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Crossing the quantum-chaotic divide

February 21, 2017

Chaos is all around us, a fact that weather forecasters know all too well.

Their job is notoriously difficult because small changes in air pressure or temperature, which ultimately drive winds and weather systems, can have huge consequences on a global scale. This sensitivity to tiny differences is commonly called the butterfly effect, and it makes weather patterns chaotic and hard to predict.

Chaos pops up in many other places, too, and scientists have studied its role in physics for more than a century. But only since the 1980s have physicists investigated the connections between chaos and quantum mechanics—the most fundamental theory we have about the building blocks of the universe.

One wrinkle in studying quantum chaos is that quantum physics itself seems to forbid chaotic behavior. The rules that govern the quantum world are actually too simple to give rise to the same kind of unpredictability as the weather. This prompted researchers to examine the differences between ordinary chaotic systems and their quantum counterparts more closely, a task that has been stalled because scientists lack the mathematical tools to quantify chaos in a quantum setting.

Now, researchers from the Joint Quantum Institute (JQI) and the Condensed Matter Theory Center (CMTC) at the University of Maryland have used a promising diagnostic tool to characterize one of the simplest systems that physicists use to study chaos. This new diagnostic tracks the emergence of quantum interference effects and shows that they eventually destroy ordinary chaotic behavior. The work, performed by JQI and CMTC graduate student Efim Rozenbaum and two collaborators, was published online in Physical Review Letters on Feb. 21.

The researchers applied the new diagnostic to a standard example of chaos that has been studied for decades. It’s known as the kicked rotor—basically a spinning rod, fixed at its center, that gets flicked at regular intervals, like a coin kept spinning on a tabletop. For weak flicks, the rod’s orientation and rotation trace out smooth trajectories over time—slightly different initial orientations and speeds don’t alter the trajectories much. But for strong flicks, the system becomes chaotic, and small changes lead to very different trajectories. The images at the top of this story and below show the changes these trajectories undergo as the kicking strength increases.

"The important question has been how this picture manifests itself, or whether it even exists, in quantum mechanics," says JQI Fellow Victor Galitski, a coauthor of the new study.

One number—the Lyapunov exponent—captures just how chaotic the ordinary kicked rotor is. It’s bigger for stronger kicks, describing how quickly individual trajectories diverge. But quantum physics has no sharply defined trajectories because of a fundamental limit known as the Heisenberg uncertainty principle. This uncertainty infests all quantum systems and in the case of the kicked rotor prevents a perfect knowledge of the rotor’s initial orientation and rotation speed.

But Galitski and his collaborators learned of an intriguing quantity that had recently been used in another area of physics—the quantum behavior of black holes—that seemed to have all the features needed for a quantum analog of the Lyapunov exponent. They set out to calculate this new quantity for the simplest possible system.

The team studied the quantum kicked rotor at differing levels of “quantumness”—allowing them to adjust, numerically, the strength of the Heisenberg uncertainty principle. In one limit, there was no uncertainty at all, corresponding to the ordinary kicked rotor. The other limit corresponded to a fully quantum rotor.

The most striking finding was how consistently and quickly quantum effects destroyed chaos. Even for a rotor that was only weakly quantum, a handful of kicks caused quantum effects to win out and chaos to vanish. This transition was captured by a new diagnostic tool, closely related to the quantum Lyapunov exponent, which would grow quickly before ceasing the explosive growth suggestive of chaos. The team says that the time for this sharp cutoff corresponds to how long it takes for quantum interference effects to build up.

They hope to use this new diagnostic in other settings, like making connections between thermodynamics and the recently discovered quantum phenomenon of many-body localization.

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