#  Low-Frequency Excitation of Singlet-Triplet Transitions.
Application to Nuclear Hyperpolarization

Laurynas Dagys  Christian Bengs  Malcolm H. Levitt  Department of Chemistry,
University of Southampton, SO17 1BJ, UK

###### Abstract

Coupled pairs of nuclear spins-1/2 support one singlet state and three triplet
states. Transitions between the singlet state and one of the triplet states
may be driven by an oscillating low-frequency magnetic field, in the presence
of couplings to a third nuclear spin, and a weak bias magnetic field. This
phenomenon allows the generation of strong nuclear hyperpolarization of
${}^{13}\mathrm{C}$ nuclei, starting from the nuclear singlet polarization of
a ${}^{1}\mathrm{H}$ spin pair, associated with the enriched para spin isomer
of hydrogen gas. Hyperpolarization is demonstrated for two molecular systems.

††preprint: APS/123-QED

##  I Introduction

Magnetic resonance experiments usually involve the application of a strong
magnetic field (typically, many Tesla) combined with radiofrequency pulses
(for nuclear magnetic resonance, NMR) or microwave pulses (for electron spin
resonance, ESR) that are resonant with the magnetic Zeeman transitions of the
system. In these high-field conditions, the parts of the spin Hamiltonian that
do not commute with the Zeeman Hamiltonian are usually removed. This _secular
approximation_ leads to a major simplification of spin dynamical theory and is
one of the cornerstones of modern NMR theory [1]. One consequence is that all
transitions involving combinations or multiples of the Larmor frequencies are
not observed in NMR or ESR spectra. Some exceptions to this paradigm exist,
such as “overtone” transitions in the NMR of nuclei with large quadrupolar
moments [2], and the use of non-secular hyperfine couplings in solid-effect
dynamic nuclear polarization [3].

In the case of NMR, the secular approximation breaks down in very small
magnetic fields, such that the Larmor frequencies are comparable in magnitude
to the spin-spin interactions [4]. Here we show that the mixing of states by
non-secular spin-spin couplings in the low-field regime allows selected
“forbidden” transitions to be induced by oscillating magnetic fields. In
suitable circumstances, this phenomenon allows the generation of strong
hyperpolarization in the reaction products of hydrogen gas enriched in the
_para_ spin isomer. We demonstrate the generation of two molecules in solution
with high levels of ${}^{13}\mathrm{C}$ polarization. One of these substances,
fumarate, is a natural metabolite which has been used for the characterisation
of cancer in magnetic resonance imaging (MRI) [5].

Figure 1:  Approximate energy level diagram of a system of two $I$-spins and
one $S$-spin, in the near-equivalence limit ($|J_{13}-J_{23}|\ll|J_{12}|$).
The circles represent the population distribution for a fully populated
singlet state between the two $I$-spins. The Larmor frequencies in the bias
field $B_{\mathrm{bias}}$ are indicated:
$\omega_{I}^{0}=-\gamma_{I}B_{\mathrm{bias}}$ for the $I$-spins, and
$\omega_{S}^{0}=-\gamma_{S}B_{\mathrm{bias}}$ for the $S$-spins. The WOLF
pulse is applied with a frequency $\omega_{\mathrm{WOLF}}$ matching the
transition frequency $\omega_{ST}$ between the indicated pair of states
(equation 7).

##  II Theory

Consider an ensemble of nuclear three-spin-1/2 systems, each consisting of two
nuclei $I_{1}$ and $I_{2}$ of one isotopic type with magnetogyric ratio
$\gamma_{I}$, and a third nucleus $S_{3}$ of a different isotopic type with
magnetogyric ratio $\gamma_{S}$. In the isotropic solution state, the three
nuclei mutually interact by scalar spin-spin coupling terms mediated by the
bonding electrons. These couplings consist of a homonuclear coupling $J_{12}$,
and two heteronuclear couplings $J_{13}$ and $J_{23}$. We assume that the two
heteronuclear couplings are unequal, $J_{13}\neq J_{23}$.

