effect, which describes the behavior of the supercurrent when Cooper pairs tunnel through a layer of insulating material from one superconductor to another. The insertion of one or two such Josephson junctions gives rise to a measurable voltage change that depends on the strength of the magnetic field [254].

Magnetometers based on the Josephson effect have been used for the detection of NMR signals since the late 1960s (see reference [31] and Sec. 1.3 for an overview). In the context of liquid-state NMR, work in the early 2000s that demonstrated SQUID detection of thermally prepolarized samples in microtesla fields [36, 37] paved the way for J-spectroscopy, as discussed in this review. More recently, SQUID-detected NMR was extended to 2D-NMR studies and combined with hyperpolarization techniques [255, 256, 257, 258].

### 5.1.3. Atomic magnetometers

SQUIDs are among the most sensitive magnetometers; however, even these are not without competition. Already in 1969 Claude Cohen-Tannoudji and co-workers in Paris used an atomic magnetometer to detect magnetization from a gas of polarized $ ^{3}\text{He} $ nuclei [42]. In an atomic magnetometer, atomic spins are polarized by optical pumping. The spins then evolve under the influence of the magnetic field and the resulting evolution is detected (nowadays, most frequently, optically). The practical use of atomic magnetometers for NMR detection had to wait for some three decades. The potential of atomic magnetometry in zero-field spectroscopy and imaging was recognized by Alexander Pines and co-workers at Berkeley, who initiated a collaboration with the local atomic-magnetometry group led by one of us (D. Budker). The developments proceeded from the initial demonstration of detection of hyperpolarized gaseous xenon with an optical atomic magnetometer [43] and low-field remote-detection imaging [259, 260] to detection of J-couplings in the ZULF regime [44] and combining ZULF with parahydrogen hyperpolarization techniques [66, 127]. Atomic magnetometers have also been successfully used by Michael Romalis and collaborators for detection of nuclear quadrupole resonance (NQR) [261], sometimes referred to as ‘the original zero-field NMR’ for its lack of necessity of a bias field.

Since around the early 2000s, the seasoned field of atomic magnetometry has experience a significant boost of activities, in part associated with developments in diode-laser technology, and in part due to the widening use of techniques to obtain narrow (at times, sub-Hz) electron spin resonance lines, with a corresponding boost in the sensitivity to magnetic fields [41, 262]. These techniques include magneto-optical rotation in alkali-metal vapors contained in antirelaxation-coated cells as well as the use of high-density alkali vapors in the so-called spin-exchange-relaxation-free (SERF) regime, see a detailed discussion of the techniques in the book [263].

In recent years, atomic magnetometers with sufficiently high sensitivity (10–20 fT/√Hz) and small size (a few milliliters) have become available commercially from a number of manufacturers (for example, TwinLeaf and FieldLine), with the vast majority of the ‘market share’ for ZULF NMR applications held by QuSpin. In fact, QuSpin sensors, in

<figure><img src="image_35.png" /><figcaption>Figure 31: Experimental apparatus for ZULF NMR comprising a commercial alkali-vapor-cell magnetometer (QZFM, QuSpin Inc.) for detection, and a permanent magnet for remote prepolarization of the sample. The inset shows a magnified view of the sample and sensor. For experiments utilizing parahydrogen, the permanent magnet and guiding-field solenoid are removed and the sample is placed in a custom NMR tube assembly that permits bubbling of parahydrogen through the solution. The end caps on each of the cylindrical magnetic shielding layers are omitted for clarity. Reproduced from reference [45], Copyright (2020), with permission from Elsevier.</figcaption></figure>

combination with high-quality magnetic shielding (e.g., from TwinLeaf) have greatly ‘democratized’ ZULF NMR, making it accessible to scientists without expertise in atomic physics or optics [45, 264, 265]. The development of these sensors was driven mostly not by NMR but rather applications in magnetencephalography (MEG), magnetocardiography (MCG) and other ultralow-frequency-sensing applications. Examples of compact NMR apparatus built using these off-the-shelf components are illustrated in Figs. 31, 43 and 44. The development of SERF magnetometers continues, sometimes specifically directed towards ZULF NMR applications [266]. Reaching the reported ~1 fT/√Hz sensitivity level in the 1 kHz bandwidth encompassing the ZULF NMR frequencies of commonly studied molecules will greatly boost the range of applications, eliminating the need for isotopically labeled compounds.

While SERF magnetometers typically operate at fields up to several microtesla, other commercial atomic magnetometers are becoming available with centimeter-scale dimensions and sensitivities on the order of a few 100 fT/√Hz. These can already be used for NMR detection and have the advantage of enabling operation in unshielded environments (see also Sec. 7.6).

Despite the availability of commercial sensors, some researchers continue to prefer building their own atomic magnetometers, driven either by the desire for particular sensor features (e.g., full control of feedback loops for better stability, for example against temperature drifts, larger bandwidth, fast recovery following an NMR control-pulse sequences, or producing synthetic gradiometers of sensor arrays), or lower cost for a home-made sensor with sufficient sensitivity for less demanding applications.
We also mention here a particular modality for the detection of nuclear spins (typically, of noble gases) with co-located alkali atoms (see, for example, [267] and references therein). An advantage of this approach is that the interaction of the alkali spins and the nuclear spins under study is enhanced due to the Fermi-contact interactions. This has two benefits. First, the nuclear spins can be efficiently hyperpolarized in collisions with the alkali atoms by optical pumping. Second, the strong coupling of the nuclear and alkali spins enables efficient probing of the nuclear spin state by interrogating the alkali atoms. One possible application of such ‘comagnetometers’ is the detection of nonmagnetic perturbations such as apparatus rotations or effective fields due to exotic interactions that may arise due to background dark-matter fields [268] or due to fifth-force interactions [269, 267]. Alkali-noble gas comagnetometers can be configured to work in the self-oscillating regime—a spin maser [270].

### 5.1.4. Color centers in diamond

Conventional NMR relies on the creation of spin polarization and the detection of the associated magnetization. However, it has also been shown that systems with no net time-averaged polarization produce an NMR signal through statistical fluctuations of the nuclear magnetization. This effect is known as ‘spin noise’ [271, 272] (see also Sec. 3.7). The spin noise signal scales as $\sqrt{N}$ (with N being the number of nuclear spins in the detection volume), whereas Boltzmann polarization scales linearly with N, meaning that in strong fields and for typical sample sizes, the spin-noise signal is significantly weaker than that arising from the magnetization. Whereas spin-noise detection can offer distinct advantages even in relatively conventional settings—for example, the spins do not have to be excited using RF irradiation [214]—the greatest advantages emerge for microscopic samples [273]. In the extreme case of N = 1, there is no difference in signal strength for polarized and unpolarized samples, and in fact the sample can be thought of as being fully polarized [274].

In proof-of-principle experiments, NV centers just a few nanometers from the diamond surface have been used for NMR spectroscopy of nanoscale samples [275, 276]. In recent years, this approach has been harnessed to detect signals from single molecules [277].

As spin noise is independent of magnetic field strength, statistical polarization can be detected at arbitrary fields (including ZULF) without the need to ‘hyperpolarize’ the sample. This leads to the ongoing experimental effort to detect nanoscale ZULF NMR with a stochastically polarized sample producing a signal detected with a single NV center [215]. Measurement of J-spectroscopy in liquids would be a natural goal, enabling measurement of metabolic products from a single cell, or other mass-limited samples. However, for liquid samples measured with single NV centers, a new problem arises, which is that spectra are broadened by diffusion [278]. The spin noise signal arises from correlated fluctuations, and high-resolution spectra require long correlation times (analogous to spin coherence times in conventional NMR). While spin fluctuations in a given molecule may remain correlated for many seconds, this is only

beneficial if the molecule stays within the sensing volume of the NV—if one molecule diffuses away and is replaced by another, the spins fluctuations of the new molecule are generally uncorrelated with those of the previous one. A number of approaches for spatial confinement of analytes are being explored, including encapsulation in host–guest complexes, intercalation in layered materials, and reversible binding to a functionalized diamond surface. Molecules encapsulated in endofullerines [279] could be an interesting option; single (electronic) spins within endofullerine have been detected with NV centers [280]. A more general, albeit technologically demanding, strategy may be to control the movement of analytes by nanofluidic (di)electrophoresis.

Another interesting area is the application to solid-state systems, where molecular diffusion is considerably slower. Particularly appealing candidates include those that are largely inaccessible to conventional NMR, including thin films, two-dimensional materials, multiferroics and topological insulators. One notable demonstration of the potential applications of NMR detected with single NV centers is the measurement of $^{14}$N and $^{11}$B in atomically thin hexagonal boron nitride [281], where the authors observed changes in the $^{11}$B nuclear quadrupolar coupling when studying monolayer, bilayer and bulk samples, establishing the capability of NMR measurements to observe nanoscale effects.

### 5.1.5. Anisotropic magnetoresistive sensors

Among the magnetic sensors that do not rely on Faraday induction are solid-state sensors that are based on anisotropic magnetoresistance (AMR). These devices, with sizes down to micrometers, feature sufficiently high bandwidth to cover the range of frequencies (up to tens of kilohertz) of interest to ZULF NMR. However, their drawback is a sensitivity that is typically several orders of magnitude lower compared to atomic magnetometers (Sec. 5.1.3).

Reference [282] reported NMR detection using an AMR sensor. A ‘remote-detection’ arrangement (see Sec. 5.1.8) was used in which protons in flowing water were prepolarized in the field of a superconducting high-field NMR magnet, adiabatically inverted, and subsequently detected in a magnetically shielded region downstream from the magnet. A fast-adiabatic-passage inverter was used to periodically modulate the magnetization of the protons in the flowing water.

A potential future use of AMR sensors is for NMR detection in microfluidic ‘lab-on-a-chip’ applications. An estimate of the optimized sensitivity [282] indicates that $\approx 6 \times 10^{13}$ protons in a volume of $1,000 \mu m^3$, prepolarized in a 1 T magnetic field, can be detected with a signal-to-noise ratio of 3 in a 1 Hz bandwidth. This level of sensitivity is competitive with that of microcoils in superconducting magnets and with the sensitivity of microfabricated atomic magnetometers.

Further progress in AMR-sensor technology may result in higher-sensitivity devices; however, an important issue to be dealt with, especially in the ZULF-NMR context, is the stray magnetic field produced by the sensor that can affect the sample.
### 5.1.6. Radioactively detected ZULF NMR

An interesting noninductive NMR-detection modality involves detection of the products of radioactive decay of NMR-active unstable nuclei [283]. β-NMR relies on preferential emission of the β-decay electron or positron in a direction along or opposite to the spin orientation. This effect has its origin in nonconservation of parity in weak interactions. In γ-NMR, another type of decay asymmetry is exploited—for quadrupolar nuclei ($I > 1/2$), the probability of γ-emission is different along the alignment axis and perpendicular to it. Note that in this case, in contrast to β-decay, there is no preferred spatial direction, only a preferred (bidirectional) axis.

As the detection efficiency for radioactive decay products is high (in principle, down to a single particle), radioactively detected NMR has the potential to increase the sensitivity compared to both inductive and noninductive NMR detection techniques [283]. Furthermore, with radioactive nuclei introduced into a sample, intrinsically background-free detection is possible. The ‘flip side’ (that comes together with the high selectivity in NMR spectroscopy and imaging) is that the method is restricted to available radioactive nuclei that need to be produced and then utilized before the radioactivity has decayed.

β-NMR has been performed in the ZULF regime for the study of quadrupolar nuclei [284, 285]. A ‘marriage’ of ZULF with γ- or β-NMR [2] may open up interesting possibilities for practical materials studies, including batteries, thin films, and biological samples. Proof-of-concept studies are ongoing at CERN in collaboration with researchers at Mainz.

