_{DD}^{(i,j)} &= -\hbar \hat{\mathbf{I}}_i \cdot \mathbf{D}^{(i,j)} \cdot \hat{\mathbf{I}}_j = \hbar \sum_{p,q} \hat{I}_{p,i} \hat{I}_{q,j} D_{pq}^{(i,j)}, \\
D_{pq}^{(i,j)} &= -\frac{\mu_0}{4\pi} \frac{\gamma_i \gamma_j \hbar}{r_{ij}^3} \frac{1}{2} \left( \frac{3r_{ij}^{(p)} r_{ij}^{(q)}}{r_{ij}^2} - \delta_{pq} \right), \quad (2) \\
p, q &\in \{x, y, z\}.
\end{aligned}
$$

Here, $\mathbf{D}$ represents the second-rank tensor of the dipole-dipole interaction between spins $i$ and $j$ which depends on the orientation of $\mathbf{r}_{ij}$—the vector joining the centers of the two nuclei with the distance $r_{ij} = |\mathbf{r}_{ij}|$ between them—with respect to the quantization axis $z$; $\mu_0$ denotes the magnetic permeability of vacuum; the quantity $\delta_{pq}$ is the Kronecker delta.

Recently the spin-1 nuclei $^{14}$N and $^{2}$H were observed in ZULF NMR [58, 59, 60, 61]. In these cases, the quadrupolar interaction between the nuclear spin and the gradient of the molecular electric field at the position of the nucleus is relevant. It is given by the Hamiltonian

$$ 
\begin{aligned}
\hat{H}_Q^{(i)} &= \frac{eQ^{(i)}}{2I_i(2I_i - 1)} \hat{\mathbf{I}}_i \cdot \mathbf{V}^{(i)} \cdot \hat{\mathbf{I}}_i = \\
&= \frac{eQ^{(i)}}{2I_i(2I_i - 1)} \sum_{p,q} \hat{I}_{p,i} \hat{I}_{q,j} V_{pq}^{(i)}, \quad (3) \\
p, q &\in \{x, y, z\}.
\end{aligned}
$$

Here $eQ^{(i)}$ denotes the nuclear quadrupole moment of the $i^{th}$ spin and the second-rank tensor $\mathbf{V}^{(i)}$ corresponds to the gradient tensor of the electric field evaluated at the nucleus $i$.

The relative strengths of the interactions described by $\hat{H}_Z$ and $\hat{H}_J$, $\hat{H}_{DD}$ and $\hat{H}_Q$ determine the dynamics of a spin system. This is because the operators (for instance, $\hat{H}_J$ and $\hat{H}_Z$) generally do not commute with one another, unless all of the spins are magnetically equivalent. This situation can be dealt with by using a perturbation approximation when one Hamiltonian is much larger than the other. As an example, in conventional NMR the differences in Larmor frequencies are much larger than the J-couplings, so that the Zeeman Hamiltonian is the leading one ($|\gamma B_0(\delta_i - \delta_j)| \gg 2\pi|J_{ij}|$ for homonuclear spins or $|(\gamma_i - \gamma_j)|B_0 \gg 2\pi|J_{ij}|$ for heteronuclear spins. This defines the high-field regime (see Sec. 1.2). Eigenstates of the spin system are to a good approximation those of $\hat{H}_Z$ plus the commuting part of $\hat{H}_J$. The noncommuting part of $\hat{H}_J$ can be considered as a first-order perturbation; in some NMR textbooks, the dropping of the noncommuting term entirely is called the secular or weak-coupling approximation.

The ZULF NMR scenario is opposite to the regime described above, in the sense that it involves J-couplings that are larger than Zeeman frequencies. Here the ZULF spin eigenstates are those of $\hat{H}_J$ plus—if it is nonzero—the commuting part of $\hat{H}_Z$. If the noncommuting part of $\hat{H}_Z$ is relatively small, this can now be applied as a first-order perturbation [62].

Compact analytical expressions for the commuting and noncommuting parts of $\hat{H}_Z$ and $\hat{H}_J$ in ZULF NMR are found for the case where the spin system contains two spin-1/2 nuclei. Ignoring chemical shifts, these are the permutation-symmetric and permutation-antisymmetric operators

$$ 
\begin{aligned}
\hat{H}_{Z,\text{commuting}} &= -\hbar \frac{\gamma_1 + \gamma_2}{2} \mathbf{B}_0 \cdot (\hat{\mathbf{I}}_1 + \hat{\mathbf{I}}_2), \quad (4a) \\
\hat{H}_{Z,\text{noncommuting}} &= -\hbar \frac{\gamma_1 - \gamma_2}{2} \mathbf{B}_0 \cdot (\hat{\mathbf{I}}_1 - \hat{\mathbf{I}}_2), \quad (4b)
\end{aligned}
$$

with the commutation property defined by

$$ 
\begin{aligned}
[\hat{H}_J, \hat{H}_{Z,\text{commuting}}] &= 0, \quad (5a) \\
[\hat{H}_J, \hat{H}_{Z,\text{noncommuting}}] &\neq 0. \quad (5b)
\end{aligned}
$$

Operator expressions similar to those in Eq. (4a) and Eq. (4b) can be generated and used to determine the near-zero-field energy states of simple molecules, such as those of the $XA_n$ type (Sec. 2.4) [62]. Perturbation theory can also be applied to more complex spin systems with one dominant J-coupling and several smaller couplings, such as those between distant spins, as discussed in Sec. 2.5 and references [63, 47].

As a final remark, the noncommuting term or permutation-antisymmetric term [Eq. (4b)] can have a major role in redistributing spin order among ZULF spin eigenstates. The spin-1/2 pair provides insights into this process as the dynamics can be solved exactly, without the use of perturbation approximations, as discussed in Sec. 2.3. More general behavior under pulsed fields is the topic of Sec. 4.
## 2.2. Density-operator formalism

### 2.2.1. Spin density operator

The state of an NMR ensemble is described by a spin density operator $\hat{\rho}$, defined as

$$ \hat{\rho} = |\psi\rangle\langle\psi|, \qquad (6) $$

where $|\psi\rangle$ is the wavefunction of a single member of the ensemble and the overbar denotes ensemble averaging.

In the following discussion, we use the Liouvillian formalism, where coherent evolution of the spin system in time is given, for the particular case of a time-independent Hamiltonian, by the equation

$$ \hat{\rho}(t) = \exp\left(-\frac{i\hat{H}t}{\hbar}\right) \hat{\rho}(0) \exp\left(\frac{i\hat{H}t}{\hbar}\right). \qquad (7) $$

The trace of the product of a spin operator $\hat{Q}$ and $\hat{\rho}$

$$ \mathrm{Tr}[\hat{Q}\hat{\rho}] = \overline{\langle\psi|\hat{Q}|\psi\rangle}, \qquad (8) $$

has several uses.<fnref n="1" /> First, the trace allows the definition of a matrix representation of the density operator, with elements

$$ \rho_{uv} = \langle v|\hat{\rho}|u\rangle = \mathrm{Tr}[|u\rangle\langle v|\hat{\rho}|], \qquad (9) $$

where $|u\rangle$ and $|v\rangle$ are orthonormal basis states of $|\psi\rangle$. Second, one can use the trace to determine the expectation value of $\hat{Q}$ at a given time $t$,

$$ \langle \hat{Q}(t) \rangle = \mathrm{Tr}[\hat{Q}\hat{\rho}(t)], \qquad (10) $$

and also the projection $p(\hat{Q}, \hat{\rho})$ of $\hat{\rho}$ onto $\hat{Q}$,

$$ p(\hat{Q}, \hat{\rho}) = \frac{\mathrm{Tr}[\hat{Q}^{\dagger}\hat{\rho}]}{\mathrm{Tr}[\hat{Q}^{\dagger}\hat{Q}]}, \qquad (11) $$

where $\hat{Q}^\dagger$ is the adjoint of $\hat{Q}$.

These textbook formulae [54] are used in the following sections.

### 2.2.2. Observable quantity

Similar to conventional NMR, ZULF NMR relies on detection of magnetic dipole fields originating from nuclear magnetization. The measured quantity is the magnetic field at the location of the detector.

In general, the field at the detector depends on the sample magnetization and the relative position of the detector with respect to the sample [64]. However, provided the detector is displaced from the sample along the sensitive axis of the sensor, the signal is determined purely by the magnetization component in that direction. In the following, for simplicity, we assume this situation.

The field is proportional to the total magnetization of the sample, with the magnetization operator $\hat{M} = (\hat{M}_x, \hat{M}_y, \hat{M}_z)$ given by

$$ \hat{M} \propto \sum_{i=1}^{N} \gamma_i \hat{I}_i, \qquad (12) $$

where the summation runs over all spins in the sample.

Equations (7)–(12) can be used to calculate time-dependent NMR signals. For example, the signal observed along an axis $p$ ($p \in \{x, y, z\}$) is

$$ s_p(t) \propto \mathrm{Tr}[\hat{M}_p \hat{\rho}(t)] = \sum_{u,v} \langle u | \hat{M}_p | v \rangle \rho_{uv}(0) \exp(-i\omega_{uv}t). \qquad (13) $$

Here the double summation is made over all pairs of eigenstates of the Hamiltonian, $\{|u\rangle, \dots |v\rangle\}$, including self pairs, $\omega_{uv} \equiv (\langle u|\hat{H}|u\rangle - \langle v|\hat{H}|v\rangle)/\hbar$ denotes corresponding transition frequencies in angular units.

The presented approach is general and provides means for calculating ZULF NMR spectra of various compounds. For that one needs to determine the initial density matrix and the form of $\langle u | \hat{M}_p | v \rangle$. In some cases, this can be done exactly and analytically, in others perturbative approaches have to be implemented. Determining the spectra of complex molecules might require numerical calculation.

In Sec. 2.4 we turn our attention to systems with $XA_n$ spin topology, which can be solved exactly within the framework introduced above. Before doing so, we discuss in Sec. 2.3 a vector-model description of XA spin systems, with a goal of providing an intuitive geometrical approach to understanding key features of ZULF NMR.

## 2.3. Vector-model description of XA spin systems

Many of the basic features of ZULF NMR that might seem unusual to the uninitiated can be understood by analyzing the dynamics of the density operator for a spin-$1/2$ pair. Of the $2^2 \times 2^2 = 16$ possible orthogonal basis operators, the following five operators are of interest:

$$ \hat{A}_1 = \hat{I}_{1x} \hat{I}_{2x} + \hat{I}_{1y} \hat{I}_{2y}, \qquad (14a) $$

$$ \hat{A}_2 = \frac{\hat{I}_{1z} - \hat{I}_{2z}}{2}, \qquad (14b) $$

$$ \hat{A}_3 = \hat{I}_{1x} \hat{I}_{2y} - \hat{I}_{1y} \hat{I}_{2x}, \qquad (14c) $$

$$ \hat{A}_4 = \hat{I}_{1z} \hat{I}_{2z}, \qquad (14d) $$

$$ \hat{A}_5 = \frac{\hat{I}_{1z} + \hat{I}_{2z}}{2}. \qquad (14e) $$

These are five of the six zero-quantum (ZQ) operators for the spin pair (the last one is the identity operator). A ZQ operator is invariant under rotation of the whole system about the z-axis ($\hat{A}_j = \exp[-i\theta(\hat{I}_{1z} + \hat{I}_{2z})]\hat{A}_j \exp[i\theta(\hat{I}_{1z} + \hat{I}_{2z})]$ for arbitrary real $\theta$) and has zero overall coherence order. A useful property of the basis in Eq. (14) is the cyclic commutation of the three

<footnote n="1">¹In matrix algebra, the trace for a product of two (or more) operators is invariant with respect to a cyclic permutation of the operators.</footnote>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Bloch sphere representation of the fictitious-spin-½ subspace of zero-quantum operators for a spin-½ pair: (a) gray arrows correspond to basis vectors representing the cyclically commuting spin operators $\hat{A}_1, \hat{A}_2$ and $\hat{A}_3$, defined by Eq. (14); (b) precession dynamics in the zero-quantum density operator subspace, illustrated for three distinct magnetic-field regimes (see also Fig. 1). The purple arrow indicates the total field vector, of magnitude $\sqrt{(2\pi J_{12})^2 + (\gamma_1 - \gamma_2)^2 B_z^2/4}$ given by the vector sum of the components in red.</figcaption></figure>

operators $\hat{A}_1, \hat{A}_2$, and $\hat{A}_3$, which obey the relationship [53]

$$[\hat{A}_1, \hat{A}_2] = i\hat{A}_3, \qquad (15a)$$

$$[\hat{A}_2, \hat{A}_3] = i\hat{A}_1, \qquad (15b)$$

$$[\hat{A}_3, \hat{A}_1] = i\hat{A}_2. \qquad (15c)$$

In addition, all operators in Eq. (14) commute with $\hat{A}_4$ and $\hat{A}_5$. Therefore, the only noncommuting operators of the ZQ basis are $\hat{A}_1, \hat{A}_2$, and $\hat{A}_3$.

A consequence of the commutation properties in Eq. (15) is that the ZQ operator basis of the spin pair behaves overall like the operator basis of a single fictitious spin-½, where operators $\hat{A}_1, \hat{A}_2$, and $\hat{A}_3$ evolve in a way that is analogous to single-spin Cartesian operators $\hat{I}_z, \hat{I}_x$, and $\hat{I}_y$, respectively. Therefore, as is the case for a single spin-½, the ZULF spin dynamics of a heteronuclear pair can be represented conveniently and graphically using a Bloch sphere (Fig. 5).

A point $\mathbf{r}_\rho$ in the sphere corresponds to a given projection of $\hat{\rho}$ onto each fictitious-spin-½ basis operator:

$$\mathbf{r}_{\rho} = \left( \frac{\mathrm{Tr}[\hat{A}_1^\dagger \hat{\rho}]}{\mathrm{Tr}[\hat{A}_1^\dagger \hat{A}_1]}, \frac{\mathrm{Tr}[\hat{A}_2^\dagger \hat{\rho}]}{\mathrm{Tr}[\hat{A}_2^\dagger \hat{A}_2]}, \frac{\mathrm{Tr}[\hat{A}_3^\dagger \hat{\rho}]}{\mathrm{Tr}[\hat{A}_3^\dagger \hat{A}_3]} \right). \qquad (16)$$

Similarly, a spin Hamiltonian

$$\hat{\mathcal{H}} = \sum_j \omega_j \hat{A}_j \qquad (17)$$

can be represented as the 3-vector

$$\mathbf{r}_{\mathcal{H}} = \left( \frac{\mathrm{Tr}[\hat{A}_1^\dagger \hat{\mathcal{H}}]}{\mathrm{Tr}[\hat{A}_1^\dagger \hat{A}_1]}, \frac{\mathrm{Tr}[\hat{A}_2^\dagger \hat{\mathcal{H}}]}{\mathrm{Tr}[\hat{A}_2^\dagger \hat{A}_2]}, \frac{\mathrm{Tr}[\hat{A}_3^\dagger \hat{\mathcal{H}}]}{\mathrm{Tr}[\hat{A}_3^\dagger \hat{A}_3]} \right) \equiv (\omega_1, \omega_2, \omega_3). \qquad (18)$$

Under these conditions, the time evolution of $\hat{\rho}$ is equivalent to precession of $\mathbf{r}_\rho$ about $\mathbf{r}_\mathcal{H}$, in analogy to the classical vector model of NMR. The motion can be summarized by the equation

$$\frac{d\mathbf{r}_{\rho}}{dt} = \mathbf{r}_{\rho} \times \mathbf{r}_{\mathcal{H}}, \qquad (19)$$

for which two extreme cases are illustrated in Fig. 5b. One of these (right side of Fig. 5b) corresponds to the familiar high-field NMR picture, where a magnetic field is applied along the z-axis. The Zeeman Hamiltonian can be identified with

$$\hat{\mathcal{H}}_z = \omega_2 \hat{A}_2 + \omega_5 \hat{A}_5, \qquad (20)$$

for $\omega_2 = (\gamma_1 - \gamma_2)B_z$ and $\omega_5 = (\gamma_1 + \gamma_2)B_z$. The time-integrated form of the precession equation can be written in this case as

$$\hat{A}_1 \xrightarrow{\hat{\mathcal{H}}_z t} \hat{A}_1 \cos(\omega_2 t) - \hat{A}_3 \sin(\omega_2 t), \qquad (21)$$

giving the familiar high-field result that the vector $\mathbf{r}_\rho$ undergoes precession at a frequency proportional to the difference in gyromagnetic ratio for the two spins. The opposite extreme (left side of Fig. 5b) is evolution at zero field, where the Hamiltonian comprises only the J-coupling:

$$\hat{\mathcal{H}}_J = \omega_1 \hat{A}_1 + \omega_4 \hat{A}_4, \qquad (22)$$

with $\omega_1 = \omega_4 = 2\pi J_{12}$. Here the precession equation is

$$\hat{A}_2 \xrightarrow{\hat{\mathcal{H}}_J t} \hat{A}_2 \cos(\omega_1 t) + \hat{A}_3 \sin(\omega_1 t). \qquad (23)$$

We note that an initial density operator produced by thermal polarization or hyperpolarization of the spin system, as discussed in Sec. 3, may contain a component proportional to $\hat{M}_z \equiv (\gamma_1 - \gamma_2)\hat{A}_2 + (\gamma_1 + \gamma_2)\hat{A}_5$, as defined in Eq. (12). Applying Eq. (23) gives

$$(\gamma_1 \hat{I}_{1z} + \gamma_2 \hat{I}_{2z}) \xrightarrow{\hat{\mathcal{H}}_J t} \frac{1}{2}(\gamma_1 - \gamma_2)(\hat{I}_{1z} - \hat{I}_{2z}) \cos(2\pi J_{12} t) \\ + (\gamma_1 - \gamma_2)(\hat{I}_{1x}\hat{I}_{2y} - \hat{I}_{1y}\hat{I}_{2x}) \sin(2\pi J_{12} t) \\ + \frac{1}{2}(\gamma_1 + \gamma_2)(\hat{I}_{1z} + \hat{I}_{2z}), \qquad (24)$$

where the $\hat{A}_2$ component evolves back and forth into $\hat{A}_3$ at the frequency of the inter-pair coupling, $J_{12}$ [65, 48]. In a heteronuclear spin pair, $\gamma_1 \neq \gamma_2$, the operator $\hat{A}_2$ corresponds to a net magnetic moment along the z-axis; therefore, part of the initial magnetization oscillates at a single, nonzero frequency, as illustrated in Fig. 4. The rationale for using heteronuclear compounds for zero-field NMR can be understood based on the fact that the magnitudes of both time-dependent terms in Eq. (24) are zero for a homonuclear spin system, i.e., for $\gamma_1 = \gamma_2$.

Returning to Eq. (21), the vector model can be used to determine the pair evolution under pulsed z magnetic fields. Evolution of scalar spin order, defined for the $I_1-I_2$ pair as $\hat{\mathbf{I}}_1 \cdot \hat{\mathbf{I}}_2 = \hat{I}_{1x}\hat{I}_{2x} + \hat{I}_{1y}\hat{I}_{2y} + \hat{I}_{1z}\hat{I}_{2z} \equiv (\hat{A}_1 + \hat{A}_4)$, is one important example, as a form of spin order that can be long-lived compared to the
longitudinal relaxation time $T_1$ [48] at low fields, and one that is often a product of spin-hyperpolarization procedures involving parahydrogen [66] (see Sec. 3.4). The scalar order is a stationary form of spin order at zero field, but according to Eq. (21) it evolves under $\hat{\mathcal{H}}_Z$ as

$$ \hat{\mathbf{I}}_1 \cdot \hat{\mathbf{I}}_2 \xrightarrow{\hat{\mathcal{H}}_Z t} (\hat{\mathbf{I}}_{1x} \hat{\mathbf{I}}_{2x} + \hat{\mathbf{I}}_{1y} \hat{\mathbf{I}}_{2y}) \cos[(\gamma_1 - \gamma_2)B_z t] \\ - (\hat{\mathbf{I}}_{1x} \hat{\mathbf{I}}_{2y} - \hat{\mathbf{I}}_{1y} \hat{\mathbf{I}}_{2x}) \sin[(\gamma_1 - \gamma_2)B_z t] \\ + \hat{\mathbf{I}}_{1z} \hat{\mathbf{I}}_{2z}. \quad (25) $$

Although none of the terms on the right side of Eq. (25) correspond to a net magnetic moment of the sample, the one on the second line (proportional to $\hat{A}_3$) may evolve into the observable moment $\hat{A}_2$ under $\hat{\mathcal{H}}_J$, as described above, upon switching the total field back to zero. Cyclic permutation of x, y, and z indices in Eq. (25) allows similar results to be obtained for d.c. fields applied along x- and y-axes.

In summary, Eq. (19) provides a framework for discussing transformations between zero-quantum forms of spin order in which conventional NMR concepts and parlance (e.g., flip angle) can be used. In addition to providing a formalism for describing the nature of spin evolution in the extremes of high and zero magnetic fields, it can provide analytical solutions for spin evolution in the low-field spin-dynamics regime, also illustrated in Fig. 5b. Although the formalism is only valid for a spin-1/2 pair, it can often be successfully applied as a first-order approximation of more complex spin systems, including those where there is a dominant J-coupling between a pair of heteronuclear spins and smaller J-couplings with all other nuclei (e.g., [¹³C₁]-benzene).