###  II.1 WOLF pulses

A small magnetic field $B_{z}$ is applied along the laboratory frame $z$-axis.
This field is the sum of a time-independent “bias” field $B_{\mathrm{bias}}$,
and a time-dependent oscillating field denoted $B_{\mathrm{WOLF}}$, such that
$B_{z}(t)=B_{\mathrm{bias}}+B_{\mathrm{WOLF}}(t)$. The oscillating field is
given by

$B_{\mathrm{WOLF}}(t)=B_{\mathrm{WOLF}}^{0}\cos(\omega_{\mathrm{WOLF}}t),$ (1)

where $\omega_{\mathrm{WOLF}}$ is the frequency of the oscillating field, and
$B_{\mathrm{WOLF}}^{0}$ is its peak amplitude. Since the applied oscillating
field is small in magnitude and low in frequency, we refer to the oscillating
magnetic field as a WOLF (_Weak Oscillating Low Frequency_) pulse.

Note that the $B_{\mathrm{bias}}$ and $B_{\mathrm{WOLF}}$ fields are applied
in the _same direction_. This geometry differs from most NMR experiments,
where oscillating fields are applied transverse to the main field, with few
exceptions [2, 6, 7].

If the magnetic fields are low enough that chemical shifts are negligible, the
spin Hamiltonian may be written as a sum of five terms

$H(t)=H_{A}+H_{B}+H_{C}+H_{D}+H_{\mathrm{WOLF}}(t).$ (2)

These are given by

$\displaystyle H_{A}$
$\displaystyle=-B_{\mathrm{bias}}\left(\gamma_{I}(I_{{1}z}+I_{{2}z})+\gamma_{S}S_{{3}z}\right)$
$\displaystyle+2\pi
J_{12}\bm{I}_{1}\cdot\bm{I}_{2}+\pi(J_{13}+J_{23})(I_{{1}z}+I_{{2}z})S_{{3}z},$
$\displaystyle H_{B}$
$\displaystyle=\pi(J_{13}-J_{23})(I_{{1}z}-I_{{2}z})S_{{3}z},$ $\displaystyle
H_{C}$ $\displaystyle=\tfrac{1}{2}\pi(J_{13}+J_{23})$
$\displaystyle\times(I^{+}_{1}S^{-}_{3}+I^{+}_{2}S^{-}_{3}+I^{-}_{1}S^{+}_{3}+I^{-}_{2}S^{+}_{3}),$
$\displaystyle H_{D}$ $\displaystyle=\tfrac{1}{2}\pi(J_{13}-J_{23})$
$\displaystyle\times(I^{+}_{1}S^{-}_{3}-I^{+}_{2}S^{-}_{3}+I^{-}_{1}S^{+}_{3}-I^{-}_{2}S^{+}_{3}),$
$\displaystyle H_{\mathrm{WOLF}}(t)$
$\displaystyle=-B_{\mathrm{WOLF}}(t)\left(\gamma_{I}(I_{{1}z}+I_{{2}z})+\gamma_{S}S_{{3}z}\right).$
(3)

The terms $H_{A}$, $H_{C}$, and $H_{\mathrm{WOLF}}$ are symmetric with respect
to exchange of the two $I$-spins, while the terms $H_{B}$ and $H_{D}$ are
antisymmetric. If the Larmor frequency of the nuclei in the bias field is
larger than the spin-spin couplings, and the difference between the
heteronuclear couplings is smaller than the homonuclear coupling
($|J_{13}-J_{23}|\ll|J_{12}|$, i.e. near-equivalence [8]), then the $H_{A}$
term dominates. The terms $H_{C}$ and $H_{\mathrm{WOLF}}$ commute with $H_{A}$
and may be regarded as secular, while the terms $H_{B}$ and $H_{D}$ do not
commute with $H_{A}$ and are non-secular. The terms $H_{B}$ and $H_{C}$ both
give rise to small eigenvalue shifts, which are neglected here, for the sake
of brevity. The exchange-antisymmetric non-secular term $H_{D}$ is the most
important one for the purposes of this paper.

