The word “chaos” usually causes a lack of order: a busy day, a teenage bedroom, a tax season. And a physical understanding of chaos is not far off. It’s something that’s extremely hard to predict, like the weather. Chaos allows a small jerk (fluttering of a butterfly’s wings) to grow into a major consequence (a half-world typhoon), which explains why weather forecasts longer than a few days in the future can be unreliable. The individual molecules of air, which are constantly bouncing, are also chaotic – it is almost impossible to determine where any single molecule could be at any given time.

Now you may be wondering why anyone would be interested in the precise location of a particular air molecule. But you might care about the properties shared by a whole bunch of molecules, like their temperature. Perhaps unintuitively, the chaotic nature of the molecules allows them to fill a room and reach a single temperature. Individual chaos ultimately leads to collective order.

Being able to use a single number (temperature) to describe a bunch of particles bouncing in some crazy, unpredictable way is extremely convenient, but it doesn’t always happen. Thus, a team of theoretical physicists from JQI set out to understand when this description applies.

“The ambitious goal here is to understand how the chaos and universal tendency of most physical systems to achieve thermal equilibrium stems from the fundamental laws of physics,” says JQI associate Victor Galitski, who is also a professor of physics at the University of Maryland (UMD)).

As a first step toward this ambitious goal, Galitski and two colleagues set out to understand what happens when many particles come together, each of which is chaotic. For example, the movement of one puck in an air hockey game, which is constantly bouncing off walls, is chaotic. But what happens when a lot of these pucks are released on the table? And further, what would happen if pucks obeyed the rules of quantum physics?

In a paper published recently in the journal Physical Review Letters, the team studied this problem of air hockey in the quantum empire. They found that the quantum version of the problem (where in fact quantum particles are like atoms or electrons) was neither ordered nor chaotic, but somewhat both, according to one common way of measuring chaos. Their theory was general enough to describe a series of physical settings, including molecules in a tank, a game of quantum air hockey, and electrons bouncing in a disordered metal, like a copper wire in your laptop.

“We always thought it was a problem that had been solved in some textbooks a long time ago,” says Yunxiang Liao, JQI’s postdoctoral fellow and first author in the paper. “It turns out it’s a harder problem than we imagined, but the results are also more interesting than we imagined.”

One of the reasons why this problem has been unresolved for so long is that when quantum mechanics enters the picture, the usual definitions of chaos do not apply. Classically, the butterfly effect — tiny changes in initial conditions that cause drastic changes down the line — is often used as a definition. But in quantum mechanics, the very notion of initial or final position doesn’t really make sense. The uncertainty principle says that the position and velocity of a quantum particle cannot be precisely known at the same time. Thus, the trajectory of the particles is not well defined, which makes it impossible to monitor how different initial conditions lead to different outcomes.

One of the tactics for studying quantum chaos is to take something classically chaotic, like a puck bouncing around an air hockey table, and handle it quantum mechanically. Certainly classic chaos should be translated. And indeed, there is. But when you insert more than one quantum puck, things become less clear.

Classically, if they can bounce off each other, exchanging energy, they will all eventually reach one temperature, exposing the collective order of basic chaos. But if they do not run into each other, and instead pass like ghosts, their energies will never change: the hot will stay hot, the cold will stay cold, and they will never reach the same temperature. Since, on the other hand, there is no interaction, the collective order cannot arise from chaos.

The team introduced this game of air hockey to the quantum-mechanical realm, expecting the same behavior – chaos for one quantum particle, but not a common order when there are many of them. To test this hunch, they chose one of the oldest and most commonly used (though not the most intuitive) tests of quantum chaos.

Quantum particles can’t just have energy, the levels available are ‘quantized’, which basically means they are limited to certain values. As far back as the 1970s, physicists discovered that if quantum particles behaved in a predictable way, their energy levels were completely independent of each other — possible values did not tend to shrink or expand on average. But if the quantum particles were chaotic, the energy levels seemed to avoid each other, spreading in characteristic ways. This repulsion of energy levels is often used today as one of the definitions of quantum chaos.

Since their hockey pucks didn’t interact, Liao and her co-workers didn’t expect to agree on a temperature, meaning they won’t see any signs of basic chaos in one puck. They thought energy levels would not care for each other at all.

Not only did they find theoretical evidence of the rejection of some level, a feature of quantum chaos, but they also discovered that some levels came together rather than refused, a new phenomenon they could not explain. It turned out that this deceptively simple problem was neither arranged nor chaotic, but some unusual combination of the two that had not been seen before.

The team was able to discover this hybrid using an innovative mathematical approach. “In previous numerical studies, researchers were able to include only 20 or 30 particles,” says Liao. “But using our mathematical approach from random matrix theory, we could include about 500. And this approach also allows us to compute analytical behavior for a very large system.”

Armed with this mathematical framework and aroused interest, researchers are now expanding their calculations to gradually allow hockey tiles to interact little by little. “Our preliminary results show that thermalization can occur by spontaneously breaking reversibility – the past is mathematically different from the future,” says Galitski. “We see that small disturbances are increasing exponentially and destroying all the remaining signatures of the order. But this is a different story.”

The equation reveals the characteristics of quantum chaos

**More information:**

Yunxiang Liao et al. Statistics at the level of the quantum chaos particle body,

*Physical review letters*(2020). DOI: 10.1103 / PhysRevLett.125.250601

Provided by the Joint Quantum Institute

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