In this paper we elucidate, theoretically and experimentally, molecular motifs which permit Long-Lived Polarization Protected by Symmetry (LOLIPOPS). The basic assembly principle starts from a pair of chemically equivalent nuclei supporting a long-lived singlet state and is completed by coupling to additional pairs of spins. LOLIPOPS can be created in various sizes; here we review four-spin systems, introduce a group theory analysis of six-spin systems, and explore eight-spin systems by simulation. The focus is on AA′X_{n}X′_{n} spin systems, where typically the A spins are ^{15}N or ^{13}C and X spins are protons. We describe the symmetry of the accessed states, we detail the pulse sequences used to access these states, we quantify the fraction of polarization that can be stored as LOLIPOPS, we elucidate how to access the protected states from A or from X polarization and we examine the behavior of these spin systems upon introduction of a small chemical shift difference.

## I. INTRODUCTION

Recent work has shown that hyperpolarized magnetic resonance spectroscopy (HP-MRS) can trace *in vivo* metabolism of biomolecules and is therefore extremely promising for diagnostic imaging,^{1–5} the study of metabolic pathways and kinetics^{4} or even the study of protein dynamics.^{6} The primary advantage of HP-MRS over other molecular imaging modalities such as PET/CT, is that, in addition to anatomic localization, HP-MRS can report on biochemical transformations and their kinetics. Nevertheless, general application of HP-MRS is hindered by a fundamental limitation: the signal lifetime for hyperpolarization, which is dictated by the spin-lattice (T_{1}) relaxation, is typically on the order of seconds.

Symmetry-protected states with no dipole allowed transitions^{7,8} offer a solution to this challenge.^{9–17} Spin population on these states can have lifetimes dramatically longer than T_{1}. A prototypical example is para-hydrogen, with the nuclear spins in the singlet state |$S \equiv (\alpha \beta - \beta \alpha)/\sqrt 2$|$S\u2261(\alpha \beta \u2212\beta \alpha )/2$; parahydrogen can be separated from the other three states (the orthohydrogen states) and can persist for weeks at room temperature.^{18} About a decade ago, long lived singlet states were found and accessed on pairs of chemically inequivalent spins^{9,10} with a chemical shift difference (Δω) much larger than their mutual *J*-coupling, and signal lifetimes of up to 25 minutes were observed.^{19} However, to maintain long-lived singlet states in these systems, the chemical shift difference has to be suppressed either by shuttling the sample into low field^{9} (∼mT) or by strong spin-locking,^{10} neither of which is particularly appealing for *in vivo* MRI experiments. Systems without a chemical shift difference can support long lived states at high magnetic fields without any additional manipulations. The first direct access to a singlet between chemically equivalent spins (other than parahydrogen) was achieved using the reversible hydration of diacetyl (CD_{3}^{13}C=O^{13}C = OCD_{3}).^{17} No spin locking was required to sustain the long-lived carbon singlet state at high magnetic fields; however, the chemical transformations limited the generality of this technique. Newer approaches in two-spin systems include a singlet state between a single pair of nearly equivalent spins^{20} (Δω ≪ *J*), where Pileio *et al.*^{21} demonstrated the so-called “MSM” (Magnetization to Singlet to Magnetization or M2S-S2M) pulse sequence to achieve interconversion between bulk magnetization and the singlet state population (Figure 2(a)). An alternative sequence called “SLIC” (Spin Lock Induced Crossing) was demonstrated by DeVience *et al.*^{22} to achieve the same purpose using continuous wave (CW) irradiation.

This paper examines larger spin systems (multiple spin pairs) which as we detail later provide considerable advantages over two-spin systems. With modified parameters,^{23–25} both M2S and SLIC can also excite long-lived states between chemically equivalent (Δω = 0) yet magnetically inequivalent spins and create long-lived polarization protected by symmetry (what we call here LOLIPOPS). The simplest example is the AA′XX′ 4-spin system, where A spins are low-γ nuclei such as ^{13}C or ^{15}N providing the core lifetime enhancement, and X are surrounding protons. We observed long-lived signals that arises from states such as the “singlet-singlet” (|$SS \equiv {\textstyle{1 \over 2}}( {\alpha \beta - \beta \alpha })_A ( {\alpha \beta - \beta \alpha })_X $|$SS\u226112(\alpha \beta \u2212\beta \alpha )A(\alpha \beta \u2212\beta \alpha )X$) which has no dipole-allowed transitions to the other states.^{26} Here we develop several themes. We show that this approach is generally extendable to larger spin systems (i.e., AA′X_{n}X_{n}′ with increasing *n*), detail how the disconnected state changes with increasing n, and explain how the pulse sequences access the long lived states in larger spin systems. We also show what fraction of the polarization can be stored with increasing *n*, including the surprising result that from a signal-to-noise ratio (SNR) perspective, adding more protons can actually *improve* the signal and the robustness of pumping, even though the long-lived state derives its longevity from the low-γ A component. Finally, we also explore the behavior of these systems upon introduction of a small chemical shift difference.

To address these questions, we first review how to access LOLIPOPS in AA′XX′ 4-spin systems with two examples, diacetylene (DIAC, Figure 1(a)) and a newly studied molecule, the ^{15}N^{15}N′HH′ 4-spin system 3,6-dichloro-^{15}N_{2}-pyridazine (DCP, Figure 1(b)). In either DIAC or DCP LOLIPOPS can be accessed from either A(^{13}C in DIAC, ^{15}N in DCP) or X (^{1}H in DIAC and DCP) polarization, yet proton excitation results in a significant signal enhancement because of the difference in gyromagnetic ratio (as previously shown in other molecules).^{24} In DIAC the resonance conditions for (singlet)_{A}-(singlet)_{H} or the (singlet)_{A}-(triplet)_{H} are very similar, but in DCP they are significantly different, so each can be selectively populated, and either gives long-lived magnetization.

Next, we use group theory to identify all disconnected states in AA′X_{2}X_{2}′ 6-spin systems. From this analysis, two distinct resonance conditions to access the LOLIPOPS are found analytically and verified by simulation and experiment on ^{13}C_{2}-diphenyl acetylene (DPA, Figure 1(c)). Then, extension to even larger spin systems (AA′X_{3}X_{3}′) is made with numerical simulation and demonstrated using the 8-spin system in 2,3-dimethylmaleic anhydride (DMMA, Figure 1(e)).

Finally, the behavior of LOLIPOPS with a chemical shift difference is exemplified by the 6-spin system in ^{13}C_{2}-meta methyl diphenyl acetylene (mDPA, Figure 1(d)), where the chemical shift difference breaks the symmetry but the important properties allowing for polarization transfer are retained. Measured parameters (e.g., *J*-couplings and chemical shift difference) for all compounds are summarized in Table I.

