Generation of wave turbulence in dipolar gases driven across their phase transitions

By dynamically driving a dysprosium dipolar Bose-Einstein condensate across the supersolid-superfluid phase transition, researchers discovered that the resulting robust nonequilibrium state exhibits self-similar wave turbulence, a phenomenon significantly enhanced by the supersolid's ability to sustain higher momenta associated with the roton minimum.

The Gist

Imagine you have a giant, invisible bowl filled with a special kind of "super-cold" gas. This isn't just any gas; it's made of atoms that act like tiny magnets (dipolar atoms). Because they are magnets, they don't just bump into each other like billiard balls; they pull and push on each other from a distance, creating a complex dance.

In this paper, scientists are playing with this gas to see what happens when they force it to change its "personality" or state of matter. Specifically, they are watching it switch between two exotic states:

1. Superfluid: A state where the gas flows like a frictionless liquid, with no resistance at all.
2. Supersolid: A weird, magical state where the gas is both a solid (it has a rigid crystal structure, like a honeycomb) and a superfluid (it can flow through itself) at the same time.

The Experiment: Shaking the Bowl

The researchers didn't just let the gas sit there. They "shook" the bowl by rapidly changing the strength of the interactions between the atoms. Imagine you have a bowl of Jell-O. If you gently wiggle it, it jiggles. If you shake it violently, it splashes and breaks apart.

In this experiment, they shook the gas back and forth across the line where it changes from a Supersolid to a Superfluid (and vice versa). They wanted to see how the gas reacted to this chaotic shaking.

The Discovery: A Cosmic Storm of Waves

When they shook the gas, they didn't just see it slosh around. They saw something much more profound: Wave Turbulence.

Think of turbulence like a storm in the ocean. You have big waves crashing, which break into smaller waves, which break into even smaller ripples, until the energy is spread out everywhere.

• The "Big Waves": These are the large, organized movements of the gas.
• The "Ripples": As the gas gets shaken, the energy cascades down, creating a chaotic mix of tiny, high-speed waves.

The scientists found that no matter how they started the experiment (whether the gas was a solid crystal or a flowing liquid) or how fast they shook it, the gas eventually settled into a predictable, chaotic pattern. This pattern is called a "quasi-steady state." It's like a storm that never stops, but the way the waves behave follows a strict mathematical rule.

The "Magic" Ingredient: The Supersolid

Here is the most interesting part. The researchers discovered that starting with the Supersolid (the crystal-like state) made the turbulence happen faster and stronger.

The Analogy:
Imagine you are trying to break a block of ice into sand.

• Scenario A (Superfluid): You have a smooth, slippery sheet of water. If you hit it, it just ripples. It takes a while for the energy to spread out into tiny droplets.
• Scenario B (Supersolid): You have a block of ice with a honeycomb pattern already carved into it. If you hit this, the cracks in the honeycomb act as weak points. The energy travels through these cracks instantly, shattering the block into sand much faster.

The "honeycomb" in the Supersolid gas is called a roton minimum. It's a built-in weakness in the structure that allows the energy to spread out into high-speed waves much more efficiently. The Supersolid acts like a pre-charged battery for turbulence.

Why Does This Matter?

You might ask, "Why do we care about shaking cold atoms?"

1. Universal Laws: The scientists found that the rules governing this chaotic gas are the same as the rules governing ocean waves, light in fiber optics, and even the plasma in stars. By studying this tiny, controlled gas, we can understand the universal laws of chaos that apply to the entire universe.
2. New Materials: Understanding how these exotic states (like Supersolids) behave under stress helps us design new materials in the future.
3. Real-World Application: The study showed that even if the gas loses some atoms (like water evaporating from a cup), the turbulence still happens. This means real-world experiments in labs can actually see this phenomenon, not just computer simulations.

The Bottom Line

This paper is like watching a controlled explosion of order into chaos. The scientists took a super-cold gas, shook it until it forgot how to be a solid or a liquid, and watched it turn into a self-sustaining storm of waves. They discovered that if the gas starts as a "crystal," it turns into a storm much faster, revealing a hidden shortcut in the laws of physics that governs how energy moves through the universe.

Technical Summary

1. Problem Statement

Turbulence is a ubiquitous hydrodynamic phenomenon characterized by energy cascades across length scales. While vortex turbulence in superfluids has been extensively studied, wave turbulence (mediated by nonlinear wave mixing) in systems with long-range anisotropic interactions remains largely unexplored. Specifically, the impact of exotic quantum phases, such as supersolids (SS)—which exhibit both crystalline rigidity and superfluidity—on nonequilibrium turbulent dynamics is unknown.

The authors address the following questions:

• Can wave turbulence be generated in dipolar Bose-Einstein condensates (BECs) when dynamically driven across phase transitions (e.g., Supersolid $\leftrightarrow$ Superfluid)?
• How do the unique properties of the supersolid phase (specifically the roton minimum and extended momentum distribution) influence the onset and characteristics of turbulence compared to standard superfluids?
• Is the resulting turbulent state universal and robust against experimental imperfections like three-body losses?

2. Methodology

The study employs a theoretical framework based on numerical simulations of the extended 3D Gross-Pitaevskii equation (GPE) for a dipolar gas of $^{164}\text{Dy}$ atoms.