The eigenstates and eigenvalues of $H_{A}$ are sketched in figure 1. The
eigenstates are direct products of the singlet and triplet states of the
$I$-spin pair with the Zeeman states of the $S$-spin, such that the triplet
product states are symmetric and the singlet product states are anti-symmetric
under permutation of spins 1 and 2:

$\displaystyle(12)|T_{M}m_{S}\rangle$ $\displaystyle=(+1)|T_{0}m_{S}\rangle,$
(4) $\displaystyle(12)|S_{0}m_{S}\rangle$
$\displaystyle=(-1)|S_{0}m_{S}\rangle,$

where $M\in\\{-1,0,1\\}$ and the Zeeman eigenequations are

$\displaystyle(I_{1z}+I_{2z})|T_{M}m_{S}\rangle$
$\displaystyle=M|T_{M}m_{S}\rangle,$ (5)
$\displaystyle(I_{1z}+I_{2z})|S_{0}m_{S}\rangle$ $\displaystyle=0,$
$\displaystyle S_{3z}|T_{M}m_{S}\rangle$
$\displaystyle=m_{S}|T_{M}m_{S}\rangle,$ $\displaystyle
S_{3z}|S_{0}m_{S}\rangle$ $\displaystyle=m_{S}|S_{0}m_{S}\rangle.$

The symbols $\alpha$ and $\beta$ are used to denoted $m_{S}=\pm\tfrac{1}{2}$
respectively.

Consider the “forbidden” transition between the $|S_{0}\beta\rangle$ and
$|T_{-1}\alpha\rangle$ state, indicated in figure 1. All Hamiltonian terms in
equation II.1 have a zero matrix element connecting these two states, except
for the non-secular antisymmetric term $H_{D}$, with matrix elements given by

$\langle S_{0}\beta|H_{D}|T_{-1}\alpha\rangle=\langle
T_{-1}\alpha|H_{D}|S_{0}\beta\rangle=2^{-1/2}\pi(J_{13}-J_{23}).$ (6)

The difference between the corresponding diagonal elements of $H_{A}$ is given
by

$\displaystyle\omega_{ST}$ $\displaystyle=\langle
T_{-1}\alpha|H_{0}|T_{-1}\alpha\rangle-\langle
S_{0}\beta|H_{0}|S_{0}\beta\rangle$ (7)
$\displaystyle=B_{\mathrm{bias}}(\gamma_{I}-\gamma_{S})+\frac{\pi}{2}(4J_{12}-J_{13}-J_{23}),$

where the notation $\omega_{ST}$ indicates “singlet-triplet transition”. Note
that this transition frequency includes a combination of $I$-spin and $S$-spin
Larmor frequencies.

The term $H_{D}$ induces a slight mixing of these two $H_{A}$ eigenstates. The
small degree of state mixing would have very little effect, if it were not for
the time-dependence introduced by the coupling to the oscillating magnetic
field, $H_{\mathrm{WOLF}}(t)$. If the oscillation frequency matches the
singlet-triplet transition frequency
($|\omega_{\mathrm{WOLF}}|\simeq|\omega_{ST}|$), the periodic time-dependence
drives coherent transitions between these two eigenstates. For example,
suppose the initial state of the system consists of a strongly populated state
$|S_{0}\beta\rangle$, and a completely depleted state $|T_{-1}\alpha\rangle$.
The density operator is described by

$\displaystyle\langle S_{0}\beta|\,\rho(0)\,|S_{0}\beta\rangle$
$\displaystyle=1,$ $\displaystyle\langle
T_{-1}\alpha|\,\rho(0)\,|T_{-1}\alpha\rangle$ $\displaystyle=0.$ (8)