## 2.4. $X_A_n$ spin systems at zero field

### 2.4.1. Energy-level structure

In the ZULF regime, the evolution of a spin is primarily governed by the J-coupling Hamiltonian. The eigenbasis of a heteronuclear spin system is therefore different from that in the high-field case. We first identify the energy-level structure of such a system.

In isotropic liquids at truly zero field, a heteronuclear spin system evolves exclusively due to the indirect spin-spin interaction. The simplest system that supports such evolution is a system with the $X_A_n$ spin topology, which consists of a set of n magnetically equivalent spins A (typically ¹H nuclei) with pairwise coupling $J_{AA}$ among them. The A-spins are furthermore coupled to a heteronuclear spin X (e.g., ¹³C, ¹⁵N, ³¹P) with the coupling constant $J_{XA}$. The corresponding spin operators are $\hat{\mathbf{S}}$, with the quantum number S, for the X-spin, and $\hat{\mathbf{I}}_{A,i}$, with quantum number $I_{A,i}$, for each of the n A-spins. The general coupling Hamiltonian is of the form

$$ \begin{aligned} \hat{\mathcal{H}}_{J,XA_n} &= 2\pi\hbar \left( \sum_i J_{XA} \hat{\mathbf{S}} \cdot \hat{\mathbf{I}}_{A,i} + \sum_{i,j>i} J_{AA} \hat{\mathbf{I}}_{A,i} \cdot \hat{\mathbf{I}}_{A,j} \right) \\ &= 2\pi\hbar \left( J_{XA} \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_A + \frac{J_{AA}}{2} \left[ \hat{\mathbf{K}}_A^2 - n \hat{\mathbf{I}}_A^2 \right] \right), \end{aligned} \quad (26) $$

where $\hat{\mathbf{K}}_A = \sum_i \hat{\mathbf{I}}_{A,i}$ is the total spin operator of the A-nuclei with the eigenvalue $K_A$, and $\hat{\mathbf{I}}_A^2 = \hat{\mathbf{I}}_{A,1}^2$. Note for an XA system ($n=1$) there is no $J_{AA}$ and the second term in Eq. (26) should be dropped. In this case $\hat{\mathbf{K}}_A = \hat{\mathbf{I}}_A$, it is zero anyway.

In order to evaluate the scalar product in Eq. (26), one can introduce the total angular momentum operator of the heteronuclear pair: $\hat{\mathbf{F}}_{XA} = \hat{\mathbf{K}}_A + \hat{\mathbf{S}}$. Because of the expansion

$$ \hat{\mathbf{F}}_{XA}^2 = (\hat{\mathbf{K}}_A + \hat{\mathbf{S}})^2 = \hat{\mathbf{K}}_A^2 + \hat{\mathbf{S}}^2 + 2\hat{\mathbf{K}}_A \cdot \hat{\mathbf{S}}, \quad (27) $$

we can write

$$ \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_A = \frac{1}{2} (\hat{\mathbf{F}}_{XA}^2 - \hat{\mathbf{K}}_A^2 - \hat{\mathbf{S}}^2). \quad (28) $$

The eigenstates of the Hamiltonian are the eigenstates of $\hat{\mathbf{F}}_{XA}^2$ and $\hat{\mathbf{F}}_{XA,z}$ and they can be written as $|F_{XA}(K_A, S), m_{F_{XA}}\rangle$, where $K_A, S, F_{XA}$, and $m_{F_{XA}}$ are the quantum numbers corresponding to the eigenvalues of $\hat{\mathbf{K}}_A, \hat{\mathbf{S}}, \hat{\mathbf{F}}_{XA}$, and $\hat{\mathbf{F}}_{XA,z}$, respectively.

The energy of the state $|F_{XA}(K_A, S), m_{F_{XA}}\rangle$ is given by

$$ \begin{aligned} E_{F_{XA}(K_A, S), m_{F_{XA}}}^{(0)} &= \langle F_{XA}(K_A, S), m_{F_{XA}} | \hat{\mathcal{H}}_{XAn}^{(0)} | F_{XA}(K_A, S), m_{F_{XA}} \rangle \\ &= \pi \hbar \langle F_{XA}(K_A, S), m_{F_{XA}} | [J_{XA} (\hat{\mathbf{F}}_{XA}^2 - \hat{\mathbf{K}}_A^2 - \hat{\mathbf{S}}^2) + J_{AA} (\hat{\mathbf{K}}_A^2 - n \hat{\mathbf{I}}_A^2)] | F_{XA}(K_A, S), m_{F_{XA}} \rangle \\ &= \pi \hbar J_{XA} [F_{XA}(F_{XA} + 1) - K_A(K_A + 1) - S(S + 1)] + \pi \hbar J_{AA} [K_A(K_A + 1) - nI_A(I_A + 1)], \end{aligned} \quad (29) $$

where $m_{F_{XA}}$ is the magnetic quantum number in the XA basis and superscript (0) indicates that the system evolution is governed only by the internal interactions (unperturbed system). The equation allows one to calculate energy-level diagrams of the $X_A_n$ spin system. Diagrams for XA, $X_A^2$, $X_A^3$ and $X_A^4$ spin systems are shown in Fig. 6. In addition, a diagram for $X_A^6$ is shown in reference [67].

It should be noted that, as in high-field NMR, the coupling

$J_{AA}$ between the magnetically equivalent A-spins can almost always be ignored, even if it has a nonzero value. This is because observable NMR transitions can only occur between states of a common $K_A$ (see Sec. 2.4.2) and while the energy differences between $K_A$ manifolds can depend on $J_{AA}$, the transition frequencies within them do not.

At zero field, each $F_{XA}$ state is $(2F_{XA} + 1)$ times degenerate due to the magnetic sublevel structure. The permutation
symmetry (i.e., magnetic equivalence) of the A-spins can also cause an additional degeneracy, as multiple states can have the same $K_A$. We call this second 'degeneracy' the multiplicity, $g_{K_A}$. As an example, there are two $K_A = 1/2$ manifolds for a methyl group (CH₃), and $g_{1/2} = 2$, while $g_{3/2} = 1$. As a consequence, the zero-field eigenstates $|F_{XA}(K_A, S), m_{F_{XA}}\rangle = |1(1/2, 1/2), m_{F_{XA}}\rangle$ and $|0(1/2, 1/2), 0\rangle$ of an XA₃ spin system ($S = I_i = 1/2$) are doubled (see Fig. 6). Values of $g_{K_A}$ in other ($A_n$) spin systems are tabulated in Table 1.

### 2.4.2. Selection rules

As discussed in Sec. 2.2.2, NMR experiments detect magnetic fields associated with the magnetization of the nuclear ensemble. For a given state of the ensemble, we can find the magnetization vector by taking the trace of the product of the magnetization operator and the density operator, see Eq. (13). We may ask what elements in the density matrix lead to nonzero magnetization. The answer is that non-oscillating magnetization is associated with the populations of eigenstates, while oscillating and rotating magnetization is associated with off-diagonal matrix elements (coherences) between certain non-degenerate eigenstates. Selection rules establish the relation between the coherences and magnetization, as well as which coherences are excited by application of an external magnetic field. More formally, this can be described as follows.

In order to determine the amplitude of the ZULF-NMR signal given in Eq. (13), one has to evaluate the specific matrix elements of the initial density operator $\rho_{uv}(0)$ and the matrix element of the magnetization operator between the corresponding eigenstates. For the XA$_n$ system,

$$ \langle u | \hat{M}_p | v \rangle = \langle u | \gamma_A \hat{F}_{XA,p} + (\gamma_X - \gamma_A) \hat{S}_p | v \rangle, \quad (30) $$

where $\hat{F}_{XA,p} \equiv \hat{K}_{A,p} + \hat{S}_p$.

A convenient approach to evaluate $\langle u | \hat{F}_{XA,p} | v \rangle$ and $\langle u | \hat{S}_p | v \rangle$ is to use the algebra of tensor operators. A spherical tensor operator $\hat{T}_{\kappa,q}^{(k)}$ is a superposition of spin operators for the subset of spins $k$ (e.g., X- or A-spins) with well-defined total angular momentum (or 'tensor rank') $\kappa$ and projection quantum numbers $q$ with respect to global rotations of the system (i.e., rotation of all spins). The quantum numbers $\kappa$ and $q$ mix the XA$_n$ eigenstates in a predictable and relatively simple way, using the rules of addition of angular momentum. As an example, in the case of $\langle u | \hat{S}_p | v \rangle$ ($k = X$), the relevant spherical tensor operators are those with rank $\kappa = 1$:

$$ \hat{S}_z \equiv \frac{\hat{T}_{1,0}^{(X)}}{\sqrt{2}}, \quad (31a) $$

$$ \hat{S}_x \equiv \frac{\hat{T}_{1,1}^{(X)} + \hat{T}_{1,-1}^{(X)}}{2}. \quad (31b) $$

The quantity $\hat{S}_p |v\rangle$ is found using a textbook approach [68]:

$$ \begin{aligned} \hat{T}_{\kappa,q}^{(X)} |F_{XA}(K_A, S), m_F \rangle = & (-1)^{2F_{XA}+\kappa} \sqrt{(2F_{XA}+1)(2\kappa+1)} \\ & \times \sum_{F'_{XA}} \begin{Bmatrix} K_A & S & F_{XA} \\ \kappa & F'_{XA} & S \end{Bmatrix} C_{F'_{XA}m_F, F_{XA}m_F, kq}^{F'_{XA}m'_F} \\ & \times |F'_{XA}(K_A, S), m'_F \rangle, \end{aligned} \quad (32) $$

where $m_F$ is the magnetic quantum number, $C_{F'_{XA}m_F, F_{XA}m_F, kq}^{F'_{XA}m'_F}$ is a Clebsch-Gordan coefficient, and the term in curly brackets is a Wigner 6j-symbol, and prime indicates the partner state in a coherence/transition. Using Eq. (32), the matrix elements are therefore

$$ \begin{aligned} \langle v | \hat{S}_z | u \rangle &= \langle F'_{XA}(K'_A, S'), m'_F | \hat{S}_z | F_{XA}(K_A, S), m_F \rangle \\ &= (-1)^{2F_{XA}+1} \sqrt{\frac{6F_{XA}+3}{2}} \begin{Bmatrix} K_A & S & F_{XA} \\ 1 & F'_{XA} & S \end{Bmatrix} \\ & \times C_{F'_{XA}m'_F, F_{XA}m_F, 10}^{F'_{XA}m'_F} \delta_{S'S} \delta_{K'_A K_A}, \end{aligned} \quad (33a) $$

$$ \begin{aligned} \langle v | \hat{S}_x | u \rangle &= \langle F'_{XA}(K'_A, S'), m'_F | \hat{S}_z | F_{XA}(K_A, S), m_F \rangle \\ &= (-1)^{2F_{XA}+1} \sqrt{\frac{6F_{XA}+3}{2}} \begin{Bmatrix} K_A & S & F_{XA} \\ 1 & F'_{XA} & S \end{Bmatrix} \\ & \times \left( C_{F'_{XA}m'_F, F_{XA}m_F, 1-1}^{F'_{XA}m'_F} + C_{F'_{XA}m_F, F_{XA}m_F, 11}^{F'_{XA}m'_F} \right) \delta_{S'S} \delta_{K'_A K_A}. \end{aligned} \quad (33b) $$

From these formulas, the selection rules for the changes of the quantum numbers $K_A, S$ and $m_{F_{XA}}$, associated with transitions induced by particularly oriented magnetic fields, can be determined. Specifically, a necessary condition for an allowed transition, arising from the 6j-term, is for $|u\rangle$ and $|v\rangle$ to have common values of $S$ and $K_A$ ($\Delta S = 0, \Delta K_A = 0$), which implies that $F_{XA}$ changes by 0 or 1 ($\Delta F_{XA} = 0, \pm 1$). In addition, the Clebsch-Gordan coefficient ensures that $m_F$ is conserved for the transitions induced by magnetic field oriented along $z$ ($\Delta m_{F_{XA}} = 0$), but it may change by $\pm 1$ for a field along $x$ or $y$ ($\Delta m_{F_{XA}} = \pm 1$). These selection rules are summarized graphically in Fig. 7.

The expression on the right-hand side of Eq. (33a) typically evaluates to between 0.4 and 0.5 for XA$_n$ spin systems with $S = 1/2$ and $I = 1/2$ (an exact result $|\langle v | S_z | u \rangle| = 1/2$ is in fact obtained for half-integer $K_A$ and $m_F = \pm 1/2$). Selected values are provided in Table 2 for reference, where the mean value is also given to compare the overall strength of transitions between $F_{XA}$ and $F'_{XA}$. Allowed transitions are therefore detected with approximately equal weighting. It follows that the intensities of the zero-field spectral lines—each of which results from a superposition of coherences with the same energy difference—are approximately proportional to the multiplicity factor $g_{K_A}$ ($2m_F + 1$). An experimental example of this is shown in Fig. 10, which is discussed in Sec. 2.4.4. Additional dependence on $\rho_{uv}$ is discussed in Sec. 2.4.3.

We also note that the selection rules can be understood intuitively by realizing that magnetic dipole transitions are in fact processes where a single photon with zero orbital angular momentum is absorbed or emitted in the course of the transition. As the photon is a spin-1 particle, this immediately leads to the selection rule that the total angular momentum and its projection can only change by 0 or $\pm 1$. The selection rules for the 'intermediate' quantum numbers can also be understood 'without equations' in analogy to how this is done in atomic physics, see, for example, the tutorial discussion in reference [69].

### 2.4.3. Zero-field-NMR signal for a thermally polarized system

In Eq. (13), the values of the matrix elements $\rho_{uv}(0)$ depend on the initial polarizations of the spins (various methods to pro-
<figure><img src="image_8.png" /><figcaption>Figure 6: Energy-level diagrams for small $XA_n$ spin-1/2 systems at zero magnetic field ($n \le 4$, $I = 1/2$, $S = 1/2$). States are grouped by total I-spin angular momentum quantum number $K_A$ as shown by the shaded regions. Outside each region, a solid horizontal line indicates the reference energy for each $K_A$ manifold due to the homonuclear spin coupling $J_{AA}$. Dotted lines indicate the energy-level shifts within each manifold due to the heteronuclear coupling $J_{XA}$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Table 1: Multiplicity of the $K_A$ energy-level manifolds in a spin system comprising $n$ magnetically equivalent spin-1/2 particles. These map directly to the multiplicity of the zero-field eigenstates for $XA_n$ spin system, where $I_A = 1/2$. The value of $g_{K_A}$ in any row ($A_n$) of the table may be derived in a manner reminiscent of Pascal's triangle, by addition of the multiplicities $g_{K_A+1/2}$ and $g_{K_A-1/2}$ from the previous row ($A_{n-1}$). Alternatively, the sequence formulae at the bottom can be used to calculate $g_{K_A}$ along the diagonal arrows. Note that in addition each $K_A$ angular momentum manifold is $(2K_A + 1)$-degenerate due to the projection quantum number. The total number of levels is given by summation of $g_{K_A}$ multiplied by the degeneracy, and equals an expected result: $\sum_{K_A} g_{K_A} (2K_A + 1) = 2^n$.</figcaption></figure>
<table><caption>Table 2: Numerical values for the right-hand side of Eq. (33a) using $S = 1/2$ or $S = 1$, which represents the relative amplitudes of nominally allowed transitions between magnetic sublevels $|F_{XA}(K_A, S), m_F \rangle$ and $|F'_A(K_A, S), m_F \rangle$ under $\hat{S}_z$. The mean (i.e., the average taken over all sublevels from $m_F = -F_{XA}$ to $m_F = F_{XA}$) is provided to indicate the relative probability of the transition $F_{XA} \leftrightarrow F'_A$ at zero magnetic field.</caption><thead><tr><th colspan="5">Angular momentum</th><th colspan="7">Projection</th></tr><tr><th>K<sub>A</sub></th><th>S</th><th>F<sub>XA</sub></th><th>F'<sub>XA</sub></th><th>0</th><th>1/2</th><th>1</th><th colspan="4">|m<sub>F</sub>|</th><th>Mean</th></tr></thead><tbody><tr><td>1/2</td><td>1/2</td><td>0</td><td>1</td><td>0.5</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.5</td></tr><tr><td>1</td><td>1/2</td><td>1/2</td><td>3/2</td><td></td><td>0.471</td><td></td><td></td><td></td><td></td><td></td><td>0.471</td></tr><tr><td>3/2</td><td>1/2</td><td>1</td><td>2</td><td>0.5</td><td></td><td>0.433</td><td></td><td></td><td></td><td></td><td>0.455</td></tr><tr><td>2</td><td>1/2</td><td>3/2</td><td>5/2</td><td></td><td>0.490</td><td></td><td>0.4</td><td></td><td></td><td></td><td>0.445</td></tr><tr><td>5/2</td><td>1/2</td><td>2</td><td>3</td><td>0.5</td><td></td><td>0.471</td><td></td><td>0.373</td><td></td><td></td><td>0.438</td></tr><tr><td>3</td><td>1/2</td><td>3/2</td><td>5/2</td><td></td><td>0.495</td><td></td><td>0.452</td><td></td><td>0.350</td><td></td><td>0.432</td></tr><tr><td>7/2</td><td>1/2</td><td>3</td><td>4</td><td>0.5</td><td></td><td>0.484</td><td></td><td>0.433</td><td></td><td>0.331</td><td>0.428</td></tr><tr><td>1/2</td><td>1</td><td>1/2</td><td>3/2</td><td></td><td>0.236</td><td></td><td></td><td></td><td></td><td></td><td>0.236</td></tr><tr><td>1</td><td>1</td><td>0</td><td>1</td><td>0.408</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.408</td></tr><tr><td>1</td><td>1</td><td>1</td><td>2</td><td>0.289</td><td></td><td>0.25</td><td></td><td></td><td></td><td></td><td>0.263</td></tr><tr><td>3/2</td><td>1</td><td>1/2</td><td>3/2</td><td></td><td>0.373</td><td></td><td></td><td></td><td></td><td></td><td>0.373</td></tr><tr><td>3/2</td><td>1</td><td>3/2</td><td>5/2</td><td></td><td>0.3</td><td></td><td>0.245</td><td></td><td></td><td></td><td>0.272</td></tr><tr><td>2</td><td>1</td><td>1</td><td>2</td><td>0.387</td><td></td><td>0.335</td><td></td><td></td><td></td><td></td><td>0.353</td></tr><tr><td>2</td><td>1</td><td>2</td><td>3</td><td>0.316</td><td></td><td>0.298</td><td></td><td>0.236</td><td></td><td></td><td>0.277</td></tr><tr><td>5/2</td><td>1</td><td>3/2</td><td>5/2</td><td></td><td>0.374</td><td></td><td>0.306</td><td></td><td></td><td></td><td>0.334</td></tr><tr><td>5/2</td><td>1</td><td>5/2</td><td>7/2</td><td></td><td>0.312</td><td></td><td>0.292</td><td></td><td>0.226</td><td></td><td>0.279</td></tr><tr><td>3</td><td>1</td><td>2</td><td>3</td><td>0.378</td><td></td><td>0.356</td><td></td><td>0.282</td><td></td><td></td><td>0.331</td></tr><tr><td>3</td><td>1</td><td>3</td><td>4</td><td>0.327</td><td></td><td>0.317</td><td></td><td>0.283</td><td></td><td>0.216</td><td>0.279</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>Figure 7: Transition rules for x-, y-, and z-oriented magnetic-field pulses in the representative $XA_2$ spin system.</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<table><caption>Table 3: Allowed changes in angular momentum quantum numbers for p-axis-observable transitions in the zero-field NMR spectra of $XA_n$ and $(XA_n)B_m$ systems.</caption><thead><tr><th rowspan="2">System</th><th rowspan="2">p</th><th colspan="3">Total angular momentum</th><th>Projection</th></tr><tr><th>ΔK<sub>A</sub></th><th>ΔS</th><th>ΔF<sub>XA</sub>, ΔF</th><th>Δm<sub>F<sub>XA</sub></sub>, Δm<sub>F</sub></th></tr></thead><tbody><tr><td rowspan="2">XA<sub>n</sub></td><td>x, y</td><td>0</td><td>0</td><td>0, ±1</td><td>±1</td></tr><tr><td>z</td><td>0</td><td>0</td><td>±1</td><td>0</td></tr><tr><td rowspan="2">(XA<sub>n</sub>)B<sub>m</sub></td><td>x, y</td><td>0</td><td>0</td><td>0, ±1</td><td>±1</td></tr><tr><td>z</td><td>0</td><td>0</td><td>±1</td><td>0</td></tr></tbody></table>

duce polarization are discussed in Sec. 3). As in conventional NMR, the conceptually simplest method is to polarize the spin system thermally. This is done by placing the sample in a strong field $B_{pol}$ (typically at least 1 T), generated, for example, with a permanent magnet situated outside of the detection region. After a time of at least $3T_1$, the spin ensemble is close to thermal equilibrium, $\rho_{uv}(0) = \rho_{th}$. For $B_{pol}$ oriented along the z-axis, the

thermal equilibrium density operator of an $XA_n$ system is

$$ \begin{aligned} \hat{\rho}_{\text{th}} &= \frac{1}{Z} \exp\left(\frac{\hat{H}}{k_B T}\right) \\ &\approx \frac{1}{Z} \exp\left[-\frac{\hbar B_{\text{pol}}}{k_B T} \left(\gamma_X \hat{S}_z + \sum_i \gamma_A \hat{I}_{z,i}\right)\right] \quad (34) \\ &\approx \frac{1}{Z} \mathbb{I} - \frac{1}{Z} \frac{\hbar B_{\text{pol}}}{k_B T} \left[(\gamma_X - \gamma_A)\hat{S}_z + \gamma_A \hat{F}_{XA,z}\right], \end{aligned} $$

where $B_{pol} = |B_{pol}|$ and $Z = \Pi_i(2I_i + 1)$ is the partition function. If both spin species X and A are spin-½, then $Z = 2^{n+1}$. It should be noted that there are two approximations used in Eq. (34). The first one is neglecting the J-coupling contribution to the high-field Hamiltonian and the second one is the assumption of operation in the low-polarization limit, in which only the first nontrivial expansion coefficient is retained.

