Violation of detailed balance in non-equilibrium spin waves

Traditional inelastic neutron scattering (INS) characterizes excitations, such as phonons and magnons, in condensed matter in thermodynamic equilibrium. However, the most intriguing and puzzling many-body effects in open quantum systems emerge from dissipative dynamics that are inherently out of equilibrium. Here, using a novel combination of laser pumping and INS, we experimentally observe, and support with theoretically modeling, long-lived non-equilibrium spin excitations in a two-dimensional (2D) square lattice Heisenberg antiferromagnet that manifest themselves as a violation of detailed balance in the dynamic structure factor. The non-equilibrium spin excitations reach steady states under a periodic driving force, displaying similarities to non-equilibrium steady states in a periodically driven dissipative system. We also show that the violation of detailed balance reflects the quantum mechanical nature of the underlying dynamical system, where out-of-time-ordered correlations of creation and annihilation operators do not satisfy commutation relations. The \emph{in operando} INS technique developed here to observe non-equilibrium spin excitations in a prototypical 2D quantum magnet can be applied to other quantum spin systems such as one-dimensional spin chains and topological many-body spin systems where non-equilibrium phenomena are ubiquitous and further novel discoveries are expected.