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The Question of Quantum Chaos

What We Know / by Veronique Greenwood / December 14, 2009

Chaos is everywhere in the natural world, present in the coiling of smoke rings, the fronds of ferns, and the beating of our hearts. But at the level of quantum physics, chaos as we now define it is unquantifiable.

Photo courtesy of George Loannaidis

Working in a cramped MIT laboratory in 1961, meteorologist Edward Lorenz stumbled upon a new science. Wanting a closer look at the data of a weather simulation he was running, Lorenz restarted it in the middle. Within a few minutes, everything changed and the data he had expected to see had morphed into strange new patterns. A stunned Lorenz checked his inputs. He had rounded the starting values by about .0001, which should have been insignificant. And yet it was not.

At the time, scientists thought small changes in starting values should make only a small difference in most systems. But sometimes such tiny shifts will cause a very different outcome, completely out of proportion with the size of the change—this hypersensitivity to initial conditions is what Lorenz dubbed the “butterfly effect” and what we now call chaos.

Chaos, which underlies systems as diverse as fractals, ferns, and weather, causes behavior so complex and unexpected that, though it is fundamental to the natural order, it was only recently that scientists began to characterize it. Chaotic movement is unstable and unpredictable, but completely deterministic, meaning that it’s controlled by its starting conditions.

A scant five decades after Lorenz’s seminal experiment, chaos is informing the new study of resilience and complexity. Chaos has been observed at nearly every level of the natural world, from the movement of the planets to the patterns of wind to the beating of the human heart. In fact, almost everything in nature is chaotic.

But at the level of atoms, our definition of chaos has run into a problem.

Chaos is usually defined by a system’s movement: Set a pendulum swinging, track exactly where it goes, and its motion will reveal whether it is chaotic. Atoms, however, are governed by the uncertainty principle, which means that their location cannot be known precisely. What’s more, the laws of quantum mechanics say that hypersensitivity to initial conditions, which is considered the primary characteristic of a chaotic system, is physically impossible for atoms—at least in the way it’s understood at the classical level.
This presents a serious quandary because quantum mechanics is considered the most basic set of universal laws. Chaos must have some connection with the quantum level, but how it manifests itself, or how to quantify it, has thus far eluded physicists. Work published recently in Nature helps shed light on this problem as researchers working with cooled atoms searched for what they call signatures of chaos.

If such hypersensitivity to initial conditions cannot happen in a quantum system, other red flags of classical chaos might still be detectable. This could indicate that chaos in some form could exist at the level of atoms, or, at the very least, would imply a connection between quantum events and classical chaos. “Though you will never be able to find hypersensitivity to initial conditions in the quantum system, you are able to tell if the outward signs produced by classical chaotic systems are the same in quantum systems,” says Poul Jessen of Arizona State University, the lead researcher on the Nature paper.

In order to see these signatures, physicists have taken the conditions that cause chaotic behavior in human-scale systems and applied them on the atomic level. Jessen and his collaborators recently succeeded in making a quantum “kicked top” out of cesium atoms for the first time.

Kicked tops are an excellent example of chaotic systems when it comes to classic physics. You start an object twirling—say, a gyroscope—and then give it a series of kicks and twists as it spins. The initial condition that decides whether a gyroscope moves stably or chaotically is the direction of its axis when it starts spinning.

In order to visualize the gyroscope’s behavior, the different values of its angular momentum are plotted on the surface of a globe. Some initial orientations of its axis cause the momentum to swerve in a “chaotic sea,” covering most of the surface of the globe. But other orientations cause the spin to settle into stable, regular motion in one of three main “islands” in the sea.

In their experiment, researchers substituted atoms for gyroscopes and looked at how angular momenta affected the atoms’ quantum states. What they found was intriguing: Some spins of the quantum top locked the atoms into a stable set of islands, while other values let the atoms’ quantum states wander erratically.

The number and location of the islands, when plotted, corresponded eerily to the classical model. So while the atoms’ behavior could not technically be called chaotic because they cannot show hypersensitivity, they mimicked the evolution of the classical, chaotic system almost exactly. Other measurements indicated that the system might have some sensitivity to disturbances, another interesting link to chaotic behavior.

These observations alone provided good evidence that something related to chaos was happening. But the most fascinating result was that one of the strangest properties of atoms, entanglement, shot up in areas corresponding to the chaotic sea. When two quantum-scale objects, like atoms or nuclei, are entangled, performing an action on one instantaneously affects the other even if vast distances separate the entangled objects. Einstein famously called entanglement “spooky action at a distance,” and it forms the basis of modern attempts to built quantum computers.

Could entanglement be a signature of chaos? Jessen and his collaborators think so. According to Jessen’s coauthor Shohini Ghose of Wilfrid Laurier University, theoretical papers have discussed potential links between entanglement and chaos for some time, but this is the first experiment to demonstrate the relationship. “[The emergence of entanglement as a signature of chaos] “is very interesting to those who want to have a lot of entanglement in a system—it’s the fuel that helps with quantum computing. So if chaos helps to increase entanglement, that’s a good thing,” Ghose says. “But it also makes it harder to predict.” Jessen adds, “ Chaos in entanglement could be why quantum computing is so difficult.”

Taken together, the presence of these signatures indicates a link between the quantum system’s behavior and classical chaos. For Jessen, this result is a first step on a long but exciting journey, a trip he shares with many other physicists who study the quantum-classical boundary. “We would like to show how classical chaos can emerge from quantum physics,” he says. “We would like to understand the transition from the quantum world to the much larger world where quantum systems behave almost classically.”

But quantifying the quantum-classical linkage is sticky issue. Hypersensitivity to starting conditions defines chaos on the classical scale, but in order to explain the presence of chaos-like behavior on the quantum level, we must turn to something that underlies hypersensitivity—for instance, the way a system interacts with its environment, or the forces that are exerted on it.

“It really is a problem of definition,” Ghose says. “This is what we as a community ultimately have to pin down: what we mean by quantum chaos.”

What this experiment has shown is that the actions that cause a difference in a system’s behavior on a classical level—a difference we would call chaos—cause a difference in a quantum system’s behavior as well. According to Ghose, that indicates that some underlying property is at the roots of chaos, a property that can produce hypersensitivity but whose definition shouldn’t be limited to it. “The smell of a rose is a property of a rose, but it comes from something deeper, from the chemicals in a rose. To me, sensitivity is a property of chaos, but there is some deeper, underlying reason why that is a property. And that underlying reason is really what connects the quantum and classical worlds,” Ghose says. “That tells me that when we look at classical chaos we should look at those underlying properties more. Hypersensitivity is a consequence of these constraints, or lack of constraints, on motion.”

The next step for the team is undertaking similar experiments with larger spins, which will bring them closer to the classical level. Where the two regimes intersect, what they will find? Perhaps another breadcrumb in the trail toward what connects the classical and the quantum—and perhaps another chance to probe the nature of chaos.

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