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Preparation of Nuclear Spin Singlet States Using Spin-Lock Induced Crossing
Stephen J. DeVience*
Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA

Ronald L. Walsworth†
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA; Center for Brain Science, Harvard University, 52 Oxford Street, Cambridge, Massachusetts 02138, USA; and Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA

Matthew S. Rosen‡
A. A. Martinos Center for Biomedical Imaging, 149 Thirteenth Street, Charlestown, Massachusetts 02129, USA; Harvard Medical School, 25 Shattuck Street, Boston, Massachusetts 02115, USA; and Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA (Received 5 July 2013; published 22 October 2013) We introduce a broadly applicable technique to create nuclear spin singlet states in organic molecules and other many-atom systems. We employ a novel pulse sequence to produce a spin-lock induced crossing (SLIC) of the spin singlet and triplet energy levels, which enables triplet-singlet polarization transfer and singlet-state preparation. We demonstrate the utility of the SLIC method by producing a long-lived nuclear spin singlet state on two strongly coupled proton pairs in the tripeptide molecule phenylalanineglycine-glycine dissolved in D2 O and by using SLIC to measure the J couplings, chemical shift differences, and singlet lifetimes of the proton pairs. We show that SLIC is more efﬁcient at creating nearly equivalent nuclear spin singlet states than previous pulse sequence techniques, especially when triplet-singlet polarization transfer occurs on the same time scale as spin-lattice relaxation.
DOI: 10.1103/PhysRevLett.111.173002 PACS numbers: 33.25.+k

There is great current interest in the controlled preparation and coherent manipulation of singlet states for nuclear spin pairs in molecules and other many-atom systems (e.g., spin networks in solids), as spin singlet states are largely decoupled from relaxation, driven by ﬂuctuating dipolar ﬁelds, that limits single-spin state lifetimes [1,2]. For example, in liquid state experiments singlet states in nuclear spin pairs can exhibit lifetimes much longer than the single-spin polarization lifetime (T1 ) [3–14]. One application is the storage of spin polarization during magnetic resonance imaging studies, where relaxation often occurs faster than biological transport. The spins of a hyperpolarized molecule can be prepared in the spin singlet state before injection and converted back to detectable polarization once the molecule reaches its target. In addition, nuclear spin singlet states can be used as a resource for spectroscopic interrogation of couplings within many-spin systems, including J couplings, dipolar, and hyperﬁne couplings in both organic molecules and spin networks in solids [15–17]. Such singlet states exist naturally when nuclear spins are strongly J coupled relative to their resonance frequency differences, Á, i.e., J ) Á, as is often the case for nuclear spins far from a molecule’s chiral center. However, due to the differences in spin singlet and triplet state symmetries, it is not possible to transfer polarization from the triplet to the singlet state by directly driving a radio frequency transition, which limits the control of singlet-state preparation and manipulation. Tayler 0031-9007=13=111(17)=173002(5)

and Levitt demonstrated that triplet-singlet polarization transfer can instead be achieved using a series of -pulse trains in which the pulse timing is synchronized to the J-coupling strength between nuclei [7]. This ‘‘M2S’’ sequence takes advantage of the small amount of mixing between singlet and triplet states that is present whenever Á > 0. Feng and Warren also showed that the M2S sequence can create singlet states in certain heteronuclear systems even when Á ¼ 0 [14]. These results hold promise for creating hyperpolarized singlet states without the need for a symmetry-breaking chemical reaction or continuous spin locking [13]. However, in all results to date, the polarization transfer to the spin singlet state only occurs during the ﬁnal third of the M2S sequence time, and before this stage the spin polarization occupies states subject to conventional spin-lattice relaxation. In this Letter, we show that better triplet-singlet polarization transfer efﬁciency can be achieved by replacing the M2S pulse trains with a continuous spin lock whose nutation frequency is matched to the J coupling between the target nuclear spins. At this spin-locking strength, the energy levels of the singlet state and one triplet state become equal in the rotating frame, which we call the ‘‘spin-lock induced crossing’’ (SLIC). Polarization transfer can occur for the duration of spin locking, which minimizes polarization loss from triplet-state relaxation and thus provides better efﬁciency for singlet-state creation than the M2S technique. SLIC is analogous to the Ó 2013 American Physical Society

