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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
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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Quantum geometry is conventionally understood as a descriptive structure characterizing the space of quantum states. In this work, we examine an alternative but fully compatible interpretation in which geometric quantities—such as Berry curvature and the quantum metric—function effectively as constraint layers shaping the set of admissible dynamical trajectories. Without introducing new degrees of freedom or modifying established theoretical frameworks, we analyze how geometric terms enter semiclassical equations of motion and transport responses in quantum systems. In particular, anomalous velocity contributions and geometry-dependent transport coefficients indicate that geometry operates not merely as a passive descriptor, but as an active structural condition on system evolution. This shift from geometry as description to geometry as constraint does not alter the formalism, but reorganizes its interpretation, reducing conceptual fragmentation by unifying diverse geometry-dependent effects under a common structural principle. It highlights that the evolution of quantum systems is not only governed by dynamical variables, but also by the geometric structure that restricts the space of accessible trajectories. This perspective suggests that transport phenomena may be more fruitfully analyzed through constraint-space formulations rather than purely state-space descriptions. We clarify the scope and limits of this reinterpretation, emphasizing that it introduces no new physical interactions or ontological commitments. Instead, it provides a conceptually sharper and operationally informative framework for understanding the role of geometry in quantum dynamics and transport.
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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.
This paper presents a geometric–informational model of quantum decoherence formulated in terms of the Forman–Ricci curvature on a weighted graph representing correlations between entangled subsystems. In this framework, decoherence emerges as a curvature–driven process that progressively flattens the informational geometry of the system. The model derives a gradient–flow dynamics from an action principle with dissipation and is quantitatively compared with the experimental results of Bordes et al. (2024) on entangled gamma photons. The proposed formulation establishes a conceptual bridge between the Lindblad formalism and informational geometry, offering a scale–independent interpretation of quantum decoherence as the geometric relaxation of correlation structures, consistently describing both optical and gamma entanglement regimes.
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