Molecule . | J_{CC(NN)}
. | ΔJ_{CH(NH)}
. | J_{HH}
. | Δω
. |
---|---|---|---|---|

DIAC | 154 | 60.3 | 0 | … |

DCP | −24 | 0.5 | 9 | … |

DPA | 181.8 | 5.82 | 0 | … |

DMMA | 13.2 | 0 | … | |

mDPA | 181.8 | 5.82 | 0 | 50.6^{a} |

Molecule . | J_{CC(NN)}
. | ΔJ_{CH(NH)}
. | J_{HH}
. | Δω
. |
---|---|---|---|---|

DIAC | 154 | 60.3 | 0 | … |

DCP | −24 | 0.5 | 9 | … |

DPA | 181.8 | 5.82 | 0 | … |

DMMA | 13.2 | 0 | … | |

mDPA | 181.8 | 5.82 | 0 | 50.6^{a} |

^{a}

Measured at 8.45 T.

## II. ACCESSING LOLIPOPS IN AA′XX′ 4-SPIN SYSTEMS

### A. Accessing LOLIPOPS with MSM

The energy levels in a typical AA′XX′ 4-spin system have been thoroughly discussed in previous studies^{23,25,27} but are reviewed here to facilitate analytical treatment of larger spin systems. The Hamiltonian for such a 4-spin system where A and X are different nuclei is

where *S* denotes the low-γ A nucleus (^{15}N or ^{13}C) and *I* denotes the high-γ X nucleus (^{1}H); *ω*_{A} and *ω*_{X} are the respective Larmor frequencies. Equation (1) assumes “weak coupling” between the A and X spins, appropriate when |ω_{A} − ω_{X}| ≫ *J*_{AX} and valid in all cases we will discuss in this paper; if A and X are the same nucleus, strong coupling can be reintroduced by intense rf irradiation, and then additional terms are capable of contributing to the evolution.^{28} In DCP (Figure 1(b)) *J*_{AX} (=0.69 Hz) and *J*_{AX}′ (=0.15 Hz) are the two “between-pair” *J*-couplings connecting one proton and one nitrogen whereas *J*_{AA} (=−24 Hz) and *J*_{XX} (=9 Hz) are the “in-pair” *J*-couplings. For further analysis the between-pair *J*-coupling interactions are best divided into their sum and their difference, and we obtain

It is the *H*_{ΔJ} term that breaks the magnetic equivalence and is used to access LOLIPOPS. As shown by Pople *et al.* in 1957,^{27} the Hamiltonian of Eq. (2) can be block-diagonalized by choosing symmetry-adapted basis functions.^{27,29,30} For describing the dynamics during the MSM sequence, we assemble the 16 nuclear states by combining the singlet state |$S = (\alpha \beta - \beta \alpha)/\sqrt 2 $|$S=(\alpha \beta \u2212\beta \alpha )/2$ and triplet states |$T_1 = \alpha \alpha,\,T_0 = (\alpha \beta\break - \beta \alpha)/\sqrt 2,\,T_{ - 1} = \beta \beta $|$T1=\alpha \alpha ,T0=(\alpha \beta \u2212\beta \alpha )/2,T\u22121=\beta \beta $ on the AA′ spin pair with the singlet and triplet states on the XX′ spin pair. An example is the “(singlet)_{A} − (singlet)_{X}” state (|${\textstyle{1 \over 2}}( {\alpha \beta - \beta \alpha } )_A( {\alpha \beta - \beta \alpha })_X $|$12(\alpha \beta \u2212\beta \alpha )A(\alpha \beta \u2212\beta \alpha )X$). The assignment to an irreducible representation depends on the molecule and its point group (or more generally the permutation group).^{31} In DCP (C_{2v} symmetry) the (singlet)_{A} as well as the (singlet)_{X} are antisymmetric with respect to rotation about the principle C_{2} axis (z-axis) and thus are of B_{2} symmetry. To assign the correct irreducible representation to the combined (singlet)_{A} − (singlet)_{X} state we use the product table of C_{2v} and we assign B_{2} × B_{2} = A_{1} symmetry. In contrast, the “(singlet)_{A} − (triplet)_{X}” state, *ST*_{0}, has B_{2} symmetry (B_{2} × A_{1} = B_{2}), as does the *T*_{0}*S* state. The mixing rules state that there is no mixing between states with different symmetry, so there is no off-diagonal term that connects *SS* and *ST*_{0}.

Next, let us consider specifically the matrix elements of *H*_{ΔJ,} which breaks the magnetic equivalence within the AA′ and XX′ pairs.

These matrix elements are only non-zero for states with anti-aligned spins, that is, the *S* and *T*_{0} states of proton or nitrogen such that m_{1} + m_{2} = m_{3} + m_{4} = 0. (Consider the *S* and *T*_{0} states for the AA′ spin pair, for example, ⟨*T*_{0}|*S*_{z1} − *S*_{z2}|*S*⟩ = ⟨*T*_{0}|*T*_{0}⟩ = 1.) Accordingly, the Hamiltonian in Eq. (1) can be divided into sub-matrices confined to a given symmetry and projection number Σm = 0. The important pairs of states that qualify are the *SS* state coupled to the *T*_{0}*T*_{0} (“triplet-triplet”) state, both of A_{1} symmetry, and the *ST*_{0} and *T*_{0}*S* (“triplet-singlet”) states, both of B_{1} symmetry.^{23,27}. Finally, the two important sub-matrices of the Hamiltonian in Eq. (1) are

where Δ*J*_{AX} = *J*_{AX} − *J*_{AX′}. Here and in later matrices, we label the corresponding irreducible representation in the upper left corner.

The goal is to drive transitions between these states in order to access the long lived states *SS* and *ST*_{0} (singlet on the A pair). Δ*J*_{AX} can drive these transitions if we can compensate for the diagonal terms *J*_{AA} ± *J*_{XX}. In the MSM sequence (Figure 2(a)), this is achieved with a multiple echo pulse train where the frequency of the 180° pulses matches the resonance frequency of these two-level systems, which is dictated by the *J*-couplings.

Specifically, as shown previously^{23,24} the resonance condition is reflected by the inter-pulse delay, *τ* in the multiple echo pulse train and the number of echo pulses, *n* (Figure 2(a)). They are different for accessing the *SS* (Eq. (6)) and *ST*_{0} (Eq. (7)) states in DCP.