### 5.1.7. Nuclear spin optical rotation

When linearly polarized light propagates co-linearly with the magnetization axis of a magnetized medium, the polarization of the light rotates as a consequence of the Zeeman effect. A classical example is the Faraday effect, discovered in heavy-atom (e.g., lead) glasses as early as the 1840s. The Faraday effect is usually discussed in conjunction with electrons. Nuclear spin optical rotation (NSOR) is also observable in the case of polarized nuclei [286]. The authors of [286] studied optical rotation in polarized water and liquid xenon and, in the case of the latter, found the magnitude of optical rotation of $(5.8 \pm 0.6) \cdot 10^{-6} \text{ rad cm}^{-1}\text{M}^{-1}$, given for fully polarized $^{129}$Xe with molar concentration M (in units mol/l). This magnitude of the effect is a factor of five or so smaller than naive estimates [286], however, is adequately reproduced by more sophisticated theory taking account of the cancellation of contributions from various excited states [287].

While an aesthetically beautiful and conceptually simple method to detect nuclear magnetization and, correspondingly, NMR, the small magnitude of the NSOR effect limits its utility to highly concentrated liquid samples. The technique becomes practically useful in combination with hyperpolarization techniques. For example, NSOR of dilute analytes (90 mmol/l solutions of pyridine and pyrazine) hyperpolarized using SABRE (see Sec. 3.4) was reported in reference [288]. The work of reference [289] combined hyperpolarization using dDNP (see Sec. 3.5) with NSOR detection, where NSOR signals from pro-

tons and $^{19}$F nuclei were observed in dilute compounds. The magnitude of NSOR typically increases towards shorter wavelengths as long as the material remains transparent.

The ideas for future application of NSOR in NMR include correlating optical and, for example, inductive detection as well as MRI without the use of magnetic gradients, the latter based on the ability to localize a laser beam (in the transverse plane).

While we are not aware of application of NSOR for detection of ZULF NMR, exploring this combination in future work appears well motivated.

### 5.1.8. Remote detection

In conventional NMR experiments, the spin-polarization, information-encoding and signal-detection stages typically happen at the same physical location, within a magnet and with the sample residing in a radio-frequency coil. Simultaneous optimization of these three stages may lead to trade-offs. In ZULF NMR, the three steps are often naturally separated spatially and temporally, which provides an additional degree of flexibility in designing and optimizing experiments. In particular, NMR information can be stored in longitudinal magnetization following the encoding stage and then be detected at another physical location. This is known as remote detection [290, 291, 292] and has been employed, for instance, in the context of low-field imaging [259, 260] and detection with AMR sensors [282].

Intriguing directions for remotely detected ZULF NMR include microfluidics-based NMR-on-a-chip [65] and the concentration of, for example, gaseous samples to optimize detection. A recent development of this idea is to use long-lived spin states [293, 294] to deliver the polarization to, say, the brain, and then convert the long-lived spin states that can be used for detection of metabolism and imaging. In connection with ZULF NMR, this has been recently discussed in [295].

## 5.2. Practical considerations

The sensitivity of inductive-coil-based sensing that is used for higher-field NMR scales linearly with signal frequency (decreases to the power of 3/4 above ca. 5 MHz) [296, 251], which explains why inductive detection is less commonly used for low-frequency signals in the ZULF regime. The alternative magnetometry methods described in Sec. 5.1 scale better for low-frequency signal detection, and in particular SQUIDs and OPMs make up the majority of sensors used for ZULF NMR.

An important factor is the background magnetic field operation range of the sensor. Sensors that are based on the interaction of spin-polarized electrons with a magnetic field (e.g., OPMs and color centers in diamond) are sensitive to the size of background fields, as these may perturb or destroy the electron-spin polarization. Other types of sensors based on properties such as magnetoresistance or flux quantization in a superconducting loop may also saturate under background fields. This is notably not the case for sensing based on magnetic induction, or for the scintillators used for RD-NMR (see Sec. 5.1.6), which operate independent of the background-field amplitude.

Most detection modalities rely on recording the dipolar field outside the spin-polarized sample, although this is notably
not the case for nuclear spin-induced optical rotation (NSOR, Sec. 5.1.7) detection in which a light beam probes inside the sample itself. But for other detection approaches it is advantageous to encompass the sample with a detector or an array of detectors to maximise sensitivity. This is essentially the case in high-field NMR and MRI experiments, in which a coil can be wrapped around a sample to maximise the ‘filling factor’, and also in β-NMR, where the scintillators usually surround the sample. Detectors such as OPMs, AMR sensors, and diamond supporting color centers are small in size (typically millimeter scale), and many can be placed around a milliliter-size sample.

Another consideration for detector choice is the logistics of operation. SQUIDs are exquisitely sensitive [5] but they require cryogenic cooling and usually operate in a magnetically shielded room, meaning they may not be easily transportable. OPMs are most sensitive in the SERF regime, at temperatures of the vapor cell exceeding 100 °C, so they must be thermally isolated from most samples. Thermal isolation might also be necessary for diamond-based sensors, as they can heat under laser irradiation. Light-based sensors such as OPMs, color centers in diamond, and NSOR require optical setups and penetration of light into the experimental apparatus, although some detectors operate with a millimeter-scale laser next to the sensing medium, obviating the need for external optical components [297]. Some sensors employ flux concentrators to enhance sensitivity (this is common for AMR-based sensors, but concentrators are can also be used with OPMs [298] and diamond sensors [299]), but these produce a dc magnetic field, and the gradient of this field can cause line broadening in ZULF spectra.

### 5.3. Effects of geometry in detection

The efficiency of detection of nuclear magnetization is generally dependent on sample and sensor geometry.

For conventional inductive sensing of NMR, the signal is proportional to the emf ($\xi$) induced in the pickup coil,

$$ \xi = - \iiint \frac{\partial}{\partial t} [\mathbf{B}_1(x, y, z) \cdot \mathbf{M}(x, y, z, t)] dxdydz, \quad (65) $$

where $\mathbf{M}(x, y, z, t)$ is the nuclear magnetization and $\mathbf{B}_1(x, y, z)$ is the field produced by a unit current applied to the pickup coil [296]. The integral over the three spatial dimensions is taken over the entire space, but is inherently zero in places without sample magnetization, i.e., where $\mathbf{M}(x, y, z, t) = 0$.

One can follow the same reciprocity-based argument for noninductive sensors by replacing $\mathbf{B}_1$ with a sensing field $\mathbf{B}_{\text{sense}}$. For SQUIDs, for example<fnref n="2" />, the sensing field is equal to the field produced by a unit supercurrent in the pickup coil. Then the SQUID signal is proportional to the flux coupled into it,

$$ \phi = \iiint [\mathbf{B}_{\text{sense}} \cdot \mathbf{M}(x, y, z, t)] dxdydz. \quad (66) $$

For atomic magnetometers, $\mathbf{B}_{\text{sense}}$ is the magnetic field that would be produced by the sensing volume if it were replaced

with a medium magnetized in the direction of its sensitive axis (for spherical samples, this is a dipole field). In many cases, the magnetometer sensing volume is approximately spherical, so that $\mathbf{B}_{\text{sense}}$ can be treated as a point dipole located at the center of the cell. A similar analysis was recently performed for micron-scale NMR with ensembles of NV centers in diamond [300].

In all of these cases, the important quantity is the projection of the magnetization onto the sensing field.

#### 5.3.1. Detector arrangement

For a spherical sample and a single point-like magnetic field sensor, the optimal arrangement is to have the sensitive axis of the magnetometer aligned with the sample magnetization and to have the sensor as close as possible to the sample along the magnetization axis. If one has access to multiple sensors, a linear combination of different sensor signals allows for tailoring of the overall $\mathbf{B}_{\text{sense}}$. This is particularly useful for applications that benefit from spatial selectivity, either for imaging or rejection of background magnetic noise [262]. For example, a pair of magnetometers is often arranged as a gradiometer: the first sensor is close to a sample of interest, the second is further away (such that the contribution from the sample is negligible), and one measures the difference in signals between the two. This helps to reject common-mode noise such as background magnetic fields, while retaining the sensitivity to fields from the sample. One downside to this particular arrangement is that by adding an additional sensor, one is measuring twice as much uncorrelated noise. If the common-mode noise is not the limiting factor, this arrangement is actually less sensitive than a single sensor.

Rather than arranging magnetometers to explicitly measure magnetic field gradients, one can arrange them to maximize the signal from a sample, while still minimizing unwanted signals (such as background fields). Examples of different arrangements are shown in Fig. 32, for a spherical sample with uniform magnetization. One approach to designing such ‘gradiometer’ arrangements is to start with the expected field produced by the sample, as in Fig. 32a–f. When placing sensors, simply choose the sensitive axis to match that of the field from the sample, and ensure that there are equal numbers of sensors pointing along any given direction so that they can cancel out background fields.

Another strategy is to design a magnetometer array by considering the overall $\mathbf{B}_{\text{sense}}$ that results from adding up the sensing fields from all of the sensors. For the same detector arrangements as in Fig. 32a–f, Fig. 32g–l shows the resulting $\mathbf{B}_{\text{sense}}$. For example, Fig. 32a,g show the simplest case, where we consider either the dipole field produced by the sample, or the dipole field produced by the sensor—keeping them as close as possible and oriented the same way gives the best overlap. A simple gradiometric arrangement where both sensors are near the sample, but measuring in opposite directions, is shown in Fig. 32b,h. In the three-dimensional case where the sample is a sphere, the signal is only 3/4 that of the most sensitive case (one sensor above and one sensor below), but has the advantage that it is insensitive to uniform background magnetic fields. For

<footnote n="2">²SQUIDs are sometimes also considered to be inductive detectors but that does not alter the argument.</footnote>
<figure><img src="image_36.png" /><figcaption>Figure 32: (a–f) Magnetometers (white circles) in gradiometric detection arrays to measure the magnetic field (vector fields) produced by a magnetized sample (grey circle). (g–l) Sensing field (see text for explanation) **B**<sub>sense</sub> (vector fields) for these magnetometer arrays. (a,g) A simple, commonly used detector arrangement (see Figs. 3, 20) where the magnetometer is offset from the sample in the direction of the sensitive axis. (b,h) A simple gradiometer arrangement—note that common-mode magnetic noise is cancelled out while magnetic signals from the sample are added. (c,i) Another two-magnetometer gradiometer arrangement. Note that this arrangement is sensitive to magnetization in the vertical direction (with respect to the page) even though neither magnetometer is sensitive to fields in the vertical direction. (d,j) Higher-order gradiometer produced by combining two gradiometers of the kind shown in panel (c). (e,k) Higher-order gradiometer produced by combining two gradiometers of the kind shown in panel (b). (f,l) ‘Halbach-type’ gradiometer produced by combining the gradiometers in panels (d) and (e).</figcaption></figure>

the ‘two-dimensional’ case where the sample is a long cylinder with transverse magnetization (a common sample geometry), this is an ideal arrangement for two magnetometers. Such an arrangement has been demonstrated in [102].

Another two-sensor arrangement is shown in Fig. 32c,i. This is an illustration of the fact that, in general, having magnetometers sensitive along a given axis does *not* mean that one is measuring magnetization in that direction. It is also worth noting that **B**<sub>sense</sub> falls off much faster in the arrangements of Fig. 32h and Fig. 32i than in the single-magnetometer case (Fig. 32g).

The arrangements in Fig. 32d,j and Fig. 32e,k are higher-order gradiometers formed by doubling up the arrangements in Fig. 32b,h and Fig. 32c,l, respectively. In addition to the improved sensitivity coming from increasing the number of sensors, these arrangements offer better rejection of background fields, and also rejection of lower-order field gradients.

The arrangement in Fig. 32f,l combines the previous two gradiometers and is reminiscent of a Halbach cylinder. Just as increasing the number of permanent magnet elements in a Halbach dipole magnet increases the strength and homogeneity of the internal field, while minimizing the external field, such

magnetometer arrays are more uniformly sensitive to internal samples, and insensitive to external field sources.

In principle, such detector arrays can be used for image reconstruction as in beamforming analysis for magnetoecephalography [301]. For image reconstruction, having magnetometers sensitive to multiple axes is particularly valuable.

