Physicists investigate dynamic phenomena of a time crystal

Physicists at TU Dortmund University have periodically driven a time crystal and discovered a remarkable variety of nonlinear dynamic phenomena, ranging from perfect synchronization to chaotic behavior within a single semiconductor structure. The team has now published its latest findings in the journal Nature Communications.

For their current research, Dr. Alex Greilich’s team from the Department of Physics utilized a highly robust time crystal, previously introduced in Nature Physics last year. The crystal, made of indium gallium arsenide, was continuously illuminated with a laser during the initial experiment. This interaction caused a nuclear spin polarization, which in turn spontaneously generated oscillations, embodying the essence of a time crystal through periodic behavior under constant excitation.

In the newly published follow-up study, the team explored the dynamic phases of the time crystal. They illuminated the semiconductor periodically instead of continuously, while also varying the frequency of the periodic drive. The observed behavior of the time crystal, its frequency response, ranged from perfect synchronization to chaotic dynamics.

A diagram reveals these dynamic phenomena clearly: the visible plateaus indicate that the system’s frequency response is strictly bound to the drive frequency. However, synchronization occurs only at specific fractions of the system’s natural frequency. These fractions, in order of appearance with increasing drive frequency, correspond to the “Farey tree sequence,” a well-known hierarchical structure implemented in a crystal for the first time.

If the driving frequency is varied further, the end of the synchronization range is reached. Here, each frequency component splits into at least two branches that are symmetrical to the synchronization frequency. These frequency branches connect the synchronization plateaus and together form a kind of staircase, known in the literature as “the devil’s staircase,” indicating a path either upwards or downwards.

Both the step height and width decrease with each step. This branching leads to multiple staircases of varying steepness, which eventually converge, resulting in chaotic motion. Chaos here does not mean that the motion becomes entirely unpredictable but rather that the slightest changes can lead to completely different forms of motion. If the driving frequency is altered even further, a threshold is crossed beyond which the chaos collapses, and the motion becomes regular and periodic again.

“For the first time, all these observations have been made in a semiconductor. They represent a significant step toward a comprehensive understanding of nonlinear systems,” says Dr. Alex Greilich.

In the future, his team will continue researching how complex dynamic states in nonlinear systems arise and evolve under external periodic driving. These fundamental research findings could help tailor the properties of semiconductors, which are essential for modern electronics.

Nonlinear systems are also ubiquitous in biology. For instance, in phenomena such as heartbeats, the organized flight of birds or the chirping of crickets.