The sample is then rapidly ($\delta t \ll T_1$) shuttled into the zero-or ultralow-field detection region, where its magnetization is manipulated and detected. Under such conditions, one can assume that the high-field thermal equilibrium state is maintained, so that $\rho_{uv}(0) = \rho_{th}$. If one is interested in ZULF-NMR signals of nonzero frequency, only the term proportional to $(\gamma_X - \gamma_A)$ in Eq. (34) is important. This follows because the matrix elements of eigenoperators $\hat{F}_{XA,z}$ are zero for $|u\rangle \neq |v\rangle$, and because the transition frequency is zero for $|u\rangle = |v\rangle$. The observable NMR signal measured along the z-axis, ignoring constant offset terms and relaxation, is therefore

$$ s_z(t) \propto (\gamma_X - \gamma_A)^2 \sum_{u,v \neq u} \langle v | S_z | u \rangle^2 e^{-i\omega_{uv}t}, \quad (35) $$
or in terms of the allowed angular momentum quantum numbers [Eq. (33a)]

$$s_z(t) \propto (\gamma_X - \gamma_A)^2 \sum_{F_{XA'} \neq F_{XA}} \sum_{F_{XA',K_A,S}, m_F} \frac{6F_{XA} + 3}{2} g_{K_A} \\ \times \left( \begin{pmatrix} K_A & S & F_{XA} \\ 1 & F_{XA'} & S \end{pmatrix} C_{F_{XA}m_F, 10}^{F_{XA'}m_F} \right)^2 \\ \times e^{-2\pi i J_{XA} [F_{XA}(F_{XA}+1) - F_{XA'}(F_{XA'}+1)]t} \quad (36)$$

Similar relations for the signals measured along the x- or y-axis can be derived by combining Eq. (33b) and Eq. (34) with Eq. (13).

In Eq. (36), we see that the amplitude of the zero-field NMR signal starting from a thermally polarized sample is proportional to the square of the difference of the gyromagnetic ratios of spins X and A. Other dependencies, however, can be generated starting from the same thermally polarized initial state. Spin-species-selective inversion of the A- or X-spins prior to nonadiabatic switching converts the initial state to

$$\hat{\rho}_{\text{th}} \xrightarrow{(\pi)_A} \frac{1}{Z} \mathbb{I} - \frac{1}{Z} \frac{\hbar B_{\text{pol}}}{k_B T} [(\gamma_X + \gamma_A)\hat{S}_z + \gamma_A \hat{F}_{XA,z}], \quad (37)$$

or

$$\hat{\rho}_{\text{th}} \xrightarrow{(\pi)_X} \frac{1}{Z} \mathbb{I} - \frac{1}{Z} \frac{\hbar B_{\text{pol}}}{k_B T} [(-\gamma_X - \gamma_A)\hat{S}_z + \gamma_A \hat{F}_{XA,z}], \quad (38)$$

respectively. The result is to re-scale the component of $\rho_{uv}$ corresponding to time-dependent magnetization, from $(\gamma_A - \gamma_X)$ to $(-\gamma_A - \gamma_X)$ or $(\gamma_A + \gamma_X)$, respectively [70]. This means that the signal is larger by a factor of $|\gamma_A + \gamma_X|/|\gamma_A - \gamma_X|$ when the gyromagnetic ratios for X and A are of the same sign. The same enhancement factors can be obtained when the spin system is adiabatically transferred to zero field under a magnetic field along z and a pulsed magnetic field along the x- or y-axis is applied before measurement (see reference [48] and its supplementary materials).

### 2.4.4. Experimental spectra

Examples of experimental zero-field spectra of $XA_n$ spin systems are shown in Fig. 8, for simple compounds available in neat liquid form, which can be regarded as ‘standard ZULF-NMR samples’: $[^{13}\text{C}]$-formic acid ($H^{13}\text{COOH}$, XA), $[^{13}\text{C}]$-formaldehyde ($^{13}\text{CH}_2\text{O}$, XA$_2$), and $[^{13}\text{C}]$-methanol ($^{13}\text{CH}_3\text{OH}$, XA$_3$). In the simplest case of an XA spin system at zero field, the only possible quantum numbers are $S = 1/2$ and $K_A = 1/2$ (as per the first row in Fig. 6), giving two sets of states, $F_{XA} = 0$ and $F_{XA} = 1$. There is only one $\Delta F_{XA} = 1$ and $\Delta K_A = 0$ transition at nonzero frequency, at the value of the $J_{XA}$-coupling constant. In formic acid, this is at around 220 Hz (Fig. 8a). For the XA$_2$ spin system, there are three sets of states with distinct $K_A$ and $F_{XA}$ (see Fig. 6). Two of these ($F_{XA} = 1/2$ and $F_{XA} = 3/2$) are derived from $K_A = 1$ and $S = 1/2$ and share an allowed transition at $3J_{XA}/2$. As a result, the zero-field spectrum of $[^{13}\text{C}]$-formaldehyde consists of a single line at around $3J_{XA}/2 = 250$ Hz (Fig. 8b). The XA$_3$ system is more interesting: there are now two $K_A$ manifolds containing non-degenerate energy states. As the third row of Fig. 6 shows, the

<figure><img src="image_12.png" /><figcaption>Figure 8: Zero-field J-spectra of $[^{13}\text{C}]$-formic acid (a), $[^{13}\text{C}]$-formaldehyde (b), and $[^{13}\text{C}]$-methanol (c), as examples of XA, XA$_2$, and XA$_3$ spin systems, respectively. Reprinted (adapted) from reference [47], Copyright (2013), with permission from Elsevier.</figcaption></figure>

$K_A = 1/2$ manifold contains two doubly degenerate states with $F_{XA} = 0$ and $F_{XA} = 1$, while the $K_A = 3/2$ manifold comprises states with $F_{XA} = 1$ and $F_{XA} = 2$. This level structure leads to two transitions at nonzero frequency, at $J_{XA}$ and $2J_{XA}$ for $K_A = 1/2$ and $K_A = 3/2$ respectively, which in $[^{13}\text{C}]$-methanol occur at 140 Hz and 280 Hz, respectively (Fig. 8c). As a general rule, the zero-field NMR spectrum of a spin-$1/2$ $XA_n$ system exhibits transition frequencies at even integer multiples of $J_{XA}/2$ for odd $n$, and odd integer multiples of $J_{XA}/2$ for even $n$, up to a maximum frequency of $(n + 1)J_{XA}/2$. Fig. 9 shows the experimental zero-field spectrum of the octahedral anion $PF_6^-$ (XA$_6$), and Fig. 10 that of the XA$_9$ system ($CH_3O_3P$).

The cases considered so far involve spin-$1/2$ nuclei, but the theory also applies to nuclei with a quadrupolar moment ($S \ge 1$), for instance $^{14}\text{N}$ and $^{2}\text{H}$ (both $S = 1$). Rapid spin relaxation due to nuclear quadrupole interactions typically hinders observation of their signals, although this is not always the case. One example is the ammonium ion $^{14}\text{NH}_4^+$, which has tetrahedral symmetry, so that the electric field gradients at the quadrupolar nitrogen nucleus and, correspondingly, $\hat{H}_Q^{(14N)}$, nearly vanish, resulting in the suppression of quadrupolar relaxation. This spin system is one of the few cases where zero-field J-spectra of coupled spin systems containing quadrupolar nuclei have been detected, with line widths comparable to isotopologs containing only spins-$1/2$. As a result, ammonium is a XA$_4$ spin system that can be studied for X being either spin-1 or spin-$1/2$ [59], see Fig. 11. The transitions in Fig. 11a are readily understood from the theory presented in Sec. 2.4.2. Specifically, the total $^{1}\text{H}$ angular momentum results in three manifolds, $K_A = 0, 1, 2$, where the last two contain one or more nonzero-frequency transitions due to the $\Delta K_A = 0$ selection
<figure><img src="image_13.png" /></figure>

Figure 9: ZULF-NMR spectrum of PF₆⁻ in the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate [BMIm][PF₆]. For an XA₆ spin system, peaks are expected at 3J<sub>XA</sub>/2, 5J<sub>XA</sub>/2, and 7J<sub>XA</sub>/2. The presence of multiple resonances allows the one-bond <sup>31</sup>P–<sup>19</sup>F coupling, <sup>1</sup>J<sub>PF</sub> = 709.974(2) Hz, to be extracted with high precision. The increased noise floor at higher frequencies is related to the relatively small detector bandwidth (~200 Hz)—correcting for the roughly Lorentzian frequency-dependent response leads to digitization noise being multiplied by an exponentially increasing calibration factor. Reprinted from reference [2] Copyright (2021), with permission from Elsevier.

<figure><img src="image_14.png" /></figure>

Figure 11: The zero-field *J*-spectrum of a 60:40 mixture of <sup>15</sup>NH<sub>4</sub>Cl and <sup>14</sup>NH<sub>4</sub>Cl dissolved in an acidic aqueous solution (~6 M NH<sub>4</sub>Cl) as an example of an XA<sub>4</sub> spin systems containing both spin-1 and spin-1/2 X-nuclei. Energy-level diagram for (b) <sup>15</sup>NH<sub>4</sub><sup>+</sup> and (c) <sup>14</sup>NH<sub>4</sub><sup>+</sup> cations. Energy-level shifts due to the <sup>1</sup>H–<sup>1</sup>H coupling *J*<sub>HH</sub> are ignored. Adapted from reference [59] under terms of the Creative Commons CC-BY license 4.0.

<figure><img src="image_15.png" /></figure>

Figure 10: ZULF NMR spectrum of trimethyl phosphite (CH<sub>3</sub>O)<sub>3</sub>P. For an (XA<sub>3</sub>)<sub>3</sub> spin system, peaks are expected at J<sub>XA</sub>, 2J<sub>XA</sub>, 3J<sub>XA</sub>, 4J<sub>XA</sub>, and 5J<sub>XA</sub>, where J<sub>XA</sub> is a three-bond coupling of 10.5 Hz between <sup>1</sup>H and <sup>31</sup>P. Relative line intensities approximately follow the multiplicity pattern for the K<sub>A</sub> eigenstates of an A<sub>9</sub> system, given in Table 1. Adapted from reference [67] under terms of the Creative Commons CC-BY license 4.0.

rule. The observable transitions in <sup>15</sup>NH<sub>4</sub>Cl occur at 3|J<sub>15NH</sub>|/2 and 5|J<sub>15NH</sub>|/2 (Fig. 11b), while in <sup>14</sup>NH<sub>4</sub>Cl resonances are observed at |J<sub>14NH</sub>|, 2|J<sub>14NH</sub>|, and 3|J<sub>14NH</sub>| (Fig. 11c). The difference in the <sup>14</sup>N-H and <sup>15</sup>N-H *J*-coupling frequencies is primarily due to the different gyromagnetic ratios of the two nuclei. From multiple measurements of the same NH<sub>4</sub>Cl sample, the ratio of the *J*-couplings is |J<sub>15NH</sub>/J<sub>14NH</sub>| = 1.4012(4), as compared to |γ<sub>15N-H</sub>/γ<sub>14N-H</sub>| = 1.4027548(5). The small difference between the ratios of |J<sub>15NH</sub>/J<sub>14NH</sub>| and |γ<sub>15NH</sub>/γ<sub>14NH</sub>| manifested as a primary isotope effect of –58 mHz is likely due to the different zero point energies of <sup>14</sup>NH<sub>4</sub><sup>+</sup> and <sup>15</sup>NH<sub>4</sub><sup>+</sup>, leading to a slight difference in electron densities, although further studies are necessary to verify this claim.

## 2.5. (XA<sub>n</sub>)B<sub>m</sub> spin systems at zero field

The theory presented in Sec. 2.4.1 describes the zero-field NMR behavior for a nuclear spin system of simple topology. With additional *J*-couplings between spins and more elaborate topologies, zero-field NMR spectra become increasingly more complex and harder to analyze rigorously. In special cases, however, simplifications can be made. In this context, we now discuss (XA<sub>n</sub>)B<sub>m</sub> spin systems (the modified Pople notation that we use throughout this review is explained in Sec. 1.5).

As one-bond *J*-couplings are usually substantially larger than couplings to more distant spins, the zero-field NMR spectra of (XA<sub>n</sub>)B<sub>m</sub> spin system can be evaluated in leading order by isolating the XA<sub>n</sub> subgroup from the overall spin network. To deal with the higher-order corrections, we analyze the system using perturbation theory [63].
### 2.5.1. Energy-level structure

The interaction Hamiltonian of an $(XA_n)B_m$ spin system at zero field is given by

$$ \hat{\mathcal{H}}_{J(XA_n)B_m} = 2\pi\hbar (J_{XA} \hat{\mathbf{K}}_A \cdot \hat{\mathbf{S}} + J_{XB} \hat{\mathbf{K}}_B \cdot \hat{\mathbf{S}} + J_{AB} \hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B), \quad (39) $$

where couplings between magnetically equivalent spins are ignored, and $\hat{\mathbf{K}}_B$ is the total angular momentum operator of the B-spin subsystem.

Now, we assume that the $J_{XA}$ coupling is the dominant interaction in the system ($|J_{XA}| \gg |J_{AB}|, |J_{XB}|$). This allows us to identify a reference Hamiltonian (i.e., that of the $XA_n$ subsystem, $\hat{\mathcal{H}}^{(0)} = \hat{\mathcal{H}}_{J(XA_n)}$) to approximate the eigenstates at lowest perturbation order. The perturbation Hamiltonian $\hat{\mathcal{H}}_{\text{per}}$ is then defined as

$$ \hat{\mathcal{H}}_{\text{per}} = 2\pi\hbar (J_{XB} \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_B + J_{AB} \hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B). \quad (40) $$

The conventional formula for the first-order energy corrections to each reference eigenstate, i.e., $\Delta E_u^{(1)} = \langle u | \hat{\mathcal{H}}_{\text{per}} | u \rangle$, does not immediately provide a compact analytical result. This is so because neither scalar product term in Eq. (40) commutes with $\hat{\mathcal{H}}_{J(XA_n)}$. Therefore, to simplify the analysis, it is convenient to replace $\hat{\mathcal{H}}_{\text{per}}$ with an approximation. First, it can be assumed that $\hat{\mathbf{K}}_B$ couples directly to $\hat{\mathbf{F}}_{XA}$, instead of coupling independently to $\hat{\mathbf{K}}_B$ and $\hat{\mathbf{S}}$. Hence, B effectively interacts with the projections of $\hat{\mathbf{K}}_B$ and $\hat{\mathbf{S}}$ onto $\hat{\mathbf{F}}_{XA}$, and one can write the first-order perturbation Hamiltonian as

$$ \begin{aligned} \hat{\mathcal{H}}^{(1)} &= 2\pi\hbar(J_{XB}\hat{\mathbf{S}}^{\parallel}\cdot\hat{\mathbf{K}}_B + J_{AB}\hat{\mathbf{K}}_A^{\parallel}\cdot\hat{\mathbf{K}}_B) \\ &= 2\pi\hbar(J_{XB}^{(1)} + J_{AB}^{(1)})\hat{\mathbf{F}}_{XA}\cdot\hat{\mathbf{K}}_B, \end{aligned} \quad (41) $$

where $\hat{\mathbf{S}}^{\parallel}$ and $\hat{\mathbf{K}}_A^{\parallel}$ are components of $\hat{\mathbf{S}}$ and $\hat{\mathbf{K}}_A$ parallel to $\hat{\mathbf{F}}_{XA}$ and

$$ J_{XB}^{(1)} = J_{XB} \left[ \frac{F_{XA}(F_{XA} + 1) + S(S + 1) - K_A(K_A + 1)}{2F_{XA}(F_{XA} + 1)} \right], \quad (42a) $$

$$ J_{AB}^{(1)} = J_{AB} \left[ \frac{F_{XA}(F_{XA} + 1) + K_A(K_A + 1) - S(S + 1)}{2F_{XA}(F_{XA} + 1)} \right] \quad (42b) $$

are the effective J-coupling constants specific to each state. Eq. (41) allows simple evaluation of the first-order energy corrections by using the total-spin operator $\hat{\mathbf{F}}$

$$ \hat{\mathbf{F}} = \hat{\mathbf{F}}_{XA} + \hat{\mathbf{K}}_B, \quad (43) $$

in which the Hamiltonian (41) can be rewritten as

$$ \hat{\mathcal{H}}^{(1)} = \pi h (J_{XB}^{(1)} + J_{AB}^{(1)}) (\hat{\mathbf{F}}^2 - \hat{\mathbf{F}}_{XA}^2 - \hat{\mathbf{K}}_B^2). \quad (44) $$

With this form of the Hamiltonian and the general formula for the first-order energy corrections, $\Delta E^{(1)} = \langle u | \hat{\mathcal{H}}^{(1)} | u \rangle$, the resulting correction for the energy of the state $|u\rangle$ is

$$ \Delta E_u^{(1)} = \pi \hbar (J_{BX}^{(1)} + J_{AB}^{(1)}) [F(F+1) - F_{XA}(F_{XA}+1) - K_B(K_B+1)]. \quad (45) $$

Here, $F$ and $K_B$ are the quantum numbers corresponding to the $\hat{\mathbf{F}}$ and $\hat{\mathbf{K}}_B$ operators, respectively.

The first-order perturbation approximation is usually sufficient to describe the $(XA_n)B_m$ system, especially in situations where $|J_{XB} + J_{AB}|$ is much smaller than $|J_{XA}|$. For a better estimate, however, higher-order energy corrections can also be calculated. A second-order correction is obtained using the generic result of perturbation theory with the untruncated form of the perturbation Hamiltonian $\hat{\mathcal{H}}_{\text{per}}$ [Eq. (40)] instead of $\hat{\mathcal{H}}^{(1)}$:

$$ \Delta E_u^{(2)} = \sum_{v \neq u} \frac{|\langle u | \hat{\mathcal{H}}_{\text{per}} | v \rangle|^2}{E_u^{(0)} - E_v^{(0)}}, \quad (46) $$

where the summation runs over all zero-order eigenstates $|v\rangle$ different in energy to $|u\rangle$ and of the same value of $F$.

To calculate the numerator of Eq. (46), one needs to calculate the specific matrix elements

$$ \langle v | \hat{\mathcal{H}}_{\text{per}} | u \rangle = 2\pi\hbar (J_{XB} \langle v | \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_B | u \rangle + J_{AB} \langle v | \hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B | u \rangle), \quad (47) $$

which is not trivial because the two coupling operators $\hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_B$ and $\hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B$ do not share an eigenbasis. To avoid this problem, however, one can evaluate the $\langle u | \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_B | v \rangle$ and $\langle u | \hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B | v \rangle$ terms separately in their respective eigenbases, and add them together afterwards [63]. The following results can be used:

$$ \begin{aligned} \langle v | \hat{\mathbf{S}} \cdot \hat{\mathbf{K}}_B | u \rangle = & \sum_{F_B = |S - K_B|}^{S + K_B} \left( F_B + \frac{1}{2} \right) \sqrt{(2F_{XA} + 1)(2F'_{XA} + 1)} \\ & \times \left\{ \begin{matrix} S & K_B & F_B \\ F & K_A & F_{XA} \end{matrix} \right\} \left\{ \begin{matrix} S & K_B & F_B \\ F & K_A & F'_{XA} \end{matrix} \right\} \\ & \times [F_B(F_B + 1) - S(S + 1) - K_B(K_B + 1)], \end{aligned} \quad (48a) $$

$$ \begin{aligned} \langle v | \hat{\mathbf{K}}_A \cdot \hat{\mathbf{K}}_B | u \rangle = & \sum_{K_{AB} = |K_A - K_B|}^{K_A + K_B} \left( K_{AB} + \frac{1}{2} \right) \sqrt{(2F_{XA} + 1)(2F'_{XA} + 1)} \\ & \times \left\{ \begin{matrix} K_A & K_B & K_{AB} \\ F & S & F_{XA} \end{matrix} \right\} \left\{ \begin{matrix} K_A & K_B & F_B \\ F & S & F'_{XA} \end{matrix} \right\} \\ & \times [K_{AB}(K_{AB} + 1) - K_A(K_A + 1) - K_B(K_B + 1)]. \end{aligned} \quad (48b) $$

An illustrative example of this perturbation approach is methyl-[1-¹³C]-formate. The system can be regarded as a strongly coupled ¹³C-¹H spin pair XA (the formate group) coupled to the methyl group, $B_3$, so the overall topology of the molecule is (XA)$B_3$. Energy levels of the XA system are shown in Fig. 12a. Upon coupling to the $B_3$ subsystem, these form two manifolds ($K_B = 1/2$ and $K_B = 3/2$), each with a rich energy-level structure, as shown in Fig. 12b. Analytical expressions for the first- and second-order corrected energies are provided in Table 4. From these arise detectable shifts and/or splittings of the zero-field NMR lines as discussed further in Sec. 2.5.3.

