Generating and sustaining long-lived spin states in 15N,15N′-azobenzene

Long-Lived spin States (LLSs) hold a great promise for sustaining non-thermal spin order and investigating various slow processes by Nuclear Magnetic Resonance (NMR) spectroscopy. Of special interest for such application are molecules containing nearly equivalent magnetic nuclei, which possess LLSs even at high magnetic fields. In this work, we report an LLS in trans-15N,15N′-azobenzene. The singlet state of the 15N spin pair exhibits a long-lived character. We solve the challenging problem of generating and detecting this LLS and further increase the LLS population by converting the much higher magnetization of protons into the 15N singlet spin order. As far as the longevity of this spin order is concerned, various schemes have been tested for sustaining the LLS. Lifetimes of 17 minutes have been achieved at 16.4 T, a value about 250 times longer than the longitudinal relaxation time of 15N in this magnetic field. We believe that such extended relaxation times, along with the photochromic properties of azobenzene, which changes conformation upon light irradiation and can be hyperpolarized by using parahydrogen, are promising for designing new experiments with photo-switchable long-lived hyperpolarization.

In the following sections we consider the coherent spin dynamics, which describes the creation of 15 N singlet order in the model spin system ʹ 2 2 ʹ. We often refer to the multiple product states between nitrogen and proton states (there are a total of 64 of such product states), and importantly, some of the singlet product states of the 15 N pair, e.g. � 0 12 , �, are connected to the triplet states � 12 , �. In all such cases it is assumed, that they are connected by J-coupling interactions of the nitrogen to protons spins. These interactions govern the coherent spin dynamics responsible for creation of the 15 N singlet order by NMR pulse sequences. Additionally, there are fluctuating local interactions, which also connect the 15 N singlet manifold to the triplet one; these interactions are responsible for the relaxation of the singlet order. In this work we do not consider these interactions in the theoretical treatment.

