Coexistence of long- and quasi-long range spatial order in 1D quantum quasicrystals

Imagine a world where things are perfectly organized, but not in a repeating pattern like a brick wall or a tiled floor. Instead, they follow a complex, non-repeating rhythm, like a piece of music that never quite loops back to the start but still feels harmonious. This is the realm of quasicrystals. In the physics of the very small, scientists study how atoms arrange themselves. Usually, atoms either flow freely like a liquid or lock into a rigid, repeating grid like a solid crystal. Quasicrystals are the cool, rebellious middle child: they have a strict long-range order, but they never repeat.

To understand how these structures behave, physicists look at "Goldstone modes." Think of these as the natural ways a system can wiggle or ripple without costing any energy. In a normal crystal, if you push the atoms, they bounce back in a specific way (like a sound wave). In a quasicrystal, there's an extra kind of wiggle called a "phason," which is like a continuous sliding or rearranging of the pattern itself. Usually, if you try to pin a quasicrystal down with an external force, you expect all these wiggles to stop or get "gapped" (meaning they need energy to happen). But what if you could stop one kind of wiggle while letting the other one run free? That's the puzzle this paper tackles, exploring a strange new state of matter where order and chaos coexist in a single, one-dimensional line.

The One-Dimensional Quasi-Party

In this study, the researchers set up a digital experiment to see what happens when you trap a line of ultracold atoms inside a special kind of "light cage" (an optical cavity). Imagine these atoms as a row of dancers on a stage. Usually, if you shine a light on them, they might all start dancing to the same beat, forming a perfect, repeating line. But here, the scientists used two different "beat generators."

One generator was a standing wave from the cavity mirrors, creating a rigid, repeating beat that the dancers had to follow. The other was a scanning laser beam, which added a second, slightly different rhythm that didn't match the first one perfectly. When you mix two rhythms that don't line up (incommensurate), you get a quasicrystal pattern. The dancers arrange themselves in a beautiful, non-repeating order.

The big question was: What happens to the dancers' ability to wiggle? In a normal quasicrystal, there are two types of wiggles: phonons (the whole line shuffling back and forth) and phasons (the pattern sliding or rearranging). The researchers wanted to know if they could "gag" one of these wiggles while leaving the other free to dance.

The Great Gapping

The team ran simulations using a mathematical model of these atoms. They found a unique scenario that only happens in quasicrystals. Because the cavity mirrors explicitly break the symmetry for one specific rhythm, the "wiggle" associated with that rhythm gets stuck. It acquires an energy "gap," meaning it can't move freely anymore. The dancers are pinned down for that specific part of the pattern.

However, the other rhythm, the one coming from the scanning laser, remains free. The "wiggle" associated with this second rhythm stays "gapless." It can still ripple and fluctuate without needing extra energy.

This leads to a bizarre and beautiful result: a mix of two different types of order existing at the same time.

1. The Pinned Order: For the part of the pattern locked to the cavity mirrors, the atoms have long-range order. They are perfectly aligned and stable, like a rigid crystal, because the "gap" stops them from wobbling apart.
2. The Floating Order: For the part of the pattern driven by the scanning laser, the atoms have quasi-long-range order. Because the "gapless" wiggle is still active, the atoms are constantly fluctuating. They stay correlated over long distances, but they aren't perfectly rigid. It's like a crowd doing "the wave"—the pattern travels across the stadium, but the individual people are still shifting and moving.

Why This Matters

The paper shows that in one dimension, you can have a state of matter that is half-crystal and half-fluid, all within the same structure. The authors demonstrate that by carefully designing the interaction between the atoms and the light (specifically using a laser beam profile that creates a specific "envelope" of interaction), you can selectively silence one type of Goldstone mode while keeping the other alive.

In their simulations, they observed that the density of the atoms forms a pattern where the peaks and valleys associated with the cavity wave are locked in place, while the peaks and valleys associated with the laser wave ripple and shift. The "ripples" in the free part of the pattern are strong enough to destroy true long-range order, leaving only the "quasi" version, while the pinned part remains perfectly ordered.

This isn't just a theoretical curiosity; the authors suggest this could be realized in real experiments with ultracold bosonic atoms in optical cavities. It's a new way to think about how matter organizes itself, proving that you don't have to choose between a rigid crystal and a wobbly fluid. In the strange world of 1D quantum quasicrystals, you can have the best of both worlds: a structure that is simultaneously locked in time and dancing in the moment.