For [¹³C]-methyl formate and most other $(XA_n)B_m$ systems studied so far (see, for example, references [63, 47]), the second-order perturbation theory appears to reproduce the
<table><caption>Table 4: The lowest-order perturbation corrections to the zero-field energy levels $|F(F_{XA}, K_{B})\rangle$ of an (XA)B$_3$ spin system of $K_A = 1/2$ and $S = 1/2$.</caption><thead><tr><th>F</th><th>F<sub>XA</sub></th><th>K<sub>B</sub></th><th>E<sup>(0)</sup>/h</th><th>ΔE<sup>(1)</sup>/h</th><th>ΔE<sup>(2)</sup>/h</th></tr></thead><tbody><tr><td>5/2</td><td>1</td><td>3/2</td><td><sup>J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td><sup>3(J<sub>XB</sub>+J<sub>AB</sub>)</sup>&frasl;<sub>4</sub></td><td>0</td></tr><tr><td>3/2</td><td>1</td><td>3/2</td><td><sup>J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td>-<sup>(J<sub>XB</sub>+J<sub>AB</sub>)</sup>&frasl;<sub>2</sub></td><td><sup>15(J<sub>XB</sub>-J<sub>AB</sub>)<sup>2</sup></sup>&frasl;<sub>16J<sub>XA</sub></sub></td></tr><tr><td>1/2</td><td>1</td><td>3/2</td><td><sup>J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td>-<sup>5(J<sub>XB</sub>+J<sub>AB</sub>)</sup>&frasl;<sub>4</sub></td><td>0</td></tr><tr><td>3/2</td><td>1</td><td>1/2</td><td><sup>J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td><sup>(J<sub>XB</sub>+J<sub>AB</sub>)</sup>&frasl;<sub>4</sub></td><td>0</td></tr><tr><td>1/2</td><td>1</td><td>1/2</td><td><sup>J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td>-<sup>(J<sub>XB</sub>+J<sub>AB</sub>)</sup>&frasl;<sub>2</sub></td><td><sup>3(J<sub>XB</sub>-J<sub>AB</sub>)<sup>2</sup></sup>&frasl;<sub>16J<sub>XA</sub></sub></td></tr><tr><td>3/2</td><td>0</td><td>3/2</td><td>-<sup>3J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td>0</td><td>-<sup>15(J<sub>XB</sub>-J<sub>AB</sub>)<sup>2</sup></sup>&frasl;<sub>16J<sub>XA</sub></sub></td></tr><tr><td>1/2</td><td>0</td><td>1/2</td><td>-<sup>3J<sub>XA</sub></sup>&frasl;<sub>4</sub></td><td>0</td><td>-<sup>3(J<sub>XB</sub>-J<sub>AB</sub>)<sup>2</sup></sup>&frasl;<sub>16J<sub>XB</sub></sub></td></tr></tbody></table>

energy-level structure quite well. However, higher precision may be desired, for which one may need to numerically solve the eigenvalues of the Hamiltonian to find the exact result. This approach can be also implemented when more complex spin systems containing several inequivalent spins are considered [71]. In this context, quantum simulation provides intriguing medium-term perspectives (see Sec. 7.1.6).

### 2.5.2. Selection rules

The rules for the change of quantum numbers associated with allowed transitions in $(XA)_n B_m$ systems are a generalization of the rules presented in Sec. 2.4.2. To consider them let us first note that the total magnetization of the considered system in the $p$ direction is

$$ \langle u | \hat{M}_p | v \rangle = \langle u | \gamma_X \hat{S}_p + \gamma_A \hat{K}_{A,p} + \gamma_B \hat{K}_{B,p} | v \rangle, \quad (49) $$

where there is an additional contribution from the B-spins with respect to the $XA_n$ system. In the low-order perturbation approximation, we can keep the eigenbases of the uncoupled reference systems: $|v_{XA}\rangle|v_B\rangle = |F'_{XA}(K'_A), m'_F\rangle|K'_B, m'_B\rangle$ and $|u_{XA}\rangle|u_B\rangle = |F_{XA}(K_A), m_F\rangle|K_B, m_B\rangle$, which allows us to rewrite Eq. (49) as

$$ \langle u | M_p | v \rangle = \langle u_{XA} | \gamma_A \hat{F}_{XA,p} + (\gamma_X - \gamma_A) \hat{S}_p | v_{XA} \rangle \langle u_B | v_B \rangle + \langle u_{XA} | v_{XA} \rangle \langle u_B | \gamma_B \hat{K}_{B,p} | v_B \rangle. \quad (50) $$

This formula shows that in addition to the previously introduced selection rules for the $XA_n$ spin system: $\Delta K_A = 0$, $\Delta F_{XA} = 0$, $\pm 1$ and (trivially) $\Delta S = 0$, there is an additional rule: $\Delta K_B = 0$. In addition, we have $\Delta F = 0$, $\pm 1$ because the overall magnetization remains a vector quantity. Finally, if we choose to define the detection axis as $z$, $\Delta m_F = 0$, and for the other two directions $\Delta m_F = \pm 1$. In the $(XA)_n B_m$ spin system, the last selection rules arise instead of the rule for the $m_{F_{XA}}$ in the $XA_n$ spin system. For a detailed derivation, we refer the reader to reference [63].

Two groups of transitions are produced in the $(XA)_n B_m$ case. One of these is a set of high-frequency transitions observed near $J_{XA}$ or its full- or half-integer multiples. A second

group, absent in the $XA$ system, arises at lower frequency due to the transitions between states associated with the B-spin coupling. The frequency of the latter is determined by the $J_{XB}$ and $J_{AB}$ coupling constants.

The transition frequencies of a representative (XA)B$_3$ spin system are summarized in Table 5. As described in Sec. 2.5.3, this complex energy-level structure and the selection rules described lead to information-rich zero-field NMR spectra.

### 2.5.3. Experimental spectra

As a representative example of an $(XA)_n B_m$ spin system, we consider methyl [1-$^{13}$C]-formate ($^{1}$H-$^{13}$COOCH$_3$), a compound with (XA)B$_3$ spin topology. The experimental zero-field NMR spectrum of the pure [1-$^{13}$C]-enriched compound in liquid form is shown in Fig. 12c. To zeroth order, the $^{13}$C-$^{1}$H spin pair forms an $XA$ system with an energy-level structure shown in Fig. 12a, identical to that given in Fig. 6 and discussed in Sec. 2.4. The formyl $^{13}$C-$^{1}$H pair is chosen as the reference system because its J-coupling constant is by far the largest in the molecule (around 226.8 Hz). The three magnetically equivalent $^{1}$H nuclei in the methyl group are the B-spins with $K_B = 1/2$ and $K_B = 3/2$ manifolds, which couple to the $XA$ reference system through scalar couplings $^3J_{HC} \equiv J_{BX} = 4.0$ Hz and $^4J_{HH} \equiv J_{AB} = -0.8$ Hz [63]. The energy-level structures incorporating interaction with the B nuclei are shown in Fig. 12b. The spectrum simulated using first- and second-order perturbation theory largely agree with the experimental one, although full numerical diagonalization reveals higher-order effects that cause small frequency shifts and lift degeneracies between some of the energy levels. The numerical approach is also adopted for larger spin systems where concise analytical expressions for the perturbation corrections are not generally available (see, for example, references [72, 67]).

The spectra of the aromatic compounds shown in Fig. 13, further illustrate important general points regarding ZULF NMR spectra of multi-spin systems. First, different spin topologies generally result in dramatically different spectral patterns (Fig. 13a). Second, manifolds with same total angular momentum can give rise to similar partial spectral patterns, such as in Fig. 13b, where a phenyl (-C$_6$H$_5$) group is coupled to the $K_A = 1/2$ manifold of a $^{13}$CHO and a $^{13}$CH$_3$ group, respectively. Third, even though certain spectral patterns (or partial patterns) for different compounds may be similar (Fig. 13b), chemical information can still be obtained from the absolute frequencies, as is done with chemical shifts in high-field NMR.

## 2.6. $XA_n$ spin systems in ultralow magnetic field

### 2.6.1. Energy-level structure

We now consider an $XA_n$ system subjected to a magnetic field of strength $B_{ULF}$ along the $z$-axis. The general Hamiltonian of such a system is

$$ \hat{\mathcal{H}}_{ULF,XA_n} = \hat{\mathcal{H}}_{J,XA_n} - \hbar \left( \gamma_X \hat{S}_z + \gamma_A \sum_{i=1}^{n} \hat{I}_{z,i} \right) B_{ULF}. \quad (51) $$

In the ultralow-field regime (see Sec. 1.2), the energy-level structure is obtained using the 'reference' zero-field Hamil-
<table><caption>Table 5: Frequencies of allowed transitions for an (XA)B₃ system in zero magnetic field ($K_A = 1/2$, $S = 1/2$).</caption><thead><tr><th></th><th>Lower state<br>|F<sub>XA</sub>, K<sub>B</sub>, F⟩</th><th>Upper state<br>|F<sub>XA</sub>, K<sub>B</sub>, F⟩</th><th>Reference</th><th>Frequency<br>1<sup>st</sup> order correction</th><th>2<sup>nd</sup> order correction</th></tr></thead><tbody><tr><td>ν<sub>1</sub></td><td>|1, 1/2, 1/2⟩</td><td>|1, 1/2, 3/2⟩</td><td>0</td><td>3/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>-3(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr><tr><td>ν<sub>2</sub></td><td>|1, 3/2, 1/2⟩</td><td>|1, 3/2, 3/2⟩</td><td>0</td><td>3/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+15(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr><tr><td>ν<sub>3</sub></td><td>|1, 3/2, 3/2⟩</td><td>|1, 3/2, 5/2⟩</td><td>0</td><td>5/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>-15(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr><tr><td>ν<sub>4</sub></td><td>|0, 3/2, 3/2⟩</td><td>|1, 3/2, 1/2⟩</td><td>J<sub>XA</sub></td><td>-5/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+15(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr><tr><td>ν<sub>5</sub></td><td>|0, 1/2, 1/2⟩</td><td>|1, 1/2, 1/2⟩</td><td>J<sub>XA</sub></td><td>-1/2(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+3(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 8J<sub>XA</sub></td></tr><tr><td>ν<sub>6</sub></td><td>|0, 3/2, 3/2⟩</td><td>|1, 3/2, 3/2⟩</td><td>J<sub>XA</sub></td><td>-1/2(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+15(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 8J<sub>XA</sub></td></tr><tr><td>ν<sub>7</sub></td><td>|0, 1/2, 1/2⟩</td><td>|1, 1/2, 3/2⟩</td><td>J<sub>XA</sub></td><td>1/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+3(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr><tr><td>ν<sub>8</sub></td><td>|0, 3/2, 3/2⟩</td><td>|1, 3/2, 5/2⟩</td><td>J<sub>XA</sub></td><td>3/4(J<sub>BX</sub> + J<sub>AB</sub>)</td><td>+15(J<sub>BX</sub>-J<sub>AB</sub>)<sup>2</sup> / 16J<sub>XA</sub></td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>Figure 12: Zero-field NMR spectroscopy of methyl [¹³C]formate. Spin eigenstates and allowed transitions are shown for the XA (¹H-¹³C) pair in formate (a) when isolated from the methyl group and (b) when coupled to the three methyl protons (B₃), producing separate manifolds with quantum numbers $K_B = 3/2$ and $K_B = 1/2$. (c) The experimental spectrum (top) and expected spectra based on first- and second-order perturbation corrections (bottom), represented by solid black and dashed red vertical lines, respectively. Reprinted (adapted) from reference [63] with the permission of AIP publishing.</figcaption></figure>

tonian [the first term Eq. (51)] and the finite-field perturbation Hamiltonian $\hat{H}_{\text{per}}$ [the second term in Eq. (51)]. In the first-order approximation, the noncommuting part of $\hat{H}_{\text{per}}$ is ignored, resulting in a simpler expression for the perturbation Hamiltonian:

$$ \hat{H}^{(1)} = -\hbar\gamma_{XA,n}(F_{XA}, K_A)B_{ULF}\hat{F}_{XA,z}, \quad (52) $$

where we introduced the quantity $\gamma_{XA,n}$ as an effective gyromagnetic ratio for the zero-field eigenstate manifolds of common $F_{XA}$, $K_A$, and $S$. Values of $\gamma_{XA,n}$ can be determined through the formula

$$ \gamma_{XA,n} = \frac{1}{F_{XA}} \sum_{m_{K_A}=-K_A}^{K_A} \sum_{m_S=-S}^{S} (\gamma_A m_{K_A} + \gamma_X m_S) \left( C_{K_A m_{K_A}, S m_S}^{F_{XA}, F_{XA}} \right)^2, \quad (53) $$

and some of these are tabulated in Table 6, where we note that the identity $\gamma_{XA,n} \equiv \gamma_X$ holds for the special scenario of total magnetic equivalence: $\gamma_A = \gamma_X$. The matrix representation of $\hat{H}^{(1)}$ in the zero-field eigenbasis is therefore diagonal, and the shifted energy levels correct to first order are given by

$$ E_{F_{XA}(K_A, S)m_{F_{XA}}}^{(1)} = \pi J_{AX} [F_{XA}(F_{XA} + 1) - K_A(K_A + 1) - S(S + 1)] \\ - \gamma_{XA,n}(F_{XA}, K_A)B_{ULF}m_{F_{XA}}, \quad (54) $$

Beyond the first-order regime, the energy dependence of $m_{F_{XA}}$ sublevels on magnetic field is generally nonlinear. For example, for any $XA_n$ spin system with $S=1/2$, diagonalization of the Hamiltonian $\hat{H}_{ULF,XA_n}$ gives an exact expression for the energies of the levels as a function of the magnetic field $B_{ULF}$:

$$ E_{F_{XA}(K_A, S)m_{F_{XA}}} = -\hbar\gamma_A m_{F_{XA}} B_{ULF} + 2\pi\hbar J_{XA} \left[ -\frac{1}{4} \pm S \sqrt{\left(K_A + \frac{1}{2}\right)^2 - 2m_F x_{ULF} + x_{ULF}^2} \right], \quad (55) $$
<figure><img src="image_17.png" /><figcaption>Figure 14: Energy-level diagram for a XA₃ spin system in the ultralow-field regime; a magnetic field is applied along the z-axis through which we define the projection quantum number $m_{F_{XA}}$. Arrows correspond to transitions where magnetization is observed along a transverse axis; red arrows indicate transitions with $\Delta F_{XA} = \pm 1$; blue arrows indicate transitions between magnetic sublevels of a given $F_{XA}$ state.</figcaption></figure>

<table><caption>Table 6: Effective gyromagnetic ratios for XAₙ spin manifolds of total angular momentum quantum numbers Kₐ, S and FₓA in the ultralow-field regime. Note that γₓAₙ does not depend explicitly on n, only through the quantum numbers.</caption><thead><tr><th>K<sub>A</sub></th><th>S</th><th>F<sub>XA</sub></th><th>γ<sub>XA<sub>n</sub></sub></th></tr></thead><tbody><tr><td>0</td><td>1/2</td><td>1/2</td><td>γ<sub>X</sub></td></tr><tr><td>1/2</td><td>1/2</td><td>1</td><td>(γ<sub>X</sub> + γ<sub>A</sub>)/2</td></tr><tr><td>1</td><td>1/2</td><td>1/2</td><td>(-γ<sub>X</sub> + 4γ<sub>A</sub>)/3</td></tr><tr><td>1</td><td>1/2</td><td>3/2</td><td>(γ<sub>X</sub> + 2γ<sub>A</sub>)/3</td></tr><tr><td>3/2</td><td>1/2</td><td>1</td><td>(-γ<sub>X</sub> + 5γ<sub>A</sub>)/4</td></tr><tr><td>3/2</td><td>1/2</td><td>2</td><td>(γ<sub>X</sub> + 3γ<sub>A</sub>)/4</td></tr><tr><td>2</td><td>1/2</td><td>3/2</td><td>(-γ<sub>X</sub> + 6γ<sub>A</sub>)/5</td></tr><tr><td>2</td><td>1/2</td><td>5/2</td><td>(γ<sub>X</sub> + 4γ<sub>A</sub>)/5</td></tr><tr><td>0</td><td>1</td><td>1</td><td>γ<sub>X</sub></td></tr><tr><td>1/2</td><td>1</td><td>1/2</td><td>(4γ<sub>X</sub> - γ<sub>A</sub>)/3</td></tr><tr><td>1/2</td><td>1</td><td>3/2</td><td>(2γ<sub>X</sub> + γ<sub>A</sub>)/3</td></tr><tr><td>1</td><td>1</td><td>1</td><td>(γ<sub>X</sub> + γ<sub>A</sub>)/2</td></tr><tr><td>1</td><td>1</td><td>2</td><td>(γ<sub>X</sub> + γ<sub>A</sub>)/2</td></tr><tr><td>3/2</td><td>1</td><td>1/2</td><td>(-2γ<sub>X</sub> + 5γ<sub>A</sub>)/3</td></tr><tr><td>3/2</td><td>1</td><td>3/2</td><td>(4γ<sub>X</sub> + 11γ<sub>A</sub>)/15</td></tr><tr><td>3/2</td><td>1</td><td>5/2</td><td>(2γ<sub>X</sub> + 3γ<sub>A</sub>)/5</td></tr><tr><td>2</td><td>1</td><td>1</td><td>(-γ<sub>X</sub> + 3γ<sub>A</sub>)/2</td></tr><tr><td>2</td><td>1</td><td>2</td><td>(γ<sub>X</sub> + 5γ<sub>A</sub>)/6</td></tr><tr><td>2</td><td>1</td><td>3</td><td>(γ<sub>X</sub> + 2γ<sub>A</sub>)/3</td></tr></tbody></table>

field and the effective gyromagnetic ratio [Eq. (53)].

Three regions of the ULF-NMR spectrum of the XA₃ spin system [¹³C]-methanol, corresponding to the discussed transitions, are shown in Fig. 15. In the low-frequency part of the spectrum, three resonances, associated with the transitions between magnetic sublevels of the $F_{XA} = 1$ and $F_{XA} = 2$ states, and an additional peak at 4 Hz, originating from the solvent (water), are visible. The amplitudes of these peaks are about a factor of 3 to 4 times larger than high-frequency peaks observed at around 141 (two peaks) and 282 Hz (six peaks), corresponding to the $\Delta F_{XA} \pm 1$ transitions, as shown in Fig. 14. The presented spectrum demonstrates the abundant analytical information provided by ULF NMR.

<figure><img src="image_18.png" /><figcaption>Figure 15: Experimentally measured ULF NMR magnitude spectrum of [¹³C]-methanol, a representative XA₃ system, at a magnetic field of 129 nT. The spectrum is divided into three regions: low-frequency peaks, corresponding to the transitions between $m_{F_{XA}}$ sublevels within the same $F_{XA}$ state; two peaks around $J_{XA} = 141$ Hz corresponding to transitions in the $K_A = 1/2$ manifold, and the six peaks around $2J_{XA} = 282$ Hz corresponding to transitions in the $K_A = 3/2$ manifold. The largest low-frequency peak around 3.5 Hz does not originate from the XA₃ system, but from the residual water in the sample. The schematic energy diagram of the system is shown in Fig. 14. Adapted from reference [74].</figcaption></figure>

## 2.7. Untruncated couplings

One of the most intriguing aspects of ZULF NMR is access to terms of interaction tensors that are suppressed in high-field NMR (see also the discussion in Sec. 1.1). In ZULF NMR, the dominant interactions are local spin-spin couplings, which involve coupling tensors with a different symmetry than the Zeeman Hamiltonian $\hat{H}_Z$. At high field, interaction tensors are reduced due to the dominance of $\hat{H}_Z$. That is, only parts that commute with $\hat{H}_Z$ contribute to the observable signal. Such a truncation of interaction Hamiltonians can be understood as coherent averaging in the interaction representation of $\hat{H}_Z$ [54]. Note, however, that the truncated tensor coupling terms retain a role in relaxation.

