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

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 $\textit{et al.}$, Science Advances, $\textbf{4}$, eaat8978 (2018) and arXiv:1710.01508), 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.


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  . 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 7,13 . Nuclear singlet order may be exceptionally long-lived, with decay time constants exceeding 1 hour in special cases 16 . The phenomenon of longlived 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 4,26 , molecular transport [27][28][29][30] , and infrequent ligand binding to biomolecules [31][32][33][34] , as well as quantum information processing 41,42 . The dynamics of nuclear singlet states is also central to the exploitation of parahydrogen spin order in hyperpolarized NMR experiments [36][37][38][43][44][45][46][47] . Singlet NMR has also been applied to imaging and in vivo experiments 23,25,35,[48][49][50][51][52][53][54][55][56] , and related techniques such as spectral editing 57,58 and low-field spectroscopy 12,[59][60][61] .
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 [2][3][4] . Methods for the "near equivalent" and "intermediate coupling" regimes (where the chemical shift frequency difference is weaker or comparable to the Jcoupling), include the magnetization-to-singlet (M2S) pulse sequence 5,6 and variants such as gM2S 24 and gc-M2S 23 , the spin-lock-induced crossing (SLIC) method [9][10][11][12] , and slow passage through level anticrossings 17,18 .
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 [62][63][64] . Pulse-Pol is an attractively simple repeating sequence of six resonant pulses and four interpulse delays. The PhD thesis of Tratzmiller 63 reports numerical simulations in which PulsePol is used for magnetization-to-singlet conversion in the nearequivalent 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 [65][66][67][68] , 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 62 . In this article we demonstrate an alternative theoretical treatment of PulsePol derived from the principles of symmetry-based recoupling in magic-angle-spinning solidstate NMR [65][66][67][68] . 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 timedependence 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 timedependence is induced by the mechanical rotation of the sample. In both contexts, selection rules for the average Hamil-tonian 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 R4 1 3 , using the notation developed for symmetry-based recoupling [65][66][67][68] . As shown below, the spin dynamical selection rules associated with R4 1 3 symmetry explain the main properties of the Pulse-Pol 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.

A. Spin Hamiltonian
The rotating-frame spin Hamiltonian for a homonuclear 2spin-1/2 system in high-field solution NMR may be written as where the chemical shift Hamiltonian is given by and the individual Hamiltonian terms are: Here, ω Σ is the sum of the chemically shifted resonance offsets for the two spins, ω ∆ is their difference, and ω J = 2πJ 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 rf (t). The rotating-frame Hamiltonian for the interaction of the nuclei with a resonant time-dependent field is given by where the nutation frequency ω nut is proportional to the radiofrequency field amplitude.
The terms H Σ , H J and H rf all mutually commute. The term H ∆ , on other hand, commutes in general with neither H J nor H rf . We consider here the case of "near-equivalent" spin pairs 5,6,9 , for which |ω ∆ | |ω J |. In this case, the term H ∆ may be treated as a perturbation of the dominant terms H J and H rf .

B. Propagators
The propagator U Λ (t) generated by a Hamiltonian term H Λ is a unitary time-dependent operator solving the differential equation with the boundary condition U Λ (0) = 1. Since H rf and H J commute, the propagator U(t) under the total Hamiltonian of equation 1 may be written as follows: where the propagator U CS (t) solves the differential equation with the boundary condition U CS (0) = 1. The interactionframe chemical shift Hamiltonian H CS (t) is defined as follows: 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: The singlet and triplet states of the spin-1/2 pair are defined as follows: Since the singlet and triplet states are eigenstates of H J , with eigenvalues −3ω J /4 and +ω J /4 respectively, the propagator U J may be written as follows: The rf propagator U rf (t) corresponds to a time-dependent rotation in three-dimensional space, described by three Euler angles: with 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 Ω rf (t) = {α rf (t), β rf (t), γ rf (t)}. In general, it is possible to modulate the amplitude ω nut (t) and phase φ (t) of the rf field in time, in order to generate any desired trajectory of Euler angles Ω rf (t).

