#  Symmetry-Based Singlet-Triplet Excitation in Solution Nuclear Magnetic
Resonance

Mohamed Sabba  School of Chemistry, University of Southampton, SO17 1BJ, UK
Nino Wili  Interdisciplinary Nanoscience Center (iNANO) and Department of
Chemistry, Aarhus University, Gustav Wieds Vej 14, DK-8000 Aarhus C, Denmark
Christian Bengs  School of Chemistry, University of Southampton, SO17 1BJ, UK
Lynda J. Brown  School of Chemistry, University of Southampton, SO17 1BJ, UK
Malcolm H. Levitt<EMAIL_ADDRESS>School of Chemistry, University of
Southampton, SO17 1BJ, UK

###### Abstract

Coupled pairs of spin-1/2 nuclei support one singlet state and three triplet
states. In many circumstances the nuclear singlet order, defined as the
difference between the singlet population and the mean of the triplet
populations, is a long-lived state which persists for a relatively long time
in solution. Various methods have been proposed for generating singlet order,
starting from nuclear magnetization. This requires the stimulation of singlet-
to-triplet transitions by modulated radiofrequency fields. We show that a
recently described pulse sequence, known as PulsePol (Schwartz et al., Science
Advances, 4, eaat8978 (2018)), is an efficient technique for converting
magnetization into long-lived singlet order. We show that the operation of
this pulse sequence may be understood by adapting the theory of symmetry-based
recoupling sequences in magic-angle-spinning solid-state NMR. The concept of
riffling allows PulsePol to be interpreted using the theory of symmetry-based
pulse sequences, and explains its robustness. This theory is used to derive a
range of new pulse sequences for performing singlet-triplet excitation and
conversion in solution NMR. Schemes for further enhancing the robustness of
the transformations are demonstrated.

††preprint: APS/123-QED

##  I Introduction

_Long-lived states_ are configurations of nuclear spin state populations
which, under suitable circumstances, are protected against important
dissipation mechanisms and which therefore persist for unusually long times in
solution Pileio (2020); Carravetta, Johannessen, and Levitt (2004); Carravetta
and Levitt (2004); Sarkar, Vasos, and Bodenhausen (2007); Pileio, Carravetta,
and Levitt (2010); Tayler and Levitt (2011); Levitt (2012); Tayler and Levitt
(2013); DeVience, Walsworth, and Rosen (2013a); DeVience (2014); DeVience,
Walsworth, and Rosen (2016); DeVience  _et al._ (2021); Levitt (2019); Zhang
_et al._ (2014, 2015); Stevanato  _et al._ (2015); Pravdivtsev  _et al._
(2016); Rodin  _et al._ (2018, 2021, 2019); Sheberstov  _et al._ (2019);
Kharkov  _et al._ (2019); Mamone  _et al._ (2020); Bengs  _et al._ (2020);
Mamone  _et al._ (2021); Bengs  _et al._ (2021); Cavadini  _et al._ (2005);
Cavadini and Vasos (2008); Sarkar  _et al._ (2008); Ahuja  _et al._ (2009);
Salvi  _et al._ (2012); Buratto  _et al._ (2014a, b, 2016); Berner  _et al._
(2019); Mamone  _et al._ (2021); Eills  _et al._ (2017); Stevanato  _et al._
(2017); Bengs, Dagys, and Levitt (2020); Korenchan  _et al._ (2021); Kharkov
_et al._ (2022); Roy and Mahesh (2010); Rodin  _et al._ (2020). The seminal
example is the _singlet order_ of spin-1/2 pair systems, which is defined as
the population imbalance between the spin $I=0$ nuclear singlet state of the
spin pair, and the spin $I=1$ triplet manifold Levitt (2012, 2019). Nuclear
singlet order may be exceptionally long-lived, with decay time constants
exceeding 1 hour in special cases Stevanato  _et al._ (2015). The phenomenon
of long-lived nuclear spin order has been used for a variety of purposes in
solution nuclear magnetic resonance (NMR), including the study of slow
processes such as chemical exchange Sarkar, Vasos, and Bodenhausen (2007);
Bengs  _et al._ (2021), molecular transport Cavadini  _et al._ (2005);
Cavadini and Vasos (2008); Sarkar  _et al._ (2008); Ahuja  _et al._ (2009),
and infrequent ligand binding to biomolecules Salvi  _et al._ (2012); Buratto
_et al._ (2014a, b, 2016), as well as quantum information processing Roy and
Mahesh (2010); Rodin  _et al._ (2020). The dynamics of nuclear singlet states
is also central to the exploitation of parahydrogen spin order in
hyperpolarized NMR experiments Bowers and Weitekamp (1987); Pravica and
Weitekamp (1988); Kadlecek  _et al._ (2010); Eills  _et al._ (2017); Stevanato
_et al._ (2017); Bengs, Dagys, and Levitt (2020); Dagys, Bengs, and Levitt
(2021); Dagys and Bengs (2022). Singlet NMR has also been applied to imaging
and _in vivo_ experiments Berner  _et al._ (2019); Mamone  _et al._ (2021);
DeVience, Walsworth, and Rosen (2013b); Mamone  _et al._ (2020); Huang  _et
al._ (2022); Pileio  _et al._ (2013); Eills  _et al._ (2021); Pileio  _et al._
(2015); Laustsen  _et al._ (2012); Graafen  _et al._ (2016); Dumez  _et al._
(2014); Yang  _et al._ (2022), and related techniques such as spectral editing
Pravdivtsev, Sönnichsen, and Hövener (2020); DeVience, Walsworth, and Rosen
(2021) and low-field spectroscopy Barskiy  _et al._ (2017); Sjolander  _et
al._ (2017); DeVience  _et al._ (2021); DeVience and Rosen (2022).

Several methods exist for converting nuclear magnetization into singlet order
in the “weak coupling" regime, meaning that the difference in the chemically
shifted Larmor frequencies greatly exceeds the J-coupling between the members
of the spin pair Carravetta, Johannessen, and Levitt (2004); Carravetta and
Levitt (2004); Sarkar, Vasos, and Bodenhausen (2007). Methods for the “near
equivalent" and “intermediate coupling" regimes (where the chemical shift
frequency difference is weaker or comparable to the J-coupling), include the
magnetization-to-singlet (M2S) pulse sequence Pileio, Carravetta, and Levitt
(2010); Tayler and Levitt (2011) and variants such as gM2S Bengs  _et al._
(2020) and gc-M2S Mamone  _et al._ (2020), the spin-lock-induced crossing
(SLIC) method DeVience, Walsworth, and Rosen (2013a); DeVience (2014);
DeVience, Walsworth, and Rosen (2016); DeVience  _et al._ (2021), and slow
passage through level anticrossings Pravdivtsev  _et al._ (2016); Rodin  _et
al._ (2018).

Recently, a new candidate sequence has emerged, namely the _PulsePol_
sequence, which was originally developed to implement electron-to-nuclear
polarization transfer in the context of diamond nitrogen-vacancy magnetometry
Schwartz  _et al._ (2018); Tratzmiller (2021); Tratzmiller  _et al._ (2021).
PulsePol is an attractively simple repeating sequence of six resonant pulses
and four interpulse delays. The PhD thesis of Tratzmiller Tratzmiller (2021)
reports numerical simulations in which PulsePol is used for magnetization-to-
singlet conversion in the near-equivalent regime of high-field solution NMR.
These simulations indicate that PulsePol could display significant advantages
in robustness over some existing methods such as M2S and its variants. In this
article we report the following: (i) the confirmation of Tratzmiller’s
proposal by experimental tests; (ii) the use of symmetry-based recoupling
theory, as used in magic-angle-spinning solid-state NMR Carravetta  _et al._
(2000); Levitt (2007, 2008); Brinkmann and Levitt (2001), for elucidating the
operation of this pulse sequence and predicting new ones; (iii) the PulsePol
sequence and its variants may be used to excite singlet-triplet coherences;
(iv) the robustness of the singlet-triplet transformation may be enhanced
further by using composite pulses.

The PulsePol sequence was originally derived using average Hamiltonian theory
with explicit solution of analytical equations Schwartz  _et al._ (2018). In
this article we demonstrate an alternative theoretical treatment of PulsePol
derived from the principles of symmetry-based recoupling in magic-angle-
spinning solid-state NMR Carravetta  _et al._ (2000); Levitt (2007, 2008);
Brinkmann and Levitt (2001). This theoretical relationship is surprising since
singlet-to-triplet conversion in solution NMR appears to be remote from
recoupling in rotating solids. Nevertheless, as shown below, the problem of
singlet-triplet conversion may be analysed in a time-dependent interaction
frame in which the nuclear spin operators acquire a periodic time-dependence
through the action of the scalar spin-spin coupling. The time-dependent spin
operators in the interaction frame may be treated in similar fashion to the
anisotropic spin interactions in rotating solids, in which case the periodic
time-dependence is induced by the mechanical rotation of the sample. In both
contexts, selection rules for the average Hamiltonian terms may be engineered
by imposing symmetry constraints on the applied pulse sequences.

One common implementation of PulsePol corresponds to the pulse sequence
symmetry designated $\mathrm{R}4_{3}^{1}$, using the notation developed for
symmetry-based recoupling Carravetta  _et al._ (2000); Levitt (2007, 2008);
Brinkmann and Levitt (2001). As shown below, the spin dynamical selection
rules associated with $\mathrm{R}4_{3}^{1}$ symmetry explain the main
properties of the PulsePol sequence. Furthermore this description immediately
predicts the existence of many other sequences with similar properties. Some
of these novel sequences are demonstrated below.

PulsePol deviates from the standard construction procedure for symmetry-based
recoupling sequences in solids. The deviation is subtle but invests PulsePol
with improved robustness. Incorporating composite pulses can increase the
robustness further.

##  II Theory

###  II.1 Spin Hamiltonian

The rotating-frame spin Hamiltonian for a homonuclear 2-spin-1/2 system in
high-field solution NMR may be written as

$H(t)=H_{\mathrm{CS}}+H_{J}+H_{\mathrm{rf}}(t),$ (1)

where the chemical shift Hamiltonian is given by

$H_{\mathrm{CS}}=H_{\Sigma}+H_{\Delta}$ (2)

and the individual Hamiltonian terms are:

$\displaystyle H_{\Sigma}$
$\displaystyle=\tfrac{1}{2}\omega_{\Sigma}(I_{1z}+I_{2z}),$ (3) $\displaystyle
H_{\Delta}$ $\displaystyle=\tfrac{1}{2}\omega_{\Delta}(I_{1z}-I_{2z}),$
$\displaystyle H_{J}$ $\displaystyle=\omega_{J}\bm{I}_{1}\cdot\bm{I}_{2}.$

Here, $\omega_{\Sigma}$ is the sum of the chemically shifted resonance offsets
for the two spins, $\omega_{\Delta}$ is their difference, and $\omega_{J}=2\pi
J$ is the scalar spin-spin coupling ($J$-coupling).

The interaction of the spin pair with resonant radiofrequency fields is
represented by the Hamiltonian term $H_{\mathrm{rf}}(t)$. The rotating-frame
Hamiltonian for the interaction of the nuclei with a resonant time-dependent
field is given by

$H_{\mathrm{rf}}(t)=\omega_{\rm
nut}(t)\left\\{\cos\phi(t)(I_{1x}+I_{2x})+\sin\phi(t)(I_{1y}+I_{2y})\right\\},$
(4)

where the nutation frequency $\omega_{\rm nut}$ is proportional to the
radiofrequency field amplitude.

The terms $H_{\Sigma}$, $H_{J}$ and $H_{\mathrm{rf}}$ all mutually commute.
The term $H_{\Delta}$, on other hand, commutes in general with neither $H_{J}$
nor $H_{\mathrm{rf}}$. We consider here the case of “near-equivalent" spin
pairs Tayler and Levitt (2011); Pileio, Carravetta, and Levitt (2010);
DeVience, Walsworth, and Rosen (2013a), for which
$|\omega_{\Delta}|\ll|\omega_{J}|$. In this case, the term $H_{\Delta}$ may be
treated as a perturbation of the dominant terms $H_{J}$ and $H_{\mathrm{rf}}$.

###  II.2 Propagators

The propagator $U_{\Lambda}(t)$ generated by a Hamiltonian term $H_{\Lambda}$
is a unitary time-dependent operator solving the differential equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}U_{\Lambda}(t)$
$\displaystyle=-iH_{\Lambda}(t)U_{\Lambda}(t)$ (5)

with the boundary condition $U_{\Lambda}(0)=1$. Since $H_{\mathrm{rf}}$ and
$H_{J}$ commute, the propagator $U(t)$ under the total Hamiltonian of equation
1 may be written as follows:

$U(t)=U_{J}(t)U_{\mathrm{rf}}(t)\widetilde{U}_{\mathrm{CS}}(t),$ (6)

where the propagator $\widetilde{U}_{\mathrm{CS}}(t)$ solves the differential
equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\widetilde{U}_{\mathrm{CS}}(t)$
$\displaystyle=-i\widetilde{H}_{\mathrm{CS}}(t)\widetilde{U}_{\mathrm{CS}}(t)$
(7)

with the boundary condition $\widetilde{U}_{\mathrm{CS}}(0)=1$. The
interaction-frame chemical shift Hamiltonian $\widetilde{H}_{\mathrm{CS}}(t)$
is defined as follows:

$\widetilde{H}_{\mathrm{CS}}(t)=U_{\mathrm{rf}}(t)^{\dagger}U_{J}(t)^{\dagger}H_{\mathrm{CS}}U_{J}(t)U_{\mathrm{rf}}(t).$
(8)

Equation 8 shows that the chemical shift terms acquire a double modulation in
the interaction frame: first from the action of the J-coupling, and secondly
from the action of the applied rf field.