• System Setup:

  • Atoms: $8 \times 10^4$ Dysprosium atoms polarized along the $z$-axis.
  • Trap: Cylindrically symmetric harmonic trap with frequencies $(\omega_x, \omega_y, \omega_z) = 2\pi \times (43, 43, 133)$ Hz, creating a quasi-2D geometry.
  • Interactions: Includes short-range contact interactions, long-range dipolar interactions ($U_{dd}$), and the Lee-Huang-Yang (LHY) correction term to account for quantum fluctuations stabilizing the supersolid.
  • Equation: The dynamics are governed by:
    $$i\hbar\partial_t\Psi = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(r) + \frac{4\pi\hbar^2 a}{m}|\Psi|^2 + \int U_{dd}|\Psi'|^2 + f(\epsilon_{dd})|\Psi|^3 \right]\Psi$$
    where the last term represents the beyond-mean-field LY correction.

• Driving Protocol:

  • The system is initialized in either a Supersolid (SS) or Superfluid (SF) phase.
  • The 3D scattering length $a(t)$ is periodically modulated: $a(t) = a_i + (a_f - a_i)\sin^2(\omega_d t)$.
  • This modulation drives the system dynamically across the phase boundary (e.g., from SS to SF or vice versa), injecting energy continuously.

• Analysis Techniques:

  • Momentum Distribution: Analysis of the 2D transverse momentum distribution $n(k, t)$ to identify power-law scaling.
  • Scaling Exponents: Fitting the high-momentum tails to $n(k) \sim k^{-\gamma}$ to identify the turbulence regime.
  • Cascade Front: Tracking the propagation of the cascade front $|k_{cf}|$ to determine the energy transfer rate ($\sim t^{-\beta}$).
  • Robustness Checks: Simulations including three-body losses (imaginary term in GPE) and phenomenological damping (viscosity) to test experimental feasibility.

3. Key Contributions

1. Discovery of Robust Nonequilibrium Quasi-Steady State: The authors demonstrate that driving dipolar gases across phase transitions leads to a self-similar, nonequilibrium quasi-steady state characterized by algebraic decay in momentum distributions.
2. Identification of Wave Turbulence Signatures: The emergence of specific scaling exponents ($\gamma \approx 2.60$, $\beta \approx -0.63$) supports the existence of weak wave turbulence, distinct from vortex turbulence.
3. Role of Supersolidity: The study reveals that the supersolid phase accelerates the onset of turbulence. The extended momentum distribution associated with the roton minimum in the SS phase provides a "head start" for the energy cascade compared to the superfluid phase.
4. Universality: The turbulent behavior is shown to be independent of the initial state (SS or SF), driving frequency, and the specific direction of the phase transition.
5. Experimental Feasibility: The results hold even in the presence of significant three-body losses (up to 40% atom loss) and dissipation, confirming that wave turbulence is observable in current experimental setups.

4. Key Results

• Scaling Exponents:

  • The momentum distribution at large wavenumbers follows a power law $n(k) \sim k^{-\gamma}$.
  • The extracted exponent is $\gamma \approx 2.50 - 2.64$. While the theoretical prediction for 2D weak wave turbulence is $\gamma \approx 2$, the deviation is attributed to energy transfer in the axial ($z$) direction and the quasi-2D nature of the trap.
  • The cascade front grows as $|k_{cf}| \sim t^{-\beta}$ with $\beta \approx -0.63$, consistent with theoretical predictions for direct energy cascades in systems with quadratic dispersion relations.

• Supersolid vs. Superfluid Dynamics:

  • Faster Onset: When starting from a Supersolid, the system reaches the turbulent quasi-steady state faster than when starting from a Superfluid. This is because the SS phase already possesses high-momentum peaks (due to the roton minimum), allowing the cascade front to start at a higher $k$ value.
  • Isotropy: Despite the anisotropic nature of dipolar interactions, the momentum distribution becomes statistically isotropic at large $k$ in the long-time limit.

• Energy Contributions:

  • During the transition, kinetic energy increases significantly, indicating the transfer of energy to smaller scales (high momenta).
  • The LHY correction term is significant initially (stabilizing the SS) but becomes negligible as the crystals melt and the system enters the turbulent regime.

• Impact of Losses and Dissipation:

  • Three-body losses: Including a recombination rate ($L_3$) reduces the total atom number but does not prevent the emergence of wave turbulence. The scaling exponent $\gamma$ remains close to 2.5, though the high-momentum tails decay more steeply.
  • Viscosity: Phenomenological damping suppresses high-momentum excitations, leading to larger values of $\gamma$ and preventing the formation of the turbulent cascade if the damping is too strong.

5. Significance

• Fundamental Physics: This work provides the first theoretical evidence of wave turbulence in dipolar quantum gases driven across phase transitions. It bridges the gap between exotic quantum phases (supersolids) and universal turbulent behavior.
• Experimental Guidance: The identification of specific scaling exponents ($\gamma \approx 2.6$, $\beta \approx -0.63$) offers a clear experimental signature for detecting wave turbulence in dipolar gases (e.g., Dysprosium or Erbium) using time-of-flight absorption imaging.
• Control of Turbulence: The findings suggest that the "exotic" nature of the initial state (specifically the roton minimum) can be used as a control knob to engineer the onset and characteristics of turbulence.
• Future Directions: The study opens avenues for exploring inverse energy cascades, non-conventional anisotropic turbulence (e.g., with rotating fields), and the role of dimensionality in long-range interacting quantum systems.

In summary, the paper establishes that dipolar supersolids are fertile platforms for generating and studying wave turbulence, with the unique momentum structure of the supersolid phase facilitating a rapid transition to a universal nonequilibrium steady state.