Note that Eqs. (6) and (7) are equivalent when *J*_{XX} is zero or negligible relative to *J*_{AA}, a relatively common situation, as is the case in 2,3-^{13}C_{2}-diacetylene (DIAC, Figure 1(a))^{23,32} and all previous demonstrations.^{23,24} However, this is not generally valid. For instance, DCP (Figure 1(b)) has *J*_{XX} = 9 Hz and *J*_{AA} = −24 Hz, giving rise to different resonance conditions. Accordingly, separate measurements can be made to determine relaxation lifetimes (T_{S}) of the *ST*_{0} (Figure 3(a)) and *SS* (Figure 3(b)) states, respectively. However, the signal produced from either state (5%–8% of total ^{15}N magnetization, Figure 3) is only half as large as what can be observed with DIAC where access to both states is synchronized, so polarization transfer from proton magnetization to the long-lived states is advantageous, and this can be achieved by implementing the M2S part of the sequence on the proton channel ((M2S(^{1}H)-S2M(^{15}N)).^{24} Note that a tenfold increase is expected based on the ratio of gyromagnetic ratios of ^{1}H and ^{15}N, which is approximately achieved for the ST_{o} state compared with a MSM measurement with only ^{15}N pulses (M2S(^{15}N)-S2M(^{15}N), Figure 3), although a smaller gain (about 3-fold signal enhancement) is seen for the SS state. We believe that the reason is that ^{1}H polarization is exposed to more relaxation during the M2S pulse sequence (for DCP, the duration of M2S pulse sequence is (1.5 × 96 × 15.2 ms)_{ST} ≅ (1.5 × 44 × 33.3 ms)_{SS} = 2.2 s). The measured *T*_{S} for *SS* and *ST*_{0} are both around 37 s while the nitrogen *T*_{1} at the same field strength is about 15 s. This indicates that the long-lived nature of such states arises from the localized ^{15}N_{2}-singlet and is little affected by the proton states. Also note that at short times (short τ_{r,} the delay between M2S and its inverse S2M) the decays are dominated by fast equilibration within the ^{15}N triplet state manifold, reducing the signal in the beginning; only after that is the long-lived component observed.

### B. Accessing LOLIPOPS with SLIC

An alternative and rather elegant pulse sequence to access the long-lived states is the SLIC sequence.^{22,25} This method uses continuous irradiation at a B_{1} power that matches the diagonal terms of Eqs. (4) and (5), i.e., *J*_{AA} + *J*_{XX} to access the SS state or *J*_{AA} − *J*_{XX} to access the ST_{0} state. As shown in the previous study,^{25} the convenient basis for SLIC is the X basis for the irradiated spins and the singlet-triplet (ST) basis for the other pair. For a given spin pair the X basis consists of

The system Hamiltonian, including a CW B_{1} field (assumed to be along the x axis) can be found by combining the states of AA′ in the X basis with those of XX′ in the ST basis. The two relevant sub-matrices that contain states *SS* and *ST*_{0} are

These matrices allow us to give a relatively simple description of how SLIC works. As noted earlier, it is only the term *H*_{ΔJ} (Eq. (3)) which breaks the magnetic equivalence of the A spins or X spins-if *H*_{ΔJ} = 0 no terms interconvert singlet and triplet. For the M2S sequence, we noted that *H*_{ΔJ}|*T*_{0}^{A}⟩ ∝ *|S*^{A}⟩ and in fact the term creates useful couplings, for example, between *S*^{A}*S*^{X}* and T*_{0}^{A}*T*_{0}^{X}. Those two states are not degenerate, so the sequence of precisely timed 180° pulses is needed to prevent the energy difference between the two states from averaging away the effect. In the SLIC case, note that *H*_{ΔJ}*|X*_{−1}^{A}⟩ ∝ *|S*^{A}⟩ and so in this case as well *H*_{ΔJ} creates useful couplings, for example, between *S*^{A}*S*^{X}* and X*_{−1}^{A}*T*_{o}^{X}*.* However, a fundamental difference in the SLIC case is that the *X*_{−1}^{A}*T*_{o}^{X} state can be *shifted* by a weak irradiation field, and thus choosing the value correctly permits resonant transfer.

Specifically, if we choose ω_{1} = 2π(*J*_{AA} + *J*_{XX}) then we induce resonance between the first (*S*^{A}*S*^{X}) and the last state (|$X_{ - 1}^A T_0^X $|$X\u22121AT0X$) in the matrix of Eq. (9). If we choose ω_{1} = 2π(*J*_{AA} − *J*_{XX}) then we induce resonance between the first (|$S^A T_0^X $|$SAT0X$) and the last state (|$X_{ - 1}^A S^X $|$X\u22121ASX$) in the matrix of Eq. (10). The off diagonal element πΔ*J*_{AX}/√2 now converts this last element into singlet population at a frequency of Δ*J*_{AX}/√2. Singlet state populations can of course also be induced by choosing ω_{1} = −2π(*J*_{AA}* − J*_{XX})*,* in which case population on state|$X_1^A T_0^X $|$X1AT0X$is converted into the *S*^{A}*S*^{X} state population. Finally, note that the duration of the SLIC pulse to create population inversion is given as|$\tau _{SL} = 1/(\Delta J_{AX} \sqrt 2)$|$\tau SL=1/(\Delta JAX2)$.

An important observation that was not discussed in the previous paper^{25} arises when the CW-B_{1} field and the preceding 90_{y} excitation pulse are applied to X spins. Then AA′ spins should be in the ST basis whereas XX′ spins are in the X basis. For the same Hamiltonian we obtain the following sub-matrices that have the same off-diagonal element, πΔ*J*_{AX}/√2:

and

Note in Eqs. (9) and (11), spin A and X swap roles, yet *S*^{A}*S*^{X} is the same state. Therefore, the same fraction of X (instead of A) magnetization is transferred into the *S*^{A}*S*^{X} state according to Eq. (11). In contrast, in the asymmetric manifold, the SLIC sequence populates either *S*^{A}*T*_{0}^{X} when A spins are irradiated (Eq. (10)) or *T*_{0}^{A}*S*^{X} when X spins are under irradiation (Eq. (12)). Assuming only states with a singlet component on the AA′ spin pair are long-lived; the net effect is that a smaller fraction of X spin magnetization is stored as long-lived polarization compared with the fraction of A spin magnetization that can be stored by SLIC irradiated on A. This “asymmetric” nature of SLIC with respect to A and X spins is distinct from the “symmetric” M2S sequence, which can transfer equal fraction of A or X spin magnetization into the A spin singlet in an AA′XX′ 4-spin system. As discussed later, this gives rise to different conversion efficiencies and ultimate different detection SNR of the two sequences with all AA′X_{(n)}X_{(n)}′ spin systems.