Since the J-coupling is time-independent, the propagator $U_{J}$ has the
following form:

$\displaystyle
U_{J}(t)=\exp\\{-iH_{J}t\\}=\exp\\{-i\omega_{J}t\bm{I}_{1}\cdot\bm{I}_{2}\\}.$
(9)

The singlet and triplet states of the spin-1/2 pair are defined as follows:

$\displaystyle|{S_{0}}\rangle$
$\displaystyle=2^{-1/2}(|{\alpha\beta}\rangle-|{\beta\alpha}\rangle),$
$\displaystyle|{T_{+1}}\rangle$ $\displaystyle=|{\alpha\alpha}\rangle,$
$\displaystyle|{T_{0}}\rangle$
$\displaystyle=2^{-1/2}(|{\alpha\beta}\rangle+|{\beta\alpha}\rangle),$
$\displaystyle|{T_{-1}}\rangle$ $\displaystyle=|{\beta\beta}\rangle.$ (10)

Since the singlet and triplet states are eigenstates of $H_{J}$, with
eigenvalues $-3\omega_{J}/4$ and $+\omega_{J}/4$ respectively, the propagator
$U_{J}$ may be written as follows:

$\displaystyle U_{J}(t)=$
$\displaystyle\exp\\{+i\tfrac{3}{4}\omega_{J}t\\}|{S_{0}}\rangle\langle{S_{0}}|$
$\displaystyle+\exp\\{-i\tfrac{1}{4}\omega_{J}t\\}\sum_{M}|{T_{M}}\rangle\langle{T_{M}}|.$
(11)

The rf propagator $U_{\rm rf}(t)$ corresponds to a time-dependent rotation in
three-dimensional space, described by three Euler angles:

$\displaystyle U_{\rm rf}(t)=$ $\displaystyle R(\Omega_{\rm rf}(t))$ (12)
$\displaystyle=$ $\displaystyle R_{z}(\alpha_{\rm rf}(t))R_{y}(\beta_{\rm
rf}(t))R_{z}(\gamma_{\rm rf}(t)),$

with

$\displaystyle R_{\chi}(\theta)=\exp\\{-{i\mkern 1.0mu}\theta I_{\chi}\\}.$
(13)

The action of the modulated radiofrequency field on the spin system may
therefore be described in terms of a time-dependent set of three Euler angles
$\Omega_{\rm rf}(t)=\\{\alpha_{\rm rf}(t),\beta_{\rm rf}(t),\gamma_{\rm
rf}(t)\\}$.

In general, it is possible to modulate the amplitude $\omega_{\rm nut}(t)$ and
phase $\phi(t)$ of the rf field in time, in order to generate any desired
trajectory of Euler angles $\Omega_{\rm rf}(t)$.

###  II.3 Spherical Tensor Operators

It is convenient to define two spherical tensor spin operators of rank-1,
denoted $\mathds{T}_{1}^{g}$ and $\mathds{T}_{1}^{u}$, where the superscripts
denote their parity under exchange of the two spin-1/2 particles:

$\displaystyle(12)\mathds{T}_{1m}^{g}(12)^{\dagger}$
$\displaystyle=\mathds{T}_{1m}^{g},$
$\displaystyle(12)\mathds{T}_{1m}^{u}(12)^{\dagger}$
$\displaystyle=-\mathds{T}_{1m}^{u},$ (14)

where $m\in\\{+1,0,-1\\}$ and $(12)$ denotes the particle exchange operator.
The _gerade_ spherical tensor operator is constructed from the total angular
momentum and shift operators for the spin system:

$\displaystyle\mathds{T}_{1\,+1}^{g}$
$\displaystyle=-2^{-1/2}(I_{1}^{+}+I_{2}^{+}),$
$\displaystyle\mathds{T}_{1\,0}^{g}$ $\displaystyle=I_{1z}+I_{2z},$
$\displaystyle\mathds{T}_{1\,-1}^{g}$
$\displaystyle=2^{-1/2}(I_{1}^{-}+I_{2}^{-}).$ (15)

The _ungerade_ spherical tensor operator of rank-1 plays a prominent role in
the current theory. It has the following components:

$\displaystyle\mathds{T}_{1\,+1}^{u}$
$\displaystyle=|{T_{+1}}\rangle\langle{S_{0}}|,$
$\displaystyle\mathds{T}_{1\,0}^{u}$
$\displaystyle=|{T_{0}}\rangle\langle{S_{0}}|,$
$\displaystyle\mathds{T}_{1\,-1}^{u}$
$\displaystyle=|{T_{-1}}\rangle\langle{S_{0}}|.$ (16)

Each component is given by a shift operator between the singlet state and one
of the three triplet states. The adjoint operators are given by

$\displaystyle\mathds{T}_{1\,+1}^{u\dagger}$
$\displaystyle=|{S_{0}}\rangle\langle{T_{+1}}|,$
$\displaystyle\mathds{T}_{1\,0}^{u\dagger}$
$\displaystyle=|{S_{0}}\rangle\langle{T_{0}}|,$
$\displaystyle\mathds{T}_{1\,-1}^{u\dagger}$
$\displaystyle=|{S_{0}}\rangle\langle{T_{-1}}|.$ (17)

Both sets of operators $\mathds{T}_{1}^{g}$ and $\mathds{T}_{1}^{u}$ transform
irreducibly under the three-dimensional rotation group:

$\displaystyle R(\Omega)\mathds{T}^{g}_{1\mu}R^{\dagger}(\Omega)$
$\displaystyle=\sum_{\mu^{\prime}=-1}^{+1}\mathds{T}^{g}_{1\mu^{\prime}}\mathcal{D}^{1}_{\mu^{\prime}\mu}(\Omega),$
(18) $\displaystyle R(\Omega)\mathds{T}^{u}_{1\mu}R^{\dagger}(\Omega)$
$\displaystyle=\sum_{\mu^{\prime}=-1}^{+1}\mathds{T}^{u}_{1\mu^{\prime}}\mathcal{D}^{1}_{\mu^{\prime}\mu}(\Omega).$

Here, $\mathcal{D}^{\lambda}_{\mu^{\prime}\mu}(\Omega)$ represents an element
of the rank-$\lambda$ Wigner rotation matrix Varshalovich, Moskalev, and
Kheronskii (1988).

The _gerade_ spherical tensor operator $\mathds{T}_{1}^{g}$ obeys the standard
relationship between its components under the adjoint transformation
Varshalovich, Moskalev, and Kheronskii (1988):

$\mathds{T}_{1\mu}^{g\dagger}=(-1)^{\mu}\mathds{T}_{1\,-\mu}^{g}.$ (19)

However, the analogous relationship does _not_ apply to the components of the
_ungerade_ spherical tensor operator $\mathds{T}_{1}^{u}$.

###  II.4 Interaction frame Hamiltonian

The chemical shift Hamiltonian terms, given in equation 3, may be written in
terms of the $m=0$ spherical tensor operator components as follows:

$\displaystyle H_{\Sigma}$
$\displaystyle=\tfrac{1}{2}\omega_{\Sigma}\mathds{T}_{1\,0}^{g},$
$\displaystyle H_{\Delta}$
$\displaystyle=\tfrac{1}{2}\omega_{\Delta}\big{(}\mathds{T}_{1\,0}^{u}+\mathds{T}_{1\,0}^{u\dagger}\big{)}.$
(20)

From equation II.2, these operators transform as follows under the propagator
$U_{J}$:

$\displaystyle U_{J}^{\dagger}(t)H_{\Sigma}U_{J}(t)$
$\displaystyle=\tfrac{1}{2}\omega_{\Sigma}\mathds{T}_{1\,0}^{g},$
$\displaystyle U_{J}(t)^{\dagger}H_{\Delta}U_{J}(t)$
$\displaystyle=\tfrac{1}{2}\omega_{\Delta}\big{(}\mathds{T}_{1\,0}^{u}\exp\\{-i\omega_{J}t\\}+\mathds{T}_{1\,0}^{u\dagger}\exp\\{+i\omega_{J}t\\}\big{)}.$
(21)

This may be combined with equations 8, 12 and 18 to obtain the following
expression for the interaction-frame chemical shift Hamiltonian:

$\widetilde{H}_{\mathrm{CS}}(t)=\sum_{m=-1}^{+1}{\
}\sum_{\mu=-1}^{+1}\widetilde{H}_{1m1\mu}(t),$ (22)

where each term has the form

$\widetilde{H}_{1m1\mu}(t)=\omega_{1m1\mu}d^{1}_{\mu
0}\big{(}-\beta_{\mathrm{rf}}(t)\big{)}\exp\\{i\big{(}m\omega_{J}+\mu\gamma_{\mathrm{rf}}(t)\big{)}\\}Q_{1m1\mu}$
(23)

and $d^{1}_{\mu 0}(\beta)$ is an element of the rank-1 reduced Wigner matrix.
The amplitudes $\omega_{1m1\mu}$ and spin operators $Q_{1m1\mu}$ take the
following values:

$\displaystyle\omega_{1\,+1\,1\mu}$
$\displaystyle=\tfrac{1}{2}\omega_{\Delta}{\ },\quad$ $\displaystyle
Q_{1\,+1\,1\mu}$ $\displaystyle=\mathds{T}_{1\,\mu}^{u},$
$\displaystyle\omega_{1\,0\,1\mu}$
$\displaystyle=\tfrac{1}{2}\omega_{\Sigma}{\ },\quad$ $\displaystyle
Q_{1\,0\,1\mu}$ $\displaystyle=\mathds{T}_{1\,\mu}^{g},$
$\displaystyle\omega_{1\,-1\,1\mu}$
$\displaystyle=\tfrac{1}{2}\omega_{\Delta}{\ },\quad$ $\displaystyle
Q_{1\,-1\,1\mu}$ $\displaystyle=(-1)^{\mu}\mathds{T}_{1\,-\mu}^{u\dagger},$
(24)

where $\mu\in\\{+1,0,-1\\}$. Note that the singlet-triplet excitation terms
have quantum number $m=\pm 1$, while the resonance offset term has $m=0$.

For the terms $\omega_{\ell m\lambda\mu}$ and $Q_{\ell m\lambda\mu}$ above,
the rank of the interaction under rotations of the spins is specified as
$\lambda=1$. The “pseudo-space-rank" $\ell=1$, on the other hand, has no
physical meaning and is introduced to establish a correspondence with the
notation used in magic-angle-spinning solid-state NMR Carravetta  _et al._
(2000); Levitt (2007, 2008); Brinkmann and Levitt (2001).

Figure 1:  Standard implementation of a $\mathrm{R}N_{n}^{\nu}$ sequence for
singlet-triplet conversion. (a) A basic $R$-element denoted $\mathcal{R}^{0}$
is selected. This element induces a rotation about the rotating-frame x-axis
through an odd multiple of $\pi$. In the current case, the element
$\mathcal{R}^{0}$ is given by the composite pulse
$\mathrm{90_{90}180_{0}90_{90}}$ with delays $\tau$ between the pulses, such
that its overall duration is $\tau_{R}=n/(NJ)$. The conjugate sequence
$\mathcal{R}^{0^{\prime}}$ is generated from $\mathcal{R}^{0}$ by a change in
sign of all phases. (b) The sequence $\mathcal{R}^{0}$ is given a phase shift
of $+\phi$, while the sequence $\mathcal{R}^{0^{\prime}}$ is given a phase
shift of $-\phi$, where $\phi=\pi\nu/N$. (c) The pair of sequences
$(\mathcal{R}^{0})_{\phi}$ and $(\mathcal{R}^{0^{\prime}})_{-\phi}$ is
repeated $N/2$ times, to give the standard implementation of a
$\mathrm{R}N_{n}^{\nu}$ sequence (d).

###  II.5 Symmetry-Based Sequences

Symmetry-based pulse sequences Carravetta  _et al._ (2000); Levitt (2007,
2008); Brinkmann and Levitt (2001) were originally developed for magic-angle-
spinning solid-state NMR, where the sample is rotated mechanically with the
angular frequency $\omega_{r}$, such that its rotational period is given by
$\tau_{r}=|2\pi/\omega_{r}|$. In the current case of singlet-triplet
excitation in solution NMR, the J-coupling plays the role of the mechanical
rotation. The relevant period is therefore given by
$\tau_{J}=|2\pi/\omega_{J}|=|J^{-1}|$.

In the current context, a sequence with $\mathrm{R}N_{n}^{\nu}$ symmetry is
defined by the following time-symmetry relationship of the rf Euler angles
$\beta_{\mathrm{rf}}(t)$ and $\gamma_{\mathrm{rf}}(t)$, which applies for
arbitrary time points $t$ Carravetta  _et al._ (2000); Levitt (2007, 2008);
Brinkmann and Levitt (2001):

$\displaystyle\beta_{\mathrm{rf}}(t+\frac{n\tau_{J}}{N})$
$\displaystyle=\beta_{\mathrm{rf}}(t)\pm\pi,$
$\displaystyle\gamma_{\mathrm{rf}}(t+\frac{n\tau_{J}}{N})$
$\displaystyle=\gamma_{\mathrm{rf}}(t)-\frac{2\pi\nu}{N}.$ (25)

A complete $\mathrm{R}N_{n}^{\nu}$ sequence has duration $T=n\tau_{J}$, and is
cyclic, in the sense that the net rotation induced by the rf field over the
complete sequence is through an even multiple of $\pi$.

The _symmetry numbers_ $N$, $n$ and $\nu$ take integer values. In the case of
$\mathrm{R}N_{n}^{\nu}$ sequences, $N$ must be even, while $n$ and $\nu$ are
unconstrained. As discussed below, the symmetry numbers define the selection
rules for the spin dynamics under the pulse sequence.

The $\mathrm{R}N_{n}^{\nu}$ Euler angle symmetries in equation II.5 do not
define the pulse sequence uniquely. Nevertheless, there is a standard
procedure Carravetta  _et al._ (2000); Levitt (2007, 2008); Brinkmann and
Levitt (2001) for generating these Euler angle symmetries, which is sketched
in figure 1. The procedure is as follows:

  * •

Select a rf pulse sequence known as a _basic R-element_ , designated
$\mathcal{R}^{0}$. This sequence may be arbitrarily complex, but must induce a
net rotation of the resonant spins by an odd multiple of $\pi$ about the
rotating-frame x-axis. If the duration of the basic element $\mathcal{R}^{0}$
is denoted $\tau_{R}$, this implies the condition

$U_{\mathrm{rf}}(\tau_{R})=R_{x}(p\pi),$ (26)

where $p$ is an odd integer.

  * •

The duration of the basic element $\tau_{R}$ is given by
$\tau_{R}=(n/N)J^{-1}$, where $n$ and $N$ are the symmetry numbers of the
$\mathrm{R}N_{n}^{\nu}$ sequence.

  * •

Reverse the sign of all phases in $\mathcal{R}^{0}$. This leads to the
_conjugate element_ designated $\mathcal{R}^{0^{\prime}}$.

