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Competing triangular and stripe supersolid orders in a dipolar quantum gas

This study experimentally demonstrates the formation and transition between competing triangular and stripe supersolid orders in a two-dimensional dipolar quantum gas, characterizing their critical behavior and observing both phases in coherent and incoherent regimes to establish a versatile platform for investigating intertwined symmetry-breaking phenomena.

Original authors: Karthik Chandrashekara, Christian Gölzhäuser, Lily Platt, Jianshun Gao, Julian Kusch, Lennart Hoenen, Manon Ballu, Wyatt Kirkby, Lauriane Chomaz

Published 2026-08-21

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CC BY 4.0

Original authors: Karthik Chandrashekara, Christian Gölzhäuser, Lily Platt, Jianshun Gao, Julian Kusch, Lennart Hoenen, Manon Ballu, Wyatt Kirkby, Lauriane Chomaz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the coldest reaches of the universe, where atoms slow down enough to act as a single, unified entity, nature reveals states of matter that defy our everyday intuition. Among the most fascinating of these are supersolids, a paradoxical phase where a substance behaves like a solid crystal with a rigid, repeating pattern, yet simultaneously flows like a superfluid with zero friction. For decades, physicists have sought to create and understand these exotic materials, which combine the order of a lattice with the fluidity of a river. While scientists have successfully made supersolids in one-dimensional lines, the two-dimensional world promised an even richer landscape of possibilities, where different crystal shapes could compete and transform into one another. Understanding how these structures form and interact is not just a matter of curiosity; it offers a pristine laboratory to study how different types of order emerge and clash in complex systems, a phenomenon that underlies much of the behavior seen in everything from magnets to active biological matter.

A team of researchers at the University of Heidelberg has now stepped into this complex two-dimensional realm, successfully creating and distinguishing between two competing forms of supersolid order in a gas of highly magnetic atoms. By carefully manipulating the strength of the atoms' interactions and the angle at which their magnetic dipoles point, the scientists guided the gas through a transition from a uniform fluid into a state where the atoms arrange themselves into distinct patterns. They observed the gas spontaneously organizing into two different shapes: a triangular grid of droplets and parallel stripes. Crucially, they found that these structures could exist in two different states of motion. In some conditions, the atoms maintained a perfect, synchronized rhythm across the entire cloud, flowing without resistance while holding their shape. In other conditions, the atoms locked into place, losing their ability to flow and becoming a rigid, insulating crystal.

The experiment began with a cloud of roughly 140,000 dysprosium atoms, cooled to a temperature of just 62 nanokelvins, a state so cold that the atoms behave as a single quantum wave. These atoms were trapped in a shallow, surfboard-shaped container created by laser beams. To control the atoms, the researchers applied a magnetic field that they could tilt and strengthen. The strength of the field determined how strongly the atoms repelled each other, while the tilt angle changed the direction of their magnetic orientation. By slowly adjusting these two knobs over a period of 100 milliseconds and then holding the system steady for 30 milliseconds, the team allowed the gas to settle into its new, self-organized forms. They then took snapshots of the gas density, revealing a landscape that shifted dramatically depending on the settings.

When the magnetic dipoles were aligned straight up, the gas formed a triangular array of droplets, resembling a honeycomb of tiny islands. As the researchers tilted the magnetic field, the pattern began to stretch and distort. At intermediate angles, the system entered a chaotic, critical regime where the atoms could not decide whether to form triangles or stripes. In this zone, the patterns fluctuated wildly from one experiment to the next, showing signs of intense competition between the two structural orders. When the tilt was increased further, the triangular islands collapsed into long, parallel stripes running across the trap. This transition was not merely a visual change; the researchers developed a statistical method to measure the "order" of the patterns, confirming that the gas was indeed switching between two distinct structural phases.

Perhaps the most significant discovery was how these structures behaved when it came to flow. By releasing the trap and letting the gas expand, the researchers could see if the atoms were moving in unison. In the triangular and stripe phases near the beginning of the transition, the atoms interfered with each other to create clear, reproducible patterns, proving that the entire cloud was moving as a single, coherent fluid—a true supersolid. However, as the density modulation became stronger and the atoms packed tighter, this global coherence vanished. The interference patterns became messy and unpredictable, indicating that the atoms had lost their collective rhythm and were stuck in place, forming an insulating crystal. This allowed the team to map out a complete phase diagram where the same material could be a flowing supersolid or a rigid insulator, depending on how tightly it was packed and how the magnetic field was tilted.

The findings provide a versatile platform for exploring how different types of symmetry breaking intertwine. The researchers showed that by simply tilting a magnetic field, they could navigate a rich landscape of states, moving from a uniform fluid to a triangular supersolid, then to a stripe supersolid, and finally to an insulating crystal. The transition between the triangular and stripe shapes was marked by enhanced fluctuations, a signature that the system was struggling to choose between two competing orders. While the exact nature of the critical point between these phases remains a subject for further study, the experiment has firmly established that these competing orders exist and can be controlled. This work opens the door to investigating how superfluid properties interact with crystalline structures in two dimensions, offering new insights into the fundamental laws that govern complex quantum fluids.