## III. LOLIPOPS IN LARGER SPIN SYSTEMS: AA′X_{2}X_{2}′ AND AA′X_{3}X_{3}′

Symmetry effects can also be applied to identify disconnected states in an AA′X_{2}X_{2}′ spin systems. ^{13}C_{2}-DPA, which is of *D*_{2h} symmetry, can be modeled as such a spin system (Figure 1(c)) with two carbons (^{13}C_{2}) and four protons (^{1}H_{4}). The other protons are neglected for they have negligible couplings to the ^{13}C spins; this assumption will be justified later. To further simplify the derivation, we have assumed zero *J*-coupling between protons on opposite aromatic rings (*J*_{XX} = 0), which synchronizes resonance conditions for multiple 2-level systems (i.e., same *τ* and *n* for MSM or same ω_{1} for SLIC). The remaining *J*-coupling between protons on the same ring is denoted as *J*_{gauche}.

First, we treat carbons and protons separately to find the symmetry-adapted basis for each species. As before, for the ^{13}C pair, we use the *ST* basis with states *T*_{+}_{1}, *T*_{0} and *T*_{−1} of *A*_{g}-symmetry and the singlet of *B*_{1u}-symmetry in the *D*_{2h} point group of DPA (see Fig. 1). On the other hand, the four protons form 16 states in total. To build up the proton states we use a *ST* basis for the individual spin pairs on either side of the molecule coupled by *J*_{gauche}. The combination of the individual singlet and triplet states results in the 16 desired states which we characterize and sort according to their symmetry as listed in Table II.

Symmetry . | ^{1}H states
. |
---|---|

A_{g} (spin 2) | T_{−1}T_{−1}, |$\,\,( {T_{ - 1} T_0 + T_0 T_{ - 1} })/\sqrt 2$|$(T\u22121T0+T0T\u22121)/2$, |$( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} + 2T_0 T_0 })/\sqrt 6$|$(T\u22121T+1+T+1T\u22121+2T0T0)/6$, |$\,( {T_1 T_0 + T_0 T_1 })/\sqrt 2$|$(T1T0+T0T1)/2$, T_{1}T_{1} |

B_{1u} (spin 1) | |$( {T_{ - 1} T_0 - T_0 T_{ - 1} })/\sqrt 2$|$(T\u22121T0\u2212T0T\u22121)/2$, |$( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} })/\sqrt 2$|$(T\u22121T+1\u2212T+1T\u22121)/2$, |$( {T_1 T_0 - T_0 T_1 })/\sqrt 2$|$(T1T0\u2212T0T1)/2$ |

B_{2u} (spin 1) | |$( {T_{ - 1} S + ST_{ - 1} })/\sqrt 2$|$(T\u22121S+ST\u22121)/2$, |$( {T_0 S + ST_0 })/\sqrt 2$|$(T0S+ST0)/2$, |$( {T_1 S + ST_1 })/\sqrt 2$|$(T1S+ST1)/2$ |

B_{3g} (spin 1) | |$( {T_{ - 1} S - ST_{ - 1} })/\sqrt 2$|$(T\u22121S\u2212ST\u22121)/2$, |$( {T_0 S - ST_0 })/\sqrt 2$|$(T0S\u2212ST0)/2$, |$( {T_1 S - ST_1 })/\sqrt 2$|$(T1S\u2212ST1)/2$ |

A_{g} (spin 0) | |$( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} - T_0 T_0 })/\sqrt 3$|$(T\u22121T+1+T+1T\u22121\u2212T0T0)/3$, SS |

Symmetry | ^{13}C States |

B_{1u} (spin 0) | S |

A_{g} (spin 1) | T_{+1}, T_{0}, T_{−1} |

Symmetry . | ^{1}H states
. |
---|---|

A_{g} (spin 2) | T_{−1}T_{−1}, |$\,\,( {T_{ - 1} T_0 + T_0 T_{ - 1} })/\sqrt 2$|$(T\u22121T0+T0T\u22121)/2$, |$( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} + 2T_0 T_0 })/\sqrt 6$|$(T\u22121T+1+T+1T\u22121+2T0T0)/6$, |$\,( {T_1 T_0 + T_0 T_1 })/\sqrt 2$|$(T1T0+T0T1)/2$, T_{1}T_{1} |

B_{1u} (spin 1) | |$( {T_{ - 1} T_0 - T_0 T_{ - 1} })/\sqrt 2$|$(T\u22121T0\u2212T0T\u22121)/2$, |$( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} })/\sqrt 2$|$(T\u22121T+1\u2212T+1T\u22121)/2$, |$( {T_1 T_0 - T_0 T_1 })/\sqrt 2$|$(T1T0\u2212T0T1)/2$ |

B_{2u} (spin 1) | |$( {T_{ - 1} S + ST_{ - 1} })/\sqrt 2$|$(T\u22121S+ST\u22121)/2$, |$( {T_0 S + ST_0 })/\sqrt 2$|$(T0S+ST0)/2$, |$( {T_1 S + ST_1 })/\sqrt 2$|$(T1S+ST1)/2$ |

B_{3g} (spin 1) | |$( {T_{ - 1} S - ST_{ - 1} })/\sqrt 2$|$(T\u22121S\u2212ST\u22121)/2$, |$( {T_0 S - ST_0 })/\sqrt 2$|$(T0S\u2212ST0)/2$, |$( {T_1 S - ST_1 })/\sqrt 2$|$(T1S\u2212ST1)/2$ |

A_{g} (spin 0) | |$( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} - T_0 T_0 })/\sqrt 3$|$(T\u22121T+1+T+1T\u22121\u2212T0T0)/3$, SS |

Symmetry | ^{13}C States |

B_{1u} (spin 0) | S |

A_{g} (spin 1) | T_{+1}, T_{0}, T_{−1} |

Subsequently, the 4 carbon states (*S*, *T*_{+1}, *T*_{0}, and *T*_{−1}) and the 16 proton states are directly combined to ultimately form the disconnected energy subspaces and symmetries are assigned based on the point-group multiplication of *D*_{2h} symmetry. The same rules applied to the 4-spin system are still valid (mixing can only occur between states with the same symmetry as well as the same total projection numbers ∑*m*_{H} and ∑*m*_{C}). Previous experiments^{23,24} relied on the assumption that DPA supported two-level subspaces similar to those found in the 4-spin systems (Eqs. (4) and (5)) without describing their specific nature which can now be accomplished with the machinery introduced here. The states we are seeking are interconnected by πΔ*J* = π(*J*_{AX} − *J*_{AX}′)(where *J*_{AX} and *J*_{AX}′ denote the near and far carbon-proton couplings, respectively). The task at hand is to find non-zero matrix element for Δ*J* in the developed basis. Following a procedure identical to the one used in going from Eq. (1) to (2) and using the spin notation of Figure 1(c), we rewrite the terms containing the AX couplings of the AA′X_{2}X_{2}′ Hamiltonian as

