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 z-axis 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.
Preparation of macroscopic quantum superposition states of a cavity field via coupling to a superconducting charge qubit