  Equilibrium quantum many-body systems in the vicinity of phase transitions
generically manifest universality. In contrast, limited knowledge has been
gained on possible universal characteristics in the non-equilibrium evolution
of systems in quantum critical phases. In this context, universality is
generically attributed to the insensitivity of observables to the microscopic
system parameters and initial conditions. Here, we present such a universal
feature in the equilibration dynamics of the Sachdev-Ye-Kitaev (SYK)
Hamiltonian -- a paradigmatic system of disordered, all-to-all interacting
fermions that has been designed as a phenomenological description of quantum
critical regions. We drive the system far away from equilibrium by performing a
global quench, and track how its ensemble average relaxes to a steady state.
Employing state-of-the-art numerical simulations for the exact evolution, we
reveal that the disorder-averaged evolution of few-body observables, including
the quantum Fisher information and low-order moments of local operators,
exhibit within numerical resolution a universal equilibration process. Under a
straightforward rescaling, data that correspond to different initial states
collapse onto a universal curve, which can be well approximated by a Gaussian
throughout large parts of the evolution. To reveal the physics behind this
process, we formulate a general theoretical framework based on the
Novikov--Furutsu theorem. This framework extracts the disorder-averaged
dynamics of a many-body system as an effective dissipative evolution, and can
have applications beyond this work. The exact non-Markovian evolution of the
SYK ensemble is very well captured by Bourret--Markov approximations, which
contrary to common lore become justified thanks to the extreme chaoticity of
the system, and universality is revealed in a spectral analysis of the
corresponding Liouvillian.