  * •

Give all components of the basic element $\mathcal{R}^{0}$ a phase shift of
$+\pi\nu/N$. This gives the phase-shifted basic element, denoted
$\mathcal{R}^{0}_{+\pi\nu/N}$.

  * •

Give all components of the conjugate element $\mathcal{R}^{0^{\prime}}$ a
phase shift of $-\pi\nu/N$. This gives the element
$\mathcal{R}^{0^{\prime}}_{-\pi\nu/N}$.

  * •

The complete $\mathrm{R}N_{n}^{\nu}$ sequence is composed of $N/2$ repeats of
the element pair, as follows:

$\mathrm{R}N_{n}^{\nu}=\\{\mathcal{R}^{0}_{+\pi\nu/N}\mathcal{R}^{0^{\prime}}_{-\pi\nu/N}\\}^{N/2}.$
(27)

The complete $\mathrm{R}N_{n}^{\nu}$ sequence has an overall duration of

$\displaystyle T=N\tau_{R}=nJ^{-1}.$ (28)

###  II.6 Selection Rules

The propagator for a complete $\mathrm{R}N_{n}^{\nu}$ sequence is given from
equation 6 by

$U(T)=U_{J}(T)U_{\mathrm{rf}}(T)\widetilde{U}_{\mathrm{CS}}(T).$ (29)

From the definition of a $\mathrm{R}N_{n}^{\nu}$ sequence, the complete
sequence propagators $U_{J}(T)$ and $U_{\mathrm{rf}}(T)$ are both proportional
to the unity operator and may be ignored. The operator
$\widetilde{U}_{\mathrm{CS}}(T)$ corresponds to propagation under a time-
independent effective Hamiltonian:

$\widetilde{U}_{\mathrm{CS}}(T)=\exp\\{-i\overline{H}_{\mathrm{CS}}T\\}.$ (30)

In the near-equivalence limit
($|\omega_{J}|\gg|\omega_{\Delta}|,|\omega_{\Sigma}|$), the effective
Hamiltonian $\overline{H}_{\mathrm{CS}}$ may be approximated by the first term
in a Magnus expansion Haeberlen and Waugh (1968); Mansfield (1971); Haeberlen
(1976):

$\displaystyle\overline{H}_{\mathrm{CS}}$
$\displaystyle\simeq\overline{H}_{\mathrm{CS}}^{(1)},$ (31)

where

$\overline{H}_{\mathrm{CS}}^{(1)}=\sum_{m=-1}^{+1}\sum_{\mu=-1}^{+1}\overline{H}_{1m1\mu}^{(1)}.$
(32)

In common with many recent papers Carravetta  _et al._ (2000); Levitt (2007,
2008); Brinkmann and Levitt (2001), this article uses a numbering of the
Magnus expansion terms which differs from the older literature Haeberlen and
Waugh (1968); Mansfield (1971); Haeberlen (1976) by one.

The individual average Hamiltonian terms are given by

$\overline{H}_{1m1\mu}^{(1)}=T^{-1}\int_{0}^{T}\widetilde{H}_{1m1\mu}(t){\
}dt,$ (33)

where the interaction frame terms $\widetilde{H}_{1m1\mu}(t)$ are given in
equation 23.

The Euler angle symmetries in equation II.5 lead to the following selection
rules for the first-order average Hamiltonian terms of $\mathrm{R}N_{n}^{\nu}$
sequences Carravetta  _et al._ (2000); Levitt (2007, 2008); Brinkmann and
Levitt (2001):

$\displaystyle\overline{H}^{(1)}_{\ell m\lambda\mu}(t_{0})=0\quad{\rm
if}\;mn-\mu\nu\neq\frac{N}{2}Z_{\lambda},$ (34)

where $Z_{\lambda}$ is any integer with the same parity as $\lambda$. This
selection rule may be visualised by a diagrammatic procedure Levitt (2007,
2008).

In the current case, $\lambda=1$ for all relevant interactions, so that
$Z_{\lambda}$ is any odd integer. Hamiltonian components for which $mn-\mu\nu$
is an odd multiple of $N/2$ are symmetry-allowed and may contribute to the
effective Hamiltonian. A symmetry-allowed term with quantum numbers
$\\{m,\mu\\}$ and ranks $\ell=\lambda=1$ is given in general by

$\overline{H}^{(1)}_{1m1\mu}=\kappa_{1m1\mu}\omega_{1m1\mu}Q_{1m1\mu},$ (35)

where the amplitudes $\omega_{1m1\mu}$ and spin operators $Q_{1m1\mu}$ are
given in equation II.4.

The scaling factor $\kappa_{\ell m\lambda\mu}$ of a symmetry-allowed term is
given by

$\displaystyle\kappa_{\ell m\lambda\mu}=\exp(-{i\mkern
1.0mu}\mu\frac{\pi\nu}{N})K_{m\lambda\mu},$ (36)

where $K_{m\lambda\mu}$ is defined with respect to the basic element
$\mathcal{R}^{0}$:

$\displaystyle
K_{m\lambda\mu}=\tau^{-1}_{R}\int_{0}^{\tau_{R}}d^{\lambda}_{\mu
0}(-\beta^{0}_{\rm rf}(t))\exp\\{i(\mu\gamma^{0}_{\rm
rf}(t)+m\omega_{J}t)\\}dt.$ (37)

Here $\beta^{0}_{\rm rf}$ and $\gamma^{0}_{\rm rf}$ represent the Euler angles
describing the rotation induced by the rf field under the basic element
Carravetta  _et al._ (2000); Levitt (2007, 2008); Brinkmann and Levitt (2001).

Symmetry-based pulse sequences are designed by selecting combinations of
symmetry numbers $N$, $n$ and $\nu$ such that all desirable average
Hamiltonian terms $\overline{H}^{(1)}_{\ell m\lambda\mu}$ are symmetry-allowed
while all undesirable terms are symmetry-forbidden. In most cases, the basic
element $\mathcal{R}^{0}$ is selected such that the scaling factors
$\kappa_{\ell m\lambda\mu}$ are maximised for the desirable symmetry-allowed
terms.

Figure 2:  Energy levels and approximate eigenstates of a J-coupled two-
spin-1/2 system in the near-equivalence limit. (a) A $\mathrm{R}N_{n}^{\nu}$
sequence, with symmetry numbers chosen to select terms $\\{m,\mu\\}=\\{\pm
1,\pm 1\\}$ and suppress all others, induces a transition between the
$|{S_{0}}\rangle$ and $|{T_{+1}}\rangle$ states. Suitable symmetries are given
in table 1. One example is $\mathrm{R}4_{3}^{+1}$. (b) If the symmetry number
$\nu$ is changed in sign, average Hamiltonian terms with quantum numbers
$\\{m,\mu\\}=\\{\pm 1,\mp 1\\}$ are selected. In this case there is selective
excitation of the transition between the $|{S_{0}}\rangle$ and
$|{T_{-1}}\rangle$ states. One example is $\mathrm{R}4_{3}^{-1}$.

$\mathrm{R}N_{n}^{\nu}$ | $\kappa_{1111}$
---|---
$\mathrm{R}4_{1}^{-1}$ | $-0.264$
$\mathrm{R}4_{3}^{1}$ | $-0.512$
$\mathrm{R}4_{5}^{-1}$ | $0.307$
$\mathrm{R}4_{7}^{1}$ | $0.038$
$\mathrm{R}4_{9}^{-1}$ | $-0.029$
$\mathrm{R}6_{1}^{-2}$ | $-0.104$
$\mathrm{R}6_{5}^{2}$ | $-0.291$
$\mathrm{R}6_{7}^{-2}$ | $0.360$
$\mathrm{R}6_{8}^{-1}$ | $0.253$
$\mathrm{R}6_{10}^{1}$ | $0.068$

$\mathrm{R}N_{n}^{\nu}$ | $\kappa_{1111}$
---|---
$\mathrm{R}8_{1}^{-3}$ | $-0.137$
$\mathrm{R}8_{3}^{-1}$ | $-0.371$
$\mathrm{R}8_{5}^{1}$ | $-0.498$
$\mathrm{R}8_{7}^{3}$ | $-0.495$
$\mathrm{R}8_{9}^{-3}$ | $0.385$
$\mathrm{R}10_{1}^{-4}$ | $-0.110$
$\mathrm{R}10_{2}^{-3}$ | $-0.215$
$\mathrm{R}10_{3}^{-2}$ | $-0.309$
$\mathrm{R}10_{4}^{-1}$ | $-0.389$
$\mathrm{R}10_{6}^{1}$ | $-0.491$
$\mathrm{R}10_{7}^{2}$ | $-0.511$

Table 1: A selection of $\mathrm{R}N_{n}^{\nu}$ symmetries that are
appropriate for symmetry-based singlet-triplet conversion in solution NMR.
These symmetries select $\overline{H}^{(1)}_{\ell m\lambda\mu}$ terms with
quantum numbers $\\{\ell,m,\lambda,\mu\\}$ given by $\\{1,\pm 1,1,\pm 1\\}$.
Changing the sign of $\nu$ selects the terms $\\{1,\pm 1,1,\mp 1\\}$ instead.
Scaling factors $\kappa_{1111}$ are given for the basic R-element in equation
48, in the limit of radiofrequency pulses with negligible duration.

###  II.7 Transition-selective singlet-triplet excitation

Table 1 shows some sets of symmetry numbers $\\{N,n,\nu\\}$ under which the
average Hamiltonian terms with quantum numbers $\\{\ell,m,\lambda,\mu\\}$ $=$
$\\{1,\pm 1,1,\pm 1\\}$ are symmetry-allowed, while all other terms are
symmetry-forbidden and are suppressed in the average Hamiltonian. In
particular, all resonance-offset terms, which have $m=0$, are symmetry-
forbidden in the first-order average Hamiltonian, for the symmetries in table
1.

For example, consider the symmetry $\mathrm{R}4_{3}^{1}$. The term
$\\{\ell,m,\lambda,\mu\\}=\\{1,1,1,1\\}$ is symmetry-allowed since the
expression $nm-\nu\mu$ evaluates to $3\times 1-1\times 1=2$, which is an odd
multiple of $N/2=2$. The term $\\{\ell,m,\lambda,\mu\\}=\\{1,1,1,-1\\}$, on
the other hand, is symmetry-forbidden, since $nm-\nu\mu$ evaluates to $3\times
1-1\times(-1)=4$, which is an _even_ multiple of $2$. Similarly, the
resonance-offset term $\\{\ell,m,\lambda,\mu\\}=\\{1,0,1,-1\\}$ is symmetry-
forbidden, since $nm-\nu\mu$ evaluates to $3\times 0-1\times(-1)=1$, which is
not an integer multiple of $2$.

All symmetries in table 1 select Hamiltonian components with quantum numbers
$\\{\ell,m,\lambda,\mu\\}=\\{1,\pm 1,1,\pm 1\\}$, while suppressing all other
terms. In this case the first-order average Hamiltonian is given through
equations II.4 by

$\displaystyle\overline{H}_{\mathrm{CS}}^{(1)}$
$\displaystyle=\kappa_{1\,+11\,+1}\omega_{1\,+11\,+1}Q_{1\,+11\,+1}$
$\displaystyle+\kappa_{1\,-11\,-1}\omega_{1\,-11\,-1}Q_{1\,-11\,-1}$
$\displaystyle=\frac{1}{2}\omega_{\Delta}\big{\\{}\kappa_{1+11+1}\mathds{T}^{u}_{1\,+1}+(\kappa_{1+11+1}\mathds{T}^{u}_{1\,+1})^{\dagger}\big{\\}}.$
(38)

The first-order average Hamiltonian therefore generates a selective rotation
of the transition between the singlet state $|{S_{0}}\rangle$ and the lower
triplet state $|{T_{+1}}\rangle$, as shown in figure 2(a):

$\displaystyle\overline{H}^{(1)}_{\rm CS}=\tfrac{1}{2}\omega^{\rm
ST}_{\mathrm{nut}}\big{(}e^{-i{\phi_{\mathrm{ST}}}}|{S_{0}}\rangle\langle{T_{+1}}|+e^{+i{\phi_{\mathrm{ST}}}}|{T_{+1}}\rangle\langle{S_{0}}|\big{)}$
(39)

The singlet-triplet nutation frequency and phase depend upon the scaling
factors as follows

$\displaystyle\omega^{\rm
ST}_{\mathrm{nut}}=\omega_{\Delta}|\kappa_{1+11+1}|=\omega_{\Delta}|\kappa_{1-11-1}|,$
(40) $\displaystyle\phi_{\mathrm{ST}}={\rm arg}(\kappa_{1-11-1})={\rm
arg}(-\kappa^{*}_{1111}).$ (41)

If a set of symmetry numbers $\\{N,n,\nu\\}$ selects the terms
$\\{\ell,m,\lambda,\mu\\}=\\{1,\pm 1,1,\pm 1\\}$, then the set of symmetry
numbers $\\{N,n,-\nu\\}$ selects the terms $\\{\ell,m,\lambda,\mu\\}=\\{1,\pm
1,1,\mp 1\\}$. As indicated in figure 2b, the change in sign of $\nu$ leads to
a selective rotation of the singlet state and the _upper_ triplet state.