Next, we single out |$H_{\Delta J} = {\textstyle{1 \over 2}}\Delta J_{AX} ( ( {S_{z1} - S_{z2} } )( I_{z3} + I_{z4}$|$H\Delta J=12\Delta JAX((Sz1\u2212Sz2)(Iz3+Iz4$ −*I*_{z5} − *I*_{z6})) being the term breaking the magnetic equivalence and find its matrix elements given by

As before, states of differing symmetry do not interconnect and the matrix elements of S_{z}_{1} *− S*_{z}_{2} vanish unless the spins are pointed in opposite directions therefore the ^{13}C_{2} state can only be *S* or *T*_{0}. In addition, the total projection number of the four protons (∑*m*_{H}), which decomposes into *m*_{12} + *m*_{34} (total projections of spins 1, 2 and spins 3, 4), can only take values of ±1 (where either *m*_{12} or *m*_{34} is ±1 and the other is zero) or 0 (when *m*_{12} = ±1 and *m*_{34} = ∓1). We finally find 20 combinations of proton and carbon states (*S* and *T*_{0}) that satisfy all stated rules. They form ten isolated 2-level systems listed in Table III. For example, the Hamiltonian matrix of one of these systems is

Γ . | ∑m_{H}
. | 2-level systems . | |
---|---|---|---|

A_{g} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} } )T_0 $|$12(T+1T0+T0T+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 - T_0 T_{ + 1} } )S$|$12(T+1T0\u2212T0T+1)S$ |

0 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} })T_0 $|$12(T\u22121T+1+T+1T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} } )S$|$12(T\u22121T+1\u2212T+1T\u22121)S$ | |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )T_0 $|$12(T\u22121T0+T0T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )S$|$12(T\u22121T0\u2212T0T\u22121)S$ | |

B_{1u} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 - T_0 T_{ + 1} } )T_0 $|$12(T+1T0\u2212T0T+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} } )S$|$12(T+1T0+T0T+1)S$ |

0 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} } )T_0 $|$12(T\u22121T+1\u2212T+1T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} } )S$|$12(T\u22121T+1+T+1T\u22121)S$ | |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )T_0 $|$12(T\u22121T0\u2212T0T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )S$|$12(T\u22121T0+T0T\u22121)S$ | |

B_{2u} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S + ST_{ + 1} })T_0 $|$12(T+1S+ST+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S - ST_{ + 1} })S$|$12(T+1S\u2212ST+1)S$ |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S + ST_{ - 1} })T_0 $|$12(T\u22121S+ST\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S - ST_{ - 1} })S$|$12(T\u22121S\u2212ST\u22121)S$ | |

B_{3g} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S - ST_{ + 1} })T_0 $|$12(T+1S\u2212ST+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S + ST_{ + 1} })S$|$12(T+1S+ST+1)S$ |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S - ST_{ - 1} })T_0 $|$12(T\u22121S\u2212ST\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S + ST_{ - 1} })S$|$12(T\u22121S+ST\u22121)S$ |

Γ . | ∑m_{H}
. | 2-level systems . | |
---|---|---|---|

A_{g} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} } )T_0 $|$12(T+1T0+T0T+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 - T_0 T_{ + 1} } )S$|$12(T+1T0\u2212T0T+1)S$ |

0 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} })T_0 $|$12(T\u22121T+1+T+1T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} } )S$|$12(T\u22121T+1\u2212T+1T\u22121)S$ | |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )T_0 $|$12(T\u22121T0+T0T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )S$|$12(T\u22121T0\u2212T0T\u22121)S$ | |

B_{1u} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 - T_0 T_{ + 1} } )T_0 $|$12(T+1T0\u2212T0T+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} } )S$|$12(T+1T0+T0T+1)S$ |

0 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} - T_{ + 1} T_{ - 1} } )T_0 $|$12(T\u22121T+1\u2212T+1T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} } )S$|$12(T\u22121T+1+T+1T\u22121)S$ | |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )T_0 $|$12(T\u22121T0\u2212T0T\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )S$|$12(T\u22121T0+T0T\u22121)S$ | |

B_{2u} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S + ST_{ + 1} })T_0 $|$12(T+1S+ST+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S - ST_{ + 1} })S$|$12(T+1S\u2212ST+1)S$ |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S + ST_{ - 1} })T_0 $|$12(T\u22121S+ST\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S - ST_{ - 1} })S$|$12(T\u22121S\u2212ST\u22121)S$ | |

B_{3g} | +1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S - ST_{ + 1} })T_0 $|$12(T+1S\u2212ST+1)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} S + ST_{ + 1} })S$|$12(T+1S+ST+1)S$ |

−1 | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S - ST_{ - 1} })T_0 $|$12(T\u22121S\u2212ST\u22121)T0$ | |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} S + ST_{ - 1} })S$|$12(T\u22121S+ST\u22121)S$ |

Equation (15) is isomorphic to the Hamiltonian in Eqs. (4) and (5), assuming *J*_{HH} = 0. Note that the coupling between protons on the same ring (*J*_{gauche}) has no effect on these systems since it shifts the two states identically. This is one reason why neglecting the couplings to the more distant protons in DPA is reasonable; the other is that this simpler model agrees extremely well with experiment.

State functions are written as combination of proton and ^{13}C states, e.g., |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} })T_0 $|$12(T+1T0+T0T+1)T0$ has ^{1}H_{4} in state |${\textstyle{1 \over {\sqrt 2 }}}| {T_{ + 1} T_0 + T_0 T_{ + 1} }\rangle $|$12|T+1T0+T0T+1\u27e9$ and ^{13}C_{2} in state |*T*_{0}⟩. The irreducible representations (Γ) of each 2-level system as well as their sum of proton projection number (∑*m*_{H}) are also provided. Long-lived signal comes from the states with ^{13}C_{2} spins in the singlet |*S*⟩, which are shown in the right column. All eight 2-level systems that have∑*m*_{H} = ±1 are accessed by the MSM sequence with regular resonance condition.

