  We characterize the early stages of the approach to equilibrium in isolated
quantum systems through the evolution of the entanglement spectrum. We find
that the entanglement spectrum of a subsystem evolves with at least three
distinct timescales. First, on an o(1) timescale, independent of system or
subsystem size and the details of the dynamics, the entanglement spectrum
develops nearest-neighbor level repulsion. The second timescale sets in when
the light-cone has traversed the subsystem. Between these two times, the
density of states of the reduced density matrix takes a universal, scale-free
1/f form; thus, random-matrix theory captures the local statistics of the
entanglement spectrum but not its global structure. The third time scale is
that on which the entanglement saturates; this occurs well after the light-cone
traverses the subsystem. Between the second and third times, the entanglement
spectrum compresses to its thermal Marchenko-Pastur form. These features hold
for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum
circuit models.