In either case the dynamics of the system may be described by a two-level
treatment. Define the single-transition operators Wokaun and Ernst (1977);
Vega (1978) for the transitions between the singlet state and the outer
triplet states:

$\displaystyle I_{x}^{\mathrm{ST}(\pm)}$
$\displaystyle=\frac{1}{2}\big{(}|{T_{\pm
1}}\rangle\langle{S_{0}}|+|{S_{0}}\rangle\langle{T_{\pm 1}}|\big{)},$
$\displaystyle I_{y}^{\mathrm{ST}(\pm)}$
$\displaystyle=\frac{1}{2i}\big{(}|{T_{\pm
1}}\rangle\langle{S_{0}}|-|{S_{0}}\rangle\langle{T_{\pm 1}}|\big{)},$
$\displaystyle I_{z}^{\mathrm{ST}(\pm)}$
$\displaystyle=\frac{1}{2}\big{(}|{T_{\pm 1}}\rangle\langle{T_{\pm
1}}|-|{S_{0}}\rangle\langle{S_{0}}|\big{)}.$ (42)

These operators have the cyclic commutation relationships Wokaun and Ernst
(1977); Vega (1978):

$\big{[}I_{x}^{\mathrm{ST}(\pm)},I_{y}^{\mathrm{ST}(\pm)}\big{]}=iI_{z}^{\mathrm{ST}(\pm)}.$
(43)

For the symmetries in table 1, the first-order average Hamiltonian in equation
39 may be written as follows:

$\overline{H}^{(1)}_{\rm CS}=\omega^{\rm
ST}_{\mathrm{nut}}\big{(}I_{x}^{\mathrm{ST}(+)}\cos\phi_{\mathrm{ST}}+I_{y}^{\mathrm{ST}(+)}\sin\phi_{\mathrm{ST}}\big{)}.$
(44)

Assume that the density operator of the spin ensemble is prepared with a
population difference between the lower triplet state and the singlet state.
This arises, for example, if the system is in thermal equilibrium in a strong
magnetic field. This state corresponds to a density operator term of the form:

$\rho(0)\sim I_{z}^{\mathrm{ST}(+)}$ (45)

omitting numerical factors and orthogonal operators. Suppose that an integer
number $p$ of complete $\mathrm{R}N_{n}^{\nu}$ sequences is applied, with
symmetry numbers selected from table 1. The excitation interval is given by
$\tau=pT$, where $T=N\tau_{R}$ is the duration of a complete
$\mathrm{R}N_{n}^{\nu}$ sequence. From the cyclic commutation relationships in
equation 43, the density operator at the end of the sequence is given by

$\displaystyle\rho(\tau)$ $\displaystyle\simeq
I_{z}^{\mathrm{ST}(+)}\cos(\omega^{\rm ST}_{\mathrm{nut}}\tau)$
$\displaystyle-I_{x}^{\mathrm{ST}(+)}\sin(\omega^{\rm
ST}_{\mathrm{nut}}\tau)\cos(\phi_{\mathrm{ST}})$
$\displaystyle+I_{y}^{\mathrm{ST}(+)}\sin(\omega^{\rm
ST}_{\mathrm{nut}}\tau)\sin(\phi_{\mathrm{ST}}).$ (46)

This suggests the following phenomena:

  1. 1.

_Excitation of Singlet-Triplet Coherence_. If the interval $\tau$ is chosen
such that $\omega^{\rm ST}_{\mathrm{nut}}\tau$ is approximately an odd
multiple of $\pi/2$, the resulting density operator contains terms
proportional to the transverse operators $I_{x}^{\mathrm{ST}(+)}$ and
$I_{y}^{\mathrm{ST}(+)}$, indicating the excitation of singlet-triplet
coherence Sheberstov  _et al._ (2019). In practice, the evolution time
$\tau^{*}$ is restricted to integer multiples of the basic element duration
$\tau_{R}$. In the absence of dissipative effects, the excitation of a
singlet-triplet coherence is optimized by completing the following number of
R-elements:

$\displaystyle n^{*}$ $\displaystyle\simeq{\rm round}(\pi/(4\omega^{\rm
ST}_{\mathrm{nut}}\tau_{\rm R}))$ (ST coherence excitation)

  2. 2.

_Generation of Singlet Order._ If the interval $\tau$ is chosen such that
$\omega^{\rm ST}_{\mathrm{nut}}\tau$ is approximately an even multiple of
$\pi/2$, the term $I_{z}^{\mathrm{ST}(+)}$ is inverted in sign. This indicates
that the populations of the singlet state and the outer triplet state are
swapped. This leads to the generation of singlet order, which is a long-lived
difference in population between the singlet state and the triplet manifold
Pileio (2020); Carravetta, Johannessen, and Levitt (2004); Carravetta and
Levitt (2004); Sarkar, Vasos, and Bodenhausen (2007); Pileio, Carravetta, and
Levitt (2010); Tayler and Levitt (2011); Levitt (2012); Tayler and Levitt
(2013); DeVience, Walsworth, and Rosen (2013a); DeVience (2014); DeVience,
Walsworth, and Rosen (2016); DeVience  _et al._ (2021); Levitt (2019); Zhang
_et al._ (2014, 2015); Stevanato  _et al._ (2015); Pravdivtsev  _et al._
(2016); Rodin  _et al._ (2018, 2021, 2019); Sheberstov  _et al._ (2019);
Kharkov  _et al._ (2019); Mamone  _et al._ (2020); Bengs  _et al._ (2020);
Mamone  _et al._ (2021); Bengs  _et al._ (2021); Cavadini  _et al._ (2005);
Cavadini and Vasos (2008); Sarkar  _et al._ (2008); Ahuja  _et al._ (2009);
Salvi  _et al._ (2012); Buratto  _et al._ (2014a, b, 2016); Berner  _et al._
(2019); Mamone  _et al._ (2021); Eills  _et al._ (2017); Stevanato  _et al._
(2017); Bengs, Dagys, and Levitt (2020); Korenchan  _et al._ (2021); Kharkov
_et al._ (2022); Roy and Mahesh (2010); Rodin  _et al._ (2020). In the absence
of relaxation, the conversion of magnetization into singlet-order is optimised
by completing the following number of R-elements:

$\displaystyle n^{*}\simeq{\rm round}(\pi/(2\omega^{\rm
ST}_{\mathrm{nut}}\tau_{\rm R}))\quad\mbox{(SO generation)}$ (47)

It follows that the application of a $\mathrm{R}N_{n}^{\nu}$ sequence to a
near-equivalent 2-spin-1/2 system in thermal equilibrium leads either to the
excitation of singlet-triplet coherences, or to the generation of singlet
order, depending on the number of R-elements that are applied. Experimental
demonstrations of both effects are given below.

There are technical complications if the number of applied R-elements does not
correspond to an integer number of complete $\mathrm{R}N_{n}^{\nu}$ sequences.
In such cases the operators $U_{J}$ and $U_{\mathrm{rf}}$ in equation 6 lead
to additional transformations. If the total number of completed R-elements is
_even_ , the main consequence is an additional phase shift of excited
coherences, which is often of little consequence. If the number of applied
R-elements is _odd_ , on the other hand, then the propagator $U_{\mathrm{rf}}$
swaps the $|{T_{+1}}\rangle$ and $|{T_{-1}}\rangle$ states, exchanging the
$I_{z}^{\mathrm{ST}(\pm)}$ operators.

###  II.8 Implementation

####  II.8.1 Standard Implementation

The standard implementation of a $\mathrm{R}N_{n}^{\nu}$ sequence is sketched
in figure 1 and described by equation 27.

There is great freedom in the choice of the basic element $\mathcal{R}^{0}$
upon which the sequence is constructed. In this paper we concentrate on the
implementation shown in figure 1, in which the basic element is a three-
component composite pulse Levitt and Freeman (1979), with two $\tau$ delays
inserted between the pulses:

$\mathcal{R}^{0}=\left(90_{90}-\tau-180_{0}-\tau-90_{90}\right)$ (48)

where degrees are used here for the flip angles and the phases. This composite
pulse generates an overall rotation by $\pi$ around the rotating-frame x-axis
Levitt and Freeman (1981), and hence is an eligible basic element
$\mathcal{R}^{0}$ for the construction of a $\mathrm{R}N_{n}^{\nu}$ sequence.

The scaling factor $\kappa_{1111}$, and hence the nutation frequency of the
singlet-triplet transition, depends on the choice of basic element. In the
case of the basic element in equation 48, the scaling factor is readily
calculated in the limit of “$\delta$-function" pulses, i.e. strong rf pulses
with negligible duration. The scaling factors $\kappa_{1\pm 11\pm 1}$ are
given for general $N$, $n$ and $\nu$ by

$\kappa_{1\pm 11\pm 1}=\
2^{1/2}\frac{N}{n\pi}(-1)^{(N\pm(n-\nu))/(2N)}\sin^{2}(n\pi/2N).$ (49)

Scaling factors for a set of $\mathrm{R}N_{n}^{\nu}$ symmetries appropriate
for singlet-triplet excitation are given in table 1. Scaling factors with the
largest magnitude are offered by sequences with the symmetries
$\mathrm{R}4_{3}^{1}$, $\mathrm{R}8_{5}^{1}$, $\mathrm{R}8_{7}^{3}$, and
$\mathrm{R}10_{7}^{2}$.

Since the scaling factors in equation 49 are real, the effective nutation axis
of the singlet-triplet transition has a phase angle of zero,
$\phi_{\mathrm{ST}}=0$. This result applies to the basic-R element in equation
48, in the $\delta$-function pulse limit.

The implementation of a $\mathrm{R}N_{n}^{\nu}$ sequence by the procedure in
figure 1 provides selective excitation of the transition between the singlet
state of a near-equivalent spin-1/2 pair and one of the outer triplet states.
However, the sequence performance is not robust with respect to rf field
errors. It is readily shown that a deviation of the rf field from its nominal
value induces a net rotation around the z-axis which accumulates as the
sequence proceeds. This causes a degradation in performance in the case of
radiofrequency inhomogeneity or instability.

Figure 3:  The construction of a riffled $\mathrm{R}N_{n}^{\nu}$ sequence for
singlet-triplet conversion. (a) Two basic $R$-elements are used; The elements
$\mathcal{R}_{A}^{0}$ and $\mathcal{R}_{B}^{0}$ have identical properties
under suitable approximations, but have opposite responses to pulse
imperfections. In the current case, $\mathcal{R}_{A}^{0}$ is given by the
composite pulse $\mathrm{90_{90}180_{0}90_{90}}$ with delays $\tau$ between
the pulses, such that its overall duration is $\tau_{R}=n/(NJ)$. The element
$\mathcal{R}_{B}^{0}$ is identical but with a $180{}^{\circ}$ phase shift of
the central pulse (dark shade). The conjugate sequence
$\mathcal{R}_{B}^{0^{\prime}}$ is generated from $\mathcal{R}_{B}^{0}$ by a
change in sign of all phases. (b) The sequence $\mathcal{R}_{A}^{0}$ is given
a phase shift of $+\phi$, while the sequence $\mathcal{R}_{B}^{0^{\prime}}$ is
given a phase shift of $-\phi$, where $\phi=\pi\nu/N$. (c) The pair of
sequences $(\mathcal{R}_{A}^{0})_{\phi}$ and
$(\mathcal{R}_{B}^{0^{\prime}})_{-\phi}$ is repeated $N/2$ times, to give a
riffled $\mathrm{R}N_{n}^{\nu}$ sequence (d). PulsePol is an example of a
riffled $\mathrm{R}N_{n}^{\nu}$ sequence (see text).

####  II.8.2 Riffled Implementation

In magic-angle-spinning NMR, error compensation is often achieved by the use
of supercycles, i.e. repetition of the entire sequence with variations in the
phase shifts, or in some cases, cyclic permutations of the pulse sequence
elements Brinkmann, Edén, and Levitt (2000); Brinkmann, Schmedt auf der Günne,
and Levitt (2002); Kristiansen  _et al._ (2004); Brouwer  _et al._ (2005);
Kristiansen  _et al._ (2006). PulsePol achieves very effective compensation
for rf pulse errors by a much simpler method, namely a phase shift of just one
pulse by $180{}^{\circ}$. This simple modification may be interpreted as a
modified procedure for constructing sequences with $\mathrm{R}N_{n}^{\nu}$
symmetry, but with built-in error compensation.

Consider two different basic elements, denoted here $\mathcal{R}^{0}_{A}$ and
$\mathcal{R}^{0}_{B}$, as shown in figure 3a. In the depicted case, the two
basic elements differ only in that the central 180∘ pulse is shifted in phase
by 180∘:

$\displaystyle\mathcal{R}^{0}_{A}$
$\displaystyle=\left(90_{90}-\tau-180_{0}-\tau-90_{90}\right)$
$\displaystyle\mathcal{R}^{0}_{B}$
$\displaystyle=\left(90_{90}-\tau-180_{180}-\tau-90_{90}\right)$ (50)

Under ideal conditions, both of these basic elements provide a net rotation by
an odd multiple of $\pi$ about the rotating-frame x-axis, and hence are
eligible starting points for the $\mathrm{R}N_{n}^{\nu}$ construction
procedure. Furthermore, _in the $\delta$-function pulse limit_, the Euler
angle trajectories generated by these sequences are identical. This implies
that, in the case of ideal, infinitely short pulses, the elements
$\mathcal{R}^{0}_{A}$ and $\mathcal{R}^{0}_{B}$ are completely
interchangeable. The modified $\mathrm{R}N_{n}^{\nu}$ construction procedure
sketched in figure 3 exploits this freedom by alternating the phase shifted
“A" basic element $(\mathcal{R}^{0}_{A})_{+\pi\nu/N}$ with the phase-shifted
conjugate “B" element $(\mathcal{R}^{0^{\prime}}_{B})_{-\pi\nu/N}$.

The alternation of two different basic elements, as shown in figure 3,
resembles the “riffling" technique for shuffling a pack of cards, in which the
pack is divided into two piles, and the corners of the two piles are flicked
up and released so that the cards intermingle. The procedure in figure 3
therefore leads to a _riffled $\mathrm{R}N_{n}^{\nu}$ sequence_.

Under ideal conditions, and for pulses of infinitesimal duration, the
“standard" and “riffled" construction procedures have identical performance.
However, an important difference arises in the presence of rf field amplitude
errors. The errors accumulate in the “standard" procedure, but cancel out in
the “riffled" procedure. Hence the procedure shown in figure 3 achieves more
robust performance with respect to rf field errors than the standard procedure
of figure 1. However, it should be emphasised that this form of error
compensation does not apply to all basic R-elements, and that even in the
current case, strict $\mathrm{R}N_{n}^{\nu}$ symmetry is only maintained in
the limit of $\delta$-function pulses. Nevertheless, within these caveats and
restrictions, this error-compensation procedure is powerful and useful. As
discussed below, error-compensation by riffling is responsible for the robust
performance of PulsePol.

To see how a PulsePol sequence Schwartz  _et al._ (2018); Tratzmiller (2021);
Tratzmiller  _et al._ (2021) arises from the riffled $\mathrm{R}N_{n}^{\nu}$
construction procedure, start with the pair of basic R-elements given in
equation II.8.2. Consider the symmetry $\mathrm{R}4_{3}^{1}$, which is
appropriate for transition-selective singlet-triplet excitation, as shown in
table 1. This symmetry implies that each R-element has duration
$\tau_{R}=(3/4)J^{-1}$, and hence that the delays between the pulses are given
by $\tau=\tau_{R}/2=(3/8)J^{-1}$, in the $\delta$-function pulse limit.

