  Quantum simulators have recently enabled experimental observations of quantum
many-body systems' internal thermalization. Often, the global energy and
particle number are conserved, and the system is prepared with a well-defined
particle number - in a microcanonical subspace. However, quantum evolution can
also conserve quantities, or charges, that fail to commute with each other.
Noncommuting charges have recently emerged as a subfield at the intersection of
quantum thermodynamics and quantum information. Until now, this subfield has
remained theoretical. We initiate the experimental testing of its predictions,
with a trapped-ion simulator. We prepare 6-21 spins in an approximate
microcanonical subspace, a generalization of the microcanonical subspace for
accommodating noncommuting charges, which cannot necessarily have well-defined
nontrivial values simultaneously. We simulate a Heisenberg evolution using
laser-induced entangling interactions and collective spin rotations. The
noncommuting charges are the three spin components. We find that small
subsystems equilibrate to near a recently predicted non-Abelian thermal state.
This work bridges quantum many-body simulators to the quantum thermodynamics of
noncommuting charges, whose predictions can now be tested.