### 5.3.2. Sample geometry

The magnetic field produced by nuclear magnetization also depends on the shape of the sample. For a uniformly magnetized sphere, the resulting magnetic field at position **r** outside of the sample is equal to that of a point dipole (having the same total magnetic moment, **m**) located at the center of the sphere:

$$ \mathbf{B} = \frac{\mu_0}{4\pi} \left( \frac{3\mathbf{m} \cdot \mathbf{r}}{r^5} - \frac{\mathbf{m}}{r^3} \right). \qquad (67) $$

Noting the $r^{-3}$ dependence in Eq. (67), one might come up with the idea of placing a sensor inside the sample, given that the field inside a uniformly magnetized sphere is $\mathbf{B} = 2\mu_0\mathbf{M}/3$, uniform over the volume of the sphere [302]. The reason why this does not work is related to the so-called sphere of Lorentz. The idea is that actually measuring the field inside the sample
necessarily requires removing some of the magnetized material, and removing this material has an effect on the magnetic field. Making use of the superposition principle, the trick is to subtract the magnetic field of a 'removed sphere' from that of the larger magnetized volume. For the case of a uniformly magnetized sphere:

$$ \mathbf{B}_{\text{M}} = \mathbf{B}_{\text{sample}} - \mathbf{B}_{\text{sphere}} = 2\mu_{0}\mathbf{M}/3 - 2\mu_{0}\mathbf{M}/3 = 0, \quad (68) $$

so anywhere you measure inside of a magnetized sphere, the magnetic field (from the sphere) is zero.

For an infinitely long cylinder magnetized transverse to its symmetry axis, the magnetic field within a small spherical cavity inside the cylinder is $\mathbf{B}_{\text{M}} = \mu_{0}\mathbf{M}/2 - 2\mu_{0}\mathbf{M}/3 = -\mu_{0}\mathbf{M}/6$, with nuclear magnetization $\mathbf{M} = n\hbar\gamma_{I}\mathbf{I}$, where n is the number density of nuclear spins, P is the spin polarization, and $\gamma_{I}$ is the nuclear gyromagnetic moment of spins I. For highly magnetized samples, peak shifts arising from this bulk magnetization effect have been used to estimate sample polarization [303].

The magnetic field outside of a uniformly transversely magnetized cylinder is

$$ \mathbf{B} = \frac{\mu_0 a^2}{2} \left( \frac{2\mathbf{M} \cdot \mathbf{r}}{r^4} \mathbf{r} - \frac{\mathbf{M}}{r^2} \right), \quad (69) $$

where a is the radius of the cylinder. If it is possible to place a sensor at a distance $r < \sqrt{3}a$, then it is still preferable to measure the signal outside rather than inside the sample.

For an infinitely long cylinder magnetized along its symmetry axis, the magnetic field within a small spherical cavity is $\mathbf{B}_{\text{M}} = \mu_{0}\mathbf{M}/2 - 2\mu_{0}\mathbf{M}/3 = \mu_{0}\mathbf{M}/3$. Note that the magnitude of this field is twice that of the cylinder with transverse magnetization.

It is important to note that for an infinitely long cylinder magnetized along the cylindrical axis, the field outside of the cylinder is zero. A graphical explanation for this is provided in Fig. 33. We can represent a point dipole such as an infinitesimal volume of magnetized material as a small loop of current, where the current is proportional to the magnetization. Similarly, the field of any two-dimensional object magnetized normal to the plane is equivalent to the field produced by a sum of such current loops. As shown in Fig. 33c, that sum is equivalent to a larger current loop, running along the perimeter of the object. Because a magnetized cylinder is equivalent to a sum of magnetized discs, the field produced is equivalent to field produced by the sum of the corresponding current loops. For our infinite longitudinally magnetized cylinder, the field is therefore equivalent to that of an infinite solenoid, which is zero outside. Many ZULF-NMR researchers made the mistake of trying to measure signals with such a sample geometry.

Another interesting geometry is a slab of material approximating an infinite plane, magnetized normal to the plane. Again, following the approach of Fig. 33, we see that the field of such a slab of material is equivalent to that of a current loop running around its perimeter. Recall that the field produced by a loop of current with radius R is $B = \mu_{0}I/2R$ at the center of the loop. As the radius tends to infinity, the field measured in the center tends to zero.

<figure><img src="image_37.png" /><figcaption>Figure 33: (a) A point dipole—such as a nuclear spin or an infinitesimal volume of magnetized material—can be considered equivalent to a small loop of current. (b) A thin slab of magnetized material, magnetized normal to the plane. (c) Representing a magnetized slab by the sum of many small current loops, we see that most of the currents cancel out, leaving only a finite current density along the edge of the slab. As the geometry approaches that of an infinite plane, the resulting magnetic field at any point outside of the material approaches zero. (d) A cylinder magnetized along its axis can be broken down into a sum of thin discs of material magnetized normal to the plane of the disc, exhibiting a field equivalent to that produced by a circumferential current loop. The sum of these current loops is equivalent to a long solenoid. As the length of such a solenoid approaches infinity, the external magnetic field goes to zero. (e) For an infinitely long strip of material, magnetized normal to the plane of the strip, the field produced is equivalent to that of a pair of currents in opposite directions on either side of the strip. (f) A slab of magnetized material can be treated as the sum of infinitesimal strips of material, magnetized normal to the plane of the strip. The field produced is equivalent to that produced by a pair of opposing current sheets. In the limit of an infinitely broad slab of magnetized material, the external magnetic field goes to zero.</figcaption></figure>
For a slab of material with in-plane magnetization, we can follow a similar thought process to that for a magnetized cylinder. In this case, the slices normal to the magnetization consist of thin strips of material, and the field of each slice is equivalent to that of two opposing lines of current above and below (neglecting the turns at the edges, which are assumed to be far away), as in Fig. 33e. Adding these up, we see that the field of an infinitely broad slab of material with in-plane magnetization is equivalent to that of two opposing sheets of current. For an infinite sheet of current with linear current density *J*, the field is *B* = *μ*₀*J*/*2, parallel to the sheet and perpendicular to the direction of the current. The sum of the fields from the two opposing current sheets is then zero.

Note that the result for an infinite slab is independent of magnetization direction, as any magnetization can be expressed as a sum of the in-plane and out-of-plane cases, which both result in zero external field. This is particularly relevant, among other cases, for microfluidic NMR measurements using NV⁻ centers in diamond, where one might be tempted to produce a wide fluid channel in order to integrate over a larger area—in reality, signals will only come from the edges of the channel, so multiple thin channels would produce more signal than one broad channel. Note also that magnetization along the axis of a long channel looks like an infinite magnetized cylinder; it is therefore best to arrange the experiment such that the magnetization to be detected is transverse to the axis of the channel.

#### 5.3.3. Applied magnetic field

In section Sec. 2, we assumed that the sensor was sensitive in the direction of sample magnetization, and was displaced along the same axis. More generally, we can replace the magnetization vector of Eq. (12) with the field produced by the sample, **B**<sub>M</sub>. Then the field direction relative to **B**<sub>sense</sub> defines selection rules, as discussed in Sec. 2.6.2. Depending on the geometry of the sample and arrangement of the detector(s), one may measure spectra with Δ*m*<sub>*F*</sub> = 0 or ±1, or a combination of the two.

# 6. Extensions of ZULF NMR

## 6.1. Two-dimensional spectroscopy

Just as for high-field NMR, ZULF NMR spectroscopy can be extended to two (or more) dimensions by introducing an additional evolution time into the experiment. Figure 34 shows an example of an early two-dimensional<fnref n="3" /> zero-field NMR experiment [304], designed to measure spin diffusion. This experiment showed that high ‘solution-like’ resolution is achievable as line broadening due to anisotropic powder averaging is absent, given that all crystallite orientations are equivalent because of the absence of a preferred direction at zero magnetic field [305].

<figure><img src="image_38.png" /><figcaption>Figure 34: Schematic diagram of the time-dependent field profile used for early two-dimensional zero-field exchange experiments to monitor spin diffusion. Reprinted from reference [304] with the author's permission.</figcaption></figure>

Zero-field multiple-quantum *J*-spectroscopy was demonstrated in [225], producing two-dimensional spectra like the one shown in Fig. 35. More specifically, Fig. 35 is an example of correlation spectroscopy for a mixture of the two ¹³C isotopomers of ethanol. As expected for correlation spectroscopy, cross-peaks are present only between peaks belonging to the same isotopomer. Furthermore, as expected in ZULF NMR, cross-peaks are only present for resonances belonging to the same spin manifold, differentiated in this case by the total angular momentum, *K*, of the methylene and methyl protons. For example, the right-hand side of Fig. 35 zooms in on one multiplet of the [1-¹³C]ethanol spectrum, dominated by a large peak at 211 Hz. This peak in fact consists of two overlapping resonances belonging to the *K* = 1/2 and *K* = 3/2 manifolds, which are not resolved in the one-dimensional spectrum, but can be identified in the two-dimensional spectrum by the presence of distinct cross-peaks.

In addition to these examples of 2D NMR in the ZULF regime, experiments at the boundary between the low- and ultralow-field regimes have been performed in SQUID-detected NMR at microtesla fields [255, 256, 257]. An in-depth review of two-dimensional methods for ZULF NMR, including field-cycling experiments between ZULF and high field, is given in [306].

## 6.2. Imaging

Imaging is one of the two major subdivisions of the NMR field, along with spectroscopy primarily discussed in the previous sections. Indeed, ZULF imaging, while being well-motivated and holding significant promise, is still in its infancy, with most of the imaging work so far done not in the ZULF conditions, albeit using ZULF-compatible equipment.

A significant body of work was devoted to imaging at fields on the order of 100 µT utilizing SQUID detection [38, 39, 307,

<footnote n="3">³Technically, as the detection phase of the experiment was carried out in a point-by-point fashion by shuttling the sample into a high-field spectrometer, one could argue that this was actually a three-dimensional experiment.</footnote>
<figure><img src="image_39.png" /><figcaption>Figure 35: A two-dimensional ZULF J-spectrum of a mixture of [1-¹³C] and [2-¹³C]ethanol, with cross-peaks appearing only between peaks corresponding to the same isotopomer. The zoomed-in portion of the [1-¹³C]ethanol spectrum shows how peaks overlapping in a 1D spectrum can be resolved in the 2D spectrum if the peaks correspond to different angular-momentum manifolds. Reprinted from reference [225], Copyright (2020), with permission from Elsevier.</figcaption></figure>

40, 308, 309, 152, 310, 136, 233, 311, 312] with a systematic account presented in a monography [313]. Of particular practical importance is MRI at low fields (1–80 mT) [314, 315, 316]. MRI with atomic magnetometer detection has been already demonstrated at such fields [317].

Recently, ¹³C imaging of 50 mM [1-¹³C]pyruvate solution was conducted at 120 µT [318], about twice the Earth's magnetic field, utilizing two different SABRE variants: SABRE-SHEATH and LIGHT (Low-Irradiation Generates High Tesla)-SABRE. The three-dimensional images of a phantom were obtained using a SQUID detector with submillimeter resolution [318]. This work demonstrates what kind of experiments can be done *in vivo* using similar methodology.

The important problem of the so-called concomitant gradients needs to be dealt with at low fields. Maxwell's equations tell us that the divergence of magnetic field is zero. This means that if we apply a gradient, say, $\partial B_z/\partial z$, there necessarily appear the $\partial B_x/\partial x$ and $\partial B_y/\partial y$ gradients so that the sum of all the three terms is zero. At high field, the transverse gradients can usually be ignored as the Larmor frequency depends on the total magnetic field and, to first order, the latter is the sum of the leading field and the $z$ component of the gradient field. However, the transverse gradients become important when the product of the

applied gradient and the size of the system becomes comparable to the (small) leading field and the image suffers significant distortions [307]. Various techniques have been developed to mitigate the concomitant-gradient problem [319].

In the case of detection using optical atomic magnetometers, one early imaging demonstration [259] used remote detection and phase (rather than frequency) encoding with point-by-point detection. While painfully slow, with typical image-acquisition times of hours, the technique has nevertheless resulted in imaging of fluid-flow patterns inside porous metals [260] and reached millimeter-scale spatial resolution [320]. An imaging modality combining prepolarization with a permanent magnet and encoding was implemented [321], which was, in fact, magnetic imaging without resonance.