Consider a WOLF pulse applied for a duration $\tau$, on the resonance
condition $\omega_{\mathrm{WOLF}}=\omega_{ST}$. As shown in the Supplemental
Material [9], the spin-state populations after the pulse are approximately
given by

$\displaystyle\langle S_{0}\beta|\,\rho(\tau)\,|S_{0}\beta\rangle$
$\displaystyle\simeq\tfrac{1}{2}\left(1+\cos(\omega^{ST}_{\mathrm{nut}}\tau)\right),$
$\displaystyle\langle T_{-1}\alpha|\,\rho(\tau)\,|T_{-1}\alpha\rangle$
$\displaystyle\simeq\tfrac{1}{2}\left(1-\cos(\omega^{ST}_{\mathrm{nut}}\tau)\right),$
(9)

where the singlet-triplet nutation frequency under the WOLF pulse is given by

$\displaystyle\omega^{ST}_{\mathrm{nut}}=2\pi\sqrt{J^{2}_{13}+J^{2}_{23}}\sin(\phi+\theta/2)J_{1}(A).$
(10)

The angles $\theta$ and $\phi$ are given by

$\displaystyle\theta$
$\displaystyle=\operatorname{arctan2}(2J_{12},J_{13}-J_{23}),$ (11)
$\displaystyle\phi$
$\displaystyle=\operatorname{arctan2}(J_{13}+J_{23},J_{13}-J_{23}),$

such that $\theta,\phi\ll 1$ in the near-equivalence regime.

The symbol $J_{1}$ denotes a Bessel function of the first kind with its
argument given by

$A=(\gamma_{I}-\gamma_{S})B_{\mathrm{WOLF}}^{0}/\omega_{ST}.$ (12)

These equations apply approximately in the limits $|J_{13}-J_{23}|\ll|J_{12}|$
and $J^{2}_{13}+J^{2}_{23}\ll\omega_{ST}^{2}$.

It is therefore possible to transport population completely from one state to
the other by applying a WOLF pulse of duration
$\pi/\omega^{ST}_{\mathrm{nut}}$. The inversion speed is maximised by choosing
the peak amplitude of the WOLF pulse to equal $B_{\mathrm{WOLF}}^{0}\simeq
2B_{\mathrm{bias}}$, at which point the Bessel function $J_{1}(A)$ reaches its
approximate maximum. Since there are practical limits on the generation of
large oscillating magnetic fields, this technique is most appropriate for low-
field magnetic resonance.

The use of oscillating magnetic fields for the selective excitation of
transitions has previously been explored in the context of ultralow-field NMR
[7]. However, those experiments follow the familiar paradigm of resonant
excitation, in which transitions are induced by modulating off-diagonal
Hamiltonian terms at a frequency that matches the transition energy. In the
current case, on the other hand, the off-diagonal non-secular terms are time-
independent, and it is the diagonal terms that are given a periodic time-
dependence by the oscillating applied field. There is a distant relationship
with selective excitation in magic-angle-spinning solid-state NMR [10].

###  II.2 Parahydrogen-induced polarization

The resonant driving of singlet-triplet transitions is particularly useful in
the context of parahydrogen-induced polarization (PHIP), a technique which is
widely used to enhance NMR signals [11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25]. In this method, hydrogen gas is enriched in the _para_
spin isomer and reacted with a substrate in the presence of a catalyst. The
proton pair of the product molecule exhibits excess population in the singlet
state. The strongly enhanced singlet spin order is converted into
hyperpolarized magnetization of heteronuclei such as ${}^{13}\mathrm{C}$ by
applying a sequence of magnetic fields. A range of suitable techniques has
been developed [26, 13, 15, 16, 17, 18, 20, 22, 21, 27].

We now show that the application of a WOLF pulse in a small bias magnetic
field leads to strong hyperpolarization of ${}^{13}\mathrm{C}$ nuclei in the
reaction products of _para_ -enriched $\mathrm{H_{2}}$ gas, with potential
advantages over other methods, as discussed below.