C. Spherical Tensor Operators
It is convenient to define two spherical tensor spin operators of rank-1, denoted T g 1 and T u 1 , where the superscripts denote their parity under exchange of the two spin-1/2 particles: where m ∈ {+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: The ungerade spherical tensor operator of rank-1 plays a prominent role in the current theory. It has the following components: 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 Both sets of operators T g 1 and T u 1 transform irreducibly under the three-dimensional rotation group: Here, D λ µ µ (Ω) represents an element of the rank-λ Wigner rotation matrix 69 .
The gerade spherical tensor operator T g 1 obeys the standard relationship between its components under the adjoint transformation 69 : However, the analogous relationship does not apply to the components of the ungerade spherical tensor operator T u 1 .

D. 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: From equation 11, these operators transform as follows under the propagator U J : This may be combined with equations 8, 12 and 18 to obtain the following expression for the interaction-frame chemical shift Hamiltonian: where each term has the form and d 1 µ0 (β ) is an element of the rank-1 reduced Wigner matrix. The amplitudes ω 1m1µ and spin operators Q 1m1µ take the following values: where µ ∈ {+1, 0, −1}. Note that the singlet-triplet excitation terms have quantum number m = ±1, while the resonance offset term has m = 0. For the terms ω mλ µ and Q mλ µ above, the rank of the interaction under rotations of the spins is specified as λ = 1. The "pseudo-space-rank" = 1, on the other hand, has no physical meaning and is introduced to establish a correspondence with the notation used in magic-angle-spinning solidstate NMR [65][66][67][68] . This element induces a rotation about the rotating-frame x-axis through an odd multiple of π. In the current case, the element R 0 is given by the composite pulse 90 90 180 0 90 90 with delays τ between the pulses, such that its overall duration is τ R = n/(NJ). The conjugate sequence R 0 is generated from R 0 by a change in sign of all phases. (b) The sequence R 0 is given a phase shift of +φ , while the sequence R 0 is given a phase shift of −φ , where φ = πν/N. (c) The pair of sequences (R 0 ) φ and (R 0 ) −φ is repeated N/2 times, to give the standard implementation of a RN ν n sequence (d).

E. Symmetry-Based Sequences
Symmetry-based pulse sequences 65-68 were originally developed for magic-angle-spinning solid-state NMR, where the sample is rotated mechanically with the angular frequency ω r , such that its rotational period is given by τ r = |2π/ω 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 τ J = |2π/ω J | = |J −1 |.
In the current context, a sequence with RN ν n symmetry is defined by the following time-symmetry relationship of the rf Euler angles β rf (t) and γ rf (t), which applies for arbitrary time points t 65-68 : A complete RN ν n sequence has duration T = nτ 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 π.
The symmetry numbers N, n and ν take integer values. In the case of RN ν n sequences, N must be even, while n and ν are unconstrained. As discussed below, the symmetry numbers define the selection rules for the spin dynamics under the pulse sequence.
The RN ν n Euler angle symmetries in equation 25 do not define the pulse sequence uniquely. Nevertheless, there is a standard procedure [65][66][67][68] 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 R 0 . This sequence may be arbitrarily complex, but must induce a net rotation of the resonant spins by an odd multiple of π about the rotating-frame x-axis.
If the duration of the basic element R 0 is denoted τ R , this implies the condition where p is an odd integer.
• The duration of the basic element τ R is given by τ R = (n/N)J −1 , where n and N are the symmetry numbers of the RN ν n sequence. • Reverse the sign of all phases in R 0 . This leads to the conjugate element designated R 0 .
• Give all components of the basic element R 0 a phase shift of +πν/N. This gives the phase-shifted basic element, denoted R 0 +πν/N .
• Give all components of the conjugate element R 0 a phase shift of −πν/N. This gives the element R 0 −πν/N . • The complete RN ν n sequence is composed of N/2 repeats of the element pair, as follows: The complete RN ν n sequence has an overall duration of T = Nτ R = nJ −1 .