Current efforts to develop ZULF MRI are focused on several directions [164], including imaging with sensor arrays (similar to how one records magnetoencephalograms, see Sec. 5.3.1) but with an important additional capability of providing spectroscopic information in addition to spatial information.

Another approach was dubbed ‘sweet-spot-scanning’, where one takes advantage of the fact that a characteristic ZULF spectrum vanishes at nonzero field. The zero-field spot
can be produced with coils in the anti-Helmholtz configuration. The ‘sweet spot’ can be moved with respect to the sample to produce its molecular image [322]. To speed up imaging, one can consider using arrays of zeros of the field for parallel imaging.

Finally, as concomitant gradients are a problem at ultralow fields but not at higher fields, to achieve spatial encoding, one can apply a gradient of the ‘dc’ magnetic-field pulse used to initiate the ZULF-NMR signal.

An alternative approach for high-field hyperpolarization-enhanced imaging is to use a singlet state formed by coupled spins prepared in the process of hyperpolarization. In this approach, the singlet state itself is a nonmagnetic singlet state. In this state, the molecules do not produce an observable NMR signal, but if the singlet state is broken, the hyperpolarization can be observed. The advantage is that the singlet state can be long-lived, which means that the signal can be imaged over a longer time [323, 324, 325].

This approach was demonstrated for a DNP-polarized ¹³C-¹³C singlet pair in maleic acid derivatives, using rf pulse sequences to ‘release’ the singlet order gradually over time [325]. Another variant called singlet-contrast imaging relies on a chemical reaction of the contrast agent to break the singlet state and release observable hyperpolarization-enhanced NMR signals [119]. This was demonstrated with a parahydrogen-polarized ¹H-¹H singlet state in fumarate, which was metabolised to malate by an enzyme to break the ¹H chemical equivalence and allow subsequent ¹H imaging. At high field, these singlet-state approaches only work for homonuclear spin pairs, where the two nuclei do not have a Larmor frequency difference. Under ZULF conditions it should be possible to extend this approach to heteronuclear spin pairs [48], which may significantly broaden the range of potential contrast agents for singlet-contrast imaging.

### 6.3. Relaxometry

Throughout the history of NMR, the magnetic field dependence of nuclear spin relaxation rates has been extensively studied and widely applied to elucidating topics ranging from protein and polymer dynamics, to contrast agents for *in vivo* MRI (including SQUID-detected ULF MRI [326]), to food spoilage [327] and petrology [328, 329]. The interest in relaxometry is largely motivated by the utility of relaxation rates in the determination, ideally quantitative, of physical and chemical properties of a system. These may include, for instance, translational or rotational diffusion rates of molecules in solution, or concentrations and relaxivities of dilute paramagnetic species [330, 87, 331].

#### 6.3.1. $T_1$ relaxometry of uncoupled nuclear spins

Nuclear spin-relaxation rates in the case of uncoupled spins can be quantitatively described by parameterizing each of the relaxation mechanisms with a corresponding correlation time $\tau_c$. For gas-phase molecules, the relevant process is rotational-state relaxation with a correlation time related to the interval between collisions (on a nanosecond timescale at room

temperature and atmospheric pressure). Correlation times of molecules in liquids are associated with molecular reorientation and are on the picosecond to nanosecond time scale. Processes with correlation times up to the inverse Larmor frequency, $\tau_c \approx 1/|\gamma B|$, affect relaxation rates most. When magnetic fields are large enough so that $\tau_c \gg 1/|\gamma B|$, the collisional relaxation mechanism becomes ineffective and other mechanisms dominate relaxation. Therefore, low-field relaxation studies are valuable for gathering information about slow processes, for instance the intermolecular dynamics of intermediate- to large-size molecules. We note that a not-too-common exception exists where chemistry allows ¹H/²H isotopic replacement to be performed; in this case, the frequency dependence of relaxation can be probed as a result of changes in gyromagnetic ratios between the two isotopes rather than changes in magnetic field [332].

It is most common to determine $\tau_c$ via a series of relaxation measurements across several decades of magnetic field, say, from millitesla to tesla, and fit the resulting ‘nuclear magnetic relaxation dispersion’ (NMRD) curve to dynamics models [333, 334, 335]. The range of magnetic fields used translates into the range of correlation times that can be probed. Various techniques have been developed to improve the convenience, speed and consistency of NMRD experiments achievable on a single instrument. One of the techniques in use is electromagnetic fast-field-cycling (FFC) pioneered in the 1990s to show adsorption correlation times of several microseconds in microporous glasses [336, 337]. More recently, FFC was extended into the 1–10 µT regime to investigate MRI contrast at such fields [338, 339]. Lately, utilization of atomic magnetometers to detect sub-kilohertz NMR signals enabled FFC measurements to be made inside magnetic shielding [340], so that NMRD data can extend the accessible range of correlation times into the millisecond range [87]. Here the NMRD measurements can also be combined with spectrally resolved ZULF-NMR detection of several chemical species at the same time, to facilitate analyses of dynamics in mixtures [341]. A remaining challenge, however, is to understand the applicability of current relaxation models at the lowest fields. The authors of reference [87] measured $T_1$-NMRD for n-octane ($C_8H_{18}$) confined in the ~5 nm diameter pore spaces of a commercial $\gamma$-alumina sample, showing relaxation rates of $1/T_1 \sim 30 \text{ s}^{-1}$ at ¹H Larmor frequencies below 10 Hz. Wangsness-Bloch-Redfield (WBR) and other conventional NMR relaxation theories cannot strictly describe these scenarios since several key assumptions are not fulfilled, including the assumptions of $T_1 \gg \tau_c$ (for surface jumps in porous materials $\tau_c$ can be the order of 10 ms and so can $T_1$) and $\tau_c \ll 1/|\gamma B|$ in the Redfield model [87]. Nevertheless, there are several known examples where Redfield theory does seem to work in cases where it should not, for instance reproducing the predictions of stochastic Liouville models of relaxation at low field [342]. It will be interesting to see the progression of these theories and their predictions of whether correlation times accessible by measurements in the ZULF regime are limited by spin-diffusion dynamics or chemical dynamics. For now, there remains much to be learned by the application of ZULF-NMRD techniques beyond proof of principle, for in-
stance, to microporous materials of industrial importance [341] or relevant biomolecules [331], biological fluids (including, for example, blood [343]) and plants [344].

### 6.3.2. Relaxometry of coupled nuclear spins

So far, our discussion of ZULF relaxometry has focused on uncoupled ¹H spin systems. In heteronuclear spin systems, there are some other field dependencies to review, because in much the same way that the coherent behavior of spins is different between ZULF and high-field NMR regimes, incoherent spin evolution or relaxation in ZULF is also strongly influenced by the spin eigenbasis. One case is heteronuclear spin singlet ($F = 0$) eigenstates in the ZULF regime (homonuclear singlet-state relaxation has been extensively studied over the past decade by many groups [345]). Examples include the singlet-pair state of ¹H¹³C in [¹³C]-formic acid (H¹³COOH) [48, 341], the E-internal rotational symmetry state of ¹³CH₃ groups [62, 346], and the pseudo-singlet eigenstates of ¹³C-substituted aromatic compounds, such as [¹³C]-benzene [72] and [2–¹³C]-benzoquinone [207]. Such singlet states are immune to relaxation by spin interactions that are perfectly correlated across the spin system, for example, the intramolecular ¹H–¹³C or intermolecular dipole–dipole couplings [347]. Pseudo-singlet eigenstates may also persist with long lifetimes in nonzero field, in the ultralow-field regime, where the Zeeman interaction is small compared to *J*-couplings [341, 348] (see Fig. 37). These states may offer a potential ‘safe haven’ for hyperpolarized spin order, as discussed for [1–¹³C]-fumarate in reference [295].

As total angular momentum states are eigenstates of nuclear spin Hamiltonians for molecules in ZULF (see Sec. 2.4.1), detectable coherences may involve transitions to and from singlet ($F = 0$) eigenstates. These can exhibit decoherence-free properties that are partly responsible for the ultra-narrow spectral linewidths observed in ZULF NMR, on the order of millihertz [349].

A separate important consideration for ZULF NMR is the joint relaxation of spins that are strongly coupled, where one nucleus can act as a relaxation sink for other spins in the system [350, 351, 352, 353]. This is commonly seen for directly-bonded ¹H–¹³C spin pairs where the ¹H spin dominates the ¹³C $T_1$ relaxation, but is particularly detrimental when quadrupolar nuclei with sub-millisecond relaxation times are part of the spin system. Nuclear spins with a high quadrupole moment, including the halogens ³⁵Cl, ³⁷Cl, ⁷⁹Br and ⁸¹Br, however, do not usually exhibit this effect because they relax so rapidly that they effectively self-decouple from the spin system [354], as indicated by the narrow-line spectra shown in Fig. 38. Low quadrupole-moment nuclei such as ²H, ¹⁴N and ¹⁷O [355] on the other hand, depending on their position in the molecule, may introduce fast relaxation of coupled ¹H spins, for example in [2,2,2-d₃]-ethanol [354, 61]. This is unfortunate as ¹H²H isotopic substitution is commonplace in high-field solution-state NMR to simplify ¹H scalar coupling patterns and mitigate intramolecular dipole–dipole relaxation. To benefit from the same tactic in ZULF NMR it would be necessary to suppress the deleterious effect of the quadrupolar relaxation sink. A possible solution

in some molecules may be to perform the ZULF-NMR experiment at a magnetic field where ¹H and quadrupolar spins (e.g., ¹⁴N) are weakly coupled, but other spins (e.g., ¹H and ¹⁹F) remain strongly coupled. An alternative may be to dynamically decouple ²H spins using suitable pulse sequences [61].

The joint-relaxation effect is not only problematic for detecting ZULF-NMR spectra, but also for driving coherent spin-system evolution in the ultralow-field regime with pulse sequences or field manipulations. This is particularly relevant for PHIP/SABRE experiments, where it is common to use ultralow fields to transform ¹H singlet order into magnetization of a non-proton spin. Partial or full deuteration of the target molecule can be highly beneficial for enhancing polarization levels outside of the strong-coupling regime as this reduces dipolar relaxation during the PHIP/SABRE process, but can cause problems for the polarization transfer step in the strong coupling regime. Current experimental practice shows that it is often better to avoid the strong coupling regime altogether by carrying out the polarization transfer step at low field (i.e., not ultralow field, so that the ²H spins are not strongly coupled to spin-1/2 nuclei), using a resonant AC-pulse sequence to drive the transfer. This was demonstrated for polarizing the ¹³C spin in [1-¹³C]-succinic acid-d₂ by applying a $B_0$ field on the order of 10 µT and using pulse sequences such as WOLF (Weak Oscillating Low Field) [356], STORM (Singlet-Triplet Oscillations through Rotating Magnetic Fields) [357], or a combination of the two [358]. Similar approaches from the technical standpoint employing alternating magnetic fields have been demonstrated for SABRE-polarized acetonitrile [359].

### 6.4. Partial molecular alignment and orientation

Independently of truncation by a dominant magnetic field (see Sec. 2.7), terms of the nuclear spin Hamiltonian can be suppressed by isotropic molecular tumbling. Because of this, in typical liquid-state NMR experiments, the magnetic dipole–dipole coupling term, Eq. (2)—a valuable source of molecular information—is averaged to zero. However, under partial alignment, this information is recovered in the form of the so-called residual dipolar couplings (RDCs).

ZULF NMR measurements of RDCs were performed in [2–¹³C]-acetonitrile aligned in stretched polyvinyl acetate gels [57]. These experiments constituted the first investigation of dipolar couplings as a perturbation on the spin–spin *J*-coupling in the absence of applied magnetic field. Future work will address the temperature- and electric-field-dependent phase transitions in liquid crystals.