The principle of the experiment is shown in figure 1. The spin state
populations (indicated by balls) are given for the case that the spins $I_{1}$
and $I_{2}$ are protons originating from the _para_ -enriched hydrogen and the
spin $S_{3}$ is a ${}^{13}\mathrm{C}$ nucleus in the product molecule. The
populations of the $|S_{0}\alpha\rangle$ and $|S_{0}\beta\rangle$ states are
strongly enhanced due to their provenance as the nuclei of the _para_
-enriched $\mathrm{H_{2}}$ spin isomer. Since these two populations are equal
there is no polarization of the $S$-spin at this stage. However, if the WOLF
pulse transports the population from the $|S_{0}\beta\rangle$ state to the
$|T_{-}\alpha\rangle$ state, as shown in equation II.1, the resulting
population distribution has excess population in the $|S_{0}\alpha\rangle$ and
$|T_{-}\alpha\rangle$ states. Since the $S$-spin is in the state
$|\alpha\rangle$ in both cases, this corresponds to a high degree of
${}^{13}\mathrm{C}$ polarization. A ${}^{13}\mathrm{C}$ polarization of the
order of unity represents an enhancement of the ${}^{13}\mathrm{C}$ NMR
signals by around 5 orders of magnitude, relative to ordinary NMR based on
thermal equilibrium polarization in a strong magnetic field.

We propose the acronym WEREWOLF (Whopping Enhancement Requires Excitation by
Weak Oscillating Low Fields) for _para_ -hydrogen-induced polarization of
heteronuclei using WOLF pulses for the singlet-triplet population transfer.

Figure 2:  ${}^{1}\mathrm{H}$-decoupled ${}^{13}\mathrm{C}$ spectra of (a) fumarate and (b) maleate at a field of 9.41 T. Single-transient WEREWOLF-hyperpolarized ${}^{13}\mathrm{C}$ spectra are compared with conventional ${}^{13}\mathrm{C}$ NMR spectra acquired at thermal equilibrium, averaged over 360 transients (for fumarate) or 512 transients (for maleate). The strong ${}^{13}\mathrm{C}$ peaks are from hyperpolarized naturally occurring ${}^{13}\mathrm{C}$ nuclei at the molecular sites indicated by filled circles. The ${}^{13}\mathrm{C}$ peaks at 160 ppm and 76 ppm in (a) correspond to unreacted disodium acetylene dicarboxylate. The small signal at 172 ppm in (b) is attributed to succinate generated by secondary hydrogenation.  Figure 3:  Schematic diagram of the experimental setup. (a) Mu-metal shield and associated components. During the WOLF pulse, the Helmholtz coil generates the bias field $B_{\mathrm{bias}}$ whereas the solenoid coil produces the oscillating field $B_{\mathrm{WOLF}}(t)$ (b) Gas-handling apparatus including a thick-wall NMR tube equipped with a capillary for the _para_ -enriched $\mathrm{H_{2}}$ gas. MCU - micro-controller unit. More details are given in the Supporting Information [9].  Table 1: Spin-spin coupling parameters for (1-${}^{13}\mathrm{C}$)fumarate and (1-${}^{13}\mathrm{C}$)maleate, adapted from reference [27].  Compound |  $J_{12}$(Hz) |  $J_{23}$(Hz) |  $J_{13}$(Hz)
---|---|---|---
Fumarate | 15.9 | 5.8 | 3.3
Maleate | 12.3 | 12.9 | 2.5

##  III Methods

###  III.1 Materials

The substances fumarate (_E_ -butenedioate) and maleate (_Z_ -butenedioate)
were used for demonstrations of WEREWOLF. The chemical structures are shown in
figure 2. Both compounds are formed by the catalytic hydrogenation of
acetylene dicarboxylate, using a ruthenium-based catalyst for fumarate [19],
and a rhodium-based catalyst for maleate [21]. Chemical details are given in
the Supplemental Material [9].

