Long-Lived Singlet State in an Oriented Phase and its Survival across the Phase Transition Into an Isotropic Phase

Long-lived singlet states (LLSs) of nuclear spin pairs have been extensively studied and utilized in the isotropic phase via liquid-state nuclear magnetic resonance. However, there are hardly any reports of LLSs in the anisotropic phase that allows contribution from the dipolar coupling in addition to the scalar coupling, thereby opening many exciting possibilities. Here we report observing LLS in a pair of nuclear spins partially oriented in the nematic phase of a liquid crystal solvent. The spins are strongly interacting via the residual dipole-dipole coupling. We observe LLS in the oriented phase living up to 3 times longer than the usual spin-lattice relaxation time constant ($T_1$). Upon heating, the system undergoes a phase transition from nematic into isotropic phase, wherein the LLS is up to 5 times longer lived than the corresponding $T_1$. Interestingly, the LLS prepared in the oriented phase can survive the transition from the nematic to the isotropic phase. As an application of LLS in the oriented phase, we utilize its longer life to measure the small translational diffusion coefficient of solute molecules in the liquid crystal solvent. Finally, we propose utilizing the phase transition to lock or unlock access to LLS.

Figure 1

LLS cannot only be prepared in POP (left) but also can be carried across the phase transition into IP (right).

Figure 2

Rotations in $\{|T_0⟩,|S_0⟩\}$ subspace by using CPMG echo sequence. (a) CPMG echo sequence. The delay $\tau$ and the number of iterations of CPMG echo are chosen to match the resonance condition [Eq. (5) and (6)]. (b) Evolution from the initial state $|T_0⟩$ under the resonant echo sequence.

Figure 3

Evolution of singlet-triplet basis states under various elements of the M2S pulse sequence (see Fig. 4). The column titles are the states that evolve under the operations listed in row titles. Here $\tau$ given by Eq. (5). The iteration numbers $n_1=\lfloor \pi /(2\theta )\rceil$ and $n_2=\lfloor \pi /(4\theta )\rceil$, respectively, correspond to the resonant conditions for $\pi$ and $\pi /2$ rotations in $\{|T_0⟩,|S_0⟩\}$ subspace. The global phases are indicated by $\varphi$, ${\varphi }^{\prime }$, ${\varphi }^{″}$, and ${\varphi }^{‴}$.

Figure 4

(a) M2S-S2M pulse sequence for preparing, storing, and detecting LLS. Here $\tau$, $n_1$, and $n_2$ were calculated using Eqs. (5) and (6). The two PFGs and the $\pi /2$ pulse in between are used to suppress artifacts. After storing the LLS for duration $T$, it is converted to detectable $I_{1x}+I_{2x}$ magnetization using the S2M sequence. (b) Simulated evolution of the populations in states ${\rho }_1=I_{1x}+I_{2x}$, ${\rho }_2=I_{1y}-I_{2y}$, ${\rho }_3=I_{1z}-I_{2z}$, $|T_0⟩⟨T_0|$, and $|S_0⟩⟨S_0|$ during the M2S-S2M pulse sequence assuming no relaxation and parameters $\mathrm{\Omega }=50\mathrm{Hz}$, $J=10\mathrm{Hz}$, $\mathcal{D}=600\mathrm{Hz}$, $T=0.03\mathrm{s}$. Note that the LLS (${\rho }_5$) persists throughout the storage interval without any spin lock and is converted back to ${\rho }_1$ by the S2M sequence.

Figure 5

(a),(b) One-pulse ${}^1\mathrm{H}$ NMR spectrum of 2-chloroacrylonitrile [CAN; molecular structure shown in the inset of (a)] in (a) POP at 294 K and (b) IP at 305 K (inset shows the full spectrum including the solvent peaks). (c),(d) the spectrum after S2M in (c) POP at 294 K after a storage time of 7 s and (d) IP at 305 K after a storage time of 45 s.

Figure 6

Measured values of $T_1$, $T_{\mathrm{LLS}}$, and their ratio $T_{\mathrm{LLS}}/T_1$ for CAN at different temperatures across both phases. $T_{\mathrm{LLS}}$ was obtained by systematically varying the storage delay $T$ in Fig. 4, while $T_1$ was measured using the standard inversion-recovery method [44]. In POP, $T_{\mathrm{LLS}}$ is approximately 3 times longer than $T_1$, whereas it is 5 times longer in IP.

Figure 7

(a) Pulse sequence for preparing LLS in POP using M2S sequence shown in Fig. 4 and detecting it after the phase transition into IP using modified CL sequence. Before LLS preparation in POP, the first bipolar PFG introduces a $z$-dependent phase shift, which is refocused by the second bipolar PFG after the phase transition into IP. This way, the “stimulated echo” signal is solely due to the LLS that survived the phase transition. In all experiments, sinusoidal PFG of duration $\delta =320\text{μ}\mathrm{s}$ were used with strength 2.5 G/cm. Since the system is transitioning from strong coupling to weak coupling, we use a 2 kHz WALTZ-16 spin lock to sustain LLS during storage. (b)–(d) Illustrating spin states (b) after the first $(\pi /2{)}_y$ pulse when both spins are pointing along $\hat{x}$, (c) after the first bipolar PFG, and (d) $I_{1y}-I_{2y}$ state after the phase transition into IP.

Figure 8

(a) Spectra corresponding to LLS that has survived the phase transition at various storage intervals as mentioned. Here the top trace is the reference, the same as in Fig. 5. (b) Decay of LLS signal intensities in (a) versus storage interval $T$.

Figure 9

(a) Pulse sequence used for estimation of diffusion coefficient using LLS. The parameters $\tau$, $n_1$ and $n_2$ are optimized at each temperature for maximum final signal. Position encoding and decoding are realized by bipolar PFGs (of duration $\delta =320\text{μ}\mathrm{s}$) to minimize eddy currents. For STE diffusion experiment at each temperature, PFG strength $G$ was varied from 2.5 G/cm to 47.5 G/cm in 19 steps, with diffusion interval $\mathrm{\Delta }=3.3\mathrm{s}$. For LLS diffusion experiment in POP, $\mathrm{\Delta }=10\mathrm{s}$ and $G$ was the same as in STE; whereas in IP, $\mathrm{\Delta }=30\mathrm{s}$ and $G$ was varied from 1 G/cm to 20 G/cm 20 steps. (b) The measured translational diffusion coefficient of CAN in MBBA plotted versus temperature using STE sequence [3] and PGM2S2M sequence shown in (a).

Figure 10

Proposed PADLOCK method using a pair of $A$ spins and covalently farther $B$ spins as shown. In POP, due to the heteronuclear dipolar coupling, the magnetic equivalence is broken, and LLS in the $AA$’$BB$’ system can be accessed. In IP, the dipolar coupling vanishes leaving only scalar coupling between magnetically equivalent AA spins, which potentially can hold LLS for indefinitely long times. Thus, the phase transition is the key to accessing or locking LLS.