To observe RDCs or other anisotropic interactions, motion of molecules should be restricted. This can be done in several ways. One method is based on the use of alignment media, such as stretched polymer gels [360] or lyotropic liquid crystals [361]. These approaches have been carefully studied as the RDCs extracted from the high-field NMR spectra of such partially oriented samples are directly related to the distances between atoms and their mobility in molecules [362, 363]. Another method is based on the use of diastereomorphous interactions between chiral analyte and chiral alignment media, which in turn can be observed through the RDCs [364].
<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

Figure 36: (a) Field dependence of ¹H magnetization decay rates 1/T₁ for n-octane and n-decane in porous γ-alumina and anatase titania. For n-decane in γ-alumina, colors indicate the NMR detection field, using fast-field switching (blue) or fixed field (red). These correspond to the colors also used in (b) and (c); (b) Pulse sequence details. The signal is detected with an atomic magnetometer; (c) Frequency- and (d) time-domain ¹H-NMR signal for n-decane in γ-alumina after relaxation at 2 µT (85 Hz Larmor frequency). The reconstructed time-domain signal (blue curve) equates to the Lorentzian line shape fitted in (c). Reproduced from reference [87] under terms of the Creative Commons CC-BY license.

<figure><img src="image_42.png" /></figure>

Figure 37: Relaxation rates of spin order originating from ¹H magnetization in [¹³C]-formic acid (¹³CHOOH) samples: (a) pure liquid, (b)–(d) pure liquid inside silica pores of mean diameter (b) 50 nm, (c) 30 nm, and (d) 10 nm. The dotted vertical line in each plot indicates the magnetic field where the ¹H Larmor frequency equals the ¹H-¹³C scalar coupling constant, around 220 Hz. Above and below this field are the high-field and ZULF relaxation regimes, respectively. The slower relaxation within the ZULF regime results from the long-lived singlet state of formic acid, which outlives longitudinal magnetization by a factor of between 2 and 3 even in the porous confinement. Dependence of the vertical error bars on the magnetic field is due to a systematic uncertainty in the estimation of the relaxation times, as described in the original paper. Reprinted from reference [341], Copyright (2018), with permission from Elsevier.
<figure><img src="image_43.png" /><figcaption>Figure 38: The zero-field *J*-coupling NMR spectrum of a mixture of [2-<sup>13</sup>C<sub>1</sub>]-acetic and bromo-[2-<sup>13</sup>C<sub>1</sub>]-acetic acids. This shows an example of halogen self-decoupling in the ZULF regime, where scalar couplings between <sup>13</sup>C and Br are slow compared to the Br relaxation rate, with the result that the presence of Br does not broaden the resonance line. Reprinted from reference [354], Copyright (2019), with permission from Elsevier.</figcaption></figure>

Apart from alignment media, external electric fields can be used to partially order molecules in solutions [365, 366]. Among the benefits of the latter approach is that allows gradual and reversible introduction of anisotropy, facilitating sensitive experiments. In addition, with electric fields molecules can be oriented, not only aligned.

Distinguishing between alignment and orientation is impor-
tant. *Alignment* of molecules implies that they are preferentially
ordered along a director axis, without distinction between ‘up’
and ‘down’ directions along that axis. *Orientation*, on the other
hand, implies a preferential direction along the director axis,
and molecules are ordered either parallel or antiparallel relative
to this axis.

6.5. ZULF NMR of solids

Zero-field NMR is ideally suited for studies of oriented
samples including the use of electric field to achieve partial
alignment and orientation (see Sec. 7.2.4 for application in the
direct determination of molecular chirality). In the ZULF NMR
of solid-state samples, evolution under dipole couplings does
not depend on the orientation of crystallites as there is no pre-
ferred direction in zero field, and the resolution of ZULF NMR
spectra of powders can be comparable to that of high-field
NMR spectra of single-crystal samples. Furthermore, ZULF
NMR provides ways of observing all six terms of the ‘dipolar
alphabet’ (see reference [159] and Sec. 2.7), whereas even in
principle only one term for nuclei of different species is visible
in high-field NMR (Fig. 16). ZULF-NMR measurements of
the dipolar alphabet are thus expected to give additional (com-
pared to high-field NMR data) dynamical information on spin
evolution in the solid state.

One way to reintroduce dipolar couplings normally aver-
aged in a liquid is to use stretched polymer gels. Using different
degrees of polymerization, one can effectively smoothly transi-
tion between a liquid and a solid. In the context of ZULF NMR,
such experiments were performed by the authors of [57] who
reported measurements of residual dipolar couplings in [2-<sup>13</sup>C]-
acetonitrile aligned in stretched polyvinyl acetate gels. They
investigated dipolar couplings as a perturbation on the indirect
spin-spin *J*-coupling in the absence of an applied magnetic field
and observed terms of the dipole-dipole coupling Hamiltonian
that are invisible in conventional high-field NMR.

A conceptual approach to moving from ZULF NMR of flu-
ids to that of solids, which complements the cross-gel tech-
nique, is the study of ZULF NMR of viscous liquids as a func-
tion of viscosity [367]. In that work, ethylene glycol was used,
a liquid with a sharp dependence of viscosity on temperature in
the convenient range of 30–100°C. Prospects for ZULF NMR
of solids were recently revisited [368]. The conclusions of the
authors are optimistic despite the significant challenges asso-
ciated with relatively broad spectral lines and the presence of
dipole-dipole interactions.

7. Applications of ZULF NMR

7.1. Physics

There are several directions where large-scale hyperpolar-
ization is particularly important in fundamental science. Ex-
amples include spin-polarized nuclear targets and NMR-based
searches for dark matter and the fifth force (e.g., spin gravity,
Sec. 7.1.4), and parity violation (Sec. 7.1.3). There are several
important parameters for this work, including, first and fore-
most, the total number of polarized spins. The density of po-
larized spins is characterized by molar polarization, which is a
product of concentration and polarization. Other relevant pa-
rameters include the degree of polarization and the relaxation
rates.

7.1.1. *Polarized nuclear targets*

Polarized targets for accelerator-based nuclear experiments prepared using PHIP (Sec. 3.4) and specifically taking advantage of the ZULF regime were discussed in [369]. There are multiple potential advantages for using hyperpolarized liquids in this regime. First of all, liquids undergo thermal convection and liquid targets are therefore particularly attractive when the target needs to take significant energy deposition from high-energy beams. For some experiments, near-zero magnetic fields are a regime of choice for extracting the relevant information from the scattering process. Finally, if one prepares a coherent superposition of ZULF states, the target polarization would undergo oscillation at the frequencies of *J*-coupling resonances. This latter feature can be thought of as a fast reversal, extremely useful for precision measurement and isolation of polarization-dependent effects. The use of nonhydrogenative parahydrogen-based hyperpolarization (SABRE) was also discussed in [369] and is currently being pursued in conjunction with experiments at the MAMI accelerator facility at Mainz (B. Collins and S. Duckett, private communication, 2023).
### 7.1.2. Dark matter searches

While ZULF-NMR spectroscopy was originally developed with no apparent connection to fundamental physics, exciting applications in this area were introduced later [370, 371].

Among these are searches for ‘ultralight bosonic dark matter’ that produced significant exclusions of the possible dark-matter parameter space [49, 372] (now further improved by other experiments). Dark-matter fields oscillating with periods as long as months or even years produce signals that are a priori hard to distinguish from slowly drifting spurious magnetic fields. With ZULF NMR, this problem is solved using co-magnetometry [370, 371]: different NMR transitions of the same molecule can have different relative sensitivities to dark matter (or other nonmagnetic entities) and magnetic fields, allowing them to be distinguished. This has enabled the search for dark-matter fields oscillating with periods as long as years [372].

### 7.1.3. Parity violation

One of the attractive features of ZULF NMR is the sensitivity to untruncated couplings (see Sec. 2.7). An intriguing application is the detection of ‘apparent chirality’ or, more precisely, parity violation in intrinsically nonchiral systems such as atoms or diatomic molecules. This phenomenon originates in fundamental weak interactions responsible for nuclear beta decay. Parity violation manifests in a small mixing of eigenstates of opposite parity in atoms and molecules, an effect observed and well studied in atoms since 1970s, but so far eluding observation in molecules [373, 269].

Of particular interest are the nuclear-spin-dependent parity nonconserving contributions to the molecular Hamiltonian. These result in indirect nuclear spin-spin coupling and can be distinguished from the parity-conserving couplings in molecules of appropriate symmetry, including diatomic molecules [374]. The estimated magnitude of the coupling is in the millihertz range for molecules such as TIF [375]. Couplings of this magnitude can be observed with ZULF NMR in liquid- or gas-phase samples (for molecules existing in gas/liquid form). However, it appears that the most promising approach is molecular-beam experiments with electrically oriented molecules [375].

A related application, detection of molecular chirality, is discussed in Sec. 7.2.4.

### 7.1.4. Search for spin-gravity coupling

An ongoing experiment at Kraków and Mainz is using ZULF NMR to search for the hypothetical nuclear gravitational dipole moment (GDM) that would violate a number of fundamental symmetries [376]. GDM may arise from exotic beyond-standard-model fields sourced by the Earth, in which case the ‘GDM’ terminology would be a misnomer and one should rather talk about an exotic ‘monopole-dipole’ coupling (having nothing to do with gravity), where monopole refers to the Earth and dipole refers to the spin. Another source is a modification of gravity beyond general relativity, in which case the term GDM would be accurate.

<figure><img src="image_44.png" /><figcaption>Figure 39: Fundamental-physics experiments such as searches for the exotic spin-gravity couplings benefit from molecular nuclear spin co-magnetometry: the splittings of different Zeeman *J*-multiplets of the same molecule generally have different dependencies on the applied (ultralow) magnetic field and the new-physics coupling. This allows the effects to be disentangled via analysis of the overall spectrum [371]. Reprinted from reference [2], Copyright (2021), with permission from Elsevier.</figcaption></figure>

Figure 39 depicts the idea of the experiment. The interaction of the GDM with the gravitational field is analogous to the interaction of a nuclear magnetic dipole moment with a magnetic field. The key to isolating the subtle GDM effect from systematics (e.g., magnetic field drifts) is the single-component co-magnetometry technique [371] that has already proven itself in a search for dark matter [372]. The ‘trick’ is to simultaneously record multiple groups of spectral lines that depend differently on magnetic field and exotic physics.

Experiments of this type operate at the boundaries of sensitivity of ZULF-NMR experiments and require samples with the highest possible numbers (‘a bucket’) of polarized spins. The other requirement is the presence of an $XA_n$ group (e.g., $XA_3$). Recent work at Kraków employed on the order of a milliliter of neat $^{13}$C-methanol prepolarized in the field of a 1.5 T Halbach magnet. Modern hyperpolarization techniques such as those based on parahydrogen (see Sec. 3.4) may not yet offer a significant improvement, due to dilution; however, brute-force polarization in a stronger field may yield several orders of magnitude in the number of polarized spins, enabling a new level of sensitivity in GDM searches.

### 7.1.5. Quantum control

While NMR of spin ensembles was once one of the central paradigms for the development of quantum computing, there is less effort in this direction today because it is not easy to scale an NMR platform and because working with pseudo-pure states necessitates the use of phase-cycling protocols with exponential time scaling [377]. Nevertheless, NMR remains an active playground for research in quantum control, if technically as a ‘classical simulation of a quantum computer’.

As ZULF NMR represents a radically different modality compared to traditional high-field NMR, it is interesting to examine what opportunities may be afforded for quantum control in the ZULF regime. Let us say we wish to look at the simple
two-spin qubit comprised by *J*-coupled ¹H and ¹³C spins. Either of the spins can be manipulated by application of dc pulses; however, such pulses necessarily affect both spins at the same time. There are various ways to overcome this limitation and effectively address only one of the spins (see Sec. 4.2.2). One technique takes advantage of the ratio of gyromagnetic ratios of ¹H and ¹³C being close to 4. Thus, for example, applying a 4π-pulse to protons is an approximation of a π-pulse on the carbons. The residual differences can be compensated by employing composite pulse sequences [378].

Experimental demonstration of universal quantum control in ZULF NMR was presented in reference [240]. The work utilized composite pulses for both arbitrary one-spin rotations and a two-spin controlled-not (CNOT) gate in the heteronuclear ¹H-¹³C two-spin system. Using randomized benchmarking inspired by quantum information processing protocols and partial quantum process tomography, single-spin control for ¹³C was achieved with an average fidelity of 0.9960(2) and two-spin control via a controlled-not (CNOT) gate with a fidelity of 0.9877(2). Applications to materials science, chemical analysis, and fundamental physics were envisioned; however these have not been developed as yet. We are confident that quantum-information processing and quantum optimal control [379] can inspire methods to improve ZULF NMR spectroscopy.