F. Selection Rules
The propagator for a complete RN ν n sequence is given from equation 6 by From the definition of a RN ν n sequence, the complete sequence propagators U J (T ) and U rf (T ) are both proportional to the unity operator and may be ignored. The operator U CS (T ) corresponds to propagation under a time-independent effective Hamiltonian: In the near-equivalence limit (|ω J | |ω ∆ |, |ω Σ |), the effective Hamiltonian H CS may be approximated by the first term in a Magnus expansion [70][71][72] : where In common with many recent papers [65][66][67][68] , this article uses a numbering of the Magnus expansion terms which differs from the older literature 70-72 by one. The individual average Hamiltonian terms are given by where the interaction frame terms H 1m1µ (t) are given in equation 23.
The Euler angle symmetries in equation 25 lead to the following selection rules for the first-order average Hamiltonian terms of RN ν n sequences [65][66][67][68] : where Z λ is any integer with the same parity as λ . This selection rule may be visualised by a diagrammatic procedure 66,67 .
In the current case, λ = 1 for all relevant interactions, so that Z λ is any odd integer. Hamiltonian components for which mn − µν is an odd multiple of N/2 are symmetry-allowed and may contribute to the effective Hamiltonian. A symmetryallowed term with quantum numbers {m, µ} and ranks = λ = 1 is given in general by where the amplitudes ω 1m1µ and spin operators Q 1m1µ are given in equation 24. The scaling factor κ mλ µ of a symmetry-allowed term is given by where K mλ µ is defined with respect to the basic element R 0 : Here β 0 rf and γ 0 rf represent the Euler angles describing the rotation induced by the rf field under the basic element [65][66][67][68] .

G. Transition-selective singlet-triplet excitation
All symmetries in table I select Hamiltonian components with quantum numbers { , m, λ , µ} = {1, ±1, 1, ±1}, while suppressing all other terms. In this case the first-order average Hamiltonian is given through equations 24 by The first-order average Hamiltonian therefore generates a selective rotation of the transition between the singlet state |S 0 and the lower triplet state |T +1 , as shown in figure 2(a): The singlet-triplet nutation frequency and phase depend upon the scaling factors as follows If a set of symmetry numbers {N, n, ν} selects the terms { , m, λ , µ} = {1, ±1, 1, ±1}, then the set of symmetry numbers {N, n, −ν} selects the terms { , m, λ , µ} = {1, ±1, 1, ∓1}. As indicated in figure 2b, the change in sign of ν 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 73,74 for the transitions between the singlet state and the outer triplet states: These operators have the cyclic commutation relationships 73,74 : For the symmetries in table I, the first-order average Hamiltonian in equation 39 may be written as follows: 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: omitting numerical factors and orthogonal operators. Suppose that an integer number p of complete RN ν n sequences is applied, with symmetry numbers selected from table I. The excitation interval is given by τ = pT , where T = Nτ R is the duration of a complete RN ν n sequence. From the cyclic commutation relationships in equation 43, the density operator at the end of the sequence is given by This suggests the following phenomena: 1. Excitation of Singlet-Triplet Coherence. If the interval τ is chosen such that ω ST nut τ is approximately an odd multiple of π/2, the resulting density operator contains terms proportional to the transverse operators I  21 . In practice, the evolution time τ * is restricted to integer multiples of the basic element duration τ R . In the absence of dissipative effects, the excitation of a singlet-triplet coherence is optimized by completing the following number of R-elements: 2. Generation of Singlet Order. If the interval τ is chosen such that ω ST nut τ is approximately an even multiple of π/2, the term I ST(+) z 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  . In the absence of relaxation, the conversion of magnetization into singlet-order is optimised by completing the following number of R-elements: It follows that the application of a RN ν n sequence to a nearequivalent 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 Relements 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 RN ν n sequences. In such cases the operators U J and U 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 rf swaps the |T +1 and |T −1 states, exchanging the I ST(±) z operators.
H. Implementation