The phase shifts $\pm\pi\nu/N$ are equal to $\pm 45{}^{\circ}$ in the case of
$\mathrm{R}4_{3}^{1}$ symmetry. Hence the pair of phase-shifted elements is
given by

$\displaystyle(\mathcal{R}^{0}_{A})_{+45}$
$\displaystyle=\left(90_{135}-\tau-180_{45}-\tau-90_{135}\right)$
$\displaystyle(\mathcal{R}^{0^{\prime}}_{B})_{-45}$
$\displaystyle=\left(90_{-135}-\tau-180_{-225}-\tau-90_{-135}\right)$ (51)

This pair of elements may be concatenated, and the pair of elements repeated,
to complete the riffled implementation of $\mathrm{R}4_{3}^{1}$:

$\mathrm{R}4_{3}^{1}\mathrm{\,[riffled]}=(\mathcal{R}^{0}_{A})_{+45}(\mathcal{R}^{0^{\prime}}_{B})_{-45}(\mathcal{R}^{0}_{A})_{+45}(\mathcal{R}^{0^{\prime}}_{B})_{-45}$
(52)

If the riffled $\mathrm{R}4_{3}^{1}$ sequence is given a $-45{}^{\circ}$ phase
shift, we get:

$\displaystyle\Big{[}(\mathcal{R}^{0}_{A})_{+45}(\mathcal{R}^{0^{\prime}}_{B})_{-45}\Big{]}_{-45}=(\mathcal{R}^{0}_{A})_{0}(\mathcal{R}^{0^{\prime}}_{B})_{-90}$
$\displaystyle=$ $\displaystyle\ \left(90_{90}-\tau-180_{0}-\tau-90_{90}\cdot
90_{0}-\tau-180_{90}-\tau-90_{0}\right)$

which is PulsePol Schwartz  _et al._ (2018); Tratzmiller (2021); Tratzmiller
_et al._ (2021). The $-45{}^{\circ}$ phase shift is of no consequence for the
interconversion of singlet order and magnetization.

The riffled construction procedure may be deployed for the other symmetries in
table 1. For example, the riffled implementation of $\mathrm{R}8_{7}^{3}$,
using the basic elements in equation II.8.2, is as follows:

$\displaystyle\mathrm{R}8_{7}^{3}\mbox{\ [riffled]}=$
$\displaystyle\left[(\mathcal{R}^{0}_{A})_{+67.5}(\mathcal{R}^{0^{\prime}}_{B})_{-67.5}\right]^{4}$
$\displaystyle=$
$\displaystyle\big{[}90_{157.5}-\tau-180_{67.5}-\tau-90_{157.5}\ \cdot$
$\displaystyle 90_{-157.5}-\tau-180_{112.5}-\tau-90_{-157.5}\big{]}^{4}$ (54)

where the superscript indicates 4 repetitions and the interpulse delays are
given by $\tau=\tau_{R}/2=(7/8)J^{-1}$, in the $\delta$-function pulse limit.
Some sequences of this type have been proposed in the form of “generalised
PulsePol sequences" Tratzmiller  _et al._ (2021); Tratzmiller (2021).

The performance of these sequences may be made even more robust by using
composite pulses for the 90∘ or 180∘ pulse sequence elements Levitt and
Freeman (1979, 1981); Levitt (1986); Shaka and Pines (1987); Odedra,
Thrippleton, and Wimperis (2012). Some examples are demonstrated below.

##  III Experimental

###  III.1 Sample

Table 2: Chemical structure of ${}^{13}\mathrm{C}_{2}$-DAND (1,2,3,4,5,6,8-heptakis(methoxy-_d_ 3)-7-((propan-2-yl-_d_ 7)oxy)naphthalene-4a,8a-[${}^{13}\mathrm{C}_{2}$ ]) with its relevant NMR parameters in a magnetic field of $9.39$ T. The singlet-triplet mixing angle is defined as $\theta_{\rm ST}=\tan^{-1}{(\omega_{\Delta}/{2\pi J})}$ Bengs  _et al._ (2020).  |
---|---
$J_{\rm CC}/{\rm Hz}$ | $54.39\pm 0.10$
$\Delta\delta/{\rm ppb}$ | $75.0\pm 2.0$
$\omega_{\Delta}/(2\pi)/{\rm Hz}$<EMAIL_ADDRESS>T] | $7.50\pm 0.20$
$\theta_{\rm ST}/{}^{\circ}$ | $7.85\pm 0.22$

Experiments were performed on a solution of a ${}^{13}\mathrm{C}_{2}$-labelled
deutero-alkoxy naphthalene derivative (${}^{13}\mathrm{C}_{2}$-DAND), whose
molecular structure with its relevant NMR parameters is shown in table 2.
Further details of the synthesis of (${}^{13}\mathrm{C}_{2}$-DAND) are given
in the reference by Hill-Cousins et al Hill-Cousins  _et al._ (2015). This
compound exhibits a very long ${}^{13}\mathrm{C}_{2}$ singlet lifetime in low
magnetic field Stevanato  _et al._ (2015). The current experiments were
performed on 30 mM of ${}^{13}\mathrm{C}_{2}$-DAND dissolved in 500
$\,\mu\mathrm{L}$ isopropanol-$\emph{d}_{8}$. The two ${}^{13}\mathrm{C}$
sites have a J-coupling of 54.39$\pm$0.10 Hz and a chemical shift difference
of 7.50$\pm 0.2$ Hz in a magnetic field of $9.39$ T. The solution was doped
with 3 mM of the paramagnetic agent (2,2,6,6-tetramethylpiperidin-1-yl)oxyl
(TEMPO) in order to decrease the $T_{1}$ relaxation time, allowing faster
repetition of the experiments, and was contained in a standard Wilmad 5 mM
sample tube.

###  III.2 NMR Equipment

All spectra were acquired at a magnetic field of 9.39 T. A 10 mm NMR probe was
used, with the radiofrequency amplitude adjusted to give a nutation frequency
of $\omega_{\rm nut}/(2\pi)\simeq$12.5 kHz, corresponding to a 90∘ pulse
duration of $20\,\mu\mathrm{s}$.

Figure 4:  High-field NMR pulse sequences used in this work. (a) After a
singlet-order destruction sequence (SOD) Rodin  _et al._ (2019) and a waiting
interval to establish thermal equilibrium, a $\mathrm{R}N_{n}^{\nu}$ sequence
is applied to thermal equilibrium magnetization, exciting coherences between
the singlet state and one of the outer triplet states. (b) Procedure for
estimating singlet order generation. A $\mathrm{R}N_{n}^{\nu}$ sequence is
applied to generate singlet order, followed by a $\mathrm{T_{00}}$ singlet-
order-filtering sequence Tayler and Levitt (2013); Tayler (2012), and a second
$\mathrm{R}N_{n}^{\nu}$ sequence to regenerate $z$-magnetization. The NMR
signal is induced by applying a composite $90{}^{\circ}$ pulse (grey
rectangle).

###  III.3 Pulse Sequences

####  III.3.1 Singlet-Triplet Excitation

The excitation of coherences between the singlet state and the outer triplet
states of ${}^{13}\mathrm{C}_{2}$-DAND was demonstrated using the pulse
sequence in figure 4a. On each transient, a singlet destruction block Rodin
_et al._ (2019) is applied followed by a waiting time of $\sim 5T_{1}$ to
establish thermal equilibrium. This ensures an initial condition free from
interference by residual long-lived singlet order left over from the previous
transient. After thermal equilibration in the magnetic field, a
$\mathrm{R}N_{n}^{\nu}$ symmetry-based singlet-triplet excitation sequence of
duration $\tau_{\mathrm{exc}}$ is applied and the NMR signal detected
immediately afterwards. Fourier transformation of the signal generates the
${}^{13}\mathrm{C}$ NMR spectrum.

$\omega_{nut}/(2\pi)$ |  $12.5$ kHz
---|---
$\tau_{90}$ | 20 $\mu$s
$\tau_{R}$ | 13800 ${\,\mu\mathrm{s}}$
$\tau$ | 6860 ${\,\mu\mathrm{s}}$
$n_{R}^{\mathrm{exc}}$ | 4
$\tau_{\mathrm{exc}}$ | 55.2 ms
Table 3:  Experimental parameters for the $\mathrm{R4_{3}^{\pm 1}}$ sequences
used to obtain the results in figure 5(c,d). The parameters have the following
meaning: $\omega_{\rm nut}$ is the radiofrequency pulse amplitude, expressed
as a nutation frequency; $\tau_{90}$ is the duration of a $90{}^{\circ}$
pulse; $\tau_{R}$ is the duration of a single R-element; $\tau$ is the
interval between pulses within each R-element (see figure 1);
$n_{R}^{\mathrm{exc}}$ is the number of R-elements in the excitation sequence;
$\tau_{\mathrm{exc}}$ is the duration of the excitation sequence.  Figure 5:
Enhanced singlet-triplet coherent excitation. (a) Conventional
${}^{13}\mathrm{C}$ spectrum of ${}^{13}\mathrm{C}_{2}$-DAND using a single
90∘ pulse for excitation, showing strong signals from the triplet-triplet
coherences; (b) Vertical expansion (by a factor of 12) of the conventional
${}^{13}\mathrm{C}$ spectrum. Additional signals are visible from minority
isotopomers, with the outer peaks barely visible. The strong central peak is
truncated. (c) Spectrum obtained by applying four elements of a riffled
$\mathrm{R}4_{3}^{1}$ sequence, showing a strongly enhanced outer peak. The
construction procedure in figure 3 was used, starting from the basic elements
in equation II.8.2. (d) Spectrum obtained by applying four elements of a
$\mathrm{R}4_{3}^{-1}$ sequence, showing the enhancement of the other outer
peak. All spectra were obtained with a total of 256 transients and the same
processing parameters. No line broadening is applied.

####  III.3.2 Singlet Order Generation

The generation of singlet order is assessed by the pulse sequence scheme in
figure 4b. After destruction of residual singlet order and thermal
equilibration, a M2S or $\mathrm{R}N_{n}^{\nu}$ sequence of duration
$\tau_{\mathrm{exc}}$ is applied to generate singlet order. This is followed
by a $\mathrm{T_{00}}$ singlet filter sequence Tayler and Levitt (2011). This
consists of a sequence of rf pulses and pulsed field gradients that dephase
all signal components not associated with nuclear singlet order. The singlet
order is reconverted to z-magnetization by a second $\mathrm{R}N_{n}^{\nu}$
sequence of equal duration to the first, or by a S2M sequence (time-reverse of
the M2S sequence) Pileio, Carravetta, and Levitt (2010); Tayler and Levitt
(2011). The recovered z-magnetization is converted to transverse magnetization
by a composite 90∘ pulse and the NMR signal detected in the following
interval. The signal amplitude serves as a measure of the singlet order
generated by the excitation sequence, and the efficiency of recovering
magnetization from the singlet order. The maximum theoretical efficiency for
passing magnetization through singlet order is $2/3$ Levitt (2016).

The $\mathrm{R}N_{n}^{\nu}$ sequences may be constructed by either the
standard or the riffled procedures. M2S and S2M sequences may be substituted
for the first and last $\mathrm{R}N_{n}^{\nu}$ sequences, respectively. The
$90{}^{\circ}$ readout pulse in figure 4b was implemented as a symmetrized BB1
composite pulse Wimperis (1994); Cummins, Llewellyn, and Jones (2003). Details
of the composite pulse, the SOD sequence, and the $\mathrm{T_{00}}$ pulse
sequence modules are given in the Supporting Information.

Figure 6:  ${}^{13}\mathrm{C}$ spectra obtained after (a) a single
$90{}^{\circ}$ pulse, or (b-f) after filtering the ${}^{13}\mathrm{C}$ NMR
signal through singlet order, using the scheme in figure 4b. (a) Standard
${}^{13}\mathrm{C}$ spectrum obtained with a single $90{}^{\circ}$ pulse. (b)
Singlet-filtered spectrum obtained with M2S for singlet order excitation and
S2M for reconversion to magnetization. (c) Singlet-filtered spectrum obtained
with a pair of $\mathrm{R}4_{3}^{1}$ sequences. (d) Singlet-filtered spectrum
obtained with a pair of $\mathrm{R}8_{7}^{3}$ sequences. Both (c) and (d) use
the standard implementation of $\mathrm{R}N_{n}^{\nu}$ sequences, as in figure
1, using the basic element in equation 48). (e) Singlet-filtered spectrum
obtained with a pair of riffled $\mathrm{R}4_{3}^{1}$ sequences. (f) Singlet-
filtered spectrum obtained with a pair of riffled $\mathrm{R}8_{7}^{3}$
sequences. Both (e) and (f) use the riffled implementation of
$\mathrm{R}N_{n}^{\nu}$ sequences, as in figure 3, using the basic elements in
equation II.8.2. All pulse sequence parameters are given in the Supporting
Information.

##  IV Results

###  IV.1 Transition-selective singlet-triplet excitation

In systems of near-equivalent spin-1/2 pairs, the chemical shift difference
induces a slight mixing of the singlet state $|{S_{0}}\rangle$ with the
central triplet state $|{T_{0}}\rangle$. This effect lends signal intensity to
the single-quantum coherences between the singlet state and the outer triplet
states $|{T_{\pm 1}}\rangle$, which generate the outer lines of the AB
quartet. These peaks are feeble for two independent reasons: (i) the coupling
of the singlet-triplet coherences to observable transverse magnetization is
weak in the near-equivalence limit, and (ii) the singlet-triplet coherences
are excited only weakly by conventional single-pulse excitation. The first of
these factors is an intrinsic property of a singlet-triplet coherence. The
second factor, on the other hand, may be overcome by using a suitable
excitation sequence to generate the desired coherence with full amplitude.
Many such schemes have been devised Sheberstov  _et al._ (2019). This effect
is useful since the frequencies of these peaks provide an accurate estimate of
the internuclear J-coupling, which can be difficult to estimate in the near-
equivalence regime.

Figure 5a shows the ${}^{13}\mathrm{C}$ NMR spectrum of the
${}^{13}\mathrm{C}_{2}$-DAND solution. The strong central doublet is due to
the two triplet-triplet coherences. The outer peaks of the AB quartet, which
correspond to the weakly allowed singlet-triplet coherences, are barely
visible in the spectrum, even after vertical expansion (figure 5b).

