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arXiv:2607.21569

A geometric framework for spin relaxation

Sophia N. Fricke, Kieren A. Harkins, Ashok Ajoy, Jeffrey A. Reimer

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• Revised Oct 12, 2026 (v2)
v1 · Jul 23, 2026v2 · Oct 12, 2026
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Abstract

Spin relaxation is conventionally described by two independent phenomenological rates - longitudinal ($R_1$) and transverse ($R_2$) - whose separation obscures a deeper structural unity. Here we develop a geometric framework in which dissipation is represented by a single covariant relaxation tensor acting in Liouville space, from which $R_1$ and $R_2$ emerge as complementary projections. This tensor structure is not merely formal but is experimentally accessible through pulse sequences that probe noncommuting directions in spin space. Using hyperpolarized $^{13}C$ spins in diamond with nitrogen-vacancy centers, we show that commuting pulse trains yield effective relaxation matrices that are approximately diagonal, while noncommuting sequences produce off-diagonal components that vary with transmitter frequency offset and pulse ordering, providing evidence that relaxation is a directional process governed by a tensor rather than a pair of scalar rates. Complementary measurements of geometric phase demonstrate that noncommuting dynamics introduce ordering-dependent effects that are separable from dissipation, consistent with the interpretation of relaxation as geometric transport on the state manifold. This framework unifies Bloch, Redfield, and Lindblad descriptions within a coordinate-independent formulation and provides a natural language for relaxation in driven, anisotropic, and non-equilibrium spin systems.

Journal reference
J. Magn. Reson. 2026; 392: 108189
DOI
10.1016/j.jmr.2026.108189

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Spin relaxation is conventionally described by two independent phenomenological rates - longitudinal (R1R_1R1​) and transverse (R2R_2R2​) - whose separation obscures a deeper structural unity. Here we develop a geometric framework in which dissipation is represented by a single covariant relaxation tensor acting in Liouville space, from which R1R_1R1​ and R2R_2R2​ emerge as complementary projections. This tensor structure is not merely formal but is experimentally accessible through pulse sequences that probe noncommuting directions in spin space. Using hyperpolarized 13C^{13}C13C spins in diamond with nitrogen-vacancy centers, we show that commuting pulse trains yield effective relaxation matrices that are approximately diagonal, while noncommuting sequences produce off-diagonal components that vary with transmitter frequency offset and pulse ordering, providing evidence that relaxation is a directional process governed by a tensor rather than a pair of scalar rates. Complementary measurements of geometric phase demonstrate that noncommuting dynamics introduce ordering-dependent effects that are separable from dissipation, consistent with the interpretation of relaxation as geometric transport on the state manifold. This framework unifies Bloch, Redfield, and Lindblad descriptions within a coordinate-independent formulation and provides a natural language for relaxation in driven, anisotropic, and non-equilibrium spin systems.
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While the geometry of equilibrium states and driven non-equilibrium processes is clearly understood, a geometric description for relaxation towards equilibrium is still lacking. Here, we propose a thermo-geometric measure based on the Rayleigh quotient, reformulating relaxation as a fundamental competition between entropic stiffness and frictional dissipation. Taking a van der Waals gas with two dissipation channels as an example, we explicitly demonstrate its relaxation landscape. Particularly, we find that upon approaching the critical temperature TcT_cTc​, the slow-mode relaxation rate vanishes linearly as λs∝(T−Tc)/Tcλ_s \propto (T-T_c)/T_cλs​∝(T−Tc​)/Tc​, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.
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