Technical Summary: Competing Triangular and Stripe Supersolid Orders in a Dipolar Quantum Gas

Problem and Motivation
Supersolids represent exotic quantum states where long-range phase coherence coexists with emergent spatial order. While one-dimensional (1D) dipolar supersolids have been extensively characterized, two-dimensional (2D) systems are predicted to support a richer phase diagram featuring multiple competing crystalline arrangements, such as triangular arrays, stripes, and labyrinthine states. However, experimental observation of this structural variety and the transitions between them has remained limited. A primary challenge is isolating the mechanisms governing the competition between these spatial orders and understanding how they interplay with global phase coherence. Previous experiments have largely focused on dipoles aligned with the tight confinement axis; recent theoretical predictions suggest that tilting the dipoles toward a weakly confined direction could reshape the phase diagram, bringing competing orders into the experimentally accessible regime.

Methodology
The authors utilize a degenerate quantum gas of $1.4(1) \times 10^5$ $^{164}$ Dy atoms confined in an anisotropic "surfboard-shaped" optical dipole trap. The trap frequencies are approximately $[39(1), 20(2), 97(1)]$ Hz for the in-plane ( $x, y$ ) and axial ( $z$ ) directions, respectively.

The experimental protocol involves dynamically controlling two key parameters:

Contact Interaction Strength ( $a_s$ ): Tuned via a low-field Feshbach resonance by adjusting the magnetic field magnitude ( $B$ ) from 2.36 G to 2.59 G.
Dipole Orientation ( $\alpha$ ): Controlled by tilting the uniform magnetic field relative to the tight confinement axis ( $z$ ) within the $xz$ plane, varying the angle $\alpha$ from $0^\circ$ to $70^\circ$ .

The system is prepared in an unmodulated superfluid state. The parameters $B$ and $\alpha$ are ramped simultaneously over 100 ms, followed by a 30 ms hold time. The resulting states are characterized using two imaging techniques:

In-situ absorption imaging: To analyze the column density distribution $n(x, y)$ and extract structural order.
Time-of-flight (TOF) imaging: After 50 ms of free expansion, to probe global phase coherence via interference patterns.

Theoretical guidance is provided by temperature-dependent extended Gross-Pitaevskii equation (TeGPE) simulations, which include quantum and thermal fluctuation effects.

Key Contributions and Results

Observation of Competing Spatial Orders: By tuning $a_s$ and $\alpha$ , the authors experimentally realize distinct 2D density-modulated states.
- Triangular Arrays: Observed at low tilt angles ( $\alpha < 35^\circ$ ) and near the unmodulated-to-modulated transition.
- Stripe Patterns: Observed at high tilt angles ( $\alpha > 55^\circ$ ).
- Intermediate Regime: At intermediate angles ( $30^\circ \lesssim \alpha \lesssim 45^\circ$ ), the system exhibits a transition region where structures evolve from stripes to sheared triangular arrays and finally to staggered triangular arrays as $a_s$ decreases.
Structural Order Parameter and Critical Behavior: The authors define a structural order parameter, $R = S_2/S_1$ , based on the ratio of amplitudes of dominant peaks in the power spectrum of the density profiles.
- $R \approx 1$ corresponds to triangular order, while $R \approx 0$ indicates stripe order.
- Statistical analysis of $R$ across 30 independent realizations reveals enhanced variance and non-Gaussian fluctuations (non-zero higher-order cumulants) in the intermediate tilt regime ( $\alpha \approx 30^\circ - 45^\circ$ ). This behavior is identified as a signature of criticality associated with the structural phase transition between triangular and stripe orders.
Global Phase Coherence and Supersolidity: Using TOF interference patterns, the authors distinguish between phase-coherent supersolids and phase-incoherent insulating states.
- Supersolid Regime: Both triangular and stripe states exhibit reproducible interference patterns near the onset of density modulation, confirming global phase coherence.
- Insulating Regime: Farther from the modulation onset (higher density modulation contrast), interference patterns become irreproducible and vanish upon averaging, indicating a loss of global phase coherence and the emergence of an insulating state.
- Critical Regime: In the intermediate tilt region, interference patterns fluctuate shot-to-shot. The authors attribute this to strong structural fluctuations rather than a fundamental loss of phase coherence, though the quantitative interpretation is complicated by these fluctuations.

Significance
The paper establishes a versatile experimental platform capable of accessing multiple phases of the 2D supersolid phase diagram. By successfully identifying both triangular and stripe phases and the critical transition between them, the work demonstrates that dipole tilting is an effective control knob for reshaping the phase diagram of dipolar quantum gases.

The authors claim that their results provide a setting to investigate intertwined symmetry-breaking phenomena, specifically the interplay between global phase coherence and rearranging spatial order. The ability to observe these states in both phase-coherent (supersolid) and phase-incoherent (insulating) regimes offers a unique opportunity to study how superfluid properties intertwine with crystalline order. Furthermore, the system's rich landscape of transitions (structural and phase coherence) provides a foundation for future studies into the order and nature of these quantum phase transitions, potentially revealing distinct collective excitations and hydrodynamic responses associated with different crystalline arrangements.

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