Greatly enhanced excitation of the outer AB peaks is achieved by the pulse
sequence in figure 4a, using an excitation sequence of symmetry
$\mathrm{R}4_{3}^{1}$ constructed by the riffled procedure (figure 3), and
with the number of R-elements satisfying equation 1. The strong enhancement of
the outer AB peaks, relative to the spectrum induced by a single 90∘ pulse, is
self-evident in figure 5c. Note that changing the sign of the symmetry number
$\nu$ switches the excitation to the opposite singlet-triplet transition
(figure 5d). The experimental pulse sequence parameters are given in table 3.

Figure 7:  Experimental ${}^{13}{\rm C}$ signal amplitudes (white dots) for
the protocol in figure 4b, using riffled $\mathrm{R}N_{n}^{\nu}$ sequences for
both the excitation and reconversion of singlet order. The following
symmetries were used: (a) $\mathrm{R}4_{3}^{1}$, (b) $\mathrm{R}8_{7}^{3}$ and
(c) $\mathrm{R}10_{3}^{2}$. The number $n_{R}$ of R-elements in the
$\mathrm{R}N_{n}^{\nu}$ sequences for singlet excitation and reconversion are
varied simultaneously (top horizontal axis). The corresponding total duration
of each sequence is shown on the lower horizontal axis. All sequences were
implemented by the riffled procedure in figure 3, using the basic elements in
equation II.8.2. The signal amplitudes are normalized relative to that
generated by a single 90∘ pulse. Light blue trajectories show numerical
simulations (excluding relaxation) with the pulse sequence parameters given in
the SI.

###  IV.2 Magnetization-to-singlet conversion

The experimental performance of some magnetization-to-singlet conversion
schemes was tested on a TEMPO-doped solution of ${}^{13}\mathrm{C}_{2}$-DAND
using the pulse sequence protocol in figure 4b. A selection of singlet-
filtered NMR spectra is shown in figure 6(b-f). In all cases the pulse
sequence parameters were optimised for the best performance. The optimised
parameters are given in the Supporting Information.

Figure 6a shows the unfiltered ${}^{13}\mathrm{C}$ NMR spectrum of
${}^{13}\mathrm{C}_{2}$-DAND. Figure 6b shows the spectrum obtained by
applying a M2S sequence to generate singlet order, suppressing other spin
order terms, and regenerating magnetization from singlet order by applying a
S2M sequence. Approximately 50% of the spin order is lost by this procedure,
as may be seen by comparing the spectra in figure 6a and b. The theoretical
limit on passing magnetization through singlet order is $2/3\simeq 67\%$.

The results obtained by using $\mathrm{R}N_{n}^{\nu}$ sequences with different
sets of symmetry numbers are shown in figure 6c and d. The standard
$\mathrm{R}N_{n}^{\nu}$ construction procedure in figure 1 was used. The
number of R-elements was selected according to equation 47. The results are
slightly inferior to the M2S sequence. Some of these spectra exhibit perturbed
peak intensities. This is unexplained.

Riffled $\mathrm{R}N_{n}^{\nu}$ sequences constructed by the procedure in
figure 3 display an improved performance, which is distinctly superior to M2S,
as shown in figure 6e and f. The improvement is attributed to the increased
robustness of the riffled procedure with respect to a range of experimental
imperfections, as discussed further below.

Note that the riffled $\mathrm{R}4_{3}^{1}$ sequence only differs from
PulsePol Schwartz  _et al._ (2018); Tratzmiller (2021); Tratzmiller  _et al._
(2021) by an overall phase shift (equations 52 and II.8.2). The increased
robustness of PulsePol with respect to M2S/S2M in the context of
singlet/triplet conversion has been anticipated by the simulations of
Tratzmiller Tratzmiller (2021).

The singlet order relaxation time $T_{S}$ is readily estimated by introducing
a variable delay before the second $\mathrm{R}N_{n}^{\nu}$ sequence in figure
4b. Some results are shown in the Supporting Information. Although $T_{S}$ is
found to be much greater than $T_{1}$, the value of $T_{S}$ is considerably
shorter than that found in previous experiments Stevanato  _et al._ (2015).
This is attributed to the TEMPO doping of the solution in the current case.

Figure 8:  Experimental ${}^{13}\mathrm{C}$ signal amplitudes of
${}^{13}\mathrm{C}_{2}$-DAND solution, obtained by the protocol in figure 4b,
as a function of relative nutation frequency $\omega_{\rm nut}/\omega_{\rm
nut}^{0}$, where $\omega_{\rm nut}^{0}$ represents the nominal nutation
frequency used for the calculation of pulse durations. The traces correspond
to the experimental amplitudes for converting magnetization into singlet order
and back again, normalized with respect to the signal generated by a single
$90{}^{\circ}$ pulse. Left column (blue): $\mathrm{R}4_{3}^{1}$ sequences;
Right column (red): $\mathrm{R}8_{7}^{3}$ sequences. (a) Standard
$\mathrm{R}N_{n}^{\nu}$ sequences using the basic element in equation 48. (b)
Riffled $\mathrm{R}N_{n}^{\nu}$ sequences using the basic elements in equation
II.8.2. (c) Riffled $\mathrm{R}N_{n}^{\nu}$ sequences with all central
$180_{0}$ pulses replaced by an ASBO-11 composite pulse Odedra, Thrippleton,
and Wimperis (2012). (d) Riffled $\mathrm{R}N_{n}^{\nu}$ sequences with all
central $180_{0}$ pulses replaced by a
$60_{180}180_{0}240_{180}420_{0}240_{180}180_{0}60_{180}$ composite pulse
Shaka and Pines (1987). The grey lines in (d) show the experimental response
of the M2S/S2M protocol. All experimental details are given in the SI.  Figure
9:  Experimental ${}^{13}\mathrm{C}$ signal amplitudes of
${}^{13}\mathrm{C}_{2}$-DAND solution, obtained by the protocol in figure 4b,
as a function of resonance offset $\Delta\omega$. The plotted points
correspond to the amplitude for converting magnetization into singlet order
and back again, normalized with respect to the signal generated by a single
$90{}^{\circ}$ pulse. Left column (blue): $\mathrm{R}4_{3}^{1}$ sequences;
Right column (red): $\mathrm{R}8_{7}^{3}$ sequences. (a) Standard
$\mathrm{R}N_{n}^{\nu}$ sequences using the basic element in equation 48. (b)
Riffled $\mathrm{R}N_{n}^{\nu}$ sequences using the basic elements in equation
II.8.2. (c) Riffled $\mathrm{R}N_{n}^{\nu}$ sequences with all central
$180_{0}$ pulses replaced by an ASBO-11 composite pulse Odedra, Thrippleton,
and Wimperis (2012). (d) Riffled $\mathrm{R}N_{n}^{\nu}$ sequences with all
central $180_{0}$ pulses replaced by a
$60_{180}180_{0}240_{180}420_{0}240_{180}180_{0}60_{180}$ composite pulse
Shaka and Pines (1987). The grey lines in (d) show the experimental response
of the M2S/S2M protocol. All experimental details are given in the SI.

Figure 7 shows the dependence of the singlet-filtered NMR signals on the
number of R-elements $n_{R}$, used for both the excitation and reconversion
sequence. The corresponding total sequence durations
$\tau_{\mathrm{exc}}=\tau_{\mathrm{recon}}=n_{R}\tau_{R}=n_{R}(n/N)J^{-1}$ are
also shown. Clear oscillations of the singlet order are observed, as predicted
by equation II.7. The singlet order oscillations induced by
$\mathrm{R}8_{7}^{3}$ are slightly slower than those for
$\mathrm{R}4_{3}^{1}$, as expected from the theoretical scaling factors
reported in table 1. The $\mathrm{R}10_{3}^{2}$ sequence induces a relatively
slow oscillation, corresponding to the small value of $\kappa_{1111}$ for this
symmetry. In all cases, numerical simulations by _SpinDynamica_ software Bengs
and Levitt (2018) show qualitative agreement with the experimental results.

The improved robustness of the riffled implementation of
$\mathrm{R}N_{n}^{\nu}$ with respect to rf amplitude variations is illustrated
by the experimental results in figure 8. These plots show the singlet-filtered
signal amplitudes as a function of rf field amplitude, using the protocol in
figure 4b. Two different pulse sequence symmetries are explored:
$\mathrm{R}4_{3}^{1}$ (blue, left column) and $\mathrm{R}8_{7}^{3}$ (red,
right column). The horizontal axis represents the rf field amplitude,
expressed as a nutation frequency $\omega_{\rm nut}$. The horizontal
coordinates are given by the ratio $\omega_{\rm nut}/\omega_{\rm nut}^{0}$,
where the nominal nutation frequency $\omega_{\rm nut}^{0}$ is used to
calculate the pulse durations, which are kept fixed. Row (a) shows that the
$\mathrm{R}4_{3}^{1}$ and $\mathrm{R}8_{7}^{3}$ sequences are both fairly
narrowband with respect to rf field amplitude when the standard
$\mathrm{R}N_{n}^{\nu}$ protocol is used (figure 1). Row b shows that their
robustness with respect to rf amplitude errors is greatly improved by the
riffled variant of the $\mathrm{R}N_{n}^{\nu}$ protocol, inspired by PulsePol
(figure 3). Their tolerance of rf amplitude errors is increased further when
the central $180{}^{\circ}$ pulses of the basic R-elements are replaced by
ASBO-11 composite pulses Odedra, Thrippleton, and Wimperis (2012) (row c). The
use of $60_{180}180_{0}240_{180}420_{0}240_{180}180_{0}60_{180}$ composite
pulses Shaka and Pines (1987) provides less improvement (row d). For
comparison, the experimental performance of the M2S/S2M protocol Pileio,
Carravetta, and Levitt (2010); Tayler and Levitt (2011) is shown by the grey
lines in row d. The performance of M2S/S2M is clearly inferior to that of the
riffled $\mathrm{R}N_{n}^{\nu}$ sequences.

Another important characteristic of pulse sequences for the generation and
reconversion of singlet order is their robustness with respect to resonance
offset, defined here as $\Delta\omega=\tfrac{1}{2}\omega_{\Sigma}$, where
$\omega_{\Sigma}$ is the sum of the chemically shifted offset frequencies, see
equation 3. A robust performance with respect to resonance offset is usually
desirable, since it renders the sequence less sensitive to inhomogeneity in
the static magnetic field, which can be particularly important in low-field
applications.

Figure 9 compares the resonance-offset dependence of several pulse sequences,
for the generation and reconversion of ${}^{13}\mathrm{C}_{2}$ singlet order
in the solution of ${}^{13}\mathrm{C}_{2}$-DAND. The left column compares
different schemes which have $\mathrm{R}4_{3}^{1}$ symmetry. The right column
compares different schemes which have $\mathrm{R}8_{7}^{3}$ symmetry. All
experimental parameters are given in the Supporting Information.

Figure 9a shows the resonance-offset dependence of $\mathrm{R}N_{n}^{\nu}$
sequences constructed by the standard protocol of figure 1, using the basic
R-element of equation 48. The resulting sequences have a strong dependence on
resonance offset, with the $\mathrm{R}8_{7}^{3}$ sequence displaying a
particularly undesirable offset dependence.

Figure 9b shows the resonance-offset dependence of riffled
$\mathrm{R}N_{n}^{\nu}$ sequences, using the pair of basic R-elements in
equation II.8.2. Riffling clearly stabilises the resonance offset dependence,
with the improvement being particularly striking for $\mathrm{R}8_{7}^{3}$.

Figures 9c and d explore the effect of substituting the central 180∘ pulse of
the basic R-elements by composite pulses. Although ASBO-11 composite pulses
Odedra, Thrippleton, and Wimperis (2012) do not change the performance of
$\mathrm{R}4_{3}^{1}$ very much, they do lead to a significant increase in the
bandwidth of $\mathrm{R}8_{7}^{3}$ (figure 9c). An even more pronounced effect
is observed upon replacing all single 180∘ pulses by 7-element
$60_{180}180_{0}240_{180}420_{0}240_{180}180_{0}60_{180}$ composite pulses
Shaka and Pines (1987) (figure 9d). The resonance-offset bandwidth of
$\mathrm{R}8_{7}^{3}$ with 7-element composite pulses Shaka and Pines (1987)
is particularly impressive.

The grey lines in figure 9d show the experimental offset dependence of the
M2S/S2M protocol Pileio, Carravetta, and Levitt (2010). All riffled
$\mathrm{R}N_{n}^{\nu}$ sequences have a clearly superior performance to
M2S/S2M. To put this in context, even the M2S/S2M protocol is regarded as
relatively robust with respect to resonance offset, being first demonstrated
on a sample in an inhomogeneous low magnetic field Pileio, Carravetta, and
Levitt (2010). Some other techniques, such as SLIC DeVience, Walsworth, and
Rosen (2013a), are far more sensitive to resonance offset than M2S.

Results for the dependence of the singlet order conversion on the pulse
sequence intervals are given in the Supporting Information.

##  V Discussion

The results shown in this paper indicate that PulsePol is a very attractive
addition to the arsenal of pulse sequences for the manipulation of nuclear
singlet order. The PulsePol sequences provide a high degree of robustness with
respect to common experimental imperfections, which is found to be superior to
existing methods such as M2S/S2M, especially when combined with composite
pulses. This robustness is likely to be particularly important for
applications to imaging and _in vivo_ experiments Berner  _et al._ (2019);
Mamone  _et al._ (2021).

In addition, PulsePol is a relatively simple repeating sequence of six pulses.
This structure has many advantages over M2S, which performs the magnetization-
to-singlet-order transformation in four consecutive steps Pileio, Carravetta,
and Levitt (2010); Tayler and Levitt (2011). For example, the PulsePol
repetitions may be stopped at any time, in order to achieve a partial
transformation of spin order. This is more difficult to achieve for M2S and
its variants.

The theoretical relationship between PulsePol and symmetry-based recoupling
sequences in solid-state NMR is unexpected. Nevertheless, this theoretical
analogy immediately allows the considerable body of average Hamiltonian theory
developed for symmetry-based recoupling to be deployed in this very different
context. This immediately allows the use of symmetry-based selection rules for
analysing existing PulsePol sequences and for designing new variants.