### 7.1.6. *Quantum simulation*

An intriguing development currently emerging in the field of quantum science and technology is the quantum simulation of NMR experiments [380, 381, 382]. In quantum simulation, quantum systems with constituents that can be, ideally, initialized, controlled and read out individually are used to emulate the quantum dynamics of other, less controllable systems. In an early example of the approach [380], a four-qubit trapped-ion quantum computer was used to simulate the zero-field NMR spectrum of the methyl group in acetonitrile [62].

The motivation for choosing ZULF NMR as an early test case was the complex dynamics even for relatively small biomolecules [382]. Given the rapid progress in several quantum technologies and on quantum algorithms for Hamiltonian learning [381], we expect that in the foreseeable future system sizes can be reached where classical numerical simulation of ZULF-NMR spectra is not practical. Experiments can then provide a direct means of verification. In the longer term, quantum simulations might guide the choice of molecules to be studied experimentally.

### 7.2. *Chemistry*

Generally speaking, for a spectroscopic method to deliver insight into the properties and behaviour of molecules, it should, first, yield detectable signatures that distinguish different molecules of interest, and, second, be applicable in 'typical' environments in which chemical reactions and transformations take place. In high-field NMR, the spectroscopic signatures are typically chemical shifts. As these can be obscured by bulk susceptibility effects related to sample inhomogeneity, 'good' samples for high-resolution NMR have to be homogeneous, or at least symmetric [383]. Variations in the magnetic

<figure><img src="image_45.png" /></figure>

Figure 40: A comparison of magnetic susceptibility broadening in low and high-field NMR for a highly heterogeneous pellet sample, shown in the photograph. The spectra on the right show the ¹H-NMR resonance in *n*-decane, which is imbibed in the porous pellets; these comprise silica of mean diameter 50 nm with a 15 wt.% surface coating of cobalt (II/III). Spectra are detected in a field of 600 nT (sum of 64 scans, prepolarization field 0.02 T) using an atomic magnetometer and at 1.4 T (single scan) using a commercial NMR instrument. Reprinted (adapted) from reference [341], with the permission of AIP publishing.

susceptibility—for example, discontinuities due to bubbles or precipitates in liquid samples—lead to distortions of the applied magnetic field and thereby degrade spectral resolution.

By contrast, ZULF NMR spectra are largely unaffected by susceptibility, due to the absence of a strong applied magnetic field. This is true even for highly heterogeneous samples [57, 341], such as porous materials supporting paramagnetic catalyst species on the surface, see Fig. 40. Such robustness against susceptibility broadening greatly helps relating information derived from ZULF NMR spectra to changes in molecular electronic structure.

In addition, as at typical frequencies used in ZULF NMR the skin effect is negligible even for metals—for example, the skin depth of aluminum at 1 kHz is about 2.5 mm, and for nonmagnetic 316 stainless steel it is more than a centimeter—applications become possible that are out of the reach for high-field NMR, including the monitoring of reactions inside metallic chemical reactors (see Sec. 7.2.2) and the study of electrochemical cells (discussed in Sec. 7.2.3).

Chemical reactions occurring in the ZULF regime exhibit different nuclear spin dynamics compared to working at high field. At high magnetic field, the Zeeman interaction dominates. When observing, for example, molecules undergoing a chemical transformation, the chemical shifts and *J*-couplings change, but these changes are on the scale of hertz to kilohertz and are small compared to the Larmor precession frequencies of hundreds of megahertz. Nonetheless, these relatively small changes are sufficient for NMR to be a powerful tool for chemical fingerprinting and identification. In zero to ultralow field the situation is different: the *J*-coupling interaction dominates and the Zeeman interaction is a perturbation to this, making chemical shifts negligible. In this case, the relative effect of changing *J*-couplings, by a chemical reaction, on the spectra is much greater, as the dominant interaction is modified. This situation is helpful for chemical identification as the spectra now depend strongly on the *J*-couplings, but can have a negative impact on the polarization of molecules undergoing transformations at zero or ultralow field.

In order to understand this negative impact, consider
nonequilibrium nuclear spin order in a molecule whose J-coupling network is subsequently modified, for example by changing the couplings and/or adding or removing spins during a chemical reaction. In a ZULF-NMR experiment, such modifications can amount to a relatively large change in the spin eigenbasis. Molecular transformations typically occur much faster than the nuclear spin system can evolve, which means that they are nonadiabatic processes with respect to the nuclear spin dynamics. Therefore, if initial spin order is distributed across a spin network and it is then subjected to a change in the J-couplings and spin topology of the network, spin order can be lost during subsequent evolution in the part of the molecule that experiences the J-coupling change [162]. This is typically not the case at high field, where changes in J-coupling values have almost no effect on the eigenbasis and, therefore, molecular changes do not lead to polarization loss. This potentially undesirable feature of ZULF NMR experiments can be addressed by maintaining a magnetic field around the sample during chemical transformations such that the polarization survives until zero-field read-out [102]. Alternatively, one can exploit the chemical reaction itself to repolarize the spin system, for example, by using parahydrogen [126, 384].

### 7.2.1. Study of chemical exchange

ZULF-NMR spectra may display a strong dependence on chemical exchange—an important kind of chemical dynamics. Zero-field NMR spectra of deuterated solids may display effects of conformational exchange, for example two-site flips, allowing one to extract information about the dynamics [385]. In other studies using $[^{15}N]$-ammonium ions $(^{15}NH_4^+)$ in aqueous solution as a model system, it was shown that pH-dependent proton exchange substantially alters ZULF-NMR *J*-spectra [162]. Well-resolved *J*-spectra can be observed for slow proton exchange rates in solution at pH 0 and below, allowing for measurements with unprecedented precision and in particular the analysis of $^{1}H-$$^{14}N$ and $^{1}H-$$^{15}N$ *J*-coupling ratios [59] (see Sec. 7.3).

### 7.2.2. Chemical reaction monitoring

Low-field NMR can be used for reaction monitoring, carrying many of the advantages of high-field NMR but with greater instrument portability and magnetic field control, for example, in relaxometry measurements [386]. The fact that low-frequency NMR signals pass through electrically conductive materials (including metals) and the absence of line broadening at low fields from magnetic susceptibility heterogeneity make the ZULF regime particularly promising for NMR-based reaction monitoring in industrial reactors. The authors of reference [126] demonstrated monitoring of biphasic chemical reactions with ZULF NMR. In this experiment, the two-step hydrogenation of dimethyl acetylenedicarboxylate to dimethyl maleate to dimethyl succinate with para-enriched hydrogen gas was studied while continuously bubbling parahydrogen through the sample (see Fig. 21). Measurements were carried out both in conventional glass NMR tubes and in a titanium tube, to more closely replicate practical operational

conditions. A consideration for process monitoring with ZULF NMR is that the sample needs to be polarized for detection.

In another example, low-field NMR relaxometry measurements were carried out on tomato samples in 1000 L, metal-lined containers, to observe the spoiling of the tomatoes over time [387]. The containers were lined with 75 µm of aluminium, which would act as an rf shield for typical high-field NMR experiments. By carrying out experiments in a $B_0$ field of ~130 mT, the proton Larmor frequency was ~5 MHz; at this frequency the skin depth of aluminium is ~40 µm, which was sufficient for signal penetration.

Finally, many common metals including some alloys of stainless steel or even aluminum [86] are weakly ferromagnetic, imposing an additional magnetic field when used in proximity to NMR samples. For ZULF NMR, this magnetic field may be large enough to perturb the signals, and also, if spatially inhomogeneous, degrade spectral resolution through gradient broadening. To avoid these problems, metal sample chambers or reactors should be kept as magnetically uniform as possible. Preferably they should be made from metals without ferromagnetic alloying elements, such as high-purity titanium. Ferromagnetic materials may be degaussed prior to experiments to remove residual magnetization, although the dc pulses used to manipulate nuclear spins in the sample may cause some degree of remagnetization.

### 7.2.3. Battery diagnostics

Despite significant progress in rechargeable-battery technologies over the past decades, nondestructive measurement modalities compatible with realistic commercial batteries are limited [388]. While mapping battery susceptibility [389] and detection of small (µA) currents during charging/discharging cycles have been demonstrated [390, 391], it was not possible to detect changes in electrolyte composition directly.

Recent work demonstrated measurements of a battery electrolyte performed with aluminium battery enclosures containing tens of µL of electrolyte [392]. Electrolyte concentrations as well as changes in composition and degradation were quantified with ZULF NMR. NMR precession measurements conducted at 2.7 µT revealed clear signatures of both protonated solvent EC/DMC and electrolyte LiPF₆. Further steps are expected in the direction of operando monitoring of electrolyte degradation during the charging-recharging cycles. While data acquisition times in [392] were on the order of a day (due in part to the small sample volumes used), with realistic improvements outlined by the authors (e.g., using a superconducting pre-polarization magnet), a practical diagnostic system for commercial batteries is within reach.

A further discussion of applications of ZULF NMR in battery research and diagnostics can be found in reference [2].

### 7.2.4. Detection of molecular chirality

One interaction that is 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 might be observed in
ZULF-NMR experiments [75]. Such experiments would enable differentiation between enantiomers without adding any additional chiral agent to the sample.

A chiral object has two forms that are mirror images of each other that cannot be transformed into one another by 3D rotation. For chiral molecules, the mirror images are called R- and S-enantiomers. Many chemicals, notably amino acids and sugars, exhibit chirality, which is often of key importance for bioactivity. It is well known that almost all amino acids in organisms are left-handed, while sugars are predominantly right-handed; their left-handed counterparts are not metabolized. Chirality also defines chemical reactivity in many organic reactions, making this property important for organic synthesis.

Chirality is, however, difficult to probe. For instance, chirality itself does not affect conventional NMR spectra. This is because all the internal interactions commonly observed in NMR spectroscopy, isotropic and nonisotropic, are identical for both enantiomers. Sometimes chiral solvents or chiral derivatizing agents are used to induce measurable enantiomer-specific chemical shifts [393] or RDCs [394] yielding the chirality of a compound relative to a chiral addition. There are several proposals suggesting that molecular chirality can be observed directly by NMR via terms in the spin Hamiltonian arising from an electric-field-induced pseudovector. These proposals involve either detection of chiral nuclear magnetic shielding through an induced oscillating molecular electric dipole moment [395, 396, 397, 398, 399, 400] or a magnetic dipole signal induced by an applied electric field [401, 402, 403]. Chiral effects might also manifest in the electric polarizability of molecules with J-coupled nuclear spins [404, 75].

The general form of the J-coupling Hamiltonian for two spins can be written as

$$ \hat{\mathcal{H}}_J = 2\pi\hbar \hat{\mathbf{I}} \cdot \mathbf{J} \cdot \hat{\mathbf{S}}. \quad (70) $$

Here $\hat{\mathbf{I}}$ and $\hat{\mathbf{S}}$ are the vector spin operators of the interacting spins, and $\mathbf{J}$ is the J-coupling tensor. It is convenient to group the terms of this tensor according to their transformation properties under 3D rotations:

$$ \mathbf{J} = J^{(0)} + J^{(1)} + J^{(2)}. \quad (71) $$

Here, the first term is the isotropic part of the J-coupling, which does not depend on the molecular orientation; it is a scalar quantity (a zero-rank tensor). $J^{(0)}$ is the part that is commonly observed in liquid-state NMR experiments. The $J^{(1)}$ and $J^{(2)}$ components, on the other hand, do depend on molecular orientation, and they always average out due to fast molecular tumbling in the liquid state. They can be measured, however, when the reorientation of the molecules is restricted. The $J^{(1)}$ component transforms as a vector upon rotations, corresponding to a first-rank tensor, whereas the $J^{(2)}$ component transforms as a second-rank tensor. The $J^{(2)}$ component has been studied and observed experimentally in NMR of molecules dissolved in liquid crystals [56] as well as, indirectly, inferred from hyperfine structure observed in molecular-beam or high-resolution microwave spectroscopy [55].