NMR excitation on the 15 N channel
Let us first consider the situation, in which the nitrogen spin magnetization is converted into the LLS.
The spin Hamiltonian is written in the doubly rotating frame, in which the resonance frequencies of 1 H and 15 N correspond to two reference frequencies, and , and all relevant J-couplings are included: The following combinations of the J-couplings are introduced: Here ̂ is the -th spin operator; the spins are numbered as follows: spins 1,2 are the two nitrogens, pairs of spins 3,4 and 5,6 are the ortho-protons of each phenyl ring (Figure 1 in the main text of the paper). In the J-coupling terms we make a secular approximation for the heteronuclear interactions; hence, only z components remain.
In addition, we need to include the term that stands for RF-excitation. The key parameter is then its strength, characterized by the corresponding spin nutation frequency, 1 { 1 H} or 1 { 15 N}, in the RF-field applied on the proton or nitrogen channel, respectively. For instance, the Hamiltonian of a SLIC pulse on the 15 N channel is given by The SLIC condition requires that the nutation frequency is matched to the corresponding J-coupling: 1 = in the tilted frame of reference, associated with the quantization axis along the -axis of the rotating frame, and by using the singlet-triplet state vectors for each pair of spins: Here ℛ � ( /2) is the operator introducing rotation of spins around the -axis by the angle of /2. In the new basis , the part of the Hamiltonian associated with the nitrogen spins is written down in a simpler way, as shown below. By | 〉 we denote the singlet state; while | +1 〉, | 0 〉 and | −1 〉 are the triplet states with the projection of the total spin equal to +1, 0 and −1, respectively.
The Hamiltonian ℋ � + ℋ � has 10 blocks, in which the singlet and triplet states of 15 N are connected. 8 of these blocks correspond to basis wave-functions with the -projection of the total magnetic moment of one of the proton spin pairs being equal to ±1, while for the other pair it is 0. In the tilted basis these blocks appear as (we also show the corresponding basis ket-vectors for the matrix elements): and 34 is the -projection of the total spin for protons 3 and 4, 56 is -projection of the total spin for protons 5 and 6. In addition, there are two more blocks, in which the amplitude of the off-diagonal elements containing Δ is twice as large: We previously determined the value of -coupling for trans-ABZ to be 2.1 Hz. However, as it can be seen from expressions (5) and (6) in all cases is added equally to the all diagonal elements; so it does not affect the spin dynamics of creation of the singlet state and thus can be ignored.
The appearance of the energy levels upon variation of 1 in the blocks of the two kinds is shown in Figure  S1. One can see that there are well-pronounced LACs found when the matching condition 1 ≈ is fulfilled. The Hamiltonian of the spin system for each of these blocks is analogous to that of a system of two strongly coupled spins-1/2 1,2 . To make this analogy mathematically correct we need to specify the effective coupling of the two levels that have an LAC. This coupling is given by the difference in the NMR frequencies of the two 15 N spins, being Δ and 2 ⋅ Δ , in the two cases shown in Figure S1. By using this analogy, we can introduce methods for converting spin magnetization into long-lived spin order and back. Previously 3 we have done so by applying the APSOC method to generate the LLS in ABZ and to determine the parameters for constant adiabaticity APSOC pulses 4,5 . Here, we introduce adiabatic SLIC pulses, i.e., RF-pulses with slowly ramped amplitude so that passage through the condition 1 = is adiabatic; following previous works we introduce constant adiabaticity SLIC pulses 4 . One should note that adiabatic passage is achieved simultaneously for the blocks of the two kinds, see eqs. (5) and (6). Thus, it becomes possible to access singlet product states � 0 12 , � simultaneously in all 10 relevant blocks, which is in contrast to other methods, such as SLIC 1 and M2S 6,7 . To understand this one should consider the fact, that SLIC duration as well as the number of spin echoes in M2S methods depend on the value of the offdiagonal element connecting a particular singlet product state � 0 12 , � to a triplet product state � ±1 12 , �; in this case it is given by constant from expressions (5) and (6). The duration of SLIC and M2S cannot be tuned so that simultaneous conversion in both types of blocks is obtained. Figure S1. Energy levels as a function of the amplitude of the resonant RF-field in the 15 N channel, which is proportional to the nutation frequency 1 of the 15 N spins around such a field. Neglecting the shift of energy levels due to the coupling, we obtain that there are 8 (subplot a) and 2 (subplot b) identical blocks in the Hamiltonian. In both cases there is a level anti-crossing at 1 = , which is exploited in the SLIC method. (a) For the 8 blocks with the -projection of total angular moment of the proton spins is equal to 1 the energy splitting at the LAC is Δ /√2. (b) For the 2 blocks with the -projection of the total moment of the proton spins being 0 the energy splitting at the LAC is 2Δ /√2. In each subplot, we show the correlation of adiabatic states upon increase of 1 , which defines the type and efficiency of spin order conversion.

NMR excitation on the 1 H channel
Now let us consider creation of the LLS using magnetization of the ortho-protons. In this situation, a SLIC pulse is described by the following term in the Hamiltonian: In the presence of this term, the Hamiltonian is factorized into several blocks with connected singlettriplet to triplet-triplet product states of the pair of 15 N spins. The triplet states in the tilted frame for the proton spin wave-function are defined as: In this situation, there are three relevant blocks, corresponding to the following sets of basis wavefunctions: For all these sets of product states, the corresponding blocks of the total Hamiltonian � � + � � are the following: The sign of the off-diagonal element is always positive for 1 block, always negative for 2 block, and for the block 3 the sign is positive for all transitions where 0 12 part changes to 0 12 and negative where 0 12 changes to 0 12 states.
In addition, there is one more block with a set of 10 wave-functions: The corresponding block of the Hamiltonian is as follows: 1. Figure S2. Energy levels as a function of the spin nutation frequency around the resonant RF-field in the 1 H channel. As in case of Figure S1 there is a number of LACs, in total 10 LACs. We obtain that there are 3 identical blocks with 2 LACs in each (subplot a) and a 10-level system with 4 LACs (subplot b). In (a) the minimal splitting at LACs is Δ /√2; in (b) the splitting is Δ for 2 LACs and √6Δ /2 for the other two LACs. Eigenstates are shown for the product states, where the first letter denotes the state of the 15 N pair, and the following number refers to the projection of the total spin of the ortho-protons along "x" axis of the 1 H rotated frame.
Again, it is useful to plot the energies as a function of the nutation frequency, 1 , in this case, of the proton nutation frequency. Figure S2 shows such plots, which allow one to identify the relevant LACs and also the correlation of adiabatic states. Again, one can see that there are several LACs found when 1 ≈ . The splitting of the corresponding states takes different values depending on the LAC. Note, that the same number of LACs is found when the SLIC pulse is applied on the 1 H and 15 N channel. All the LACs are always correspond to the nutation frequency 1 { 15 1 } = . Adiabatic passage over the LACs with minimal splitting (Δ /√2) will automatically work for all the others LACs. Consequently, the conversion elements are the same for 15 N and 1 H channels. As in the previous case, it is advantageous to use adiabatic pulses in order to exploit spin mixing at all LACs.