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Hartmann-Hahn condition for polarization transfer between two magnetically inequivalent nuclear spins, except that for SLIC the nuclei are nearly identical and their spin symmetry subspaces are inequivalent [18]. We experimentally compare M2S and SLIC using liquidstate NMR of the tripeptide molecule phenylalanineglycine-glycine (phe-gly-gly), which contains two nearly equivalent proton spin pairs in which to prepare singlet states. We ﬁnd that for these two proton spin pairs, singlet-state creation with SLIC is 19% and 75% more efﬁcient than with M2S. We also demonstrate the utility of SLIC for characterizing singlet-state lifetimes as well as small J couplings and chemical shift differences between nearly identical nuclear spins. Figure 1(a) shows the M2S experimental protocol used to create a nuclear spin singlet state from triplet-state polarization and then return the singlet state to transverse triplet-state (i.e., measurable) polarization after an evolution time, evolve . Figure 1(b) gives a simulation of spin state and coherence dynamics during singlet state preparation with M2S if relaxation is ignored [19]. The ﬁrst pulse train, n1 , converts the triplet-state polarization into
(a)

a singlet-triplet coherence with a relaxation time of T2 =3 % T1 =3 (for liquid-state NMR of small molecules), and the second pulse train, n2 , creates a singlet population with relaxation time TS [11,20]. The number of pulses required for the M2S sequence increases as the resonance frequency difference (Á) between the two nearly identical nuclear spins decreases and hence the singlet state becomes closer to ideal. In many cases, the required M2S pulse sequence time approaches or exceeds T1 of the nuclear spins, and signiﬁcant spin polarization can be lost before it is transferred to the singlet state, particularly during the ﬁrst 2=3 of the sequence. For an ideal system, the time required for maximum singlet state creation is tM2S;max % 3 1:18 ¼ : 8Á Á (1)

Figure 1(c) shows the SLIC pulse sequence used to create a nuclear spin singlet state from triplet-state polarization and return the singlet state to transverse triplet-state polarization after an evolution time (evolve ) in analogy to the M2S experiment. However, instead of pulse trains, continuous spin locking is applied on resonance with the
(c)

(b)

(d)

Time (s)

FIG. 1 (color online). Simulated comparison of M2S and SLIC techniques applied to singlet state creation, evolution, and readout for a proton spin pair with chemical shift  ¼ 3:71 ppm in the phe-gly-gly molecule. Guide lines connect points in the sequence with corresponding times in the simulation. (a) Schematic of M2S experiment. Following an initial 90 deg pulse to create transverse triplet polarization, the M2S pulse sequence creates singlet-state population by applying pulse trains with appropriate length and pulse spacing synchronized with the proton pair’s J coupling and resonance frequency difference. The system then evolves over evolve , and the M2S sequence is applied in reverse order to convert singlet population back to transverse triplet polarization for inductive NMR detection. (b) Simulation of M2S experiment. Transfer, in a series of stages, of transverse (x axis) triplet polarization to singlet-triplet coherences and ﬁnally to singlet- and triplet-state populations of equal magnitudes. Transfer to the singlet-state population only occurs in the ﬁnal 1=3 of the M2S preparation sequence. (c) Schematic of SLIC experiment. Transverse triplet polarization is created via an initial 90 deg pulse and is then transferred to singlet-state population by the application of a spin-locking ﬁeld with n ¼ J for period SL . For spin locking, the transmitter frequency is set to the average resonance frequency of the nearly equivalent spin pair. The system then evolves for evolve and an identical spin-locking ﬁeld converts singlet population back to transverse triplet polarization for detection. (d) Simulation of SLIC experiment. Transfer of transverse triplet polarization to singlet-state population begins immediately and occurs during a single spin-locking stage.

