A directly observable, Zeeman-insensitive nuclear spin coherence in solution

TL;DR

This work demonstrates clock-like, Zeeman-insensitive nuclear-spin coherence in a solution-state three-spin system, [1--13C]-fumarate, at ultralow magnetic field around $B_{LAC} ≈ 400~nT$. By combining parahydrogen-induced polarization with real-time alkali-vapor magnetometry, the authors observe a LAC between a singlet-like and a triplet-like manifold with a minimum frequency near $f_{min} ≈ 2~Hz$ and a coherence lifetime near $25~s$. The coherence is first-order immune to magnetic-field perturbations, though its lifetime remains limited by inhomogeneous broadening and high-magnetization back-action; the latter reveals a dipolar-field-induced nonlinearity and Josephson-like dynamics. Overall, the study establishes a directly observable ZEFOZ-like mechanism in a molecular liquid, with potential implications for PHIP-based imaging and extending such coherence protections to other spin networks in solution.

Abstract

Clock transitions are well known in atomic and solid-state systems, but are largely unexplored in molecular liquids. Here we demonstrate a clock-like, nuclear-spin avoided crossing in [1--$^{13}$C]-fumarate that supports long-lived and directly observable coherences at ultralow magnetic field: a three-spin transition $|S_0α\rangle \leftrightarrow |T_{+1}β\rangle$ near 400 nT exhibits a shallow crossing with a frequency minimum of 2 Hz. The transition is first-order immune to magnetic field perturbations and displays a lifetime of 25 s, around three times the longest single-spin $T_2^*$. Sensitivity to effective pseudo-fields is also demonstrated, including the internal dipolar field of the sample.

Figures (4)

• Figure 1: Energy levels of [1--13C]-fumarate in magnetic fields below 1µT. Upper plots correspond to the fictional scenario of non-coupled 1H and 13C spins, with $J_{13}=J_{23}=0$ Hz. Lower plots represent the actual scenario using experimentally measured heteronuclear J-couplings, as listed. The expansion shows that across a field range of c. 100 nT the LAC transition is much less sensitive to spectral broadening; shaded rectangles in the lower plot have a lower aspect ratio $\Delta E:\Delta B_z$.
• Figure 2: Left: Vector model representation of the $\{\ket{\phi_a},\ket{\phi_b}\}$ subspace in [1--13C]-fumarate . Upper sphere indicates the $B_z$-dependent effective field orientation in the $xz$ plane (\ref{['eq:wlacvectormodel']}) for the combined Zeeman and J-coupling Hamiltonian (\ref{['eq:Htotal']}). Lower sphere shows simulated time-evolution trajectories of $\rho^{\rm(ab)}(0)=A_z$ under $H$, which closely reflects precession about the effective axes. Right: Simulated operator trajectories when switching from an initial field of $B_z=$ 2µT to a lower field after 1s. Plots show the transformation amplitude of the operator on the left side of the arrows ("$\rightarrow$") to those indicated on the right. $I_{1z}+I_{2z}$ is proportional to the total 1H magnetization along $z$, and $\mathbf{I}_1\cdot\mathbf{I}_2$ is the traceless 1H spin-singlet order. Conversion of $A_z$ is most efficient at $B_z=+B_{\rm LAC}=+$ 0.4µT, reducing as the field offset increases, including $B_z=-B_{\rm LAC}$, while magnetization-to-singlet and singlet-to-magnetization conversions are effective equally (except opposite in sign) at $B_z=\pm B_{\rm LAC}$ due to the same initial population of the $m=\pm1/2$ manifolds. Switching to zero field yields no net magnetization.
• Figure 3: Solution-state LAC coherences in hyperpolarized [1--13C]-fumarate observed via magnetometry. Top: Experimental scheme, showing the initialization of [1--13C]-fumarate in a near-pure 1H-singlet state by reaction of the precursor with para-H2, followed by syringe transfer to a low-field setup for subsequent field-switching operations and measurement of the nuclear magnetization using an OPM. Bottom: Experimental results. On the right, magnetization curves from sample experiments are shown. To the left are two plots showing the fitted frequency and decoherence rate at different $B_z$ evolution fields. On the coherence frequency plot, the black dashed curve is the calculated eigenvalue difference between the evolution states of the Hamiltonian in \ref{['eq:HBz']}, using best-fit spin-spin coupling parameters $J_{12} = 15.92Hz$, $J_{\Sigma} = 9.34Hz$ and $J_{\Delta} = 2.514Hz$. Dashed red and gray lines indicate the predicted $B_{\rm LAC}$ field from calculating the eigenvalues of the Hamiltonian in \ref{['eq:HBz']}, and calculated from \ref{['eq:bprimedef']}, respectively. Curve/data-point colors indicate: (Black) multiple experiments, each with a single switch from 2µT to a chosen evolution field $B_z$; (Teal) a single experiment with multiple switches to a different $B_z$ field every 2s.
• Figure 4: A model representation of the 'high magnetization´´ regime through simulated trajectories of the $\bm{\rho}^{(ab)}$ Bloch vector and magnetization projection $\braket{M_z(t)}\propto (1-\braket{A_z(t)}) /2$. Each row shows evolution from the same initial condition, $\rho^{(ab)}(0) = A_z$, $\omega_{\rm LAC}/(2\pi) = 2Hz$, for different fields: (top) a constant on-resonance field, $\delta B = 0$; (center) a time-varying pseudo-field due to the macroscopic magnetization $\delta B(t) = B_{d,\rm max} \times \braket{M_z(t)}$, with uniform $B_{d,\rm max} = 50nT$; (bottom) an average of trajectories with time-dependent $\delta B$, across a uniform distribution of $B_{d,\rm max}$ between strengths $40nT$ and $60nT$. The latter best replicates the magnetization trajectory for 99%-13C-enriched [1--13C]-fumarate in \ref{['fig:experiment1']}.