All of the work reported in this paper uses the same set of basic elements,
given in equations 48 and II.8.2. There is clearly scope for using different
basic elements within the $\mathrm{R}N_{n}^{\nu}$ symmetry framework.

As discussed above, PulsePol may interpreted as a variant implementation of
$\mathrm{R}N_{n}^{\nu}$ symmetry, involving the alternation of two different
basic elements, which compensate each others’ imperfections. Such riffled
$\mathrm{R}N_{n}^{\nu}$ sequences are more robust with respect to a range of
experimental imperfections. The same principle might be applied to symmetry-
based recoupling sequences in magic-angle-spinning solids. Extensions are also
possible, involving more complex interleaved patterns of multiple basic
elements. We intend to explore such “riffled supercycles" in future work.

In magic-angle-spinning solid-state NMR, symmetry-based pulse sequences have
been used to address a wide variety of spin dynamical problems Carravetta  _et
al._ (2000); Levitt (2007, 2008); Brinkmann and Levitt (2001), including
multiple-channel sequences for the recoupling of heteronuclear systems
Brinkmann and Levitt (2001); Levitt (2007). Such extensions should be possible
in the solution NMR context as well.

Variants of M2S/S2M sequences have been applied to heteronuclear spin systems
Eills  _et al._ (2017); Stevanato  _et al._ (2017); Bengs, Dagys, and Levitt
(2020). This has important applications in parahydrogen-induced polarization
Eills  _et al._ (2017). It is likely that riffled $\mathrm{R}N_{n}^{\nu}$
sequences are also applicable to this problem.

The theory of symmetry-based recoupling in magic-angle-spinning solids was
originally formulated using average Hamiltonian theory, as sketched above. It
is also possible to obtain the key results using Floquet theory Leskes, Madhu,
and Vega (2010); Ivanov  _et al._ (2021), which may have advantages in certain
circumstances. Floquet theory should also be applicable to the current
context.

In summary, the PulsePol sequence Schwartz  _et al._ (2018); Tratzmiller
(2021); Tratzmiller  _et al._ (2021) is an important innovation that has
potential applications in many forms of magnetic resonance. It sits at the
fertile intersection of diamond magnetometry, quantum information processing,
solid-state NMR, parahydrogen-induced hyperpolarization, and singlet NMR in
solution.

###### Acknowledgements.

We acknowledge funding received by the European Research Council (grant
786707-FunMagResBeacons), and EPSRC-UK (grants EP/P009980/1, EP/P030491/1,
EP/V055593/1). We thank Sami Jannin, Quentin Stern, Chloé Gioiosa, Olivier
Cala, Laurynas Dagys, and Maria Concistré for help and discussions.