Standard Implementation
The standard implementation of a RN ν n sequence is sketched in figure 1 and described by equation 27. There is great freedom in the choice of the basic element 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 75 , with two τ delays inserted between the pulses: where degrees are used here for the flip angles and the phases. This composite pulse generates an overall rotation by π around the rotating-frame x-axis 76 , and hence is an eligible basic element R 0 for the construction of a RN ν n sequence.
The scaling factor κ 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 "δ -function" pulses, i.e. strong rf pulses with negligible duration. The scaling factors κ 1±11±1 are given for general N, n and ν by Scaling factors for a set of RN ν n symmetries appropriate for singlet-triplet excitation are given in table I. Scaling factors with the largest magnitude are offered by sequences with the symmetries R4 1 3 , R8 1 5 , R8 3 7 , and R10 2 7 . Since the scaling factors in equation 49 are real, the effective nutation axis of the singlet-triplet transition has a phase angle of zero, φ ST = 0. This result applies to the basic-R element in equation 48, in the δ -function pulse limit.
The implementation of a RN ν n 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.

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 [77][78][79][80][81] . PulsePol achieves very effective compensation for rf pulse errors by a much simpler method, namely a phase shift of just one pulse by 180 • . This simple modification may be interpreted as a modified procedure for constructing sequences with RN ν n symmetry, but with built-in error compensation. Consider two different basic elements, denoted here R 0 A and 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 • : Under ideal conditions, both of these basic elements provide a net rotation by an odd multiple of π about the rotatingframe x-axis, and hence are eligible starting points for the RN ν n construction procedure. Furthermore, in the δ -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 R 0 A and R 0 B are completely interchangeable. The modified RN ν n construction procedure sketched in figure 3 exploits this freedom by alternating the phase shifted "A" basic element (R 0 A ) +πν/N with the phase-shifted conjugate "B" element (R 0 B ) −πν/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 RN ν n 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 RN ν n symmetry is only maintained in the limit of δ -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 [62][63][64] arises from the riffled RN ν n construction procedure, start with the pair of basic Relements given in equation 50. Consider the symmetry R4 1 3 , which is appropriate for transition-selective singlet-triplet excitation, as shown in table I. This symmetry implies that each R-element has duration τ R = (3/4)J −1 , and hence that the delays between the pulses are given by τ = τ R /2 = (3/8)J −1 , in the δ -function pulse limit.
The phase shifts ±πν/N are equal to ±45 • in the case of R4 1 3 symmetry. Hence the pair of phase-shifted elements is given by This pair of elements may be concatenated, and the pair of elements repeated, to complete the riffled implementation of R4 1 3 : If the riffled R4 1 3 sequence is given a −45 • phase shift, we get: which is PulsePol [62][63][64] . The −45 • 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 I. For example, the riffled implementation of R8 3 7 , using the basic elements in equation 50, is as follows: where the superscript indicates 4 repetitions and the interpulse delays are given by τ = τ R /2 = (7/8)J −1 , in the δ -function pulse limit. Some sequences of this type have been proposed in the form of "generalised PulsePol sequences" 63,64 .
The performance of these sequences may be made even more robust by using composite pulses for the 90 • or 180 • pulse sequence elements 75,76,[82][83][84] . Some examples are demonstrated below.

A. Sample
Experiments were performed on a solution of a 13 C 2labelled deutero-alkoxy naphthalene derivative ( 13 C 2 -DAND), whose molecular structure with its relevant NMR parameters is shown in table II. Further details of the synthesis of ( 13 C 2 -DAND) are given in the reference by Hill-Cousins et al 85 . This compound exhibits a very long 13 C 2 singlet lifetime in low magnetic field 16 . The current experiments were performed on 30 mM of 13 C 2 -DAND dissolved in 500 µL isopropanol-d 8 . The two 13 C sites have a J-coupling of 54.39±0.10 Hz and a chemical shift difference of 7.50±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.

B. 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 ω nut /(2π) 12.5 kHz, corresponding to a 90 • pulse duration of 20 µs.

Singlet-Triplet Excitation
The excitation of coherences between the singlet state and the outer triplet states of 13 C 2 -DAND was demonstrated using  20 and a waiting interval to establish thermal equilibrium, a RN ν n 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 RN ν n sequence is applied to generate singlet order, followed by a T 00 singlet-order-filtering sequence 8,86 , and a second RN ν n sequence to regenerate z-magnetization. The NMR signal is induced by applying a composite 90 • pulse (grey rectangle). the pulse sequence in figure 4a. On each transient, a singlet destruction block 20 is applied followed by a waiting time of ∼ 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 RN ν n symmetry-based singlettriplet excitation sequence of duration τ exc is applied and the NMR signal detected immediately afterwards. Fourier transformation of the signal generates the 13 C NMR spectrum.