About 2% of fumarate and maleate molecules contain a naturally-occurring
${}^{13}\mathrm{C}$ nucleus at the sites shown in figure 2. The two
${}^{1}\mathrm{H}$ nuclei and the ${}^{13}\mathrm{C}$ nucleus form a three-
spin-1/2 system of the type discussed above. The $J$-coupling parameters for
the two molecular systems are given in table 1.

_Para_ -enriched hydrogen was produced by passing hydrogen gas over an iron
oxide catalyst cooled by liquid nitrogen.

###  III.2 Equipment

A sketch of the equipment is shown in figure 3. This consists of a mu-metal
chamber for magnetic shielding, equipped with a solenoid coil (for the WOLF
pulses) and a Helmholtz coil (for the bias field). A guide coil allows the
sample to traverse the chamber walls without passing through zero magnetic
field. The coils are driven by waveform generator and stable current sources.
The equipment set also includes computer-actuated valves and an NMR tube
equipped with a capillary, allowing bubbling of _para_ -enriched
$\mathrm{H_{2}}$ under microprocessor control. Full details are in the
Supplemental Material [9].

Figure 4:  Timing diagram for the WEREWOLF procedure, involving bubbling of
the sample with _para_ -enriched hydrogen gas and a sequence of magnetic
fields. (a) ${}^{13}\mathrm{C}$ radio-frequency pulse applied in high magnetic
field at the end of the procedure and NMR signal acquisition. (b) Magnetic
field along the $z$-axis, showing the ambient laboratory field during sample
heating, the reduction in field as the sample is placed in the shield, the
bias field $B_{\mathrm{bias}}$ during the WOLF pulse, the removal of the
sample from the shield and insertion into the high-field NMR magnet. (c)
Oscillating WOLF pulse field, applied for a duration $\tau$, with a frequency
$\omega_{\mathrm{WOLF}}$ and peak amplitude $B_{\mathrm{WOLF}}^{0}$. The total
field experienced by the sample is the sum of (b) and (c).

###  III.3 Experimental Procedure

Figure 4 shows a timing diagram of the WEREWOLF experiment, showing the
magnetic fields experienced by the sample as a function of time. Each
experiment starts by warming the sample mixture to $\sim
90{}^{\circ}\mathrm{C}$ in the ambient magnetic field of the laboratory
($\sim$ 110 $\mu$T), followed by insertion into the magnetic shield. _Para_
-enriched hydrogen gas is bubbled through the solution at 6 bar pressure. The
bias field is reduced to $B_{\mathrm{bias}}=2\,\mu\mathrm{T}$ while the
oscillating WOLF pulse is applied with amplitude
$B_{\mathrm{WOLF}}^{0}=2\,\mu\mathrm{T}$ for an interval $\tau$. The sample is
removed from the shield and inserted by hand into the 9.41 T NMR magnet where
a 90∘ ${}^{13}\mathrm{C}$ radio-frequency is applied. The ${}^{13}\mathrm{C}$
free-induction decay is acquired in the presence of ${}^{1}\mathrm{H}$
decoupling. Fourier transformation of the NMR signal gives the
${}^{13}\mathrm{C}$ NMR spectrum. Full details are in the Supplemental
Material [9].

Figure 5:  Hyperpolarized fumarate (a) and maleate (b) ${}^{13}\mathrm{C}$
signal intensities as a function of the WOLF pulse duration $\tau$. The WOLF
pulse frequencies are 77.3 Hz for fumarate and 74.0 Hz for maleate. The black
dots depict integrated experimental signal amplitudes, normalized to $1$ for
the maximum value. The grey lines represent numerical _SpinDynamica_ [28]
simulations for the coupling parameters in table 1. The blue lines show
analytical solutions given by equation II.1. All values of $\tau$ used for the
experiments are integer numbers of the WOLF period
$2\pi/\omega_{\mathrm{WOLF}}$, which is 12.94 ms for fumarate and 13.51 ms for
maleate.