Population differences can be driven between the two states by the MSM pulse sequence. For example, from Eq. (15) the state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} })S$|$12(T\u22121T0\u2212T0T\u22121)S$ can be accessed, which has a carbon singlet component and hence is a source of long-lived signal. According to Table III, for eight of the ten 2-level systems, (16 states) all states have ∑*m*_{H} = ±1. All of them can be accessed with MSM using the same resonance condition shown in Eqs. (6) and (7). Thus, a quarter of total spin states (16 out of 64) in a 6-spin system are used to store bulk magnetization, the same as in a 4-spin system (4 out of 16 states), and the interconversion efficiency is the same for 4-spin and 6-spin systems between carbon bulk magnetization and the long-lived signal.

Interestingly, in the AA′X_{2}X_{2}′ 6-spin system, there exists a second type of 2-level sub-space that has the following structure:

where the two states are interconnected by double the frequency found in the other eight pairs (off-diagonal element is 2*π*Δ*J* rad/s or |$2 \times {\textstyle{{\Delta J} \over 2}} = \Delta J$|$2\xd7\Delta J2=\Delta J$Hz) and both states have ∑*m*_{H} = 0 instead of ±1. This 2-level system in B_{1u} (and the ∑*m*_{H} = 0 state in A_{g}, see Table III) are not seen with MSM using the optimized transfer condition for the other states, as their double-speed evolution makes the net effect of the MSM sequence vanish, but can be seen under other circumstances.

The two different sets of states can also be found with the detected signal from the SLIC sequence. As for the AA′XX′ system we express the disconnected states of the irradiated spin species (^{13}C in this case) in the X basis.^{25} For example, the sub-matrix of the Hamiltonian that contains the long-lived state written out in equation^{15} is, for SLIC,

and the sub-matrix for the state written out in equation^{16} is

Equation (17) is isomorphous with the matrix for a 4-spin system derived above in Eq. (11) (assuming *J*_{XX} = 0).^{25} Again *J*_{gauche} shifts all levels identically and therefore has no effect on the system. Irradiation on carbon spins with ω_{1} = ±2π*J*_{CC} puts state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )S$|$12(T\u22121T0\u2212T0T\u22121)S$ in resonance with state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )X_{ - 1} $|$12(T\u22121T0+T0T\u22121)X\u22121$ or |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )X_1 $|$12(T\u22121T0+T0T\u22121)X1$ and population on |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} })S$|$12(T\u22121T0\u2212T0T\u22121)S$ (denoted as *p*_{s}) is created at a frequency of |$ \pm \Delta J/\sqrt 2 $|$\xb1\Delta J/2$. To a good approximation, |$p_s ( {\tau _{SL} }) \break \propto - {\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}{\rm cos}( {\sqrt 2 \pi \Delta J\tau _{SL} })$|$ps(\tau SL)\u221d\u221212+12 cos (2\pi \Delta J\tau SL)$,^{25} where τ_{SL}(Figure 2(b)) is the duration of the CW pulse. However, a new manifold in the 6-spin system is also put to resonance by the same irradiation (ω_{1} = ±2π*J*_{CC}). Specifically, in the Hamiltonian sub-matrix that contains state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} })S$|$12(T\u22121T+1+T+1T\u22121)S$ of B_{1u}-symmetry shown in Eq. (18) the singlet population is created twice as fast (at a frequency of |$ \pm \sqrt 2 \Delta J$|$\xb12\Delta J$) as that created in Eq. (17). Therefore, for any given τ_{SL}, population induced on the overall ^{13}C_{2}-singlet state (denoted as *P*_{S}) is a combined effect of both resonance conditions, which can be approximated as

where the factors of 8 and 2 come from eight long-lived states that are accessed with the normal condition (Eqs. (17)) and two accessed with Eqs. (18) at double the frequency. This is supported by Figure 4, where the singlet state population after the first CW is simulated^{33} and also measured against the first CW duration, τ_{SL}.

The τ_{SL}-dependence in DPA is clearly different from a single frequency sinusoidal oscillation, which is the dependence obtained from an AA′XX′ 4-spin system. Generalization is also made in Figure 4(a) to an 8-spin system (AA′X_{3}X_{3}′); consider for instance, DMMA (Figure 1(e)) where a strongly coupled ^{13}C spin pair has a methyl group on each side. Simulation shows the same portion of carbon magnetization can be converted into the singlet state population and its dependence on τ_{SL} is even closer to a square wave. The measured lifetime for the long-lived states in DMMA is shown in Figure 5. T_{S} was determined to be around 30 s, three times longer than T_{1} (∼10 s) at the same field strength.

Lastly, it is worthy to note that the same sequences can also induce long-lived polarization in homonuclear AA′B_{(n)}B_{(n)}′ systems where A and B spins have different chemical shifts. The intrinsic narrow bandwidth of the SLIC sequence makes it convenient to selectively irradiate one but not the other spins. As noted earlier, when the chemical shift difference between A and B spins is comparable to achievable rf irradiation Rabi frequencies, more interconnected 2-level systems are recovered such as that between |$S^A T_{ + 1}^B $|$SAT+1B$ and |$T_{ + 1}^A S^B $|$T+1ASB$. Then long-lived signals can be found with sequences using intense irradiation (typically with ≈10^{4} times the power used in SLIC) to induce level anti-crossing.^{34}

## IV. SIGNAL-TO-NOISE ESTIMATIONS EXPLOITING LOLIPOPS IN VARIOUS EXPERIMENTAL SCENARIOS

An important question is how the interconversion efficiency and the ultimate detection SNR of both SLIC and MSM change with increasing size of the spin system. Tables IV and V compare numerical calculations of the relative SNR for a variety of different spin architectures and pulse sequences used to access LOLIPOPS. The *J*-couplings in the simulations are chosen to match those of DPA (*J*_{AA} = 181.8 Hz, *J*_{XX} ≈ 0, Δ*J*_{AX} = 5.82 Hz) for all spin systems, and we simply change the number of spins to obtain the 4- and the 8-spin case. Table IV assumes thermal polarization and coil-dominated noise as appropriate in NMR spectrometers used for detecting and screening for such states. In that case the signal to noise scales as SNR ∝ γ_{prep}γ_{det}^{7/4} *B*^{7/4},^{35,36} because the signal is proportional to the magnetization (∝γ_{det}), to the initial polarization (∝γ_{prep}*B*) and the induced voltage (∝γ_{det}*B*), but the coil noise scales with (γ_{det}*B*)^{1/4}. Table V, on the other hand, appropriate for human hyperpolarized MRI, assumes constant fractional hyperpolarization (e.g., 10% for proton or 10% for carbon) and body-dominated noise. Body noise scales with ∝γ_{det}*B* and hyperpolarization takes out another factor ∝γ_{prep}*B* making SNR ∝ γ_{det} and independent of *B* for hyperpolarized MRI.