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 RN ν n sequence of duration τ exc is applied to generate singlet order. This is followed by a T 00 singlet filter sequence 6 . This consists of a sequence of rf pulses and pulsed field gradients that dephase all signal components not associated with nuclear singlet or-  figure 5(c,d). The parameters have the following meaning: ω nut is the radiofrequency pulse amplitude, expressed as a nutation frequency; τ 90 is the duration of a 90 • pulse; τ R is the duration of a single R-element; τ is the interval between pulses within each R-element (see figure 1); n exc R is the number of R-elements in the excitation sequence; τ exc is the duration of the excitation sequence. der. The singlet order is reconverted to z-magnetization by a second RN ν n sequence of equal duration to the first, or by a S2M sequence (time-reverse of the M2S sequence) 5,6 . 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 87 .
The RN ν n sequences may be constructed by either the standard or the riffled procedures. M2S and S2M sequences may be substituted for the first and last RN ν n sequences, respectively. The 90 • readout pulse in figure 4b was implemented as a symmetrized BB1 composite pulse 88,89 . Details of the composite pulse, the SOD sequence, and the T 00 pulse sequence modules are given in the Supporting Information.

A. 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 with the central triplet state |T 0 . This effect lends signal intensity to the single-quantum coherences between the singlet state and the outer triplet states |T ±1 , which generate the outer lines of the AB quartet. These peaks are feeble for two independent reasons: (i) the coupling of the singlettriplet coherences to observable transverse magnetization is weak in the near-equivalence limit, and (ii) the singlet-triplet coherences are excited only weakly by conventional singlepulse 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 21 . 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 C NMR spectrum of the 13 C 2 -DAND solution. The strong central doublet is due to the two triplettriplet 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 R4 1 3 constructed by the riffled procedure (figure 3), and with the number of R-elements satisfying equation 47. 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 ν switches the excitation to the opposite singlet-triplet transition (figure 5d). The experimental pulse sequence parameters are given in table III.