Maximal conversion efficiency
Hence, we have introduced the spin order conversion blocks, which can convert magnetization into the long-lived spin order. It is useful to assess the efficiency of the conversion for different schemes to have a reference for the experiment.
Thermal equilibrium is described by the following spin density matrix: Here are the Boltzmann factors of the nitrogen and proton, respectively ( is the Boltzmann constant and stands for the temperature). Note, that due to the opposite signs of the gyromagnetic ratio, the "spindown" | 〉 state is overpopulated for 15 N spins in contrast to 1 H spins, for which the "spin-up" | 〉 state is overpopulated at thermal equilibrium. Eq. (11) is valid when the Zeeman interaction of the spins with the The degree of polarization associated with operator � , as determined from a density matrix , is defined as follows: 8 This kind of an expression is sometimes called "Liouville bracket" or expectation value of an operator 8 . Thus, for 15 N the equilibrium magnetization is calculated by taking the Liouville brackets with �̂1 +̂2 � yielding ℎ = Δ 2 , while for protons the relevant operator is �̂3 +̂4 +̂5 +̂6 � and ℎ = Δ .
Hence, as expected, the equilibrium magnetization of 1 H spins is approximately 20-fold larger than the 15 N magnetization: the number of protons is twice as high and their gyromagnetic ratio is ca. 10 times greater as compared to 15 N The singlet order of a spin pair is defined as the difference between the population of the singlet state and the mean population of the triplet states of this pair: Here we calculate the maximal value of the LLS-derived NMR signal after following the sequence of events: (i) conversion of magnetization to 15 N singlet order, (ii) equilibration of the populations within the triplet and singlet 15 N manifolds, (iii) backward conversion of the LLS into magnetization.
We model the scheme with the creation and detection of the singlet order on the 15 N channel in the following way. At the first stage (i) the populations of the singlet product states � 0 12 , � in the connected blocks are swapped with those of the corresponding outer triplet product states, either � +1 12 , ⟩ or � −1 12 , ⟩.
In the SLIC method the + or − sign depends on the relative phases of the first 90-degree pulse and the SLIC pulse. For concreteness, let us assume that we populate the singlet state. In the equilibrium all the populations of the singlet product states  (5) and (6), because only in these blocks the 15 N singlet state is connected to the triplet states by J-couplings. Another important thing is that this conversion is happening independently in each of these blocks. This isolation occurs because the RF-pulses do not mix blocks between each other, expressions (5) and (6) reflect this fact. Exchange of the � 0 12 , � and � −1 12 , ⟩ populations corresponds to 100% efficiency of the M→S conversion for a particular block of the Hamiltonian. It is convenient to introduce the number of the singlet-triplet product states in which the populations were swapped. More rigorously number is given by: where 1 and 2 are the efficiencies of the M→S conversion in the blocks of the first and second types respectively. Figure S3 illustrates creation of singlet order under a SLIC pulse, applied on the 15 N channel. It can be seen that evolution of singlet order in the first type of blocks (expression (5)) is by a factor of two slower than in the second type (expression (6)). The maximal total population of the singlet state is achieved when the SLIC duration is set to:

√2
In this case singlet product states are created in 8 connected blocks of the first type (see expression (5)) and are not created at all in the 2 blocks of the second type (see expression (6)). This also agrees with previous results obtained for diphenyl acetylene 6 . So the number is the population of the singlet state of the 15 N spin pair, measured in units of 64 (or 64 ). As was discussed before, adiabatic methods allow to perform simultaneous conversion in all 10 blocks, so for them equals to 10. . The 2 dependence on the number of the blocks, in which exchange between singlet and triplet populations happens, is due to the fact that the populations are swapped twice: once in the M→S stage and once again during the S→M conversion. Note, that some residual singlet order remains after the magnetization-to-singlet and singlet-to-magnetization (M→S→M) sequence.
When the singlet order is generated by applying pulses on the 1 H channel, redistribution of the populations between singlet and triplet states of the 15 N spin states is more complicated, it is graphically represented in Figure S2. Within all connected blocks the population of the corresponding product singlet 15 N states is increased while the corresponding 15 N product triplet states are depleted (or vice versa, depending on the conversion type introduced by the M→S block). One should note that the populations of the 15 N states are exchanged with the populations of the outer triplet states of the 1  .
These considerations can be extended to all possible M→S→M conversion schemes shown in Figure 2 in the main text of the paper. We would like to note that exactly the same results can be obtained by considering unitary bounds for the spin dynamics 9 . However, to keep the discussion clearer, here we do not dwell on the details of such consideration.
For detection on the 15 N channel, the LLS-derived NMR signal is normalized to the 15 N magnetization and it is given by expression: Here corresponds to the type of nuclei, in which singlet order was created. For detection in the 1 H channel the LLS-derived NMR signal is given by expression: The number of the ortho-protons is two times greater than the number of 15 N spins, so normalization to the 1 H magnetization causes an apparent decrease of LLS population by a factor of 2 (compare eq. (15) and (16)), although it corresponds to the same polarization level. In practice, however, the scheme with detection on the 1 H channel should include suitable filters, to remove a strong background signal, not associated with the LLS.

Numerical simulations of MSM sequences for SLIC and adSLIC
To illustrate this difference in the performance of different methods we performed simulations for traditional SLIC, adiabatic SLIC with RF-excitation on the 15 N channel, and adiabatic SLIC with RF-excitation on the 1 H channel (see Figure S4). Calculations have been performed using the Spin Dynamica package 10 . In all cases we start with thermal populations of the spin states. Details of the calculation and normalization method are described in the text. Note, that in all cases trajectories for the starting magnetization, which evolves during the first SLIC/adSLIC pulses are not shown; their intensity always starts from 100% and depending on the method is decreasing while the singlet order is growing.
In these calculations, we considered the initial spin state, given by the following density matrix 0 = 1 64 (̂1 +̂2 +̂3 +̂4 +̂5 +̂6 ) (17) meaning that there is only transverse magnetization present. The time dependences of magnetization and singlet order were calculated as normalized expectation values: The singlet order generated in the course of the spin evolution has been calculated as normalized expectation value of the operator We do not show the evolution of the magnetization on the first step, because it has significantly larger intensity, its evolution is basically a converse and scaled evolution of the singlet order. At the LLS maintenance stage we introduce very fast relaxation of magnetization by equilibrating the populations of all 16 singlet product states � 0 12 , � as well as the populations of all 48 triplet product states � 12 , � of the 15 N spin pair. This equilibration imitates such relaxation, which conserves the symmetry of 15 N spin states, e.g. mutual dipole-dipole coupling between 15 N spins, and the symmetric part of the CSA tensor.
Normalization introduced in eq. (18) is convenient, as it allows to present the trajectories with different norm on the same plot; also, it gives values of the final magnetization in the same manner as in eqs. (15) and (16): normalized values per equilibrium magnetization of the observation channel. We can see ( Figure  S4a) that SLIC applied on the 15 N channel produces final magnetization of about 16% of the starting value. A comparable value of 16.(6)% is obtained from eq. (15), when the number is set to 8, which corresponds to maximal conversion efficiency for SLIC. Figure S4b shows, that by using adSLIC pulse the final 15 N magnetization, coming from the LLS signal, reaches about 25%, whereas eq. (15) predicts 26% in case of = 10. Finally, Figure S4c shows, that magnetization of protons, which originates from the LLS component reaches 12.5%, a value two times smaller than 15 N magnetization, because of the normalization by the total 1 H magnetization. It is also in agreement with eq. (16), when is set to 10. All these amplitudes of maximal LLS signal intensities are summarized in Figure 2 of the main text of the paper. One also can check that the singlet order calculated from the numerical values corresponds to that one, calculated in the previous section, but proper scaling should be taken into account: the values of should be compared to the SO trajectories shown in Figure S4. Figure S5 illustrates the square dependence of the magnetization intensity on the efficiency of the conversion. Good agreement is observed between analytical solution of eq. (15) and numerical results from simulations with SLIC pulses of different durations. Minor deviations are caused by the fact that singlet and triplet states are not the exact eigen states due to magnetic unequivalence of the 15 N pair, and also by the fact that it is impossible to perform an ideal adiabatic conversion within a finite period of time. So, all the main conclusions, which were made in the previous sections, are supported by Figure S4 and Figure S5 as well. One can see that the M→S conversion efficiency is the same for RF-excitation on the proton and nitrogen channels. In practical NMR experiments with initially thermally polarized spins, due to the 10-fold difference in the nuclear gyromagnetic ratios, this means that the LLS-derived signal is approximately 10 times stronger when the M→S block is introduced on the proton channel. These numerical simulations also show that adiabatic SLIC is more efficient than the standard SLIC method and that there is always some residual singlet order left after a 100% efficient M→S→M sequence.

