  We study sudden quantum quenches in which the initial states are selected to
be either eigenstates of an integrable Hamiltonian that is nonmappable to a
noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after
the quench is always integrable and mappable to a noninteracting one. By
studying weighted energy densities and entropies, we show that quenches
starting from nonintegrable (chaotic) eigenstates lead to an "ergodic" sampling
of the eigenstates of the final Hamiltonian, while those starting from the
integrable eigenstates do not (or at least it is not apparent for the system
sizes accessible to us). This goes in parallel with the fact that the
distribution of conserved quantities in the initial states is thermal in the
nonintegrable cases and nonthermal in the integrable ones, and means that, in
general, thermalization occurs in integrable systems when the quench starts
form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the
spectrum), while it fails (or requires larger system sizes than those studied
here to become apparent) for quenches starting at integrable points. We test
those conclusions by studying the momentum distribution function of hard-core
bosons after a quench.