B. Magnetization-to-singlet conversion
The experimental performance of some magnetization-tosinglet conversion schemes was tested on a TEMPO-doped solution of 13 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 C NMR spectrum of 13 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 67%.
The results obtained by using RN ν n sequences with different sets of symmetry numbers are shown in figure 6c and d. The standard RN ν n 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 RN ν n 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 R4 1 3 sequence only differs from Pulse-Pol 62-64 by an overall phase shift (equations 52 and 53). 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 63 .
The singlet order relaxation time T S is readily estimated by introducing a variable delay before the second RN ν n 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 16 . This is attributed to the TEMPO doping of the solution in the current case. 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 τ exc = τ recon = n R τ R = n R (n/N)J −1 are also shown. Clear oscillations of the singlet order are observed, as predicted by equation 46. The singlet order oscillations induced by R8 3 7 are slightly slower than those for R4 1 3 , as expected from the theoretical scaling factors reported in table I. The R10 2 3 sequence induces a relatively slow oscillation, corresponding to the small value of κ 1111 for this symmetry. In all cases, numerical simulations by SpinDynamica software 90 show qualitative agreement with the experimental results.
The improved robustness of the riffled implementation of RN ν n 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: R4 1 3 (blue, left column) and R8 3 7 (red, right column). The horizontal axis represents the rf field amplitude, expressed as a nutation frequency ω nut . The horizontal coordinates are given by the ratio ω nut /ω 0 nut , where the nominal nutation frequency ω 0 nut is used to calculate the pulse durations, which are kept fixed. Row (a) shows that the R4 1 3 and R8 3 7 sequences are both fairly narrowband with respect to rf field amplitude when the standard RN ν n 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 RN ν n protocol, inspired by PulsePol (figure 3). Their tolerance of rf amplitude errors is increased further when the central 180 • pulses of the basic R-elements are replaced by ASBO-11 composite pulses 84 (row c). The use of 60 180 180 0 240 180 420 0 240 180 180 0 60 180 composite pulses 83 provides less improvement (row d). For comparison, the experimental performance of the M2S/S2M protocol 5,6 is shown by the grey lines in row d. The performance of M2S/S2M is clearly inferior to that of the riffled RN ν n 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 ∆ω = 1 2 ω Σ , where ω Σ 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 C 2 singlet order in the solution of 13 C 2 -DAND. The left column compares different schemes which have R4 1 3 symmetry. The right column compares different schemes which have R8 3 7 symmetry. All experimental parameters are given in the Supporting Information. Figure 9a shows the resonance-offset dependence of RN ν n 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 R8 3 7 sequence displaying a particularly undesirable offset dependence. Figure 9b shows the resonance-offset dependence of riffled RN ν n sequences, using the pair of basic R-elements in equation 50. Riffling clearly stabilises the resonance offset dependence, with the improvement being particularly striking for  The grey lines in figure 9d show the experimental offset dependence of the M2S/S2M protocol 5 . All riffled RN ν n 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 5 . Some other techniques, such as SLIC 9 , 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 25,35 .
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 5,6 . 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 50. There is clearly scope for using different basic elements within the RN ν n symmetry framework.
As discussed above, PulsePol may interpreted as a variant implementation of RN ν n symmetry, involving the alternation of two different basic elements, which compensate each others' imperfections. Such riffled RN ν n 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 [65][66][67][68] , including multiple-channel sequences for the recoupling of heteronuclear systems 66,68 . Such extensions should be possible in the solution NMR context as well.
Variants of M2S/S2M sequences have been applied to heteronuclear spin systems [36][37][38] . This has important applications in parahydrogen-induced polarization 36 . It is likely that riffled RN ν n sequences are also applicable to this problem. The theory of symmetry-based recoupling in magic-anglespinning solids was originally formulated using average Hamiltonian theory, as sketched above. It is also possible to obtain the key results using Floquet theory 91,92 , which may have advantages in certain circumstances. Floquet theory should also be applicable to the current context.
In summary, the PulsePol sequence [62][63][64] 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.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request. The BB1 family of composite pulses originally defined by Wimperis 1 achieves broadband compensation of pulse strength errors. In the time-symmetric version 2 , which we designate BB1(β ), a composite implementation of a simple β 0 pulse with generic flip angle β takes the following form: The angle θ W in the phases of the error correcting block depends on the desired flip angle β , and is given by: For a 90 • and 180 • pulse respectively:  In general, the phases of the 11 pulses are given by: In this context, φ is a free variable which may be tailored for the compensation of resonance offset errors, pulse strength errors, or both.
We have found that the choice φ = 4 3 π − θ W (π)/2 ≈ 187.8 • works well for dual-compensation. This choice of phase appears to correspond to "ASBO-11(B 1 )" described by Odedra et al. (they give φ = 188 • ) which was found by a numerical search over φ in 1 • increments for the ASBO-11 sequence with the largest bandwidth with respect to pulse strength errors.
For φ = 4 3 π − θ W (π)/2, we obtain the set of solutions:  (12) B. T 00 filter The T 00 filter is a common block in singlet NMR experiments. It consists of a series of pulsed field gradients and radiofrequency pulses which are designed to dephase unwanted operators i.e.
those not corresponding to the T 00 symmetry of the nuclear singlet order operator. A typical implementation consists of three gradients sandwiched by two radiofrequency pulses: Here, the angle β m is the magic angle arctan √ 2 ≈ 54.74 • . In order to ensure the optimal performance of the T 00 filter, all pulses were replaced by the corresponding BB1 composite pulses as de- The parameters used in our experiments are shown in Table SI. In practice, due to hardware limitations, rest delays τ r follow each pulsed field gradient.

