Quantum scrambling across an energy barrier

Abstract

Unless you are a trained juggler, it is very hard to balance a sharpened pencil on its tip.We expect it will fall, at some time, in some unpredictable direction.In contrast, in the ammonia maser, a simple molecule can be poised in a state between its isoenergetic inversion isomers, and ammonia will predictably oscillate between the two isomers.Indeed, the oscillation is so predictable that it revolutionized precision timekeeping (1).The lesson is that while classical motions can be very sensitive to perturbations, quantum motions can be remarkably stable and insensitive to perturbation (Fig. 1A).In this issue of PNAS, Sadhasivam, Meuser, Reichman, and Althorpe show that a limit to quantum sensitivity previously proposed for black hole dynamics holds for a molecular barrier model as well (2).In the 19th century, the sensitivity of classical motion to perturbation became an intellectual starting point for statistical mechanics.Aleksandr Lyapunov suggested a way to quantify this sensitivity through the eponymous Lyapunov exponent λ L , which measures how rapidly slightly perturbed systems will diverge from each other.In a dilute gas, the Lyapunov exponent is inversely related to the mean free time between collisions.Since the world is quantum mechanical beneath its classical surface, how can we reconcile classical sensitivity with the clock-like behavior of quantum systems?One possibility is mere physical size, but quantum superposition has been observed in cold macroscopic objects.The key seems then to be not the physical size of a system, but rather some combination of the size of its accessible quantum state space (classically: phase space), its physical size, and how easy it is to hop from one part of the system's state space to another.For a sufficiently connected large state space, quantum motion becomes entangled or "scrambled" and a statistical and sensitive behavior emerges even from quantum mechanics.Classically there is no upper bound on the size of λ L because classical phase space has infinite room in it, so the proverbial "Butterfly Effect" of the weather changing at Columbia University depending on how a butterfly beats its wings over the Andaman Islands can be very efficient.In contrast, quantum chaos has turned out to be a subtle problem because the Schrödinger equation is a linear equation.Time evolution is just a linear superposition of eigenstates with different phases, and so it's all in the nature of those eigenstates-do they look "regular" or do they look scrambled?The puzzles of quantum scrambling have a history even older than modern quantum mechanics itself: in 1919, using Old Quantum Theory, Herzfeld proposed that bimolecular reactive collisions in gases could scramble atoms at a maximal rate of k B T/h (3), which would be slowed down only by a Boltzmann factor exp[-E † /k B T ], if an energy E † , the activation energy, was needed to rearrange the atoms.It was only in 1969 that out-of-time-order correlators, or OTOCs, were introduced by Larkin and Ovchinnikov to describe the