Importantly, dipole-dipole interactions cannot produce a first-rank (vector) component of the interaction tensor, even in the presence of an applied electric field. Therefore, observation of the rank-one interaction is a signature of molecular chirality or parity violation. This can be seen in several ways [374, 375]. Let us consider a general interaction between spins $\hat{\mathbf{I}}$ and $\hat{\mathbf{S}}$. The rank-zero (scalar) interaction should be proportional to the only scalar that can be built out of these vectors, $\hat{\mathbf{I}} \cdot \hat{\mathbf{S}}$, while the first-rank (vector) interaction should be proportional to $\hat{\mathbf{I}} \times \hat{\mathbf{S}}$. The latter term is a (pseudo)vector, and it can only enter the Hamiltonian as part of the scalar triple product with another vector available in the problem. In the presence of an external electric field, $\mathbf{E}$, this vector is the average direction of the molecular axis, which is along $\mathbf{E}$. With this, the triple product becomes $a\mathbf{E} \cdot \hat{\mathbf{I}} \times \hat{\mathbf{S}}$, where $a$ is a proportionality constant specific for the system under study. The quantity $\mathbf{E} \cdot \hat{\mathbf{I}} \times \hat{\mathbf{S}}$ is pseudoscalar, as it changes sign under spatial inversion (which transforms as $\mathbf{E} \to -\mathbf{E}$, $\hat{\mathbf{I}} \to \hat{\mathbf{I}}$, and $\hat{\mathbf{S}} \to -\hat{\mathbf{S}}$). In the absence of parity violation, the only way for the Hamiltonian term to be parity-even is for $a$ to be a pseudoscalar, which is possible in the case of chiral media. We note that an external magnetic field cannot substitute for an electric field because the quantity $\mathbf{B} \cdot \hat{\mathbf{I}} \times \hat{\mathbf{S}}$ is odd under time-reversal.

A practical and important consequence of this symmetry argument is that there is a distinct way, at least in principle, to distinguish the chirality and parity-violation interactions from larger terms in the Hamiltonian. The sought-after effect should behave in a distinct manner with respect to reversals, for example, it should change sign under the reversal of $\mathbf{E}$.

The existence of antisymmetric J-coupling, which transforms as a first-rank spherical tensor under symmetry operations, was predicted more than four decades ago [405]. However, so far this interaction has never been observed, despite experimental efforts to do so [406]. Calculations show that in some cases the antisymmetric component $J^{(1)}$ can be substantial, of the same order, or even larger than the corresponding isotropic part of the J-coupling. Recently, it was found in quantum calculations for a series of molecules [407] that the antisymmetric component of the spin-spin coupling tensor has a magnitude on the order of a few hertz (or possibly even orders of magnitude larger [408]) for commercially available chemical compounds. The largest values of the antisymmetric J-coupling are expected between the heteronuclear pairs containing $^{19}$F or $^{17}$O. In ZULF NMR, such J-couplings between heteronuclei can be detected, but they are truncated at high field.

The antisymmetric J-coupling tensor can be represented as a 3×3 matrix in Cartesian coordinates:

$$ J^{(1)} = \begin{pmatrix} 0 & J_{xy} & J_{xz} \\ -J_{xy} & 0 & J_{yz} \\ -J_{xz} & -J_{yz} & 0 \end{pmatrix}. \quad (72) $$

A 3D transformation given by a matrix $\mathbf{T}$ transforms any tensor as $\mathbf{T}^{-1}J^{(1)}\mathbf{T}$; the three independent components ($J_{xy}, J_{xz}, J_{yz}$) transform as coordinates of a vector upon rotation. These components behave as components of a pseudovector under reflections. The component perpendicular to the mirror plane is
conserved, whereas the other components are inverted. Therefore, different enantiomers of a molecule possess differently oriented antisymmetric components, whereby they can be distinguished. As discussed in Sec. 6.4, an electric field can orient polar molecules. In this situation, the $J^{(1)}$ component that is parallel to the molecular dipole moment is not averaged to zero and can be detected. For a pair of coupled spins, the component parallel to the orienting axis has different signs for R- and S-enantiomers. This causes opposite phases of the ZULF-NMR signal for the enantiomers [75]. Experiments to check these predictions are currently under way in several laboratories.

### 7.3. Precision metrology

An advantage of ZULF NMR is in the intrinsically narrow resonances, as the spectral lines are not broadened by field inhomogeneity. Typically, the statistical uncertainty in determining the spectral line position is given by the width of the line divided by the SNR of the signal. In the case of ZULF NMR it has been possible to push the precision of the line-position determination and the corresponding extraction of $J$-coupling parameters down to the sub-millihertz level [409]. Such levels of uncertainty opens up possibilities to study the subtle effects on spin-spin couplings of environmental effects such as temperature, solvent composition, and viscosity [367].

Another interesting direction is the study of isotopic effects in spin-spin coupling. The strength of spin-spin coupling between nuclei of light elements is mainly determined by the Fermi contact interaction (i.e., a second-order hyperfine interaction effect based on finite probability of finding an electron inside a magnetic nucleus [410]). For example, for amino groups and ammonia, the main difference in the experimentally measured $^{14}$N-H and $^{15}$N-H $J$-coupling values is expected to be due to the Fermi contact interaction. In particular, the coupling strength between $^{1}$H and $^{14}$N/$^{15}$N spins is expected to be directly proportional to the product of the corresponding gyro-magnetic ratios (assuming identical electronic structure). However, a careful analysis of the ZULF-NMR spectra of $[^{14}N]$- and $[^{15}N]$-ammonium revealed that the $|J_{15NH}/J_{14NH}|$ ratio differs from that of gyro-magnetic ratios, and this difference is statistically significant (Fig. 11). From 36,000 measurements of a $NH_4Cl$ sample containing 50:50 mixture of $^{15}$N and $^{14}$N isotopes, the ratio of the $J$-couplings, $(|J_{15NH}/J_{14NH}|)$, was determined to be within the range of 1.4009–1.4013, as compared to the literature data for gyro-magnetic ratios $|\gamma_{15NH}/\gamma_{14NH}| = 1.4027548(5)$ [411]. A specific value of $|J_{15NH}/J_{14NH}|$ depends on the specific peaks used for such estimation and ranges from 1.4009(7) to 1.40129(9) [59].

The amplitude of the primary isotope effect may be defined as

$$ \Delta J = \left( \frac{\gamma_{14NH}}{\gamma_{15NH}} \right) J_{15NH} - J_{14NH}, \quad (73) $$

quantifying the gyro-magnetic-ratio-independent effect of replacing $^{14}$N with $^{15}$N in a molecule. Experimentally, a primary isotope effect $\Delta J \approx -58$ mHz was deduced by analysis of the proton-nitrogen $J$-coupling ratios [59]. The negative sign

for $\Delta J$ is consistent with computational investigations for tetrahedral molecules [412, 413].

While $J$-couplings are well studied for bound molecules, the second-order hyperfine interactions are also manifest in unbound van der Waals complexes. This effect was first observed in a solution of hyperpolarized $^{129}$Xe and pentane [414]. Measurements were performed with SQUID detection in a field of 1 µT. The shifts of xenon and proton NMR lines were analyzed and the average value of the $J$-coupling was found to be statistically significant, -2.7(6) Hz.

### 7.4. ZULF NMR to prepare specific hyperpolarized spin states

A common use of the ZULF regime is to manipulate hyperpolarized spin states within molecules, which can then be used in applications such as biomedical imaging and protein-ligand binding studies. This is particularly relevant for PHIP-polarized molecules, as the $^{1}$H singlet order of parahydrogen is nonmagnetic and would not produce enhanced NMR signals after being incorporated into molecules. It has to be first converted into a magnetic state, and the apparatus for ultralow-field NMR is convenient for this task. Due to the large coils that encompass entire samples and provide high field homogeneity, the fidelity of state transformations is typically higher than is achieved in a high-field NMR system. The ZULF regime has also been used to manipulate polarization in dissolution DNP-polarized samples [415], but this is less common as the DNP process itself yields a magnetic state in the substrate (in contrast to PHIP).

As discussed in Sec. 3.4, level anticrossings can exist between spin states at ultralow fields where Larmor frequency differences between different atoms approximately match molecular $J$-couplings. These level anticrossings can be used to transform and manipulate the spin order of nuclei in molecules, which enables the creation of magnetization (magnetogenesis) from the nonmagnetic starting state of parahydrogen. In the case of PHIP, it is typical to use adiabatic passage through level anticrossings to transform $^{1}$H singlet order into $^{1}$H and $^{13}$C/$^{15}$N magnetization [122, 124, 416]. In SABRE experiments, it is common to carry out the SABRE process at ultralow field to match the field to a level-anticrossing condition, so that polarization is spontaneously generated on the heteronuclei [130]. Reference [417] provides a useful guide for constructing ULF NMR apparatus for the task of polarization transfer in liquids.

In NMR experiments it is usually not possible to observe the state of the nuclear spin system while driving evolution (as the detector saturates under strong driving fields), and therefore these methods to generate hyperpolarization have been optimized by experimental repetition. The experiment is repeated on a freshly polarized sample, and each time a parameter of the transfer sequence is incremented. The state that is generated by the sequence is then measured (usually at high field), and in this way the parameter space can be sampled, albeit slowly.

It was recently demonstrated that it is possible to directly measure the conversion of $^{1}$H singlet order into $^{1}$H and $^{13}$C magnetization in $[^{1-13}C]$fumarate during adiabatic passage using ZULF NMR.[413] This is illustrated in Fig. 41. Using a piercing-solenoid arrangement to provide a magnetic field at
the sample location but not at the position of the magnetome-
ter (so the magnetometer remains sensitive throughout), it is
possible to detect the magnetization of the NMR sample while
applying a field sweep from 0 to 2 µT to induce singlet-to-
magnetization conversion. The geometry of the ZULF appa-
ratus is shown in Fig. 41b. In this example, toggling pulses
are applied before and after the field sweep using composite π-
pulses (90ₓ180ᵧ90ₓ) selective to ¹H, to measure the ¹H mag-
netization without background-field contributions. Before the
field sweep, essentially no magnetization is observed during
toggling. During the linear field sweep, the sample magneti-
zation is observed to increase rapidly as the applied field passes
400 nT, corresponding to the level anticrossing. Following the
field sweep, the toggling pulses reveal a sample magnetization
of a few nanotesla as measured at the position of the mag-
netometer. These results are shown in Fig. 41c. Related ef-
forts are currently underway to characterize the spin dynamics
in the SABRE-SHEATH hyperpolarization process by SQUID-
detected ZULF NMR [258, 418].

ZULF methods are not limited to solution-state samples: al-
though less common, in the solid state, low-field thermal mix-
ing (LFTM) is used to obtain polarization transfer between het-
eronuclei. For LFTM, the external magnetic field is temporar-
ily reduced (typically for millisecond timescales) so that differ-
ent spin species within a solid become strongly coupled (i.e.,
the dipolar couplings dominate over Larmor frequency differ-
ences), and polarization can diffuse from one nuclear species to
another [419, 420, 421].

7.5. Biomedicine

Magnetic resonance has played an important role in
medicine since the development of MRI in the 1970s and 80s,
for both diagnostics and providing image contrast to see inside
the body. This form of radiology is attractive because it is
noninvasive, does not use ionizing radiation, and can be both
quantitative and chemically specific. There is a wide range of
applications such as structural imaging, measuring diffusion,
real-time imaging of organs, and more recently metabolic MRI
where metabolism is observed in real time using hyperpolarized
tracers (e.g., [1-¹³C]-pyruvate). There has been a recent drive
towards lower-field, portable MRI systems [422, 423], to make
this tool more widely accessible for clinicians, particularly
in less economically developed parts of the world. Zero and
ultralow-field NMR (as defined in this review) has not yet been
employed for animal or human imaging, but given the inherent
portability and low cost of ZULF spectrometers and recent
progress in ZULF-imaging, it may in the future be used for
biomedical imaging.