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nearly equivalent spins at a nutation frequency equal to the J coupling between spins, i.e., n ¼ J. The simulation shown in Fig. 1(d) illustrates that such spin locking transfers triplet-state polarization directly from transverse polarization into singlet-state population more quickly than in the M2S sequence. A density matrix analysis [21] shows that selecting a nutation frequency n ¼ J matches the energies of the singlet state and one of the triplet states, creating a spin-lock induced crossing. At this energy, offpﬃﬃﬃ diagonal interaction terms Á=2 2 become signiﬁcant and induce oscillatory triplet-singlet polarization transfer with pﬃﬃﬃ a period of 2=Á and maximum transfer to the singlet state at half this time: 1 0:707 pﬃﬃﬃ ¼ tSL;max ¼ : (2) Á Á 2 Comparison with Eq. (1) shows that SLIC produces singlet-state polarization about 40% faster than M2S, which results in fewer relaxation losses. To compare the effectiveness of the two sequences, we performed simulations using Bloch equations to model relaxation and singlet-triplet polarization transfer. Figure 2 plots the calculated polarization transfer efﬁciency for M2S and SLIC as a function of the product T1 Á. SLIC is found to be signiﬁcantly more efﬁcient than M2S for all ranges of parameters, and particularly for T1 Á < 1. To assess the relative utility of the SLIC and M2S sequences for producing nuclear spin singlet states, we performed NMR measurements at 4.7 T on a 20 mM solution of the phe-gly-gly molecule dissolved in D2 O, addressing a nearly equivalent proton spin pair with chemical shift  ¼ 3:71 ppm and T1 ¼ 912 Æ 7 ms [22]. For the M2S sequence [Fig. 1(a)] we found optimized parameters for singlet creation to be n1 ¼ 10, n2 ¼ 5, and  ¼ 14:4 ms, which indicates J ¼ 17:4 Æ 0:1 Hz and Á ¼ 2:8 Æ 0:3 Hz. We also found a singlet lifetime of TS ¼ 25:1 Æ 0:8 s with no spin locking applied during evolve . We measured the NMR signal intensity (x-axis magnetization, proportional to the transverse triplet-state polarization) at the end of the M2S sequence for evolve ¼ 5 s, which arises from the transfer of transverse triplet-state polarization to singlet-state population and then back to measurable transverse triplet-state polarization after evolve , and we then used the singlet lifetime to extrapolate the singlet-state population at evolve ¼ 0. During evolve , remaining triplet-state polarization is lost to relaxation, and it does not contribute to the ﬁnal signal. We compared this M2S NMR signal magnitude to a reference measurement arising from a single 90 deg pulse applied to the sample, i.e., without singlet creation. From this analysis we estimate that 24% of the initial triplet-state polarization was transferred to the singlet state and back to triplet for evolve ¼ 0, out of a theoretical maximum of 50%, yielding an efﬁciency of 69% for each application of the M2S sequence. For the SLIC technique, we determined the optimal spin-lock nutation frequency by performing a truncated SLIC pulse sequence in which the NMR signal (x-axis
FIG. 2 (color online). Simulations of ideal triplet-singlet polarization transfer efﬁciency for M2S (blue) and SLIC (red). Results are shown for TS ) T1 and TS ¼ 3T1 . M2S was modeled in two steps: ﬁrst, polarization transfer from I1x þ I2x with lifetime T2 ¼ T1 to I1y À I2y with lifetime T1 =3; second, polarization transfer from I1z À I2z with lifetime T1 =3 to singlet state S0 with lifetime TS . Only one polarization transfer needed to be modeled for SLIC, between I1x þ I2x with lifetime T2 ¼ T1 and S0 with lifetime TS . A maximum of 50% polarization transfer to the singlet state can be achieved by both sequences, which we deﬁne to be an efﬁciency of 100%. Note that both sequences are less efﬁcient for smaller TS =T1 due to singlet relaxation.

magnetization) was acquired directly after the ﬁrst spinlocking period. As a function of nutation frequency, the measured NMR signal exhibited a dip centered at n ¼ 17:5 Æ 0:3 Hz with a relative depth of % 25% [Fig. 3(a)], consistent with the SLIC condition of n ¼ J for optimal triplet-singlet polarization transfer. We then used this optimal spin-lock nutation frequency in the complete SLIC sequence with evolve ¼ 5 s [Fig. 1(c)] and optimized the spin-lock duration (SL ) to produce the strongest NMR signal and hence maximal singlet state creation. The measured dependence of the SLIC NMR signal on the spin-lock duration [Fig. 3(b)] exhibits a ﬂat maximum for SL % 280–360 ms. Using SL ¼ 300 ms provided about 34% polarization transfer from the triplet to the singlet state and back when extrapolated to evolve ¼ 0, indicating an 82% polarization transfer efﬁciency for each application of SLIC spin locking. Although the matching condition is narrow, the B1 homogeneity of our spectroscopy coil was sufﬁcient to achieve good polarization transfer. Applications in magnetic resonance imaging may require a pulse sequence with a broader bandwidth. We next applied the SLIC technique to singlet state creation in a second proton spin pair in the phe-gly-gly molecule, with chemical shift  ¼ 3:20 ppm and T1 ¼ 430 Æ 5 ms. This proton spin pair is coupled to a third proton that decreases both the singlet lifetime (TS ¼ 2:15 Æ 0:05 s with no spin locking applied) and the triplet-singlet polarization transfer efﬁciency.