##  IV Results

Figure 2 shows single-transient hyperpolarized ${}^{13}\mathrm{C}$ NMR spectra
for (a) fumarate and (b) maleate, obtained using the WEREWOLF procedure. The
WOLF pulse parameters were $\omega_{\mathrm{WOLF}}/2\pi=77.3\,\mathrm{Hz}$ and
$\tau=0.65\,\mathrm{s}$ for fumarate, and
$\omega_{\mathrm{WOLF}}/2\pi=74.0\,\mathrm{Hz}$ and $\tau=0.12\,\mathrm{s}$
for maleate.

Conventional ${}^{13}\mathrm{C}$ NMR spectra obtained by multiple signal
acquisitions on the hydrogenated samples after thermal equilibration are also
shown in figure 2. Comparison of these spectra allows the estimation of the
${}^{13}\mathrm{C}$ polarization levels achieved by WEREWOLF, which are
$p\simeq 8\%$ for fumarate and $p\simeq 19\%$ for maleate. These results are
highly competitive with previous work [29, 30, 24], especially when the
relatively low _para_ -hydrogen enrichment levels, crude apparatus, manual
sample transport, and sub-optimal reaction conditions are taken into account.

Integrated ${}^{13}\mathrm{C}$ signal amplitudes as a function of the WOLF
pulse duration $\tau$ are shown in figure 5. The coherent oscillations of the
hyperpolarized magnetization are striking. Each experimental point was
obtained from a separate experiment on a fresh sample. The normalized
experimental data is compared with numerical simulations and analytical curves
derived from equations II.1 and 10. The agreement between both curves and the
experimental data is gratifying.

##  V Conclusions

Magnetic resonance phenomena involving non-secular spin-spin couplings are
normally encountered in systems with large nuclear quadrupolar couplings [2]
or hyperfine couplings to unpaired electrons, as in solid-effect dynamic
nuclear polarization (DNP) [3]. The work described here shows that non-secular
couplings may also be exploited in purely nuclear spin systems, albeit at an
energy scale which is $\sim 9$ orders of magnitude lower than in solid-effect
DNP. Here too, non-secular effects allow “forbidden” transitions to couple to
the electromagnetic field, allowing the generation of strong nuclear
hyperpolarization.

There are several other techniques for performing the transformation of
nuclear singlet order to heteronuclear magnetization, including pulse
techniques in high field [13, 15, 17, 20, 27], field-cycling and level-
anticrossing phenomena [26, 16, 18, 21], and low-field methods [18, 21, 25].
The WOLF method described here is highly intuitive, provides a very fast
exchange of singlet and triplet populations, and has further potential
advantages over current methods. Since the method works in the presence of a
bias magnetic field, problems associated with ultralow-field NMR are avoided,
such as short coherence lifetimes in the presence of rapidly relaxing nuclear
species such as $\mathrm{{}^{2}H}$ and $\mathrm{{}^{14}N}$ [23]. The bias
field resolves the Larmor frequencies of the different nuclear isotopes,
making it possible to implement heteronuclear spin decoupling by resonant
transverse oscillating fields. Furthermore, the coherent oscillatory spin
evolution under the WOLF pulses suggests that the full palette of “pulse
tricks” developed for conventional high-field NMR may be deployed in this
regime, including error-compensating composite pulses [31] and adiabatic
frequency sweeps [32]. Applications are envisaged to other hyperpolarization
techniques, such as the SABRE (Signal Amplification by Reversible Exchange)
method [14, 22]. The method described in the current paper might be related to
a recently described variation of SABRE [25].

###### Acknowledgements.

We acknowledge funding received by the Marie Skłodowska-Curie program of the
European Union (grant number 766402), the European Research Council (grant
786707-FunMagResBeacons), and EPSRC-UK (grants EP/P009980/1, EP/P030491/1,
EP/V055593/1). We thank Soumya Singha Roy for discussions.

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