C. Singlet order destruction (SOD) element
In standard NMR experiments, the waiting delay between scans is typically set to be on the order of ×5 the longitudinal relaxation constant T 1 , which is usually enough to fully equilibrate a spin system for most practical purposes. However, in experiments which excite nuclear singlet order -which relaxes with a time constant T S , often orders of magnitude larger than T 1 -this approach is problematic.
In order to ensure the quality of experimental data, a singlet order destruction (SOD) element was incorporated in all experiments.
The SOD element consists of a T 00 filter followed by a train of J-synchronized spin echoes repeated m 1 times.
The J-synchronized block is a building block of M2S, and similar to the M2S sequence has a total echo duration τ e ideally set to: For optimal singlet order destruction, the number of repetitions should roughly accomplish a 2π/3 rotation in the |S 0 -|T 0 Bloch sphere 3 : The SOD element may be repeated m 2 times. Previous work 3 suggests m 2 ≈ 1 − 3 is sufficient for singlet order destruction. Out of an abundance of caution, we set m 2 = 7 in our experiments.
The SOD element is illustrated in Figure S2. Figure S2. Illustration of the SOD filter implemented in the experiments. The T 00 filter has the same meaning as the previous section. τ e is the total spin echo duration. m 1 is the number of times the spin echo is repeated within a single SOD element. m 2 is the total number of SOD elements. τ r is the relaxation delay.
The parameters used in the SOD element in the main text are given in Table SII. A. Description of M2S/S2M sequences The M2S sequence is prototypical hard-pulse sequence for generating singlet order from longitudinal magnetization in the near-equivalence regime 4-6 . In general, M2S takes the form: Here, τ 1 and τ 2 are interpulse delays, while n 1 and n 2 denote the number of repetitions. Figure S3. Illustration of the M2S sequence in this work. τ 1 is the interval between pulses in the spin echoes (of total duration τ e ), and τ 2 is the interval after the 90 y pulse.
The pulse sequence which reconverts singlet order to magnetization is the emphtime reverse, denoted S2M.
B. Parameters for sequences in Figure 6 The experimental parameters for the RN ν n and M2S sequences that appear in Figure 6 are shown in Table SIII.

III. RELAXATION EXPERIMENTS
A. T 1 measurement The time constant for the relaxation of longitudinal magnetization is typically denoted T 1 in NMR.
We have used a standard inversion recovery experiment to measure T1, as shown in Figure S4.

B. T S measurement using PulsePol
The singlet relaxation time T S can be measured using the sequences described in the main text.
The time evolution of nuclear singlet order may be fitted to the simple equation: Figure S5. Longitudinal relaxation of spin magnetization in 13 C 2 -DAND@ 9.4 T and 25 • C, following the experiment in Figure 3.. Black circles: experimental data. Dashed line: fit using Equation (16), with the parameters A = 0.984 ± 0.006 and T 1 = 3.41 ± 0.05s Figure S6. Illustration of the inversion recovery sequence used to measure T S . After the SOD filter, and generation of nuclear singlet order using the R4 1 3 sequence, the singlet order is allowed to evolve, filtered, and then read out with another R4 1 3 sequence and a 90 degree pulse. The R4 1 3 sequence is performed as per the PulsePol implementation, and has the parameters described in the main text. Figure S7. Singlet relaxation in TEMPO-doped 13 C 2 -DAND solution @ 9.4 T and 25 • C following the experiment in Figure 5. Black circles: experimental data. Dashed line: fit using Equation (17), with the parameters A = 1.03 ± 0.01 and T S = 89.4 ± 4.3s

IV. ADDITIONAL PERFORMANCE COMPARISONS
A. Dependence on delay mismatch Figure S8. Experimental 13 C signal amplitudes (white dots) for (a) R4 1 3 , (b) R8 3 7 and (c) M2S as a function of the relative inter-pulse delay mismatch ∆τ/τ 0 , where τ 0 represents the nominal inter-pulse delay. For the M2S sequence the nominal inter-pulse delay is given by τ 0 = 1/(4J), whereas for R-based sequences the nominal inter-pulse delay is given by τ 0 = n/(NJ). The R-sequences have been implemented according to the PulsePol procedure. The final 13 C signal amplitudes were referenced with respect to a single 13 Cpulse-acquire spectrum. Light blue trajectories represent numerical simulations with the pulse sequence parameters given in Tables I-II