Dynamical change under slowly changing conditions: the quantum Kruskal-Neishtadt-Henrard theorem

Abstract

Adiabatic approximations break down classically when a constant-energy contour splits into separate contours, forcing the system to choose which daughter contour to follow; the choices often represent qualitatively different behavior, so that slowly changing conditions induce a sudden and drastic change in dynamics. The Kruskal-Henrard-Neishtadt theorem relates the probability of each choice to the rates at which the phase space areas enclosed by the different contours are changing. This represents a connection within closed-system mechanics, and without dynamical chaos, between spontaneous change and increase in phase space measure, as required by the Second Law of Thermodynamics. Quantum mechanically, in contrast, dynamical tunneling allows adiabaticity to persist, for very slow parameter change, through a classical splitting of energy contours; the classical and adiabatic limits fail to commute. Here we show that a quantum form of the Kruskal-Neishtadt-Henrard theorem holds nonetheless, due to unitarity.