  In statistical mechanics, a small system exchanges conserved
quantities---heat, particles, electric charge, etc.---with a bath. The small
system thermalizes to the canonical ensemble, or the grand canonical ensemble,
etc., depending on the conserved quantities. The conserved quantities are
represented by operators usually assumed to commute with each other. This
assumption was removed within quantum-information-theoretic (QI-theoretic)
thermodynamics recently. The small system's long-time state was dubbed ``the
non-Abelian thermal state (NATS).'' We propose an experimental protocol for
observing a system thermalize to the NATS. We illustrate with a chain of spins,
a subset of which form the system of interest. The conserved quantities
manifest as spin components. Heisenberg interactions push the conserved
quantities between the system and the effective bath, the rest of the chain. We
predict long-time expectation values, extending the NATS theory from abstract
idealization to finite systems that thermalize with finite couplings for finite
times. Numerical simulations support the analytics: The system thermalizes to
the NATS, rather than to the canonical prediction. Our proposal can be
implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions,
quantum dots, and perhaps nuclear magnetic resonance. This work introduces
noncommuting conserved quantities from QI-theoretic thermodynamics into quantum
many-body physics: atomic, molecular, and optical physics and condensed matter.
