Report on 2003.04702v3

We study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states.Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state.We back-up our analysis with numerical results obtained on the XXX spin chain with a random field along the zaxis in the ergodic regime and find good qualitative and quantitative agreement with the theory.We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.