In the absence of a dominant magnetic field, we have access to a wider range of terms. These include, for instance, the antisymmetric J-coupling terms [75], several terms of the direct dipole-dipole coupling [57], and a number of postulated but as-yet unobserved exotic interactions such as those mediated by pseudoscalar (axion-like) bosons [76, 77, 78], which would lead to anomalous spin-spin tensor couplings, most of which do not commute with $\hat{H}_Z$.
<figure><img src="image_19.png" /><figcaption>Figure 16: Dipole-dipole interactions of two nuclear spins and matrix plot of the dipolar alphabet (see main text).</figcaption></figure>

As a first example, consider the dipole-dipole interaction, described by the Hamiltonian $\hat{H}_{DD}$, see Eq. (2). For a spin-½ pair, $\hat{H}_{DD}$ can be rewritten in the form of the so-called ‘dipolar alphabet’ [79, 80]:

$$ \hat{H}_{DD} = -\frac{\mu_0 \gamma_1 \gamma_2 \hbar}{4\pi r^3} (\hat{A} + \hat{B} + \hat{C} + \hat{D} + \hat{E} + \hat{F}), \quad (57) $$

where (with the internuclear vector $\mathbf{r}$ expressed in polar coordinates $\theta$ and $\phi$, and $\hat{I}_{j\pm} = \hat{I}_{jx} \pm \hat{I}_{jy}$)

$$ \begin{aligned} \hat{A} &= \hat{I}_{1z}\hat{I}_{2z}(3 \cos^2 \theta - 1), \\ \hat{B} &= (\hat{I}_{1+}\hat{I}_{2-} + \hat{I}_{1-}\hat{I}_{2+}) \frac{1 - 3 \cos^2 \theta}{4}, \\ \hat{C} &= (\hat{I}_{1+}\hat{I}_{2z} + \hat{I}_{1z}\hat{I}_{2+}) \frac{3 \sin(2\theta)e^{-i\phi}}{4}, \\ \hat{D} &= (\hat{I}_{1-}\hat{I}_{2z} + \hat{I}_{1z}\hat{I}_{2-}) \frac{3 \sin(2\theta)e^{i\phi}}{4}, \\ \hat{E} &= \hat{I}_{1+}\hat{I}_{2+} \frac{3 \sin^2 \theta e^{-i2\phi}}{4}, \\ \hat{F} &= \hat{I}_{1-}\hat{I}_{2-} \frac{3 \sin^2 \theta e^{i2\phi}}{4}. \end{aligned} \quad (58) $$

A schematic matrix representations of these terms is given in Fig. 16, with $\alpha$ and $\beta$ representing spin-up and spin-down states, respectively. Of the six terms, only $\hat{A}$ commutes with $\hat{H}_z$. Therefore, at high field and for a heteronuclear pair, $\hat{H}_{DD}$ reduces to $\hat{A}$. (For a homonuclear pair, $\hat{B}$ also has to be taken into account, as the operator $\hat{I}_{1+}\hat{I}_{2-} + \hat{I}_{1-}\hat{I}_{2+}$ is time-independent in the rotating frame, provided that the spins are magnetically inequivalent.) By contrast, ZULF NMR provides ways to observing all six terms of the dipolar alphabet [57], thereby giving access to additional information on molecular structure and dynamics. This is discussed in Sec. 6.4.

An interaction that is entirely suppressed by truncation at high magnetic field is the rank-1 antisymmetric $J$-coupling, a nuclear-spin analog of the Dzyaloshinskii-Moriya interaction [81, 82, 83]. This interaction is connected to molecular chirality, and it was proposed that it may be observed in ZULF-NMR experiments [75]. Such experiments would enable differentiation between enantiomers without adding any additional chiral agent to the sample [84]. Possible routes to observing antisymmetric $J$-coupling terms with ZULF NMR are discussed in Sec. 7.2.4 (in the context of molecular chirality) and

in Sec. 7.1.3 (in the context of parity violation in nonchiral systems).

### 3. Spin polarization for ZULF NMR

#### 3.1. Thermal polarization

As already explored in Sec. 2.4.3, nuclear spin systems can become polarized by reaching thermodynamic equilibrium in a magnetic field $\mathbf{B}_0$. In particular, for a sample consisting of particles with nuclear spin quantum number $I$, the magnetization $\mathbf{M}$ (in SI units, $\text{J T}^{-1} \text{m}^{-3}$) is given by the product of individual nuclear magnetic moments $\gamma_I \hbar I$, the ensemble spin polarization $\mathbf{P}$ (a dimensionless vector of magnitude between 0 and 1), and the spin number density $n$. Given Boltzmann statistics and in the high-temperature approximation (which implies that the thermal energy of the system, $k_B T$, is much larger than the energy gap between spin levels), the magnetization can be estimated as [79]

$$ \mathbf{M} = n \cdot \gamma_I \hbar I \cdot \mathbf{P}_{\text{thermal}} = n \cdot \gamma_I \hbar I \cdot \frac{\gamma_I \hbar (I + 1) \mathbf{B}_0}{3 k_B T}, \quad (59) $$

where $\gamma_I$ is the spin-$I$ gyromagnetic ratio. As a practical example, consider a sample of water ($\text{H}_2\text{O}$, $\gamma_H/(2\pi) \approx 42.6 \text{ MHz/T}$, $I = 1/2$) polarized at 3 T and room temperature ($T = 298 \text{ K}$). Thermal nuclear polarization is on the order of $P_{\text{thermal},z} = 10^{-5}$, and the corresponding magnetization expressed in magnetic-field units ($\mu_0 M$) is on the order of 100 pT, where $\mu_0$ is the vacuum permeability. Geometry effects on the actual field measured will be discussed in Sec. 5.3.

Under ZULF conditions, the thermal equilibrium polarization is so low that this essentially precludes direct acquisition of NMR signals. One solution is to temporarily apply a higher field to increase polarization prior to signal detection. This can be done, for example, by mechanically shuttling the sample between a permanent magnet array and the detection region [72, 85] (see Fig. 3a), or by applying millitesla fields using electromagnets in situ [37, 86, 87]. These methods are sufficient for observing highly concentrated species in solution ($\ge 100 \text{ mM}$), but for samples at millimolar concentrations or below, dedicated hyperpolarization techniques are needed [88]. Using such techniques, polarization levels above 50% ($P > 0.5 \gg P_{\text{thermal}}$) for $^1\text{H}$ [89, 90, 91], $^{13}\text{C}$ [89, 91] and above 20% ($P > 0.2 \gg P_{\text{thermal}}$) for $^{15}\text{N}$ [92, 93] have been demonstrated.

#### 3.2. From thermal polarization to hyperpolarization

Generally, hyperpolarization implies nonthermal distributions of populations among spin energy levels, thus, all forms of spin order apart from thermal-equilibrium magnetization (which may include, for example, alignment, two-spin order, dipole order, etc.) are referred to below as hyperpolarization [94]. We remark that the efficacy of a given hyperpolarization technique is often dependent on the magnetic field at which hyperpolarization is performed. Several novel approaches specifically rely on the use of the ZULF regime. Therefore, not only can applications of ZULF NMR benefit from hyperpolarization, but a deeper understanding of hyperpolarization
mechanisms can also be gained from NMR studies at zero and ultralow fields.

Below we discuss the basic principles and applica-
tions in ZULF NMR of the most common hyperpolariza-
tion techniques: spin-exchange optical pumping (Sec. 3.3),
parahydrogen-induced polarization (Sec. 3.4), dynamic nuclear
polarization (Sec. 3.5), and chemically induced dynamic nu-
clear polarization (Sec. 3.6), along with two less commonly
used approaches (Sec. 3.7 and Sec. 3.8).

3.3. Spin-exchange optical pumping

Light can be readily polarized. Circularly polarized light is composed of photons (spin-one particles) with the projection of the spin angular momentum of +ħ or −ħ in the direction of the light propagation. When polarized light is absorbed, the angular momentum is transferred to the absorbing medium, which can become spin-polarized. Typically, electrons are polarized with light and the polarization is transferred to nuclei via electron–nucleus interactions. Two examples are spin-exchange optical pumping (SEOP) and metastability-exchange optical pumping (MEOP), as reviewed in references [95, 96, 97, 98, 99]. In SEOP, the nuclear spins of noble-gas atoms are polarized through ‘spin-exchange’ collisions with alkali atoms present in the same volume, where the alkali atoms are optically pumped. In MEOP, typically used to polarize ³He nuclei, atoms are excited in an electrical discharge to a metastable state with unpaired electron spins. In this state, optical pumping occurs, and the resultant nuclear polarization persists as the atoms relax back to the ground electronic state.

In the following section, we mention hyperpolarized noble gases with precessing magnetization detected by noninductive sensors below the Earth magnetic field. While by our own definition (Sec. 1.2), this work qualifies as high-field NMR rather than ZULF NMR, these developments have occurred in parallel to those of ZULF NMR and with some of the similar (or same) hardware.

The earliest example of MEOP being used for sub-Earth-
field NMR is the work of Cohen-Tannoudji *et al.* in 1969 [42].
The authors arranged two 6 cm diameter cells adjacent to one
another, with one containing <sup>87</sup>Rb (without a buffer gas) and the
other 3 torr of pure <sup>3</sup>He. Inducing an rf discharge in the helium
cell, one can populate the first excited metastable state of he-
lium, in which these atoms can be directly pumped with light
at a wavelength of 1.083 µm, produced in the early experiments
with a xenon discharge lamp (diode lasers for this wavelength
are readily available today). The cells were held in a magnetic
shield with shim coils, which attenuated the external magnetic
field to be on the order of 100 pT. A beam of circularly polarized
light was passed through the <sup>3</sup>He cell to generate <sup>3</sup>He nuclear-
spin hyperpolarization via MEOP, and an orthogonal beam of
circularly polarized light was passed through the <sup>87</sup>Rb cell to
optically pump the <sup>87</sup>Rb. The Rb cell was used as a magne-
tometer to detect the magnetic signals from the <sup>3</sup>He nuclei pre-
cessing about a 185 pT applied field, with a sensitivity on the
order of 100 fT/√Hz. Three decades later, a series of papers
followed in which hyperpolarized <sup>3</sup>He and <sup>129</sup>Xe were detected
in low magnetic fields using SQUID detectors. In the first of

a

<figure><img src="image_20.png" /></figure>

b

<figure><img src="image_21.png" /></figure>

Figure 17: a) Schematic drawing of the SQUID-based NMR spectrometer. The sample is at the center of the orthogonal pickup and excitation saddle coils; a persistent-current solenoid provides a static magnetic field along the axis of the sample tube. Coils and sample are enclosed in a Pb tube. A two-layer µ-metal shield (not shown) reduces the ambient magnetic field to below 0.1 pT. b) SQUID-detected <sup>129</sup>Xe NMR spectra from isotopically enriched <sup>129</sup>Xe (80%) at four different magnetic fields. Reproduced from reference [100], Copyright (1997), with permission from Elsevier.

these [100], TonThat *et al*. collected hyperpolarized <sup>129</sup>Xe in a
sample tube at 4.2 K, which solidified the <sup>129</sup>Xe. The sample
was held in a mu-metal shield and was surrounded by coils to
provide a B₀ field and apply pulses. A pickup saddle coil was
used to transfer the magnetic signals to a SQUID for detection.
The experimental apparatus is shown in Fig. 17a. The <sup>129</sup>Xe
signals were detected in B₀ fields ranging from 0.019 to 8.9 mT.
The authors also measured the solid-state <sup>129</sup>Xe T₁ as a function
of magnetic field, and saw it vary from approximately 8000 s at
5 mT to ~2000 s at fields below 0.05 mT, where it eventually
levelled off.

In a subsequent paper from the same group, the authors ex-
tended the scope of this work to perform 1D and 2D imaging
and included experiments with hyperpolarized ³He [34]. A few
years later, imaging techniques were applied to obtain low-field
images of hyperpolarized ¹²⁹Xe in porous media [35]. Hyper-
polarized ¹²⁹Xe was flowed through a glass sample chamber
containing a block of aerogel, a few millimeters in each dimen-
sion. The imaging experiments were conducted in a B₀ field
strength of 2.26 mT, with gradients on the order of 1 mT/m ap-
plied for spatial encoding. The authors were able to image the
penetration of ¹²⁹Xe into the aerogel. In addition, they mea-
sured chemical shifts of xenon atoms in two different chemical
environments at low field, thanks to the large chemical shift dispersion of ¹²⁹Xe. Hyperpolarized ¹²⁹Xe was flowed through powdered polypropylene, and the xenon adsorbed to the surface exhibited a 5 Hz (166 ppm at 2.55 mT) shift compared to ¹²⁹Xe in the gas phase.

A significant step forward came in 2003, when the scope of these experiments was expanded by using hyperpolarized ¹²⁹Xe to polarize other molecules in solution for detection at low field [101] via an effect that became known as the spin-polarization-induced nuclear Overhauser effect (SPINOE). ¹²⁹Xe was polarized to around 2% and was bubbled into a protonated organic solvent (cyclopentane, acetone, or methanol). Cross-relaxation between the ¹²⁹Xe atoms and the protons in the solvent molecules is the basis of SPINOE [32], leading to enhancements of the ¹H NMR signals at the detection field of 0.95 µT of around 10⁶. Although the corresponding polarization levels of the ¹H spins are still low (on the order of 10⁻⁵), this method is a promising general approach to enhancing NMR signal, and would benefit greatly from the improved ¹²⁹Xe polarization levels that can now be obtained.

Apart from the 1969 demonstration [42], all experiments measuring SEOP-enhanced low-field NMR until 2004 were conducted using SQUID magnetometers. In 2004, laser-polarized ¹²⁹Xe in a 45 µT field was detected using an optically pumped magnetometer with an experimental sensitivity of approximately 100 fT/√Hz, with noise dominated by field fluctuations in the magnetic shielding [43]. This work was a significant step for ZULF NMR, as it marked the first use of a piercing solenoid, a solenoid that surrounds the sample and passes entirely through the magnetic shielding, with the ends of the solenoid outside. Passing a current through the solenoid generates a magnetic field inside it, while the field outside the solenoid but within the shielding is nominally zero. A magnetometer can be placed outside the solenoid next to the sample to detect magnetic fields from the sample without being affected by the much larger solenoid field. Experimentally, reduction factors in the field outside versus inside the solenoid of around 10³ are typically observed [87, 102], with even larger factors obtainable by careful winding of the solenoid.

In 2007, a study of ¹²⁹Xe relaxation times was conducted in the Berlin Magnetically Shielded Room (BMSR) [103]. The study involved measurements at various xenon pressures and an ambient magnetic field in the range of 4.5–15 nT, with the hyperpolarized ¹²⁹Xe held in spherical glass cells. In this work, SQUID magnetometers were employed to observe ¹²⁹Xe $T_2$ relaxation times of up to 8000 s in a 15 nT $B_0$ field. Figure 18 shows a plot of ¹²⁹Xe $T_2$ times for various gas pressures. In a separate set of experiments from the same time at the BMSR [104], the authors modified two SQUID systems (originally developed for biomagnetic applications) for ultralow-field NMR applications. They used hyperpolarized ¹²⁹Xe as the target nucleus to optimize their system.

In the ensuing years, OPMs were further developed and miniaturized, which enabled the combination of SEOP with ¹²⁹Xe and OPM readout on a single microfluidic device [105]. The microfluidic chip was constructed of silicon sandwiched between layers of borosilicate glass, with chambers cut from

<figure><img src="image_22.png" /><figcaption>Figure 18: (a) Precession of hyperpolarized ¹²⁹Xe nuclei in a 4.7 nT field, as measured by a SQUID magnetometer. (b) ¹²⁹Xe $T_2$ relaxation rates as a function of pressure for samples in 4.5 nT (circles) and 15 nT (crosses) magnetic fields. The three lines represent fits to the data using the equations described in the original paper. Image adapted from reference [103].</figcaption></figure>

the silicon for the SEOP of ¹²⁹Xe and for optical readout. Figure 19 shows the chip and the lasers used for optical excitation and readout. The total volume of this microfluidic chip was approximately 100 µL. Xenon was polarized in the pump chamber, and in some experiments it was detected there, and in others in the probe chamber. Application of a $\pi/2$ pulse at a magnetic field ($B_0$) of 0.8 µT resulted in the detection of NMR signals from ¹²⁹Xe spins at 9 Hz. Polarization levels of 0.7% were measured in this device.

A few years later, an improved version of this device was developed [106]. By changing the method of rubidium production on the chip (using photolysis of rubidium azide instead of reacting barium azide with ⁸⁷Rb chloride), the authors avoided the formation of barium chloride, drastically reducing contamination on the walls of the chip. This improvement enhanced ¹²⁹Xe relaxation times by a factor of approximately 5, and the device achieved ¹²⁹Xe polarizations of up to 7%. Using this platform, the authors compared *in situ* and *ex situ* signal readout methods. In the *in situ* experiments, the Xe and Rb atoms were in the same cell during signal detection, whereas in the *ex situ* measurements, the Rb vapor cell was adjacent to a separate cell containing xenon. In the *ex situ* measurement, the Rb electrons sense the dipolar field produced by the polarized xenon atoms. In the *in situ* measurement, the Rb electrons sense the Xe nuclear spins predominantly via the Fermi contact interaction [107]. The overall signal enhancement compared to the *ex situ* measurement was reported to be approximately 5300 [106].
<figure><img src="image_23.png" /></figure>

Figure 19: The microfluidic device used by Jimenez-Martinez et al., combining optical pumping and detection of ¹²⁹Xe on one platform. (a) The device, with lasers shown in red. (b) Size measurement of the chip. (c) The pumping and probing sequence. Adapted from reference [105] under terms of the Creative Commons CC-BY license 4.0.

### 3.4. Parahydrogen-induced polarization

In 1986, Clifford Bowers and Daniel Weitekamp proposed a thought experiment [108] in which they predicted the observation of enhanced ¹H NMR signals in the spectra of products of a hydrogenation reaction with *pH*₂, the spin-0 nuclear spin isomer of molecular hydrogen. Later, they experimentally demonstrated that the hydrogenation of acrylonitrile with *pH*₂ using Wilkinson's catalyst at high magnetic fields indeed resulted in the predicted effect [109]. Termed PASADENA, the effect is characterized by enhanced antiphase multiplets in the ¹H NMR spectra and can be observed both for the reaction products and for the hydride complexes of the metals used as hydrogenation catalysts [110]. PASADENA and the related ALTADENA effect discovered a year later [111] are now collectively known as parahydrogen-induced polarization (PHIP) [112, 113].

At the core of PHIP and related techniques is the quantum-mechanical connection between the symmetry of the nuclear spin state and the allowed rotational states of molecules. This effectively gives the nuclear spin degrees of freedom access to the relatively large energy scales associated with rotations and thus enables much higher nuclear spin order than would be possible with spin interactions alone. The underlying physics is well described in the literature [114, 115] and will not be addressed further here.

Perhaps the greatest advantage of PHIP is the ease with which the source of polarization, *pH*₂ gas, can be generated, stored, and used. Hydrogen gas at >99% para-enrichment can be prepared by cooling the gas to below 25 K in the presence of a paramagnetic catalyst; if the catalyst is then removed, the hydrogen can remain in the para state for days at room temperature [116].

In *hydrogenative* PHIP, hyperpolarization can be engen-

dered in a target molecule by chemically reacting *pH*₂ with a suitable molecular precursor, with three important caveats: (1) the chemical reaction should proceed on a timescale not longer than that of nuclear spin relaxation; (2) the hydrogen addition should be pairwise, i.e., the hydrogen atoms from one molecule of *pH*₂ should be incorporated into the same product molecule, so that their correlated spin state is preserved; and (3) the symmetry between the two *pH*₂ protons must be broken (either in the final product or in a reaction intermediate) for hyperpolarized signals to be observable. In solution-state experiments, the first two considerations are addressed by the use of organometallic catalysts. If the magnetic equivalence of the two *pH*₂ protons persists in the final product molecule, the spin order remains in the unobservable singlet state. This state can be long-lived [117] and can support the spin order until further chemical reactions break the magnetic equivalence and hyperpolarized NMR signals are released [118, 119].

In *non-hydrogenative* PHIP, also known as signal amplification by reversible exchange (SABRE), target molecules are polarized through temporary interactions with *pH*₂ molecules during the reversible binding of these molecules to an organometallic catalyst [120]. The *J*-couplings between the *pH*₂-nascent hydrides and the bound-substrate nuclear spins in the complex allow for the redistribution of hyperpolarization onto the ligated substrates (molecules to be polarized) during the lifetime of the complex.

