Origin of long-lived nuclear spin states and coherences in aliphatic chains revealed by relaxation theory

Delocalized long-lived states (LLSs) and collective zero-quantum long-lived coherences (LLCs) in aliphatic chains provide a promising route for preserving nuclear spin order on timescales exceeding conventional longitudinal and transverse relaxation times, respectively. Their extended lifetimes make them attractive for applications including hyperpolarization storage, ligand-observed drug screening based on the loss of longevity upon binding to a target protein, and quantum information processing exploiting collective properties of many-body spin systems. Although LLSs and LLCs have been observed experimentally in methylene networks, their origin and general structure in chains of arbitrary length have remained unclear. Here we show that these relaxation-protected modes follow directly from Redfield relaxation theory by constructing the null space of the dominant intra-pair dipole-dipole relaxation superoperator. For a chain containing N methylene groups, this subspace contains 2N - 1 independent non-trivial operators, excluding the identity operator. When the two geminal protons are chemically equivalent within each methylene group, global intra-pair parity restricts experimental access to at most 2N - 2 of these operators. We develop a general framework for constructing the corresponding long-lived operators for chains of arbitrary length and illustrate it explicitly for N = 2, 3, and 4. Numerical simulations that explicitly account for coherent evolution and inter-pair dipole-dipole relaxation closely reproduce the experimentally observed decay dynamics and further demonstrate that the resulting LLSs and LLCs exhibit lifetimes substantially longer than the corresponding T1 and T2 relaxation times, respectively.