. | AB (carbon) . | AA′XX′ . | AA′X_{2}X_{2}′
. | AA′X_{3}X_{3}′
. |
---|---|---|---|---|

MSM (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

SLIC (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

MSM (^{1}H only) | N/A | 22 | 23 | 35 |

SLIC (^{1}H only) | N/A | 16 | 14 | 10 |

. | AB (carbon) . | AA′XX′ . | AA′X_{2}X_{2}′
. | AA′X_{3}X_{3}′
. |
---|---|---|---|---|

MSM (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

SLIC (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

MSM (^{1}H only) | N/A | 22 | 23 | 35 |

SLIC (^{1}H only) | N/A | 16 | 14 | 10 |

. | AB (carbon) . | AA′XX′ . | AA′X_{2}X_{2}′
. | AA′X_{3}X_{3}′
. |
---|---|---|---|---|

MSM (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

SLIC (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

MSM (^{1}H only) | N/A | 2.0 | 2.1 | 3.1 |

SLIC (^{1}H only) | N/A | 1.4 | 1.2 | 0.9 |

. | AB (carbon) . | AA′XX′ . | AA′X_{2}X_{2}′
. | AA′X_{3}X_{3}′
. |
---|---|---|---|---|

MSM (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

SLIC (^{13}C only) | 1 | 0.5 | 0.5 | 0.5 |

MSM (^{1}H only) | N/A | 2.0 | 2.1 | 3.1 |

SLIC (^{1}H only) | N/A | 1.4 | 1.2 | 0.9 |

In all cases, the simulated signal is “filtered” by artificially adding a sequence of 90° pulses on the ^{13}C channel combined with crusher gradients (Figure 2, *n*_{f} = 15), leaving only states with ^{13}C-singlet character intact while inducing fast equilibration among the ^{13}C triplet manifold. Consequently, the final detected signal originates solely from the difference between the singlet state population and the “averaged” triplet population. (See titles of Tables IV and V for details of the calculations).

The trends in Tables IV and V can be understood as follows. We begin with the carbon-only sequences. In the AA′XX′ spin system, the maximum signal from either carbon MSM or carbon SLIC is half the signal from the AB spin system, because only half of the proton states^{23} (*S* and *T*_{0} for MSM, *S* and *X*_{1} for SLIC) participate in the transitions which are perturbed by either sequence. As shown earlier, for MSM or SLIC with the optimal timing, half of the sixteen proton states in AA′X_{2}X_{2}′ also participate in the sequence, and numerical calculations verify that this trend continues with AA′X_{3}X_{3}′.

AA′X_{n}X_{n}′ systems with n > 3 exist, but they are much less common than the n = 3 case, which usually arises from two methyl groups. For example, the molecule 2,2,3,3-tetramethyl 2,3-^{13}C butane (two tert-butyl groups fused together) would be an AA′X_{9}X_{9}′ system.

The proton-only sequences have substantial gains from the higher gyromagnetic ratio (which is more important under the assumptions of Table IV than under those of Table V) and by the assumption of constant molar concentration (so AA′X_{3}X_{3}′ has three times the starting X magnetization of AA′XX′). As shown before, for AA′XX′, MSM on protons outperforms SLIC on protons because MSM alters the population in two long lived states, *SS* and *ST*_{0}, either for A or X irradiation; in contrast, SLIC starting from protons populates *SS* and *T*_{0}*S* (where, as above, the first letter signifies the carbon component). *T*_{0}*S* is not expected to have long-lived characteristics, quickly reducing the observable population differences. The net effect is a reduction in long-lived state production by one-third. It is interesting to note that as *n* increases in the AA′X_{n}X_{n}′ series, the overall efficiency of the proton sequences *rises* somewhat for MSM and *falls* somewhat for SLIC, although it stayed constant for the carbon sequences. In fact, increasing *n* decreases the fraction of states participating in the proton-only sequences for MSM as well, but this is overcome by the magnetization, which increases proportionally with *n*, for SLIC the loss in participating states is more dramatic and cannot be compensated for by the increased magnetization. The differences between using A vs. X nucleus detection are even more dramatic for ^{15}N (γ_{1H}/γ_{15N} ≈ 10) instead of ^{13}C.

For sequences with different excitation and detection nuclei such as M2S(^{13}C)-S2M(^{1}H) and its inverse, the SNR enhancement factor is intermediate between the carbon-only and hydrogen-only cases in Tables IV and V. For example, in Table IV the SNR for AA′XX′ SLIC and MSM ^{1}H-to-^{13}C experiments can be found by multiplying the ^{13}C only results by (γ_{1H}/γ_{13C}) = 4; SNR for ^{13}C-to-^{1}H experiments is estimated by dividing the ^{1}H only results by (γ_{1H}/γ_{13C}) = 4, accounting for different initial polarization. In the context of hyperpolarized MRI the SNR for ^{‑1}H-to-^{13}C experiments will be close to those of ^{13}C only; SNR for ^{13}C-to-^{1}H experiments will be close to the values for ^{1}H only experiments, assuming the same initial polarization on ^{1}H and ^{13}C.