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FIG. 3. Experimental application of the SLIC technique to nuclear spin singlet state creation in the phe-gly-gly molecule. Results for  ¼ 3:71 ppm proton pair: (a) The NMR signal (normalized x-axis magnetization, proportional to transverse triplet polarization) following the ﬁrst SLIC spin lock as a function of nutation frequency for SL ¼ 300 ms exhibits a pronounced dip when the spin-lock frequency equals the J coupling. A Lorentzian is ﬁt to the measurement to determine n ¼ J ¼ 17:5 Æ 0:3 Hz. (b) NMR signal after the complete SLIC experiment with evolve ¼ 5 s (proportional to ﬁnal transverse triplet polarization surviving transfer to singlet and back) as a function of spin lock duration. Maximal singlet-state creation is found for SL % 280–360 ms. We ﬁt for amplitude, A, and pﬃﬃﬃ Á with the function I ¼ Asin4 ðSL Á= 2Þ, and we ﬁnd Á ¼ 2:15 Æ 0:02 Hz. Results for  ¼ 3:20 ppm proton pair: (c) NMR signal following the ﬁrst SLIC spin lock as a function of nutation frequency for SL ¼ 332 ms, with a Lorentzian ﬁt to the data yielding optimal n ¼ J ¼ 13:5 Æ 0:2 Hz. (d) NMR signal after the complete SLIC experiment with evolve ¼ 500 ms as a function ﬃﬃﬃ spin pof lock duration. Maximal singlet state creation is found for SL % 300–400 ms. From a ﬁt with the function I ¼ Asin4 ðSL Á= 2Þ þ c, we ﬁnd Á ¼ 2:13 Æ 0:06 Hz.

We followed the procedure outlined above to determine the optimal spin-lock nutation frequency n ¼ J ¼ 13:5 Æ 0:2 Hz and the optimal spin-lock duration SL;max ¼ 332 Æ 9 ms [Figs. 3(c) and 3(d)]. We then applied the complete SLIC sequence with evolve ¼ 500 ms and found about 12% polarization transfer to the singlet state and back when extrapolated to evolve ¼ 0, which represents a transfer efﬁciency of 49% for each application of SLIC spin locking. For comparison, we experimentally investigated singlet state creation with the M2S sequence. We determined optimized M2S parameters to be n1 ¼ 4, n2 ¼ 5, and  ¼ 17:9 ms, and measured that only 4% of the polarization was transferred from the triplet to the singlet state and back when extrapolated to evolve ¼ 0, which represents a 28% efﬁciency for each application of the M2S sequence. In summary, we introduced an improved and broadly applicable method, known as SLIC (spin-lock induced crossing), for the creation of long-lived singlet states of

nuclear spins in molecules and other many-atom systems. As an example, we applied our SLIC technique to two different nearly equivalent proton spin pairs in the phe-glygly molecule and demonstrated that the SLIC experiment gives 40% and 300% more signal than the previous M2S technique for the transfer of triplet-state polarization to singlet-state population and then back to measurable transverse triplet-state polarization. SLIC is more effective than M2S primarily because the transfer to the long-lived singlet state begins immediately with SLIC, without the need for an initial transfer to a singlet-triplet coherence as with M2S. Though a singlet-triplet coherence can have an extended lifetime relative to a single-spin coherence time (T2 ), it generally relaxes signiﬁcantly faster than the singlet population lifetime (TS ), leading to greater polarization loss and less efﬁcient singlet state creation for M2S than for SLIC. Additionally, the shorter length and lower peak power of SLIC ultimately results in signiﬁcantly less power dissipation in the sample. The relative advantage

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of SLIC grows for molecules with small resonance frequency difference Á between the nuclear spins, which in many cases can be much smaller than 1=T1 . Beyond liquid-state NMR, we foresee applications of SLIC in long-lived quantum memories composed of nuclear spin pairs [23], as well as selective nuclear spin state manipulation at low magnetic ﬁeld [24], and highprecision solid-state electronic spin measurements such as nitrogen vacancy (NV) diamond magnetometry [25–32]. For a nuclear spin quantum memory with singlet and triplet bases, SLIC may provide a straightforward way to prepare desired states on the singlet-triplet Bloch sphere. At low magnetic ﬁeld, nuclear spins are strongly coupled and do not exhibit resolvable spectral features for manipulation using conventional single-spin NMR techniques. However, SLIC could be used to mix and thereby manipulate nuclear spin states by exploiting their weak chemical shift differences. SLIC may also provide a tool for improving the sensitivity of NV diamond magnetometry by allowing state-speciﬁc preparation and manipulation of the nearly equivalent nitrogen electron spins that occur in the diamond lattice at about 10 times the concentration of optically polarizable and detectable NV centers. These nitrogen electron spins could be prepared in singlet states to increase the NV coherence time, or in speciﬁc triplet states to serve as ancillary magnetic ﬁeld sensors controllably coupled via dressed-state techniques to the optically detectable NV centers [33]. We acknowledge support from the NSF, Army, and DARPA-QuASAR and MURI QuISM programs.

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