## Author Declarations

### Conflict of interest

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

## References

  * Pileio (2020) G. Pileio, ed.,  _Long-Lived Nuclear Spin Order: Theory and Applications_ , 1st ed. (Royal Society of Chemistry, S.l., 2020).
  * Carravetta, Johannessen, and Levitt (2004) M. Carravetta, O. G. Johannessen, and M. H. Levitt, “Beyond the T1 Limit: Singlet Nuclear Spin States in Low Magnetic Fields,” Physical Review Letters 92, 153003 (2004).
  * Carravetta and Levitt (2004) M. Carravetta and M. H. Levitt, “Long-Lived Nuclear Spin States in High-Field Solution NMR,” Journal of the American Chemical Society 126, 6228–6229 (2004).
  * Sarkar, Vasos, and Bodenhausen (2007) R. Sarkar, P. R. Vasos, and G. Bodenhausen, “Singlet-State Exchange NMR Spectroscopy for the Study of Very Slow Dynamic Processes,” Journal of the American Chemical Society 129, 328–334 (2007).
  * Pileio, Carravetta, and Levitt (2010) G. Pileio, M. Carravetta, and M. H. Levitt, “Storage of nuclear magnetization as long-lived singlet order in low magnetic field,” Proceedings of the National Academy of Sciences 107, 17135–17139 (2010).
  * Tayler and Levitt (2011) M. C. D. Tayler and M. H. Levitt, “Singlet nuclear magnetic resonance of nearly-equivalent spins,” Physical Chemistry Chemical Physics 13, 5556–5560 (2011).
  * Levitt (2012) M. H. Levitt, “Singlet Nuclear Magnetic Resonance,” Annual Review of Physical Chemistry 63, 89–105 (2012).
  * Tayler and Levitt (2013) M. C. D. Tayler and M. H. Levitt, “Accessing Long-Lived Nuclear Spin Order by Isotope-Induced Symmetry Breaking,” Journal of the American Chemical Society 135, 2120–2123 (2013).
  * DeVience, Walsworth, and Rosen (2013a) S. J. DeVience, R. L. Walsworth, and M. S. Rosen, “Preparation of Nuclear Spin Singlet States Using Spin-Lock Induced Crossing,” Physical Review Letters 111, 5 (2013a).
  * DeVience (2014) S. J. DeVience,  _Nuclear Magnetic Resonance with Spin Singlet States and Nitrogen Vacancy Centers in Diamond_ , Ph.D. thesis, Harvard University, United States – Massachusetts (2014).
  * DeVience, Walsworth, and Rosen (2016) S. J. DeVience, R. L. Walsworth, and M. S. Rosen, “Probing scalar coupling differences via long-lived singlet states,” Journal of Magnetic Resonance 262, 42–49 (2016).
  * DeVience  _et al._ (2021) S. J. DeVience, M. Greer, S. Mandal, and M. S. Rosen, “Homonuclear J-Coupling Spectroscopy at Low Magnetic Fields using Spin-Lock Induced Crossing**,” ChemPhysChem 22, 2128–2137 (2021).
  * Levitt (2019) M. H. Levitt, “Long live the singlet state!” Journal of Magnetic Resonance 306, 69–74 (2019).
  * Zhang  _et al._ (2014) Y. N. Zhang, P. C. Soon, A. Jerschow, and J. W. Canary, “Long- Lived 1H Nuclear Spin Singlet in Dimethyl Maleate Revealed by Addition of Thiols,” Angewandte Chemie-International Edition 53, 3396–3399 (2014).
  * Zhang  _et al._ (2015) Y. Zhang, K. Basu, J. W. Canary, and A. Jerschow, “Singlet lifetime measurements in an all-proton chemically equivalent spin system by hyperpolarization and weak spin lock transfers,” Physical Chemistry Chemical Physics 17, 24370–24375 (2015).
  * Stevanato  _et al._ (2015) G. Stevanato, J. T. Hill-Cousins, P. Håkansson, S. S. Roy, L. J. Brown, R. C. D. Brown, G. Pileio, and M. H. Levitt, “A Nuclear Singlet Lifetime of More than One Hour in Room-Temperature Solution,” Angewandte Chemie International Edition 54, 3740–3743 (2015).
  * Pravdivtsev  _et al._ (2016) A. N. Pravdivtsev, A. S. Kiryutin, A. V. Yurkovskaya, H.-M. Vieth, and K. L. Ivanov, “Robust conversion of singlet spin order in coupled spin-1/2 pairs by adiabatically ramped RF-fields,” Journal of Magnetic Resonance 273, 56–64 (2016).
  * Rodin  _et al._ (2018) B. A. Rodin, A. S. Kiryutin, A. V. Yurkovskaya, K. L. Ivanov, S. Yamamoto, K. Sato, and T. Takui, “Using optimal control methods with constraints to generate singlet states in NMR,” Journal of Magnetic Resonance 291, 14–22 (2018).
  * Rodin  _et al._ (2021) B. A. Rodin, V. P. Kozinenko, A. S. Kiryutin, A. V. Yurkovskaya, J. Eills, and K. L. Ivanov, “Constant-adiabaticity pulse schemes for manipulating singlet order in 3-spin systems with weak magnetic non-equivalence,” Journal of Magnetic Resonance 327, 106978 (2021).
  * Rodin  _et al._ (2019) B. A. Rodin, K. F. Sheberstov, A. S. Kiryutin, L. J. Brown, R. C. D. Brown, M. Sabba, M. H. Levitt, A. V. Yurkovskaya, and K. L. Ivanov, “Fast destruction of singlet order in NMR experiments,” The Journal of Chemical Physics 151, 234203 (2019).
  * Sheberstov  _et al._ (2019) K. F. Sheberstov, A. S. Kiryutin, C. Bengs, J. T. Hill-Cousins, L. J. Brown, R. C. D. Brown, G. Pileio, M. H. Levitt, A. V. Yurkovskaya, and K. L. Ivanov, “Excitation of singlet–triplet coherences in pairs of nearly-equivalent spins,” Physical Chemistry Chemical Physics 21, 6087–6100 (2019).
  * Kharkov  _et al._ (2019) B. Kharkov, X. Duan, E. S. Tovar, J. W. Canary, and A. Jerschow, “Singlet excitation in the intermediate magnetic equivalence regime and field-dependent study of singlet–triplet leakage,” Physical Chemistry Chemical Physics 21, 2595–2600 (2019).
  * Mamone  _et al._ (2020) S. Mamone, N. Rezaei-Ghaleh, F. Opazo, C. Griesinger, and S. Glöggler, “Singlet-filtered NMR spectroscopy,” Science Advances 6, eaaz1955 (2020).
  * Bengs  _et al._ (2020) C. Bengs, M. Sabba, A. Jerschow, and M. H. Levitt, “Generalised magnetisation-to-singlet-order transfer in nuclear magnetic resonance,” Physical Chemistry Chemical Physics 22, 9703–9712 (2020).
  * Mamone  _et al._ (2021) S. Mamone, A. B. Schmidt, N. Schwaderlapp, T. Lange, D. von Elverfeldt, J. Hennig, and S. Glöggler, “Localized singlet-filtered MRS in vivo,” NMR in Biomedicine 34, e4400 (2021).
  * Bengs  _et al._ (2021) C. Bengs, L. Dagys, G. A. I. Moustafa, J. W. Whipham, M. Sabba, A. S. Kiryutin, K. L. Ivanov, and M. H. Levitt, “Nuclear singlet relaxation by chemical exchange,” The Journal of Chemical Physics 155, 124311 (2021).
  * Cavadini  _et al._ (2005) S. Cavadini, J. Dittmer, S. Antonijevic, and G. Bodenhausen, “Slow Diffusion by Singlet State NMR Spectroscopy,” Journal of the American Chemical Society 127, 15744–15748 (2005).
  * Cavadini and Vasos (2008) S. Cavadini and P. R. Vasos, “Singlet states open the way to longer time-scales in the measurement of diffusion by NMR spectroscopy,” Concepts in Magnetic Resonance Part A 32A, 68–78 (2008).
  * Sarkar  _et al._ (2008) R. Sarkar, P. Ahuja, P. R. Vasos, and G. Bodenhausen, “Measurement of Slow Diffusion Coefficients of Molecules with Arbitrary Scalar Couplings via Long-Lived Spin States,” ChemPhysChem 9, 2414–2419 (2008).
  * Ahuja  _et al._ (2009) P. Ahuja, R. Sarkar, P. R. Vasos, and G. Bodenhausen, “Diffusion Coefficients of Biomolecules Using Long-Lived Spin States,” Journal of the American Chemical Society 131, 7498–7499 (2009).
  * Salvi  _et al._ (2012) N. Salvi, R. Buratto, A. Bornet, S. Ulzega, I. Rentero Rebollo, A. Angelini, C. Heinis, and G. Bodenhausen, “Boosting the Sensitivity of Ligand–Protein Screening by NMR of Long-Lived States,” Journal of the American Chemical Society 134, 11076–11079 (2012).
  * Buratto  _et al._ (2014a) R. Buratto, D. Mammoli, E. Chiarparin, G. Williams, and G. Bodenhausen, “Exploring Weak Ligand–Protein Interactions by Long-Lived NMR States: Improved Contrast in Fragment-Based Drug Screening,” Angewandte Chemie International Edition 53, 11376–11380 (2014a).
  * Buratto  _et al._ (2014b) R. Buratto, A. Bornet, J. Milani, D. Mammoli, B. Vuichoud, N. Salvi, M. Singh, A. Laguerre, S. Passemard, S. Gerber-Lemaire, S. Jannin, and G. Bodenhausen, “Drug Screening Boosted by Hyperpolarized Long-Lived States in NMR,” ChemMedChem 9, 2509–2515 (2014b).
  * Buratto  _et al._ (2016) R. Buratto, D. Mammoli, E. Canet, and G. Bodenhausen, “Ligand–Protein Affinity Studies Using Long-Lived States of Fluorine-19 Nuclei,” Journal of Medicinal Chemistry 59, 1960–1966 (2016).
  * Berner  _et al._ (2019) S. Berner, A. B. Schmidt, M. Zimmermann, A. N. Pravdivtsev, S. Glöggler, J. Hennig, D. von Elverfeldt, and J.-B. Hövener, “SAMBADENA Hyperpolarization of 13C-Succinate in an MRI: Singlet-Triplet Mixing Causes Polarization Loss,” ChemistryOpen 8, 728–736 (2019).
  * Eills  _et al._ (2017) J. Eills, G. Stevanato, C. Bengs, S. Glöggler, S. J. Elliott, J. Alonso-Valdesueiro, G. Pileio, and M. H. Levitt, “Singlet order conversion and parahydrogen-induced hyperpolarization of 13C nuclei in near-equivalent spin systems,” Journal of Magnetic Resonance 274, 163–172 (2017).
  * Stevanato  _et al._ (2017) G. Stevanato, J. Eills, C. Bengs, and G. Pileio, “A pulse sequence for singlet to heteronuclear magnetization transfer: S2hM,” Journal of Magnetic Resonance 277, 169–178 (2017).
  * Bengs, Dagys, and Levitt (2020) C. Bengs, L. Dagys, and M. H. Levitt, “Robust transformation of singlet order into heteronuclear magnetisation over an extended coupling range,” Journal of Magnetic Resonance 321, 106850 (2020).
  * Korenchan  _et al._ (2021) D. E. Korenchan, J. Lu, M. H. Levitt, and A. Jerschow, “31P nuclear spin singlet lifetimes in a system with switchable magnetic inequivalence: Experiment and simulation,” Physical Chemistry Chemical Physics 23, 19465–19471 (2021).
  * Kharkov  _et al._ (2022) B. Kharkov, X. Duan, J. Rantaharju, M. Sabba, M. H. Levitt, J. W. Canary, and A. Jerschow, “Weak nuclear spin singlet relaxation mechanisms revealed by experiment and computation,” Physical Chemistry Chemical Physics 24, 7531–7538 (2022).
  * Roy and Mahesh (2010) S. S. Roy and T. S. Mahesh, “Initialization of NMR quantum registers using long-lived singlet states,” Physical Review A 82, 052302 (2010).
  * Rodin  _et al._ (2020) B. A. Rodin, C. Bengs, A. S. Kiryutin, K. F. Sheberstov, L. J. Brown, R. C. D. Brown, A. V. Yurkovskaya, K. L. Ivanov, and M. H. Levitt, “Algorithmic cooling of nuclear spins using long-lived singlet order,” The Journal of Chemical Physics 152, 164201 (2020).
  * Bowers and Weitekamp (1987) C. R. Bowers and D. P. Weitekamp, “Parahydrogen and synthesis allow dramatically enhanced nuclear alignment,” Journal of the American Chemical Society 109, 5541–5542 (1987).
  * Pravica and Weitekamp (1988) M. G. Pravica and D. P. Weitekamp, “Net NMR alignment by adiabatic transport of parahydrogen addition products to high magnetic field,” Chemical Physics Letters 145, 255–258 (1988).
  * Kadlecek  _et al._ (2010) S. Kadlecek, K. Emami, M. Ishii, and R. Rizi, “Optimal transfer of spin-order between a singlet nuclear pair and a heteronucleus,” Journal of Magnetic Resonance 205, 9–13 (2010).
  * Dagys, Bengs, and Levitt (2021) L. Dagys, C. Bengs, and M. H. Levitt, “Low-frequency excitation of singlet–triplet transitions. Application to nuclear hyperpolarization,” The Journal of Chemical Physics 155, 154201 (2021).
  * Dagys and Bengs (2022) L. Dagys and C. Bengs, “Hyperpolarization read-out through rapidly rotating fields in the zero- and low-field regime,” Physical Chemistry Chemical Physics 24, 8321–8328 (2022).
  * DeVience, Walsworth, and Rosen (2013b) S. J. DeVience, R. L. Walsworth, and M. S. Rosen, “Nuclear spin singlet states as a contrast mechanism for NMR spectroscopy,” Nmr in Biomedicine 26, 1204–1212 (2013b).
  * Huang  _et al._ (2022) C. Huang, Y. Peng, E. Lin, Z. Ni, X. Lin, H. Zhan, Y. Huang, and Z. Chen, “Adaptable Singlet-Filtered Nuclear Magnetic Resonance Spectroscopy for Chemical and Biological Applications,” Analytical Chemistry 94, 4201–4208 (2022).
  * Pileio  _et al._ (2013) G. Pileio, S. Bowen, C. Laustsen, M. C. D. Tayler, J. T. Hill-Cousins, L. J. Brown, R. C. D. Brown, J. H. Ardenkjaer-Larsen, and M. H. Levitt, “Recycling and Imaging of Nuclear Singlet Hyperpolarization,” Journal of the American Chemical Society 135, 5084–5088 (2013).
  * Eills  _et al._ (2021) J. Eills, E. Cavallari, R. Kircher, G. Di Matteo, C. Carrera, L. Dagys, M. H. Levitt, K. L. Ivanov, S. Aime, F. Reineri, K. Münnemann, D. Budker, G. Buntkowsky, and S. Knecht, “Singlet-Contrast Magnetic Resonance Imaging: Unlocking Hyperpolarization with Metabolism**,” Angewandte Chemie International Edition 60, 6791–6798 (2021).
  * Pileio  _et al._ (2015) G. Pileio, J.-N. Dumez, I.-A. Pop, J. T. Hill-Cousins, and R. C. D. Brown, “Real-space imaging of macroscopic diffusion and slow flow by singlet tagging MRI,” Journal of Magnetic Resonance 252, 130–134 (2015).
  * Laustsen  _et al._ (2012) C. Laustsen, G. Pileio, M. C. D. Tayler, L. J. Brown, R. C. D. Brown, M. H. Levitt, and J. H. Ardenkjaer-Larsen, “Hyperpolarized singlet NMR on a small animal imaging system,” Magnetic Resonance in Medicine 68, 1262–1265 (2012).
  * Graafen  _et al._ (2016) D. Graafen, M. B. Franzoni, L. M. Schreiber, H. W. Spiess, and K. Münnemann, “Magnetic resonance imaging of 1H long lived states derived from parahydrogen induced polarization in a clinical system,” Journal of Magnetic Resonance 262, 68–72 (2016).
  * Dumez  _et al._ (2014) J.-N. Dumez, J. T. Hill-Cousins, R. C. D. Brown, and G. Pileio, “Long-lived localization in magnetic resonance imaging,” Journal of Magnetic Resonance 246, 27–30 (2014).
  * Yang  _et al._ (2022) X. Yang, K.-R. Hu, J.-X. Xin, Y.-X. Li, G. Yang, D.-X. Wei, and Y.-F. Yao, “Multiple-targeting NMR signal selection by optimal control of nuclear spin singlet,” Journal of Magnetic Resonance 338, 107188 (2022).
  * Pravdivtsev, Sönnichsen, and Hövener (2020) A. N. Pravdivtsev, F. D. Sönnichsen, and J.-B. Hövener, “In vitro singlet state and zero-quantum encoded magnetic resonance spectroscopy: Illustration with N-acetyl-aspartate,” PLOS ONE 15, e0239982 (2020).
  * DeVience, Walsworth, and Rosen (2021) S. J. DeVience, R. L. Walsworth, and M. S. Rosen, “NMR of 31P nuclear spin singlet states in organic diphosphates,” Journal of Magnetic Resonance 333, 107101 (2021).
  * Barskiy  _et al._ (2017) D. A. Barskiy, O. G. Salnikov, A. S. Romanov, M. A. Feldman, A. M. Coffey, K. V. Kovtunov, I. V. Koptyug, and E. Y. Chekmenev, “NMR Spin-Lock Induced Crossing (SLIC) dispersion and long-lived spin states of gaseous propane at low magnetic field (0.05T),” Journal of Magnetic Resonance 276, 78–85 (2017).
  * Sjolander  _et al._ (2017) T. F. Sjolander, M. C. D. Tayler, A. Kentner, D. Budker, and A. Pines, “13C-Decoupled J-Coupling Spectroscopy Using Two-Dimensional Nuclear Magnetic Resonance at Zero-Field,” The Journal of Physical Chemistry Letters 8, 1512–1516 (2017).
  * DeVience and Rosen (2022) S. J. DeVience and M. S. Rosen, “Homonuclear J-coupling spectroscopy using J-synchronized echo detection,” Journal of Magnetic Resonance 341, 107244 (2022).
  * Schwartz  _et al._ (2018) I. Schwartz, J. Scheuer, B. Tratzmiller, S. Müller, Q. Chen, I. Dhand, Z.-Y. Wang, C. Müller, B. Naydenov, F. Jelezko, and M. B. Plenio, “Robust optical polarization of nuclear spin baths using Hamiltonian engineering of nitrogen-vacancy center quantum dynamics,” Science Advances 4, eaat8978 (2018).
  * Tratzmiller (2021) B. Tratzmiller,  _Pulsed Control Methods with Applications to Nuclear Hyperpolarization and Nanoscale NMR_ , Ph.D. thesis, Universität Ulm (2021).
  * Tratzmiller  _et al._ (2021) B. Tratzmiller, J. F. Haase, Z. Wang, and M. B. Plenio, “Parallel selective nuclear-spin addressing for fast high-fidelity quantum gates,” Physical Review A 103, 012607 (2021).
  * Carravetta  _et al._ (2000) M. Carravetta, M. Edén, X. Zhao, A. Brinkmann, and M. H. Levitt, “Symmetry principles for the design of radiofrequency pulse sequences in the nuclear magnetic resonance of rotating solids,” Chemical Physics Letters 321, 205–215 (2000).
  * Levitt (2007) M. H. Levitt, “Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR,” in  _eMagRes_ (John Wiley & Sons, Ltd, 2007).
  * Levitt (2008) M. H. Levitt, “Symmetry in the design of NMR multiple-pulse sequences,” J. Chem. Phys. 128, 052205–25 (2008).
  * Brinkmann and Levitt (2001) A. Brinkmann and M. H. Levitt, “Symmetry principles in the nuclear magnetic resonance of spinning solids: Heteronuclear recoupling by generalized Hartmann–Hahn sequences,” The Journal of Chemical Physics 115, 357–384 (2001).
  * Varshalovich, Moskalev, and Kheronskii (1988) D. A. Varshalovich, A. N. Moskalev, and V. K. Kheronskii,  _Quantum Theory of Angular Momentum_ (World Scientific, Singapore, 1988).
  * Haeberlen and Waugh (1968) U. Haeberlen and J. S. Waugh, “Coherent Averaging Effects in Magnetic Resonance,” Physical Review 175, 453–467 (1968).
  * Mansfield (1971) P. Mansfield, “Symmetrized pulse seqs. in high-res. NMR in solids,” J. Phys. C 4, 1444 (1971).
  * Haeberlen (1976) U. Haeberlen,  _High Resolution NMR in Solids. Selective Averaging_ (Academic, New York, 1976).
  * Wokaun and Ernst (1977) A. Wokaun and R. R. Ernst, “Selective excitation and detection in multilevel spin systems: Application of single transition operators,” The Journal of Chemical Physics 67, 1752–1758 (1977).
  * Vega (1978) S. Vega, “Fictitious spin 1/2 operator formalism for multiple quantum NMR,” The Journal of Chemical Physics 68, 5518–5527 (1978).
  * Levitt and Freeman (1979) M. H. Levitt and R. Freeman, “NMR population inversion using a composite pulse,” Journal of Magnetic Resonance (1969) 33, 473–476 (1979).
  * Levitt and Freeman (1981) M. H. Levitt and R. Freeman, “Compensation for Pulse Imperfections in NMR Spin Echo Experiments,” J. Magn. Reson. 43, 65 (1981).
  * Brinkmann, Edén, and Levitt (2000) A. Brinkmann, M. Edén, and M. H. Levitt, “Synchronous Helical Pulse Sequences in Magic-Angle Spinning NMR. Double Quantum Recoupling of Multiple-Spin Systems,” J. Chem. Phys. 112, 8539–8554 (2000).
  * Brinkmann, Schmedt auf der Günne, and Levitt (2002) A. Brinkmann, J. Schmedt auf der Günne, and M. H. Levitt, “Homonuclear Zero-Quantum Recoupling in Fast Magic-Angle Spinning Nuclear Magnetic Resonance,” J. Magn. Reson. 156, 79–96 (2002).
  * Kristiansen  _et al._ (2004) P. E. Kristiansen, M. Carravetta, W. C. Lai, and M. H. Levitt, “A robust pulse sequence for the determination of small homonuclear dipolar couplings in magic-angle spinning NMR,” Chem. Phys. Lett. 390, 1–7 (2004).
  * Brouwer  _et al._ (2005) D. H. Brouwer, P. E. Kristiansen, C. A. Fyfe, and M. H. Levitt, “Symmetry-Based 29Si Dipolar Recoupling Magic Angle Spinning NMR Spectroscopy: A New Method for Investigating Three -Dimensional Structures of Zeolite Frameworks,” J. Am. Chem. Soc. 127, 542–543 (2005).
  * Kristiansen  _et al._ (2006) P. E. Kristiansen, M. Carravetta, J. D. van Beek, W. C. Lai, and M. H. Levitt, “Theory and applications of supercycled symmetry-based recoupling sequences in solid-state nuclear magnetic resonance,” J. Chem. Phys. 124, 234510–19 (2006).
  * Levitt (1986) M. H. Levitt, “Composite pulses,” Progress in Nuclear Magnetic Resonance Spectroscopy 18, 61–122 (1986).
  * Shaka and Pines (1987) A. J. Shaka and A. Pines, “Symmetric phase-alternating composite pulses,” Journal of Magnetic Resonance (1969) 71, 495–503 (1987).
  * Odedra, Thrippleton, and Wimperis (2012) S. Odedra, M. J. Thrippleton, and S. Wimperis, “Dual-compensated antisymmetric composite refocusing pulses for NMR,” Journal of Magnetic Resonance 225, 81–92 (2012).
  * Hill-Cousins  _et al._ (2015) J. T. Hill-Cousins, I.-A. Pop, G. Pileio, G. Stevanato, P. Håkansson, S. S. Roy, M. H. Levitt, L. J. Brown, and R. C. D. Brown, “Synthesis of an Isotopically Labeled Naphthalene Derivative That Supports a Long-Lived Nuclear Singlet State,” Organic Letters 17, 2150–2153 (2015).
  * Tayler (2012) M. C. D. Tayler,  _Theory and Practice of Singlet Nuclear Magnetic Resonance_ , Ph.D. thesis, University of Southampton (2012).
  * Levitt (2016) M. H. Levitt, “Symmetry constraints on spin dynamics: Application to hyperpolarized NMR,” Journal of Magnetic Resonance 262, 91–99 (2016).
  * Wimperis (1994) S. Wimperis, “Broadband, Narrowband, and Passband Composite Pulses for Use in Advanced NMR Experiments,” Journal of Magnetic Resonance, Series A 109, 221–231 (1994).
  * Cummins, Llewellyn, and Jones (2003) H. K. Cummins, G. Llewellyn, and J. A. Jones, “Tackling systematic errors in quantum logic gates with composite rotations,” Physical Review A 67, 042308 (2003).
  * Bengs and Levitt (2018) C. Bengs and M. H. Levitt, “SpinDynamica: Symbolic and numerical magnetic resonance in a Mathematica environment,” Magnetic Resonance in Chemistry 56, 374–414 (2018).
  * Leskes, Madhu, and Vega (2010) M. Leskes, P. K. Madhu, and S. Vega, “Floquet theory in solid-state nuclear magnetic resonance,” Progress in Nuclear Magnetic Resonance Spectroscopy 57, 345–380 (2010).
  * Ivanov  _et al._ (2021) K. L. Ivanov, K. R. Mote, M. Ernst, A. Equbal, and P. K. Madhu, “Floquet theory in magnetic resonance: Formalism and applications,” Progress in Nuclear Magnetic Resonance Spectroscopy 126–127, 17–58 (2021).