  \emph{Probabilistic hysteresis} is a manifestation of irreversibility in a
small, isolated classical system [Sci. Rep. 9, 14169]: after a slow cyclic
sweep of a control parameter, the probability that a microcanonical ensemble
returns to the neighborhood of its initial energy is significantly below one. A
similar phenomenon has recently been confirmed in a corresponding quantum
system for not too small particle number $N$. Quantum-classical correspondence
has been found to be non-trivial in this case, however; the rate at which the
control parameter changes must not be extremely slow and the initial
distribution of energies must not be too narrow. In this paper we directly
compare the quantum and classical forms of probabilistic hysteresis by making
use of the Husimi quantum phase space formalism. In particular we demonstrate
that the classical ergodization mechanism, which is a key ingredient in
classical probabilistic hysteresis, can lead to a breakdown of
quantum-classical correspondence rather than to quantum ergodization. As a
result strong quantum effects in the long-term evolution are present, even
though the quantum corrections in the equations of motion are proportional to
$1/N$ and therefore would naively seem to be small. We also show, however, that
quantum ergodization is restored by averaging over energies, so that for
sufficient initial energy width and not-too-slow sweep rate the classical
results are recovered after all. Finally we show that the formal
incommutability of the classical and adiabatic limits in our system, leading to
the breakdown of quantum-classical correspondence in the quasi-static limit, is
due to macroscopic quantum tunneling through a large energetic barrier. This
explains the extremely slow sweep rates needed to reach the quantum adiabatic
limit that were reported in our previous work.