The low cost, small footprint, and ease of experimental implementation make PHIP an attractive hyperpolarization technique for combination with ZULF NMR. The majority of PHIP and SABRE experiments employing ZULF detection use a pressurizable NMR tube as the reaction vessel, in which the reaction solution is contained and *pH*₂ gas is typically bubbled in at elevated pressure (5–10 bar). After the chemical reaction has started, hyperpolarization-enhanced NMR signals can be detected either by applying a magnetic field pulse or by conducting the hydrogenation in the presence of a small bias magnetic field and generating hyperpolarized NMR signals by nonadiabatically switching off the field. In the first experiments combining ZULF NMR with PHIP, four different chemical systems were probed: the hydrogenation of [1-¹³C]dimethyl acetylene dicarboxylate to [1-¹³C]dimethyl maleate, styrene to ethylbenzene, 1-phenyl-1-propyne to 1-phenyl-1-propene, and 3-hexyne to 3-hexane [66]. In each case, after the chemical reaction, ¹H-¹³C coherences were excited by applying a square magnetic field pulse with amplitude *B*<sub>P</sub> and duration τ<sub>P</sub> such that *B*<sub>P</sub>(γ<sub>H</sub> - γ<sub>C</sub>)τ<sub>P</sub> = π/2. The pulse length τ<sub>P</sub> is chosen to maximize the amplitude of the sine term in Eq. (25), thereby optimizing the conversion between scalar spin order and observable spin order. The authors identified the spectral regions corresponding to particular coupled ¹H-¹³C groups, and simulated the zero-field PHIP spectrum of ethylbenzene, which was found to agree with the experiment.

Following this initial experimental demonstration, a theoretical description of parahydrogen-induced polarization at zero field was developed [121]. The incorporation of individual parahydrogen molecules into a target molecule is a nonadiabatic process. This means that the initial singlet order
<figure><img src="image_24.png" /><figcaption>Figure 20: (a) The zero-field NMR setup employed for the first parahydrogen-enhanced NMR experiments [66]. (b) A background spectrum showing the noise floor of the experimental apparatus. (c) The pulse sequence employed for the experiments. (d,e) The PHIP-enhanced zero-field NMR spectra of ethylbenzene-β-¹³C and ethylbenzene-α-¹³C polarized after the hydrogenation of ¹³C-labelled styrene, with simulations shown in blue and green. Several features are seen in the spectra. For example, as expected (see Sec. 2), the ¹³CH₂ group produces a spectral feature around 3J/2, while the ¹³CH₃ group produces features at J and 2J. Moreover, the spectral features for the group further from the aromatic ring are less perturbed by the ring protons compared to the features from the group closer to the ring. Reproduced from reference [66].</figcaption></figure>

of $pH_2$ is projected onto the populations and coherences of a new eigenbasis. As individual target molecules in the sample are hydrogenated at different time points, coherences among nondegenerate states are averaged to zero and only populations survive. Mathematically, this corresponds to eliminating the nondiagonal elements of the spin density matrix in the new eigenbasis. The resulting magnetization remains zero.

In general, asymmetrical coupling between the hydrogen spins and the heteronucleus—i.e., broken magnetic equivalence of the $pH_2$-sourced hydrogens in the hydrogenation product—is of interest, as then the process yields polarization of spin rank greater than zero. Application of subsequent magnetic field pulses leads to the generation of observable vector order, which can then be measured [122, 123, 124, 125].

The advantages of $pH_2$-based polarization compared to other hyperpolarization methodologies are high polarization yield and simple, low-cost infrastructure. This should in principle

allow straightforward scale-up to produce large boluses of hyperpolarized material (e.g., liters in seconds). For this reason, $pH_2$-based polarization is currently considered a main candidate for applications in fundamental physics (see Sec. 7.1 for details).

Another important application of $pH_2$-based hyperpolarization in the context of ZULF NMR is chemical analysis. One example is the demonstration that ZULF NMR can be used as a spectroscopic technique for chemical reaction monitoring, by observing the two-step hydrogenation of dimethyl acetylene dicarboxylate to dimethyl maleate to dimethyl succinate [126]. This is possible because ZULF NMR is a quantitative method, providing chemical specificity thanks to the unique *J*-spectra exhibited by different molecules. A schematic of the experimental apparatus used in this work and the key results are shown in Fig. 21. It was additionally demonstrated that the reaction monitoring could be carried out under continuous bubbling and at high pressure (>10 bar) in metal containers, something that is not possible using high-field NMR. It remains to be seen whether this technique has a role to play in industrial-scale process monitoring.

SABRE was also shown to be an effective way to hyperpolarize a chemical sample without the need for any magnets [127]. In this work, parahydrogen gas was bubbled at high pressure (5.8 bar) into an NMR tube containing a solution of ¹⁵N-pyridine and Crabtree's catalyst in anhydrous methanol at zero field. After several seconds of bubbling, during which the singlet spin order was transferred from parahydrogen to pyridine, a magnetic-field pulse of duration $τ_P$ and amplitude $B_P$ was applied that satisfied the condition $B_P(γ_H - γ_N)τ_P = π/2$. The resulting NMR spectrum was collected using an optically pumped magnetometer positioned beneath the sample. For comparison, the authors also collected a zero-field NMR spectrum of neat ¹⁵N-pyridine, thermally polarized prior to signal acquisition in a 1.6 T permanent magnet. Peak shifts of ≈1 Hz were observed between the two spectra, which was attributed to the *J*-couplings differing between the pyridine neat liquid and pyridine dissolved in methanol.

Recently, another way of observing $pH_2$-based hyperpolarization under ZULF conditions was demonstrated [128]. In this approach, which is conceptually similar to SABRESHEATH (SABRE in SHield Enables Alignment Transfer to Heteronuclei) [129, 130], hyperpolarization of heteronuclei (for example, ¹⁵N) was detected at zero field followed by a sudden switching off of the sub-microtesla magnetic field. The authors also demonstrated that SABRE reactions could be carried out for longer periods of time by pressurating the parahydrogen gas with the solvent prior to bubbling into the sample, which helps to alleviate the problem of solvent evaporation. This work paves the way for SABRE-enhanced ZULF NMR experiments that can be carried out for hours or even days with applications ranging from those in fundamental physics (Sec. 7.1) to industrial ones (Sec. 7.2).

Typical concentrations of substrates used in the SABRE development work are on the order of 10–100 mM. Specific demonstrations were performed with concentrations ranging from sub-µM (e.g., 0.5 µM in reference [131]) to molar (e.g.,
<figure><img src="image_25.png" /><figcaption>Figure 21: Left: Parahydrogen can be bubbled into a vessel (in this case a 5-mm NMR tube) containing a reaction solution at zero field, and the NMR signals are excited and observed periodically using an OPM. Right, top: The two-step hydrogenation reaction carried out. Right, middle: A zero-field NMR spectrum showing resonances that correspond to [2-<sup>13</sup>C]dimethyl maleate and [2-<sup>13</sup>C]dimethyl succinate. Right, bottom: Integrals of the peaks at the specified frequencies over time as the reaction was carried out. The signal from [2-<sup>13</sup>C]dimethyl succinate decays due to relaxation of the hyperpolarized signals. On the right the titanium tube is shown in which the reaction was carried out. Reproduced from reference [126] under terms of the Creative Commons BY-NC license 4.0.</figcaption></figure>

polarization of a neat liquid used as a solvent, see reference [132]). In general, due to the nature of SABRE mechanism, which is based on chemical exchange and formation of a specific polarization-transfer complex, polarization values drop with increasing substrate concentration [133]. This limits applicability of SABRE to searches of new physics such as spin-gravity coupling (Sec. 7.1.4) or dark matter (Sec. 7.1.2), where often a near-unity polarization and a high spin density (on the order of 10<sup>20</sup> cm<sup>-3</sup>) are necessary. For such applications, ‘brute force’ polarization of isotopically enriched compounds with as-high-as-possible magnetic field (for example, using superconducting magnets, ideally at low temperatures) could be a better choice.

One should note the difference in how the SABRE-based ZULF-NMR signals can be observed depending on whether the polarization is performed at zero or finite field. The zero-field case produces scalar spin order (spherical rank 0) by virtue of a lack of preferred field orientation, and requires a magnetic field pulse to generate observable coherences between ZULF eigenstates. The other case, where polarization occurs at finite

field (e.g., 0.1–1 µT, known as SABRE-SHEATH [130]) generates z-magnetization on the heteronucleus [129, 93, 134], and requires only a sudden switching-off of the magnetic field to generate coherences. Both of these approaches result in similar amplitudes of peaks in the zero-field *J*-spectra and the choice of approach depends on the particular goal of the experiment.

Atomic magnetometers are not the only type of sensors used to detect molecules hyperpolarized by SABRE. Low-field NMR spectra of 18 different SABRE-polarized fluorinated *N*-heterocycles were measured using a SQUID magnetometer [135]. Magnetic fields between 0.1 and 10 mT were applied during the *pH*₂ bubbling to induce spontaneous spin-order transfer from *pH*₂ to the target molecules, and stepped down to 144 µT for signal acquisition. Although this is not ultralow field per our definition, this work demonstrates the relative ease with which the SABRE experiment could be adapted for ZULF detection, as the SQUID magnetometer is also sensitive down to lower frequencies. Thanks to advances in homogeneous catalysis and progress in understanding of *pH*₂-based spin dynamics, in recent years PHIP and SABRE have become
routinely employed hyperpolarization methods. For this reason they were also used in the first demonstration of ZULF NMR carried out using commercial atomic magnetometers [45].

In principle, parahydrogen-enabled hyperpolarization allows extending all applications discussed above to imaging at magnetic fields below the Earth field. In reference [136], the authors demonstrated ¹H MRI at ~30 µT of a bubble-separated phantom by using a solution of SABRE-polarized pyridine. Strong signal enhancement of ¹H signals provided by SABRE (~2600) shortened the MRI operation time, so that an image with resolution of 1.6×1.6 mm² of a 3D-printed phantom with a field of view of 60 mm could be obtained in less than 50 minutes.

There is a considerable literature describing low- or Earth-field PHIP experiments [137, 138]. Another topic involving PHIP is polarization-transfer methods that exploit level anticrossings in the ZULF regime [133]. There have been a number of demonstrations involving both hydrogenative and non-hydrogenative PHIP: in some cases the reaction is carried out at ZULF, and in other cases the reaction is carried out at high/low field and a magnetic field cycle is applied to induce polarization transfer [139, 140, 141]. These methods are typically applied with detection in high magnetic fields [142] but are also compatible with detection under ZULF conditions [143].

### 3.5. Dynamic nuclear polarization

Dynamic nuclear polarization (DNP) is a term used to describe a set of hyperpolarization methods that rely on polarization transfer from unpaired electrons (typically present in the form of free radicals) to spin-active nuclei. This transfer typically takes place at elevated magnetic field, either at room temperature in solution (such as in Overhauser DNP) or within a low-temperature (a few K) amorphous solid, and is effected by microwave irradiation applied to electron spin transitions [144, 145, 146, 147]. DNP allows generating NMR signals increased by orders of magnitude compared to thermal equilibrium, resulting in dramatically decreased signal averaging times [148]. Below we discuss in more detail the DNP approaches that have been used in the context of ZULF NMR.

#### 3.5.1. Overhauser Dynamic Nuclear Polarization

The Overhauser effect manifests itself as polarization transfer from electrons to nuclei, driven by dipolar relaxation. For the case of a coupled electron-nuclear system, dipolar relaxation in solution gives rise not only to individual $T_1$ relaxation of spins but also to cross-relaxation, a process in which both spins are flipped simultaneously [149]. In such systems, upon saturation of electron transitions, cross-relaxation leads to a steady state with enhanced nuclear polarization, the process known as Overhauser DNP [150, 149]. Saturation here refers to equalizing populations of spin energy levels upon applying a strong resonant microwave pulse. One can think of saturating EPR transitions as a way to block pure electronic relaxation. As this relaxation pathway is blocked, relaxation occurs via other channels in which electrons and nuclei relax together. The enhancement factor ($\epsilon$) for the nuclear spin signal can be calculated as

<figure><img src="image_26.png" /><figcaption>Figure 22: Overhauser-DNP-enhanced 3D image of a moon cactus, injected with TEMPO solution. The image was acquired at a field of 96 µT in a single scan with 1-mm×1-mm in-plane resolution. D is the depth of a given 5-mm-thick layer under the bottom of the cryostat. Reproduced from reference [152], Copyright (2010), with permission from Elsevier.</figcaption></figure>

$$ \epsilon = \frac{\langle I_z \rangle}{I_0} = 1 - \rho f s \frac{|\gamma_S|}{|\gamma_I|}, \qquad (60) $$

where $\langle I_z \rangle$ and $I_0$ are average spin polarizations for nuclei (for example, protons in a solvent such as H₂O) after saturating electron transitions and at thermal equilibrium, respectively; $\rho$ is a coupling constant, which can range from -1 assuming pure scalar interaction to +0.5 in the case of pure dipolar coupling between electron and nuclear spin; $f$ is a leakage factor that shows how effectively the proton spin is relaxed by the electron spin and ranges from 0 to 1; and $s$ is the saturation factor, which is equal to 1 for complete saturation [151]. For these conditions, Eq. (60) predicts the maximum polarization enhancement for protons to be ~660 in the case of pure scalar coupling and ~330 for pure dipolar coupling between the nucleus to be polarized and the ‘electron source’, for example a nitroxide radical. Negative $\epsilon$ means that the nonequilibrium magnetization vector created with Overhauser DNP has the direction opposite to that of the thermal equilibrium magnetization in an external magnetic field [152].

Overhauser DNP enhancement with direct detection below the Earth field was demonstrated in reference [153]. As at fields below 1 µT, electron and nuclear ¹⁴N/¹⁵N-spins are strongly coupled in the nitroxide radical (the so called ‘hyperfine-field-dominant regime’), the saturated resonance frequencies are determined by both the electron-nucleus hyperfine coupling and the static magnetic field. By applying a 74.4 MHz ra-
diofrequency field at the background fields where the hyperfine
resonances occur (0.71, 2.04, 4.91, and 47.68 µT) signal en-
hancements of protons in aqueous TEMPOL solutions on the
order of a few hundreds were demonstrated. The same authors
also demonstrated *in situ* generation and detection of a proton
NMR signal (2 mM aqueous solution of TEMPOL) at 325 nT
[154]. An SNR of 32 at a linewidth of 0.7 Hz was achieved
after 16 averages, paving the way for novel applications of
Overhauser DNP in ZULF NMR.

Not only spectroscopy is possible with Overhauser DNP at
near-zero-field conditions. DNP-enhanced MRI at 96 µT was
demonstrated with a broadband SQUID sensors [152]. Imag-
ing of water phantoms and a cactus plant (Fig. 22) was per-
formed by using a hyperfine transition frequency of 120 MHz.
Enhancement factors as large as –95 for protons and –200 for
¹³C were demonstrated, which would require thermal polariza-
tion at 0.33 T and 1.1 T, respectively.

3.5.2. *Dissolution Dynamic Nuclear Polarization*

An attractive hyperpolarization method is dissolution dy-
namic nuclear polarization (dDNP). In contrast to PHIP/SABRE
or CIDNP (see Sec. 3.6 below), this technique is general. This
means that, in principle, any molecule in solution can be
hyperpolarized, as the polarization process does not rely on
specific chemical reactions or interactions. To polarize a
sample using dDNP, the molecule of interest is frozen in a
glassy or polymeric matrix with a free radical dispersed ho-
mogeneously through the solid. The sample is cooled down to
cryogenic temperatures and microwave irradiation is used to
drive polarization transfer from the electrons of the free radi-
cals to nuclear spins in the to-be-polarized molecules. There
are several mechanisms by which this transfer occurs: solid
effect, cross-effect, and thermal mixing [155]. All of these
mechanisms rely on hyperfine interactions in combination with
relaxation accompanying the microwave-driven electron spin
flips.

In the *solid effect*, transitions that are normally 'forbid-
den' in the liquid-state (such as single- or double-quantum
transitions) become 'allowed' and, thus, can be pumped when
anisotropic hyperfine interaction is present. By driving the for-
bidden transitions one can flip nuclear spins and hyperpolarize
them [156]. Importantly, electronic $T_1$ relaxation brings the
system to a new equilibrium and the process can be repeated
to polarize, via spin diffusion, other nuclei. As in the case
of Overhauser DNP, the sign of the effect can be positive or
negative, depending on whether a single- or double-quantum
transition is saturated.

In the *cross effect*, polarization transfer happens when a
difference between electron Larmor frequencies matches the
nuclear frequency: $\Delta\omega_e = \omega_n$. This is typically achieved when
the radicals have an inhomogeneously broadened EPR spec-
trum whose breadth is larger than the nuclear Larmor frequency
[157] or when special biradicals (molecules containing two
chemical groups with unpaired electrons) are used in the pro-
cess. Pairwise electron spin flips provide a field fluctuation at
the resonance with nuclear spins. Once a nucleus is polarized,
polarization can distribute to other nuclei via spin diffusion.

*Thermal mixing* works when the concentration of free radicals is so high that an EPR line is homogeneously broadened (in other words, all spins ‘talk’ to each other). It is hard to develop a mechanistic model for a situation like this, but thermodynamic arguments can be used to correctly predict and explain experimental observations in which nuclei are polarized upon saturation of the microwave transitions at specific frequencies [158]. In practice, several mechanisms can contribute to the experimental observations and it is not always possible to identify a single mechanism responsible for polarization transfer from electrons to nuclei [159, 156, 160].

For the dissolution step, after sufficient nuclear spin polar-
ization has been built up in the solid state (typically 20–60%
can be achieved on a timescale of tens of minutes), the sam-
ple is rapidly (within seconds) dissolved in a hot solvent and
flushed out of the polarizer as a room-temperature solution.

The dDNP method has been used to polarize substances such as the [1-¹³C]-pyruvic acid, [2-¹³C]-pyruvic acid, [¹³C]-formic acid, [2-¹³C]-acetic acid and [1-¹³C]-glycine molecules, or their anions, for detection at zero field using atomic magnetometers [162, 161, 163]. The enhanced spin polarization made it possible to detect molecules in solution at micromolar concentration and natural ¹³C isotopic abundance. As the dDNP method is general, it is possible to hyperpolarize multiple species at once and observe them simultaneously, as demonstrated in recent dedicated dDNP ZULF experiments [161].

Dissolution-DNP is a particularly promising method to pair
with ZULF NMR since, as of 2024, several commercial polar-
izers exist. While dDNP usually involves a relatively complex
setup based on a high-field magnet and cryogenics, its com-
bination with ZULF-NMR modalities offers opportunities in
a broad range of applications, including medical diagnostics,
monitoring catalytic reactions within metal reactors, fragment-
based drug screening, and others [164], see Sec. 7.5. Another
version of DNP, bullet-DNP, has been recently demonstrated,
where a sample is quickly transferred outside of a DNP polar-
izer in its solid form prior to dissolution. The dissolution step
takes place at room temperature inside of the measurement ap-
paratus, offering more experimental flexibility [165].

A drawback of the direct dDNP–ZULF combination is that the free radicals plus other additives (such as rare-earth ions) that are used in the polarization step remain in solution after the dissolution step, and these can induce paramagnetic relaxation of the nuclear spins at low fields [86]. However, a number of methods have been developed to circumvent this problem in the context of high-field NMR: (1) adding scavengers such as ascorbic acid to the solution, which quench the radicals [166] or chelate paramagnetic ions; (2) phase extraction of the radicals into an organic solvent [167]; (3) the use of solid matrices that contain the free radicals, which are not dissolved into solution during the dissolution step [168]; and (4) the use of nonpersistent UV-generated radicals, which are rapidly quenched upon sample warming during the dissolution step [169].

3.6. *Chemically induced nuclear polarization*

In 1965, while investigating polymerization reactions by *in situ* NMR, Joachim Bargon observed unusual high-field NMR
<figure><img src="image_27.png" /></figure>

Figure 23: ZULF-NMR spectra with *d*-DNP hyperpolarization. (a) Single-scan ZULF-NMR spectrum of a hyperpolarized mixture of sodium [¹³C]-formate and sodium [1-¹³C]-acetate (blue trace) compared with a reference spectrum of 5.2 M [¹³C]-sodium formate thermally prepolarized at 2 T after averaging 16 scans (black trace). (b) Single-scan ZULF-NMR spectrum of a hyperpolarized mixture of sodium [¹³C]-formate and [1-¹³C]-glycine (blue trace) compared with the same reference spectrum as in panel (a) (black trace). Note the scaling of the reference spectra. The noise peaks arising from the power line at 50 Hz and overtones are grayed out, and probe-laser noise peaks are marked with asterisks; C refers to the concentration of [¹³C]-formate. Reprinted (adapted) with permission from reference [161]. Copyright 2022 American Chemical Society.

spectra [170]. Whenever the polymerization of the maleic anhydride was initiated by using free radicals (i.e., molecules possessing unpaired electrons), intense absorption and emission lines appeared immediately upon the onset of reaction. If, instead, the polymerization was initiated ionically using pyridine, no such phenomena occurred. As the only seemingly related phenomenon known at that time was DNP (see Sec. 3.5) based on the Overhauser effect, the new phenomenon was named ‘chemically induced dynamic nuclear polarization’ (CIDNP). Indeed, it seemed logical that DNP-like enhanced absorption and emission lines were ‘chemically induced’ rather than induced by microwave irradiation of a system containing paramagnetic centers [171].

However, it was quickly understood that the DNP-based theory could not explain all experimentally observed features and only in 1969 did Kaptein and Oosterhoff [172] and independently Closs find the correct interpretation, namely, the ‘radical pair’ (RP) theory of CIDNP. The RP-based explanation of CIDNP took care of all previously unresolved problems; nonetheless, this currently accepted terminology of dynamic

polarization remains in use today [173].

