Probabilistic Hysteresis from a Quantum Phase Space Perspective

Abstract

\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.