### 7.5.1. *Studies with biomolecules*

Given the portability and noninvasive nature of ZULF
NMR, it is well-placed for studies of biological systems, al-
though with the caveat that at present the sensitivity needs to
be boosted using hyperpolarization. Though to date no obser-
vations have been made of molecules reporting on changes or
flux in biological systems, a number of papers have explored
ZULF NMR for the detection of different biomolecules.

One example was the detection of different isotopologues
of urea, to understand how the zero-field NMR spectra change
depending on proton exchange and isotope labelling [60].
[¹⁵N₂,¹³C]-urea and [¹⁵N₂]-urea were measured in water, at
varied pH, to understand the effect of chemical exchange on
the spectra, and simulations supported the results. Additional
experiments were carried out with varied deuteration level of
the solvent, which produced different spectral patterns due to
the changing spin topology.

Hundreds of biomolecules used in fragment-based drug
discovery can be hyperpolarized by photo-CIDNP to perform
drug-screening experiments [189, 190]. A first step in this
direction was made when benzoquinone polarized via photo-
CIDNP was detected using an OPM at low field [206]. For
these experiments the benzoquinone target was at 5 mM con-
centration, and the polarization obtained by photo-CIDNP was
approximately 0.01%. With further advances in instrumenta-
tion, such as parallel detection of multiple samples, and with
high-field prepolarization in excess of 10 T, such screening
methods have the potential for high throughput even without
hyperpolarization [265].

An important class of biomolecules is that of phosphorus-
containing compounds, since these play a vital role in biologi-
cal functions such as pH homeostasis, cell metabolism and bone
mineralization. A first step towards ZULF-NMR characteriza-
tion of organophosphorus compounds was made when a set of
such molecules were measured at zero field, after thermal pre-
polarization in a 1.8 T magnet [67]. It was shown that subtle J-
coupling differences of less than a hertz between, for example,
trimethyl phosphate and trimethyl phosphine, can be detected
to differentiate between the species.

The library of ZULF spectra of biomolecules was expanded
further by the use of dissolution DNP to polarize metabolites
for ZULF detection, including [2-¹³C]pyruvate, [2-¹³C]acetate
and [1-¹³C]glycine [162, 161]. Spectra have also been ob-
tained for samples of thermally-polarized [1-¹³C]glycine,
[1,2-¹³C₂]fumarate and [1-¹³C]-D-glucose [331].

Furthermore, metronidazole and fampridine molecules
were polarized with SABRE and detected with ZULF
NMR [424]. This is significant because these molecules
are regulatory-approved drugs, fampridine being a specific
blocker of potassium ion channels used to treat patients with
multiple sclerosis, while metronidazole is used to combat
amoebal infections.

A goal of ZULF NMR has been to detect ¹³C-hyperpolarized
metabolites used for *in vivo* metabolic imaging in animals and
humans. There are only a small number of compounds used in
this way, and two key examples are the injection of hyperpo-
larized [1-¹³C]pyruvate, which is converted into [1-¹³C]lactate
(and sometimes detectable quantities of [1-¹³C]alanine and
[¹³C]bicarbonate), and the injection of hyperpolarized [1-
¹³C]fumarate, which is converted into [1-¹³C]malate. At the
time of writing, no *in vivo* imaging or spectroscopy in the
ZULF regime has been performed, but the metabolic reac-
tions described have been observed in NMR tubes at both
zero and low field [102]. The metabolites [¹³C]pyruvate and
[¹³C]fumarate were formed via PHIP, and enzymes were used
<figure><img src="image_46.png" /></figure>

Figure 41: Observation of magnetogenesis in nuclear-spin-hyperpolarized molecules. (a) Hyperpolarized fumarate is formed by chemical reaction of [1-¹³C]-acetylenedicarboxylic acid with para-enriched H₂, and a 0 to 2 µT magnetic field sweep is applied to convert the ¹H singlet order (|S₀α⟩ + |S₀β⟩) into a state of positive ¹H and negative ¹³C magnetization (|T⁺₁β⟩ + |S₀β⟩). (b) An alkali-vapor OPM is situated next to the hyperpolarized [1-¹³C]-fumarate (1 mL, 60 mM) sample, inside a multi-layer magnetic shield. A 'piercing solenoid' coil produces the field sweep at the sample location, but this is attenuated by a factor of ~500 at the position of the OPM. (c) The signal from the OPM during the magnetic field sweep, and the time before and after the sweep while toggling pulses are applied. Reprinted from reference [143] under terms of the Creative Commons CC BY-NC-ND license 4.0.

for the metabolic reactions to obtain sufficient concentrations for detection of the downstream metabolites [1-¹³C]lactate and [1-¹³C]malate.

### 7.5.2. *Dynamic nuclear polarization for in vivo studies*

The work employing DNP for *in vivo* as well as proof-of-principle studies can be generally divided into two categories, one employing Overhauser DNP for enhancing solvent (typically, water-proton) signals [425], and another employing *d*DNP in combination with detection employing OPMs [161, 163] or SQUIDs (to be demonstrated).

Applying ZULF NMR for *in vivo* molecular imaging and/or spectroscopy in combination with DNP methods seems reasonable [164]. Since *d*DNP has been extensively used for both clinical and pre-clinical metabolic imaging for more than two decades [146, 426], sufficiently enhanced NMR signals can in principle be measured *in vivo* with ZULF-NMR. Importantly, in ZULF NMR, not only can magnetization of selected spin groups be detected (for example, not only ¹³C of carboxylic groups or ¹H NMR signals of methyl groups) as in high-field NMR, but also spin orders between various nuclei within molecules, for example in AX₂ or AX₃ groups (see Sec. 2.4). These spin orders can be prepared by varying *d*DNP preparation conditions [323, 427]. Their lifetimes are expected to be different from state to state and to have pronounced magnetic field dependence at hypogeomagnetic conditions: this can be used as an additional source of information about dynamics and function of molecules.

It is important to emphasize that currently the main driver of atomic magnetometer technology is magnetoencephalography (MEG).

ZULF-NMR spectra have been collected using the same devices (see, for example, [428]). Detection of metabolic changes associated with cognitive processes simultaneously with MEG is an interesting direction of research related to brain activity. Combination of ZULF NMR with MEG for simultaneous mapping of brain activity and brain chemistry could be used for detecting abnormalities and improving our general understanding of cognition [164].

We foresee applications of affordable and scalable ZULF NMR coupled with hyperpolarization to study chemical-exchange phenomena *in vivo* and in situations where high-field NMR detection is impractical or impossible.

### 7.6. *Portability and fieldable devices*

A ZULF-NMR setup is fundamentally simple. One needs a magnetic shield to achieve the ZULF conditions and screen out environmental noise, but such a shield is more straightforward to construct and is more lightweight than a high-field NMR magnet (which requires a homogeneous magnetic field; note, however, the developments in portable magnets, for example, [429]). The magnetic field readout can now be done with commercial magnetometers [45, 430, 331], which have a size of typically 1 cm³ and sensor standoff distances of only a few millimeters. These sensors exist thanks to other application areas such as magnetoencephalography that have motivated their path to commercialization. The greatest barrier to portability is in achieving nuclear spin polarization, though this is already possible through the use of permanent magnets (which in contrast to high-field NMR magnets do not need to be homogeneous) and sample shuttling. Significantly larger signals
are achievable using other hyperpolarization techniques, which are rapidly approaching the degree of maturity that would enable their use in such devices. *In-situ* hyperpolarization of the sample, for example using PHIP, as demonstrated in [66, 127], eliminates the need for sample shuttling, which further simplifies the setup. The particular advantage of PHIP for portability is that the polarization source (parahydrogen) can be transported in gas cylinders or in liquid form.

By eliminating the requirement of high magnetic fields for NMR spectroscopy, one can imagine the possibility of developing a lightweight, portable instrument based on ZULF NMR, potentially much smaller even than those illustrated in Fig. 43 and Fig. 44. The ultimate goal for portability and miniaturization may be 'ZULF on a chip' along the lines of 'NIST on a chip' [431], towards which the first steps have already been made [65, 106, 102]. Recent developments with chip-scale PHIP [432] should interface particularly well with ZULF-NMR detection.

The portability of ZULF devices makes them appealing for fieldable NMR, meaning carrying out spectroscopy outside the lab, in an applications setting. In many cases, the high resolution of a standard NMR system is not needed. For portable chemical characterization it is often sufficient to carry out relaxometry or diffusion measurements to measure the presence or quantity of a specific chemical. Examples would include measuring the oil content of grains, or the water content in a hydrocarbon mixture. A fieldable NMR system was shown in 2019 when a helicopter-borne NMR coil weighing 1000 kg and 6 m in diameter was used to measure oil in jerry cans to imitate measurements under sea ice [11]. A simple ZULF spectrometer operating with a 50 µT background field (to imitate that of the Earth) was also shown to have sufficient sensitivity and resolution to differentiate between water and different hydrocarbons [340]. Relaxation measurements showed $T_1$ and $T_2$ differences between water, alcohols, and different hydrocarbons, and diffusion measurements could readily distinguish decane from water. The results serve as a proof-of-principle demonstration that low-field NMR may be a suitable fieldable spectroscopy for characterizing hydrocarbon mixtures in oil wells.

## 8. Conclusions and outlook

We conclude our tour of ZULF NMR with a diagram (Fig. 42) summarizing the status and some of the ongoing development directions in this relatively young field, which, however, has already established itself as a vibrant and rapidly developing subfield of NMR. As for all other NMR branches, progress in ZULF NMR depends on the development of hyperpolarization—a must for ZULF!—, encoding, and detection techniques. Naturally, we cannot know for sure which of the many directions discussed in this review article will turn out most fruitful, and what other interesting developments and application will eventually emerge. However, we can now say with certainty that ZULF NMR will be used and developed for years to come.

What makes us particularly confident in this is the fact that recent years have seen several dozen new doctoral students

<figure><img src="image_47.png" /><figcaption>Figure 42: ZULF NMR at a glance. The techniques and extensions that have already been implemented are shown with dark font, while the ones that are in progress are shown in grey.</figcaption></figure>

specifically trained in ZULF NMR. Here a quote from Felix Bloch, one of the founding fathers of NMR, is perfectly fitting. In his Nobel Banquet speech [433] he said:

> *"Our indebtedness to youth has for me two different aspects. One originates from the daily contact with my students: Their interest and enthusiasm have been a constant stimulus and a great source of inspiration and the spirit of my young collaborators has been an important factor in the success of our work. The other aspect is of more personal nature. I am sure my fellow-scientists will agree with me if I say that whatever we were able to achieve in our later years had its origin in the experiences of our youth and in the hopes and wishes which were formed before and during our time as students."*

The authors of this article agree wholeheartedly.

## Acknowledgements

We are grateful to Prof. Malcolm Levitt, Prof. Igor Kopyug, Dr. Seyma Alcicek, and Dr. Piotr Put for helpful discussions, and to Anne Fabricant and Jingyan Xu for performing the comparison of sensitivity between ZULF- and benchtop-NMR modalities. This work was supported by the DFG (Project ID 465084791), by the DFG/ANR grant BU 3035/24-1, and by the Cluster of Excellence Precision Physics, Fundamental Interactions, and Structure of Matter (PRISMA+ EXC 2118/1) funded by the DFG within the German Excellence Strategy (Project ID 39083149). DAB acknowledges support from the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award. MCD Tayler acknowledges financial support from the Spanish Ministry of Science MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and by Generalitat de Catalunya "Severo Ochoa" Center of
<figure><img src="image_48.png" /></figure>

Figure 43: A mobile ZULF NMR spectrometer developed in Mainz. Reprinted from reference [2], Copyright (2021), with permission from Elsevier.

<figure><img src="image_49.png" /></figure>

Figure 44: An even more portable ZULF NMR spectrometer currently operating in Barcelona [264].

Excellence CEX2019-000910-S; the Spanish Ministry of Science projects, MARICHAS (PID2021-1260590A-I00), SEE-13-MRI (CPP2022-009771) plus RYC2022-035450-I, funded by MCIN/AEI /10.13039/501100011033. This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement 101063517.

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