CIDNP is commonly detected at high fields in a wide class of reactions, for example, during thermal decomposition [172] or in reversible reactions between photoexcited dyes and amino acids [174]. This is one of the hyperpolarization methods that enables direct enhancement of NMR signals in macromolecules and their building units, including aromatic amino acids, proteins [175], nucleotides [176], and oligonucleotides [177]. Photo-CIDNP observation of NMR signals from nanomolar concentrations of proteins is one of the most sensitive experiments in bio-NMR [178]. Moreover, photo-CIDNP can be used to screen chemical reactions on a microsecond timescale [173, 179], providing information on chemical rate constants [180] and structures for transient radicals [181, 182]. Recent developments in microfluidic probes [183, 184], implementation of affordable systems with light-emitting diodes (LEDs) [185], discovery of new photosensitizers [186], reductive radical quenchers for prolonging the stability of reversible photo-reactions [187], and the expanding application of the methodology for a growing list
<figure><img src="image_28.png" /><figcaption>Figure 24: Experimental setup for magnetometer-detected photo-CIDNP NMR measurements at nanotesla to microtesla fields. Reprinted (adapted) with permission from reference [206]. Copyright 2023 American Chemical Society.</figcaption></figure>

of molecules [188, 182, 189, 190] have further increased the interest in this technique. However, it should be noted that photo-CIDNP has a complex dependence on the magnetic field in which the reaction is performed. Sign alternations and/or extremes in the magnetic field dependence that characterise the properties of the radicals such as exchange interaction, g-factors, and hyperfine couplings can be observed. For common organic radicals these features occur at fields of < 50 mT. In this section, we discuss the experiments developed for the observation of photo-CIDNP at (ultra)low magnetic fields.

Traditionally, low-field photo-CIDNP experiments are indirect, involving mechanical sample shuttling between two or more magnetic fields. Samples are illuminated in low magnetic field and then mechanically transported into a high-field NMR spectrometer for detection [191, 192, 176, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204]. Low-field photo-CIDNP profiles (i.e., dependences of the polarization on the magnetic field) obtained indirectly are useful for determining the exchange interaction in photoexcited electron donor-acceptor dyads, as these contain sharp features at magnetic fields corresponding to the level anticrossings due to exchange interaction [205, 194, 197, 202, 203, 204]. However, analysis of field dependence of photo-CIDNP measured indirectly is challenging for the cases of molecules containing multiple nuclear spins [201]. Recently, it was demonstrated that photo-CIDNP signals can be detected directly at nanotesla to microtesla fields using atomic magnetometers. The setup for this measurements is shown in Fig. 24. Fast field cycling was performed to observe photo-CIDNP occurring at fields up to 100 mT [206]. It was shown that at ultralow fields, photo-CIDNP can be used to enhance the long-lived heteronuclear singlet order formed by directly bound ¹³C and ¹H spins [207], which can be readily detected with ZULF NMR.

An important feature of the RP mechanism is that nuclear spin states of the radicals in a pair can affect electron-spin evolution and thus, the recombination rate in a given RP. Below we compare the simplest schemes for the RP mechanism operational at high field (Fig. 25A) and at zero field (Fig. 25B). Evolution of the RP in solution includes the following stages:

a) *Birth of the radical pair.* A radical pair in solution is a short-lived formation consisting of two radicals in proximity to one another, experiencing at least one re-encounter before they recombine or separate. This requirement is fulfilled in liquids due to the 'cage effect': solvent molecules surrounding the two radicals trap them together for a certain time, limited by diffusion. During this time, the radicals experience multiple encounters, in the course of which nuclear-spin selection/flipping occurs. There are two types of radical pairs: those with a correlated spin state of the two electrons (geminal or G-pairs) and those with uncorrelated states (free pairs or F-pairs). The former are created in a single transfer of the electron from the excited photosensitizer to the substrate or (as in the case of thermal decomposition of the radical pair) obtained immediately after the breakage of chemical bond. After the formation of a G-pair, its spin state is the same as that of its precursor. In the case of thermal decomposition, the RP is in the singlet state, and in the case of a photochemical reaction, the RP can be in singlet (S) or triplet (T) state. F-pairs are obtained upon encounter of two 'escape' radicals, for example those that were initially formed from G-pairs and then diffused apart in solution. F-pairs usually have more impact on steady-state CIDNP [208], which is achieved, for example, in the case of continuous light irradiation. This is the case for all ZULF-NMR experiments as of the time of writing.

b) *Spin dynamics in radical pairs.* Dynamics of singlet-triplet interconversion of the electron states in the RPs is of central importance for CIDNP. This dynamics depends strongly on the magnetic field, so that different scenarios may play out at different fields. Here we compare two cases: 'spin-selection' occurs at high field and is responsible for the origin of the net polarization (Fig. 25A); 'spin-flipping' occurs at low to ultralow and zero fields and is responsible for 'multiplet' polarization of spin networks (Fig. 25B). These two mechanisms have different requirements on the spin system of the RP: spin-selection requires having at least one nuclear spin with nonzero hyperfine coupling in the RP; spin-flipping requires having at least two different nuclei with distinct hyperfine couplings in the RP. These requirements can be understood from symmetry considerations: whereas the high field sets the direction along which spin magnetization can build up, at zero field there is no preferred direction and the spin order produced by CIDNP must have spherical symmetry. This means that the terms in the density matrix describing hyperpolarization must only be rank-zero spherical tensor operators, and those can be found only in systems of multiple spins, with the simplest example being the singlet order of a two-spin system. Note that we do not discuss multiplet CIDNP that also may occur under spin-selection at high fields, neither do we consider net CIDNP occurring at low fields due to spin-flipping. There are dedicated reviews on these and other regimes of CIDNP [193, 209, 210, 211, 179].
<figure><img src="image_29.png" /><figcaption>Figure 25: **A:** Photo-CIDNP in the high-field case: the geminal radical pair (G-pair) mechanism of photo-CIDNP. ‘Geminal’ designates the primary P–Q pair produced upon photoexcitation of P followed by electron capture by Q. The photosensitizer (P) absorbs light and—after intersystem crossing occurring on a femtosecond timescale—becomes a relatively long-lived triplet radical (lifetime on the µs timescale or longer). (i) A quencher (Q) containing a nuclear spin (e.g., ¹H), in either α or β state with 50:50% probability (neglecting weak thermal polarization), encounters P, producing an RP. (ii) In most of the cases, P and Q molecules separate, producing escape radicals, but sometimes the electron triplet state is converted into the singlet state, increasing the probability of in-cage recombination. Due to hyperfine interactions, the conversion preferentially occurs for RPs with nuclei in one particular state (assumed β here, as an example). (iii) Escape radicals exist for a relatively long time on the nuclear-spin relaxation time scale, leading to equilibration of nuclear-spin populations. By contrast, in-cage recombination occurs quickly, leading to accumulation of nuclear spins in the β state. The recombination of escape radicals happens mostly in secondary radical pairs (F-pairs) that are formed upon encounter of two free P and Q radicals. The process is cyclic and, upon continuous irradiation, nuclear polarization builds up in the diamagnetic product, giving rise to the ‘emission-type’ NMR spectrum. In steady state, F-pairs may further enhance photo-CIDNP. **B:** Photo-CIDNP in the zero-field case. At least two different type of nuclei (e.g., ¹H and ¹³C) having different hyperfine couplings to an electron should be present in Q to obtain hyperpolarization. Instead of α and β states, heteronuclear singlet (S) and triplet (T, i, i = 0, ±1) influence the RP dynamics. In both cases (A and B) high-field NMR detection is shown. In (B), an adiabatic transfer from zero to high field is assumed for the last step (iii).</figcaption></figure>

Let us now proceed with the analysis of the spin-selection mechanism. We assume here that RPs are initially excited in the triplet states (stage i in Fig. 25A) and that only one nuclear spin is present in the spin system of the RP. Once the two radicals separate in space, the exchange interaction between electrons essentially vanishes due to negligible overlap of the electron wavefunctions between the two radicals. They still constitute a radical pair ‘trapped’ in the ‘cage’, but now they are a bit further apart: a separation of 5 to 10 Å is large enough to almost fully suppress short-range exchange interaction in common radicals [210]. Therefore, upon abrupt separation, the correlated triplet state is no longer an eigenstate of the high-field Hamiltonian in the radical pair, $\hat{H}_{\text{RP}}^{\text{HF}}$, and starts to evolve. The RP Hamiltonian can be written in the form [210]

$$ 
\begin{align} 
\hat{H}_{\text{RP}}^{\text{HF}} &= \hat{H}_{\text{RP}}^{\text{HF,0}} + \hat{H}_{\text{RP}}^{\text{HF,1}}, \tag{61} \\ 
\hat{H}_{\text{RP}}^{\text{HF,0}} &= \frac{g_1 + g_2}{2} \mu_B B_0 (\hat{S}_{1z} + \hat{S}_{2z}) \nonumber \\ 
&\quad -2\pi\hbar J_{\text{ex}} \left(\frac{1}{2} + 2\hat{S}_1 \cdot \hat{S}_2\right) \nonumber \\ 
&\quad +\pi\hbar A_1 (\hat{S}_{1z}\hat{I}_z + \hat{S}_{2z}\hat{I}_z), \nonumber \\ 
\hat{H}_{\text{RP}}^{\text{HF,1}} &= \frac{g_1 - g_2}{2} \mu_B B_0 (\hat{S}_{1z} - \hat{S}_{2z}) \nonumber \\ 
&\quad +\pi\hbar A_1 (\hat{S}_{1z}\hat{I}_z - \hat{S}_{2z}\hat{I}_z). \nonumber 
\end{align} 
$$

Here $\mu_B$ denotes the Bohr magneton, $g_i$ the g-factor of the

electron i, $\hat{S}_i$ the spin operator of the i-th electron, $\hat{I}_z$ the z-projection of the spin operator of the nucleus, $A_1$ the hyperfine coupling between the nuclei and the first electron, and $J_{\text{ex}}$ residual exchange interaction (we keep it here to remain general, although often it can be neglected for nonviscous solutions [212]). Both the exchange interaction and the hyperfine coupling are expressed here in hertz. The $\hat{H}_{\text{RP}}^{\text{HF,0}}$ term is diagonal in the basis of singlet–triplet electron functions, but the $\hat{H}_{\text{RP}}^{\text{HF,1}}$ term mixes the triplet $T_0$ with the singlet $S_0$ electrons states. Therefore, the overpopulated $T_0$ state starts evolving into $S_0$. The frequency of this evolution depends on the nuclear state (α or β) and is proportional to the corresponding matrix elements $\langle S_0^{\text{el}}\alpha^{\text{nuc}}|\hat{H}_{\text{RP}}^{\text{HF,1}}|T_0^{\text{el}}\alpha^{\text{nuc}}\rangle$, and $\langle S_0^{\text{el}}\beta^{\text{nuc}}|\hat{H}_{\text{RP}}^{\text{HF,1}}|T_0^{\text{el}}\beta^{\text{nuc}}\rangle$:

$$ 
\begin{align} 
\omega_{\text{TS}}^{\alpha} &= \frac{\Delta g \mu_B B_0}{2\hbar} + \frac{\pi A_1}{2}, \tag{62} \\ 
\omega_{\text{TS}}^{\beta} &= \frac{\Delta g \mu_B B_0}{2\hbar} - \frac{\pi A_1}{2}. \nonumber 
\end{align} 
$$

Assuming $\Delta g = (g_1 - g_2) > 0$ and $A_1 > 0$, it can be seen that $T_0$ converts into $S_0$ faster with nuclear spin up than with the nuclear spin down by a value dependent on the hyperfine splitting. Triplet-to-singlet conversion is never complete because typical lifetimes of the RPs are too short; however, the lifetimes can be long enough that the probability of an RP being in the singlet (triplet) state upon the re-encounter of the radicals is correlated with the nuclear spin state being up (down) [210, 193]. This is called spin sorting.
Now let us consider the spin-flipping mechanism responsible for multiplet CIDNP occurring at zero field. Zero-field condition for the RP evolution means that the energies of the hyperfine couplings greatly exceed the energy associated with the difference between g-factors in the RP; for typical organic radicals, this occurs at fields below 100 µT. Again, we consider the case when the RPs are excited in the triplet state of the electrons (stage (i) in Fig. 25B). The quencher molecule contains two distinct J-coupled nuclei (denoted as 1 and 2). The Hamiltonian in this case can be written as [192]:

$$ 
\begin{align} 
\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF}} &= \hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,0}} + \hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,1}}, \tag{63} \\ 
\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,0}} &= -2\pi\hbar J_{\text{ex}} \left(\frac{1}{2} + 2\hat{\mathbf{S}}_1 \cdot \hat{\mathbf{S}}_2\right) \nonumber \\ 
&\quad + \pi\hbar (\hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2)(A_1\hat{\mathbf{I}}_1 + A_2\hat{\mathbf{I}}_2), \nonumber \\ 
\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,1}} &= \pi\hbar (\hat{\mathbf{S}}_1 - \hat{\mathbf{S}}_2)(A_1\hat{\mathbf{I}}_1 + A_2\hat{\mathbf{I}}_2). \nonumber 
\end{align} 
$$

The nonzero hyperfine couplings $A_1$ and $A_2$ are between electron 1 and the two nuclei. The $\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,1}}$ term mixes the singlet with the three triplet states of the electrons, whereas $\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,0}}$ does not. Therefore, the presence of the $\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,1}}$ term drives the conversion of the overpopulated electron-triplet states into the singlet state, and the speed of this conversion depends on the nuclear spin state. Kaptein and Hollander showed that mixing the nuclear spin states does not affect spin dynamics in the RP [192], but the total spin of the nonequivalent nuclei is what matters. The frequency of the conversion is different for the nuclei in the singlet state and triplet states:

$$ 
\begin{align} 
\omega_{TS}^S &\propto \frac{\pi|A_1 - A_2|}{2}, \tag{64} \\ 
\omega_{TS}^T &\propto \frac{\pi|A_1 + A_2|}{2}. \nonumber 
\end{align} 
$$

Thus in the case of hyperfine couplings of opposite sign, the electron conversion into the singlet state is faster for the RPs having nuclei in the singlet state, as shown in Fig. 25B. By considering the mixing of different states induced by $\hat{\mathcal{H}}_{\text{RP}}^{\text{ZF,1}}$, it can also be shown that the states of the nuclei may change depending on the change of state of the electron. For example, the RP in the $T_{+1}^e S_0^{nuc}$ state evolves into $S_0^{el} T_{+1}^{nuc}$. Generally speaking, this occurs because the Hamiltonian of the RP [Eq. (63)] commutes with the operator of the total spin of the nuclei and electrons but not with the operators representing the total spin of either the electron or nuclei. Therefore the total spin of the RP is conserved, but not the total spin of the nuclei or electron. This peculiarity is the reason behind the name of the mechanism (spin-flipping).

c) Recombination. Nuclear hyperpolarization obtained upon recombination of the geminal RPs can be understood as follows. According to the model, an RP recombines only in the singlet state forming ‘in-cage’ diamagnetic products, P and Q, that regenerate to the initial form but may acquire nuclear polarization [210]. Only a small fraction, $\epsilon$, of geminal RPs—not even all singlet RPs—recombine in the cage as shown in Fig. 25 (extent of reaction between stages i and ii). The remaining RPs separate, forming escape radicals. In high

field, spin selection applies, see Eq. (62), and those nuclear states that convert the electron triplet state into the singlet state faster gradually become overpopulated in the in-cage products. In the example considered here $\omega_{TS}^\beta > \omega_{TS}^\alpha$, so that on average after evolving under $\hat{\mathcal{H}}_{\text{RP}}^{\text{HF}}$, the RP has a higher probability to be found in the singlet state for the RPs with the nuclei being in $\beta$ state and therefore the $\beta$ nuclear state is said to be selected in the in-cage products. This is illustrated in Fig. 25A, stage iii. Escape radicals, on the other hand, are enriched immediately after separation with the nuclear $\alpha$ states. However, the lifetime of the escape radicals is much longer than the lifetime of RPs, and over this time efficient paramagnetic relaxation occurs so that on average nuclear spins in the escape products quickly relax to the initial 50:50% distribution of $\alpha$ and $\beta$ nuclear states. Here we ignore the relaxation dynamics [179] and for clarity consider that these states do not contribute to the NMR signal. Diamagnetic products in the considered example are therefore enriched with the $\beta$ nuclear states, giving rise to the emission type of NMR spectrum (Fig. 25A right-hand side).

Geminal hyperpolarization at zero-field occurs similarly, but in this case, nuclear spins are flipped. The example shown in Fig. 25B assumes that having singlet nuclear states in the RP stimulates the conversion of the electronic triplet into the singlet state ($\omega_{TS}^S > \omega_{TS}^T$), and a fraction $\epsilon$ of these RPs recombine. Because of the nuclear spin flip, in-cage recombination products become enriched with nuclei in the triplet states. Considering ZULF studies, the obvious example of a spin pair would be a directly bound $^{13}$C and $^{1}$H spin pair; in this case, conventional NMR after an adiabatic increase of the field should reveal enhanced $^{1}$H and $^{13}$C NMR signals, one being of the absorption type, the other of the emission type (or vice versa, depending on the sign of the J-coupling) as shown on the right-hand side of Fig. 25B. Indeed, such spectra of hyperpolarized $^{1}$H and $^{13}$C were observed experimentally for photo-CIDNP of para-quinone [207]. Alternatively, one can perform direct detection of photo-CIDNP ZULF J-spectra. At the time of writing, direct observation of photo-CIDNP spectra of $^{1}$H magnetization precessing in nT fields has been demonstrated [206], but not yet photo-CIDNP ZULF J-spectra.

### 3.7. Statistical polarization

If an ensemble of $N$ independent spin-1/2 nuclei is prepared in a random fashion with each nucleus in either spin-up or spin-down state with respect to an arbitrarily chosen quantization axis, then an excess of either spin-up or spin-down nuclei on the order of $\sqrt{N}$ is expected in each realization of the ensemble. This kind of random polarization enables magnetic resonance noise spectroscopy and even imaging [213, 214], which becomes particularly advantageous for small-$N$ samples. This polarization approach that does not require any action on the part of experimentalists is especially effective in the case of single-spin NMR (see, for instance, [215] and references therein), where the signal size in the case of stochastic polarization is no smaller that that of a fully polarized sample. ZULF NMR with stochastic polarization is being pursued using detection with single-spin sensors based on color centers in diamond [216], see Sec. 5.1.4.
### 3.8. Spin polarization induced by rotating magnetic fields

An intriguing and somewhat counter-intuitive way to generate static thermal spin polarization without a static magnetic field was first demonstrated in 1957 by Whitfield and Redfield [217], who applied a circularly polarized rf field to a paramagnetic solid and showed that a stationary magnetization component perpendicular to the plane of the rf field emerges. The effect can be understood as relaxation toward the instantaneous direction of the rf field and subsequent precession of the magnetization about the rotating rf field. The result is a nonzero magnetization component perpendicular to the plane in which the rf field rotates.

Half a century later, an analogous experiment was performed for nuclear spins in a liquid [218] using audio-frequency rotating fields (9.6 kHz). By adiabatically reducing the strength of the rotating field, the steady-state polarization was transformed into laboratory-frame magnetization and detected with a SQUID. In contrast to prepolarization with static magnetic fields (see Sec. 3.1), the use of rotating fields—which have zero time average—avoids magnetising materials in the vicinity of the sample. This might be of practical relevance in a number of applications [218].

## 4. Spin evolution in ZULF NMR

Polarization of nuclear spins is generally followed by further application of magnetic fields to redistribute the spin order from one nucleus to another or to create detectable coherences. Below we discuss the techniques most widely used in ZULF NMR.

### 4.1. Initiating free evolution

Radiofrequency (rf) pulses are closely associated with the modern NMR performed in persistent high-field magnets, where small (typically on the order of ppm of the leading field) alternating magnetic fields are applied to the sample at an eigenfrequency of the observable magnetization. Countless textbooks and reviews explaining aspects of these pulses—from the most basic to most elaborate—often do not explicitly state that this form of external perturbation is the only option, since as the name ‘persistent’ suggests, the strength and orientation of the magnet are hard or impossible to change. In contrast, there is generally the opportunity to switch the magnetic field in low-field and ZULF NMR. One example is the experiment of Varian and Packard [6], described in Sec. 1.3, in which the magnetic fields for spin polarization and precession were applied along separate (ideally, orthogonal) axes. In that case, the latter field is provided by the Earth and therefore the ‘pulse sequence’ involves merely switching off the polarizing field. The procedure may be emulated in ultralow field by placing two orthogonal coils around the sample inside a magnetic shield, so as to polarize the spins along the first axis and then observe Larmor precession at much lower frequencies about the second [86]. McDermott et al. implemented a spin echo this way by nonadiabatically inverting the bias field, rather than the magnetization, during free precession [36]. Field reversal

of this type is in analogy to the gradient echoes used in MRI, where the polarity of the encoding field is changed. In place of the polarizing coil, one can also use a piercing solenoid passing through the shield walls to introduce the prepolarized sample from the laboratory outside [43, 219].