Experimental details
Experiments of the long-lived states, which exist without spin-locking and have lifetimes of the order of minutes, are challenging. It is necessary to restrict the sample within the active volume of the detection coil. In order to achieve high stability and resolution the sample was prepared as shown in Figure S6: an insert (Wilmad-LabGlass, 529-E) in an NMR tube was used, filled with a solution of acetonitrile-d 3 with 1,2-15 N-ABZ. It was degassed by three freeze-pump-thaw cycles and sealed by flame. Protonated acetonitrile was added around the insert, to achieve good homogeneity of the magnetic susceptibility within the sample ( 1 H signals of ABZ at 700 MHz spectra had linewidths of about 0.5 Hz).  Constant adiabaticity adSLIC and APSOC pulses were optimized to have constant adiabaticity using an algorithm as described in ref. 4 for an effective 2 spin-1/2 system with the difference of resonance frequencies = 2.1 Hz and -coupling equal to = 16 Hz. The maximal RF-amplitude, characterized by the nutation frequency 1 , was set to 2 for both methods. In the APSOC method, the adiabatic RF-pulses were applied with a frequency offset of 10 Hz relative to the resonance frequency of the NMR signal ( 1 H or 15 N). The resulting pulse envelopes are shown in Figure S7. Software for calculating constant adiabaticity profiles is available online for free 11 . Figure S8 demonstrates an optimization of the adSLIC pulse duration, aimed at achieving a high degree of adiabaticity. In this optimization, the duration of the adSLIC pulses was varied, while the period (LLS is sustained by using 15 N spin-locking) was fixed. It can be seen that starting from approximately 0.8 s of the adSLIC duration, the signal intensity reaches a plateau, corresponding to optimal performance of the conversion element ( Figure S8b). Analogous optimization was done for the APSOC pulse in the previous study 3 ; a pulse duration of 1.2 s was sufficient to achieve efficient conversion.

Results of the individual fittings of different experiments
Results of measuring the long-lived lifetimes in a series of experiments with different methods and conditions are summarized in Table 1 and Table 2. Table 1. Best fit parameters of the experiments without spin-locking. A bi-exponential decay was used ( ) = 0 + exp�− / � + exp{− / }. Gradient filter T00 was used in some experiments the same as in ref. 12 (otherwise the singlet order selection filter was always used). It is denoted at which channel the hard pulses were applied, and in some cases the filter was repeated twice.