## V. TRANSFER OF PROTON MAGNETIZATION INTO LOLIPOPS IN THE PRESENCE OF A SMALL CHEMICAL SHIFT DIFFERENCE: mDPA

An additional generalization is demonstrated with the example of mDPA (Figure 1(d)), where a considerable chemical shift difference (Δω = 0.56 ppm, 50 Hz at 8.45 T, almost nine times larger than Δ*J*_{CH} ≈ 6Hz, yet smaller than *J*_{CC} = 181.8 Hz) between the two carbons breaks symmetry of the spin system. If all protons are neglected then a near-equivalent 2-spin system is recovered. Therefore, the resonance condition shown by Taylor and Levitt^{20} can be applied to interconvert carbon magnetization and the singlet state polarization. Nonetheless, we show here that the strategy discussed in this study (using Δ*J*_{CH} to access the singlet state) can still be pursued giving access to the signal enhancements that come with the polarization transfer from proton magnetization. To find the correct condition (i.e., τ for MSM and B_{1} for SLIC) we resort to numerical optimization because the introduced chemical shift difference does alter these parameters slightly. Figure 6(a) shows such an optimization to the M2S sequence where all pulses are implemented on proton. The population difference between the ^{13}C_{2} *S* and *T*_{0} states is evaluated against number of 180° pulses (n/2) and inter-pulse delay (τ). The maximum conversion occurs around |${\textstyle{n \over 2}} = \pi /( {4 \times \arctan ( {\Delta J_{CH} /J_{CC} })}) = 24$|$n2=\pi /(4\xd7arctan(\Delta JCH/JCC))=24$ and |$\tau = 1/( {2\sqrt {J_{CC}^2 + \Delta \omega ^2 } }) = 2.65$|$\tau =1/(2JCC2+\Delta \omega 2)=2.65$ ms. Subsequent relaxation measurements with an accordingly adjusted M2S(^{1}H)-S2M(^{13}C) sequence are shown in Figure 6(b). M2S(^{1}H) uses the resonance conditions as listed, while S2M(^{13}C) uses |${\textstyle{N \over 2}}\break = \pi / ( {4 \times {\rm arctan}( {\Delta \omega /J_{CC} })}) = 6$|$N2=\pi /(4\xd7 arctan (\Delta \omega /JCC))=6$ and |$\tau = 1/( {2\sqrt {J_{CC}^2 + \Delta \omega ^2 } })\break = 2.65$|$\tau =1/(2JCC2+\Delta \omega 2)=2.65$ ms as expected for the near-equivalent case.^{20} For comparison the measurement from MSM (^{13}C only) is also shown. The signal invoking the ^{1}H to ^{13}C polarization transfer is clearly enhanced over the ^{13}C only experiment, given the polarization transfer despite the fact that with M2S(^{13}C) relying on the chemical shift difference alone 67% of the initial (carbon) magnetization can be stored,^{21} whereas with M2S(^{1}H) relying on Δ*J*_{CH} only 33% of the initial (proton) polarization can be stored. The measured singlet state lifetime T_{S} is around 146 s whereas T_{1} of the ^{13}C at the same field strength is only 12 s.

Similarly, we can also find the adjusted resonance condition for the SLIC sequence to transfer polarization from proton to ^{13}C_{2}-singlet state. As shown in Figure 7, maximum conversion occurs at ω_{1} ≈ 2π × 188.7 Hz, which is slightly larger than*J*_{CC}, and CW duration τ ≈ 98 ms, which is slightly shorter than the CW duration for DPA (|$d_{SL} = 1/( {\sqrt 2 \Delta J_{CH} }) \approx 120$|$dSL=1/(2\Delta JCH)\u2248120$ ms) but is still much longer than that for the near-equivalent resonance condition (|$d_{SL}\break = 1/( {\sqrt 2 \Delta \omega }) \approx 14$|$dSL=1/(2\Delta \omega )\u224814$ ms). SLIC is associated with much lower power dissipation than M2S while it is much more sensitive to field inhomogeneity, especially when the CW-pulse is long. That also contributes to the lower signal intensity of SLIC (^{1}H to ^{13}C, Figure 7(b), red) compared with that of MSM (^{1}H to ^{13}C, Figure 6(b), red). Nonetheless, this signal loss can be compensated for by using adiabatic CW pulses.^{25} For the adiabatic implementations exact values of *d*_{SL}and ω_{1} are of less critical. The same adiabatic proton pulse for DPA is used here, giving rise to an evident signal enhancement (Figure 7(b), magenta).

As we discussed previously,^{24} the possibility of using proton-only sequences to exploit a ^{13}C_{2} “singlet state” lifetime is intriguing because the technology then becomes compatible with all existing MRI scanners that only have proton channels. The last demonstration (Figure 8) shows such a proton-only relaxation measurement on mDPA. Implemented with the same adiabatic proton-only SLIC sequence,^{25} we measured a signal lifetime around 108 s, which is shorter than the measured T_{S} on carbon (130 ∼ 147 s) but is a tremendous signal enhancement of the proton T_{1} of 4 s.

## VI. CONCLUSIONS

LOLIPOPS can be created on AA′X_{(n)}X_{(n)}′ spin systems by exploiting the magnetic inequivalence between A and A′ spins created by the difference in *J*-coupling of *J*_{AX} (=*J*_{A′X′}) and *J*_{AX′} (=*J*_{A′X}). Increasing the number of coupling partners, *n*, does not have a significant impact on the amount of A-spin magnetization stored as LOLIPOPS despite the exponentially increasing number of states because the number of states connecting to the long-lived states increases at a similar rate. The fraction of X-spin magnetization that can be transferred into LOLIPOPS does decrease with increasing *n*, but this can be completely compensated for by the much higher sensitivity of X (usually proton) spin detection. We also discussed the difference between the two sequences (MSM vs SLIC) used to access the long lived states and systematically evaluated their overall efficiencies in various experimental scenarios. To give the best intuitive understanding of the sequences, we showed that MSM is best understood in a singlet-triplet basis for both the A and X spins, whereas for the description of SLIC, the basis for the irradiated spins is best chosen such that the irradiated spins are in a basis that diagonalizes the pulse-Hamiltonian (e.g., *S*_{x}_{1} + *S*_{x}_{2}) and the other spins are in a singlet triplet basis. Finally, we have demonstrated that even when a chemical shift difference is introduced into these spin systems LOLIPOPS can be accessed either directly from the A spins exploiting the chemical shift difference or from the X spins with polarization transfer exploiting the difference in out-of-pair *J*-couplings *J*_{AX} and *J*_{AX′}.

### A. Materials and methods

3,6-dichloro-^{15}N_{2}-pyridazine (DCP) was dissolved in DMSO-d6 at a concentration of ∼1 M. For MSM on DCP *τ*_{SS} = 33.3 ms and *n*_{SS} = 44 according to Eq. (6); *τ*_{ST} = 15.1 ms and n_{ST} = 96 according to Eq. (7). ^{13}C_{2}-diphenyl acetylene (DPA) and ^{13}C_{2}-meta methyl diphenyl acetylene (MDPA) were prepared at similar concentration in CDCl_{3}. The resonance conditions for both MSM and SLIC sequence on these two compounds are detailed in the discussion or can be found in the previous studies.^{24,25} Lifetime measurements of DCP, DPA, and mDPA were conducted in an 8.45 T Bruker Spectrometer with a 5 mm NMR tube. On the other hand, the measurement on DMMA was made at 16.44 T with ^{13}C at natural abundance (1.1%)^{32} dissolved in DMSO-d6 (∼3 M). Coupling parameters of DMMA were determined to be 68 Hz (*J*_{CC}), 7.6 ± 0.6 Hz (*J*_{CH}) and −5.6 ± 0.2 Hz (*J*_{CH}′), giving a CW duration of 0.0536 s for the SLIC sequence. All experiments were conducted without degassing, i.e., in the presence of O_{2}.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation through Grant Nos. CHE-1058727 and 1363008.