There is a further minimalist approach to omit the second coil, so that the total field is switched to zero after prepolarization. Fast quenching of the magnetic field amplitude with respect to the Larmor frequency, while keeping the same axis, can lead to coherent oscillations of the sample magnetization in heteronuclear compounds under dipolar and/or J-coupling interactions (see Sec. 2.3). Experimental examples that fit within the modern definition of ‘zero to ultralow field’ appear in the literature as early as 1961, where sub-Earth-field NMR studies of HPO(OH)₂ and other heteronuclear compounds are reported [220, 221]. More recent examples use much the same approaches, combined with coil-based signal detection at high field [222, 223] or magnetometer-based detection in the ZULF regime [38, 44]. In order to produce a measurable signal, the spins must have unequal magnetic moments and initial polarization, as for instance occurs starting from Boltzmann equilibrium of a heteronuclear spin pair I–S. Here polarization is proportional to $γ_I I_z + γ_S S_z$, so that when switching to zero field, the component $(I_z - S_z)$ does not commute with the J-coupling or dipolar Hamiltonians and leads to a time-dependent overall magnetization (see Sec. 1.4). Pulses or more advanced prepolarization techniques can increase the initial amplitude of the component $(I_z - S_z)$ and thus improve overall SNR.

### 4.2. Coherent control

#### 4.2.1. Pulsed-field excitation

Most of the ZULF NMR spectroscopy performed to date involves spin excitation via direct current (dc) pulses. As in the experiments described in the previous section, excitation relies upon nonadiabatic switching of the magnetic field axis to induce Larmor precession—and/or J-coupling evolution—of the spins, so the flip angle of the pulse is given by $θ = γ_I B_{pulse} τ$ for pulse length $τ$, dc field amplitude $B_{pulse}$ and spin species i.

A typical ZULF NMR instrument employs three orthogonal coils such as a triple set of Helmholtz coils [46]. These allow one to change the direction of the pulsed field, for coherence-order-based signal-selection approaches akin to the ‘phase cycling’ techniques used in conventional NMR. These and other error-correction techniques can be useful in two-dimensional and multidimensional implementations of ZULF NMR spectroscopy [224, 225]. Because more than one coil can be energized at a time, such that the pulsed field is applied along any arbitrary direction in space, original pulses can be developed for which there is no direct analog at high field. The option of arbitrary pulse axis offers other advantages, for example direct point-to-point rotations in the Bloch sphere, which may be important where speed matters in a ZULF-NMR pulse sequence.

A disadvantage of single dc pulses is their limited spin selectivity. Under a field of constant amplitude and direction, spins rotate through an angle proportional to their gyromagnetic ratio, therefore the ratio of flip angles $θ_I/θ_S$ between any
two spin species *I* and *S* is a constant ratio, $γ_I/γ_S$, and a non-integer. For instance, in a system of ¹H and ¹³C, the ratio is $γ_{1H}/γ_{13C} = 4.02$. Even rounding to the nearest integer, 4, this is an inconvenient ratio because while it is possible to rotate either spin by a net-odd multiple of $π$ radians, it is impossible to rotate both by a net-odd multiple of $π$ at the same time—for a graphical representation, see the top row of Fig. 26. For similar reasons it can also be difficult to apply selective pulses to spin species with close gyromagnetic ratios, such as ¹⁹F and ¹H ($γ_{19F}/γ_{1H} ≈ 0.94$), and the problem is worsened by having three or more spin species because conditions for spin-selective rotation occur much less frequently. As an example, Fig. 26 shows that there is no common intersection of inversion profiles under a single dc pulse for nuclei ¹H, ¹³C and ³¹P, where $γ_{31P}/γ_{1H} = 2.47$.

An approach towards improving the selectivity of dc pulses is to chain several rotations together, thus creating a composite dc pulse [61, 226]. A successful composite may be regarded as one that is sufficiently tolerant of flip-angle variations to absorb an 'inconvenient' residual fraction of the gyromagnetic ratio into the error-compensation bandwidth of the pulse [61]. Candidate pulses are dc analogs of high-field NMR composite pulses, where flip angles in the rf rotating frame map to the lengths of the dc pulses, and the rf phases correspond to the laboratory-frame axes of the pulsed fields. One example is $π/2_X - π_Y - π/2_X$, which in high field is implemented with a single coil and three pulses with relative rf phases of 0, $π/2$ and 0. In ZULF NMR (see second row of Fig. 26), the composite pulse could be applied as square pulses through separate coils oriented along *x*-, *y*- and *z*-axes, respectively [226]. An alternative, flexible approach is provided by dc-composite-pulse analogs of high-field finite-band pulses, such as Broadband Uniform-Rotation Pure-phase pulses (BURP pulses [227, 228, 229]). These pulses aim to produce finite-band excitation over a relatively narrow but tunable range of gyromagnetic ratios $γ$, whose width depends on the number of simple dc pulses in the composite pulses [229]. The bottom two rows of Fig. 26 illustrate how such pulses can produce spin-species-selective inversion in a system containing ¹H, ¹³C and ³¹P, which have relative gyromagnetic ratios 1:0.252:0.405. In general, these dc composite pulses can produce spin-selective rotations of arbitrary flip angle, including the special angles of $π/2$ and $π$. If for some reason the composite-pulse approach is inconvenient, one can alternatively temporarily apply a higher magnetic field and use conventional pulses to achieve spin selectivity, for example, audiofrequency ac pulses [70].

### 4.2.2. *Selective excitation*

Oscillating fields of nanotesla amplitude can be used to excite a small subset of transitions or even single transitions in the ZULF spectrum of a compound [230]. Such pulses that achieve selective population inversion across a transition $|A⟩ ↔ |B⟩$ can be used to assign the characteristic *F* and *K* quantum numbers (or other conserved quantum numbers, see Sec. 2.4.1) for any connected transition in the ZULF-NMR *J*-coupling spectrum [231]. For example, in an $XA_n$ system at zero field, selection rules are $ΔK_A = 0$ and $ΔF_XA = 0, ±1$. Selective popula-

<figure><img src="image_30.png" /><figcaption>Figure 26: Sequences of constant-field pulses and their spin-species selectivity for control of magnetization along the *z*-axis (red: ¹H, black: ¹³C, blue: ³¹P). Large yellow spheres represent the 3D Bloch-vector trajectory during the pulse, starting from the North pole where the magnetization vector lies along the *z*-axis, (0, 0, 1). Colored spheres indicate the final position of the magnetization vectors at the end of each pulse. The plots below the large spheres indicate the *z*-projections of the polarization of the three nuclear species. The pulse durations are scaled to $π$ rotation on ¹H.</figcaption></figure>
tion inversion across eigenstates $|A\rangle$ and $|B\rangle$ prior to measuring the zero-field spectrum leads to perturbed amplitudes of all allowed transitions to connected states $|C\rangle$, namely $|A\rangle \leftrightarrow |C\rangle$ and $|B\rangle \leftrightarrow |C\rangle$. From these changes in amplitude, one can deduce that $|A\rangle$, $|B\rangle$ and $|C\rangle$ belong to a common manifold of $K_A$. This approach also applies where energy states are split by dipolar couplings [232].

Another type of selective pulse can be used when the ZULF spectrum contains multiple transitions: a pulse of a rotating (i.e. circularly polarized) magnetic field. Fields rotating in the positive or negative sense in the x-y plane respectively excite only positive or negative $\Delta m_F$ transitions [231]. Such fields can be introduced by applying a sine current waveform to two orthogonal coils, with relative phase $\pi/2$. Rotating fields can also be used to saturate the hyperfine transitions of coupled electron-nucleus systems, leading—in analogy to optical pumping—to a mechanism of dynamic nuclear polarization via the Overhauser effect [233].

### 4.2.3. Scaling and suppression of spin interactions

The majority of the early ZULF-NMR experiments in the 1980s and 1990s were focused on obtaining high-resolution NMR spectra of powdered solid samples, by measuring the ‘untruncated’ and rotation-invariant dipole-dipole spin couplings (see Sec. 1.3). To observe only a specific subset of spin couplings while ensuring an overall invariance to sample rotation, coherent-averaging techniques were introduced and developed. In most of these, delta-function pulses were applied between periods $(\tau)$ of free evolution in ZULF. One of these sequences, XY4 $(\tau/2 - \pi_x - \tau - \pi_y - \tau - \pi_x - \tau - \pi_y - \tau/2)$ was particularly effective at suppressing the first-order average-Hamiltonian contributions to linear operator terms of the spin Hamiltonian, such as the magnetic field, while preserving bilinear operator terms such as those involved in homonuclear dipole-dipole couplings [234]. The XY4 sequence therefore provided a zero-field analog of a spin echo and Carr-Purcell-Meiboom-Gill (CPMG) decoupling, where it allowed nominally zero-field spectra to be acquired even in the presence of a small dc field $|B| \ll 1/|\gamma\tau|$. One application of coherent averaging in ZULF NMR was therefore to lower the experimental demands on field shimming, by coherently decoupling the spins from residual background magnetic fields. Related sequences for averaging higher-order terms of the Hamiltonian were found via recursive expansion of the XY4, for example the WHH or WAHUHA sequence [235].

Around the same time, a few groups showed that isotropic averaging could be performed entirely in high magnetic field by combining rapid magic-angle spinning of the sample with synchronous rf pulses, to select terms of the Hamiltonian of a given spherical tensor rank [236, 237, 238]. These ‘zero-field in high-field’ experiments have the advantage that conventional spin-selective (between isotopes, e.g., $^2$H and $^1$H) rf pulses can be employed, and they do not require field cycling. They are limited, however, by a need to use spinning speeds on the order of several tens of kHz. At the time these were considered somewhat high; today, sample speeds above 100 kHz are

considered to be routine for commercial magic-angle-spinning NMR probes.

Families of pulse sequences for true-zero-field and ‘zero-field-in-high-field’ coherent averaging were analyzed extensively by Llor, who developed an analytical formalism for Hamiltonian engineering using the theory of rotation quadrature in polyhedral symmetry groups [26, 27, 28]. The objective of Llor’s approach was to find a series of pulses that scale a given spin interaction in the first-order average Hamiltonian by a fixed numerical factor that depends only on its (spin) spherical tensor rank, in a way that is independent of principal axis system. It turned out that the Euler rotation angles of the pulses, when analyzed in a convenient frame of reference, i.e., the ‘toggling’ interaction frame in which the frame is rotated cumulatively by all of the previous pulses, could be equated to the polar coordinates of regular polyhedron vertices, for instance vertices of an octahedron—hence the name polyhedral isotropic averaging. It was then shown that, depending on the symmetry group of the polyhedron, spin interactions of tensor rank 1, 2 or higher could be independently scaled to zero, therefore finding conditions under which a given spherical component of the spin Hamiltonian is averaged completely. For example, a rank-1 decoupling pattern (or in other words, a sequence that scales the rank-1 average Hamiltonian by a factor $k_1 = 0$) should always scale the rank-2 interactions between a factor $-1/5 \le k_2 \le +2/5$. The flip angle of the corresponding pulses is $2\pi/3$. The allowed set of scaling factors is represented by the dark-shaded region in Fig. 27. A negative scaling factor corresponds to time reversal of the corresponding spin interaction. Experimentally, this can be observed in the form of a signal echo, see for example rank-2 dipole-dipole coupling reversal shown in Fig. 28.

The application of isotropic averaging under true ZULF conditions as a means to obtain ultrahigh-resolution NMR spectra of solid samples is nowadays almost obsolete, following the major increase in magic-angle-spinning speeds achievable and magnetic field strengths that make ‘zero-field-NMR in high field’ the preferred alternative. Recently, however, ZULF decoupling techniques have become of interest and have been applied successfully to liquid-state samples, where it can be desirable to average out heteronuclear scalar couplings [239, 61]. It can be convenient, for example, to decouple ‘unwanted’ spins during the course of a hyperpolarization procedure, in order to steer the polarization towards a particular spin species of interest. The decoupling of residual magnetic fields can also prolong spin lifetimes by suppressing unwanted spin-decoherence pathways, or provide a margin of error to tolerate poor shimming of the background field [61]. As another application, decoupling can also be used to implement quantum gates in the ZULF regime [240].

### 4.2.4. Magnetic field sweeps in the ZULF regime

The ultralow-field regime is used in PHIP experiments to transfer the $^1$H spin-singlet order into $^1$H or $^{13}$C magnetization. The attraction of $^{13}$C as a target nucleus includes a wide chemical-shift range for spectroscopic dispersion (at high field), combined with long relaxation times on the order of
<figure><img src="image_31.png" /></figure>

Figure 27: Allowed isotropic scaling factors $k_1$ and $k_2$ in the first-order average Hamiltonian for spin spherical rank-1 and rank-2 interactions, respectively, under true-zero-field (ZF, dark gray) and zero-field-in-high-field (ZFHF, light gray) conditions. The curved boundary corresponds to the most general of the regular-polyhedron sequences. XY4 and bilinear rotation decoupling methods developed in [234] also fall within the ZF area. After reference [27].

<figure><img src="image_32.png" /></figure>

Figure 28: An example of an isotropic spin echo produced by a polyhedral symmetry-based pulse sequence at zero field. In (a), transverse magnetization (signal on the vertical axis) of ¹H in a sample of polycrystalline adamantane decays due to ¹H-¹H dipole-dipole couplings. In (b), starting at around 70 µs and lasting until 810 µs, a sixteen-element decoupling sequence imposes scaling factors $k_1 = +1/5$ and $k_2 = -1/5$. Time reversal of the strong rank-2 interaction produces an echo at ~350 µs or around $|k_2|^{-1} = 5$ times later. In (c), when free evolution resumes, a second echo appears at 880 µs. Reprinted (figure) with permission from [26]. Copyright 1991 by the American Physical Society.

<figure><img src="image_33.png" /></figure>

Figure 29: A procedure for microtesla-field hyperpolarization of [1-¹³C]-fumarate, an XAA' spin system (X=¹³C, A=¹H). (a) Reaction scheme for production of spin-singlet polarized AA' spins in fumarate by chemical addition of parahydrogen to [1-¹³C]-acetylene dicarboxylate. (b) Representation of population transfer within the $m_F = \pm 1/2$ state manifolds. Colored balls represent the initial and final state populations during an adiabatic magnetic field sweep from -2 µT to 2 µT along the z axis. The result of the sweep is to generate opposite Zeeman polarization on the ¹³C and ¹H spin species. Adapted from references [241] and [242] under terms of the Creative Commons CC-BY license 4.0.

minutes to facilitate perfusion of the hyperpolarized compound through a living organism. There are a few mechanisms to convert the singlet order into ¹³C magnetization which involve avoided crossings that occur at magnetic field values on the order of $B \approx 2\pi J_{IS}/|\gamma_I - \gamma_S|$, where ZULF eigenstates transition between zero-field (total angular momentum) and high-field (Zeeman) eigenstates. In the simplest case we can consider a three-spin system such as the [1-¹³C]-fumarate molecule as shown in Fig. 29a (see also Fig. 41), containing two protons weakly coupled to a ¹³C spin. In Fig. 29b, the relevant ultralow-field avoided crossings are shown. In one singlet-to-magnetization transfer approach the magnetic field is increased adiabatically from negative to positive field, and the spin system passes through four avoided crossings (of which three are relevant for the evolution of state populations) [124]. In another approach, the field is increased adiabatically from zero field upwards and the spin system passes through two avoided crossings (of which one is relevant for the evolution of state populations) [122]. A third approach is to carry out the hydrogenation at the avoided-crossing field itself, which can yield up to 50% ¹³C polarization in a three-spin system [243]. The situation is more complex for molecules with additional spins in the *J*-coupling network, but these methods are still employed albeit with some need for optimization of the precise magnetic field used. These procedures are closely related to singlet-to-magnetization pulse sequences used to convert homonuclear spin-singlet order into observable coherence in high-field NMR [244].

An alternative type of system for which such ultralow-field transfers of singlet polarization to magnetization are employed is SABRE-SHEATH [129, 130]. The major difference in this case is that breaking of the magnetic-equivalence symmetry oc-
<figure><img src="image_34.png" /><figcaption>Figure 30: (a) Atomic magnetometer-detected single-scan ¹H NMR spectrum of a 1.5 mL quantity of 10 % H₂O in D₂O, recorded ~100 ms after remote prepolarization at 1 T. A four-channel "gradiometric quadrature" configuration of two commercial magnetometers was used, as described in reference [249]. The spectrum was obtained from a 10 s trace sampled at 2 kHz; no filtering or other lineshape correction was applied. SNR was calculated as the ratio of peak amplitude to standard deviation of noise in the frequency region 14-15 Hz (inset). (b) Single-scan ¹H NMR spectrum of a 0.5 mL quantity of 10 % H₂O in D₂O, recorded using a 1 T benchtop spectrometer (Magritek ULTRA, 40 MHz). The spectrum, obtained from a 6 s acquisition, shifted to 5 Hz oscillation frequency and sampled at 2 kHz; no filtering or other lineshape correction was applied. SNR was calculated as the ratio of peak amplitude to standard deviation of noise in the rightmost 5 Hz frequency window (inset).</figcaption></figure>

curs due to chemical exchange and therefore involves modulation of J-couplings during the transfer process. In typical cases, the sample is held at an avoided-crossing field during parahydrogen bubbling, so that the proton singlet order is converted into heteronuclear ($^{15}$N or $^{13}$C) magnetization. This approach has been modified in recent years to include modulating magnetic fields for obtaining signal enhancements under conditions beyond those described by level anticrossings [245, 246, 247, 248].

## 5. Detection in ZULF NMR

In high-field NMR, Faraday induction is used for signal pick-up, providing high sensitivity at signal frequencies of megahertz and above. Inductive detection was used in early ZULF experiments, but is less common now since alternative detectors that are sensitive at low signal frequencies (kilohertz and below) are available, such as optically pumped magnetometers (OPMs), superconducting quantum interference devices (SQUIDs), color centers in diamond, and others. In particular, the emergence of SQUIDs in the 1980s and advances in OPM technology in the early 2000s were critical points for the development of ZULF NMR.

These gains in sensitivity mean that, at the time of writing, detection in the ZULF regime is only one to two orders of magnitude less sensitive than detection using a high-field benchtop NMR spectrometer. We illustrate this in Fig. 30, comparing the signal-to-noise obtained for single-scan detection of 10 % H₂O in D₂O samples in the two regimes. The benchtop measurement was performed on a 0.5 mL static sample in a 1-T field; the ZULF measurement on a 1.5-mL sample pre-polarized in a 1-T field, shuttled down (~100 ms) into the ZULF region for quadrature detection using two commercial OPMs. Although these results show an approximate comparison between ZULF

versus benchtop NMR, they depend on experimental factors and details of the detector used.

In this section we describe the available detection modalities in greater detail.

### 5.1. Detection modalities

#### 5.1.1. Inductive detection

The first NMR experiments at zero external field that we are aware of were performed by detecting the unusually large line splittings due to dipolar interactions in solid hydrogen, which are on the order of hundreds of kilohertz [20]. These experiments used the ‘traditional’ method of absorption of energy from the radiofrequency field in a cavity [21] (see also Sec. 1.3), but could also be performed with modern inductive detection. However, these experiments rely on the specific properties of solid hydrogen.

The first ZULF NMR experiments based on a generally applicable method were carried out using indirect inductive detection: after a polarization period at high field the samples were allowed to evolve under ZULF conditions, and then were rapidly shuttled back to high field for coil-based signal detection [23, 24, 250]. This approach carries the advantage of the high sensitivity of high-field inductive detection, which is usually below 1 fT/Hz¹/² at frequencies above 1 MHz [251], although the challenge comes from the need for shuttling the sample a relatively long distance (order of 1 m) to the center of a high-field NMR magnet.

An alternative approach is to carry out inductive detection *in situ*. This helps to simplify the experimental apparatus and avoid the need for a SQUID or OPM [252, 253]. Using a 1400 turn 90 mm o.d., 20 mm i.d. coil wound with AWG24 (0.51 mm) wire, a detection sensitivity of 20 fT/Hz¹/² at 3.3 kHz was achieved, and this sensor was used to carry out imaging and relaxation measurements on liquid samples. The footprint was similar to that of a SQUID, although the drawback is the drop in sensitivity at lower frequencies relevant for the ZULF regime, meaning detection must be carried out at moderate B₀ field strength.

#### 5.1.2. Superconducting interferometers

The pioneering experiments in zero-field NMR [23, 24] were based on indirect inductive detection (see Sec. 5.1.1), but the necessity of repeatedly returning the sample to a high field meant that these experiments were both technologically challenging and time-consuming. An alternative emerged in the form of detection with SQUIDs, magnetometers sensitive enough to detect nuclear magnetization directly and therefore independently of precession frequency.

The operation of a SQUID is based on the effect of flux quantization in superconducting loops and the Josephson effect. The magnetic flux through a loop is quantized; that is, it is an integer multiple of the elementary flux quantum Φ₀ = 2.07 × 10⁻¹⁵ V·s (1 V·s=1 Wb is the SI unit of magnetic flux, the weber). When an external magnetic field penetrates the ring, an electric current flows to increase or decrease the magnetic flux in the ring to the nearest multiple of Φ₀. The corresponding supercurrent is measured by exploiting the Josephson