🔬 condensed matter

Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism

This paper proposes and numerically demonstrates that spontaneous quantum turbulence, characterized by a proliferation of quantum vortices, can be generated in a newborn Bose-Einstein condensate via the Kibble-Zurek mechanism during a thermal quench, establishing its nonequilibrium universality through Kibble-Zurek and Kolmogorov scaling.

Original authors: Seong-Ho Shinn, Matteo Massaro, Mithun Thudiyangal, Adolfo del Campo

Published 2026-07-08

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CC BY 4.0

Original authors: Seong-Ho Shinn, Matteo Massaro, Mithun Thudiyangal, Adolfo del Campo

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

The Big Picture: Making a "Super-Fluid" and Watching It Spin

Imagine you have a pot of water. If you cool it down slowly, it freezes into a block of ice. But if you cool it down very quickly, it might freeze unevenly, creating cracks and bubbles.

This paper is about a similar process, but with a special kind of "super-fluid" called a Bose-Einstein Condensate (BEC). A BEC is a state of matter where atoms act like a single, giant wave rather than individual particles. It's so cold that it flows without any friction (superfluidity).

The researchers asked: What happens if we create this super-fluid very quickly? They found that the speed at which you "freeze" the atoms determines how much "turbulence" (chaotic spinning) is created inside it.

The Main Characters

The Kibble-Zurek Mechanism (The "Speed Limit" Rule):
Think of this as a rule about how fast a system can react. Imagine you are trying to organize a chaotic crowd of people into neat rows. If you shout "Line up!" very slowly, everyone has time to move perfectly into place. But if you shout it instantly, people will panic, bump into each other, and form messy little clusters.
In physics, this rule predicts that if you change the conditions of a system too fast, it can't keep up. It gets "frozen" in a messy state, creating defects (like holes or swirls) that shouldn't be there.
Quantum Vortices (The "Tiny Whirlpools"):
In a normal fluid, if you stir it, you get big swirls that eventually break into smaller ones. In a quantum super-fluid, the swirls are different. They are quantized, meaning they can only spin in specific, fixed amounts. They are like tiny, indestructible tornadoes. You can't have a "half" tornado; it's either there or it isn't.
Quantum Turbulence (The "Chaotic Dance"):
This is what happens when you have a huge number of these tiny tornadoes spinning around, bumping into each other, and creating a chaotic mess. It's like a crowded dance floor where everyone is spinning wildly.

What the Scientists Did

The team used a supercomputer to simulate this process. They didn't use real atoms; they used math to model how a cloud of atoms behaves when cooled down rapidly (a process called a "thermal quench").

The Experiment: They simulated cooling the atoms down at different speeds. Some were cooled slowly, others very fast.
The Result: When they cooled the atoms fast, the system couldn't organize itself perfectly. Instead, it spontaneously created a swarm of these tiny quantum tornadoes (vortices).
The Discovery: They found that the number of these tornadoes followed a precise mathematical rule (the Kibble-Zurek rule). The faster you cool the system, the more tornadoes appear, and they appear in a predictable pattern.

The "Turbulence" Connection

The most exciting part of the paper is that these tornadoes didn't just sit there; they created Quantum Turbulence.

In normal fluids (like water in a river), turbulence follows a famous rule discovered by a mathematician named Kolmogorov. It says that energy flows from big swirls to smaller swirls in a specific way.

The researchers found that even though this is a quantum system (which is usually very weird and different from normal water), the chaotic dance of these quantum tornadoes also followed Kolmogorov's rule.

The Analogy: Imagine a forest fire. Whether it's a small campfire or a massive wildfire, the way the flames flicker and spread follows certain patterns. The scientists found that the "flames" of this quantum turbulence followed the same universal pattern as a regular fire, even though the "fuel" was made of quantum atoms.

Why This Matters (According to the Paper)

The paper claims to have proven two main things:

Spontaneous Creation: You don't need to stir the fluid or poke it with a stick to create turbulence. If you just cool it down fast enough, the turbulence creates itself naturally.
Universal Rules: The chaos follows strict mathematical laws. The number of defects and the way energy moves through the system depend on how fast you cooled it, and these dependencies are universal (they work the same way regardless of the specific details of the experiment).

Summary in One Sentence

By simulating the rapid creation of a super-fluid, the scientists showed that the system naturally generates a chaotic storm of tiny quantum whirlpools that follow the same universal laws of turbulence found in everyday fluids, proving that the "speed of freezing" dictates the amount of chaos created.

Technical Summary: Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism

Problem Statement
Understanding turbulence remains a fundamental unsolved problem in physics. While classical turbulence is well-studied, Quantum Turbulence (QT)—characterized by a tangle of quantized vortices in superfluids—presents unique challenges due to the topological nature of its defects. A central question in nonequilibrium physics is whether QT can emerge spontaneously during a phase transition, rather than being induced by external driving forces (such as continuous energy pumping or sweeping potentials). Specifically, the authors investigate whether the Kibble-Zurek Mechanism (KZM), which predicts the spontaneous formation of topological defects during a continuous phase transition driven at a finite rate, can generate a state of Spontaneous Quantum Turbulence (SQT) in a newborn Bose-Einstein Condensate (BEC).

Methodology
The study employs numerical simulations of the two-dimensional Stochastic Projected Gross-Pitaevskii Equation (SPGPE). This formalism models the non-equilibrium dynamics of a Bose gas by separating modes into a coherent region (low-energy, highly populated) and an incoherent thermal reservoir.

System Setup: The authors simulate a periodic, homogeneous 2D system ( $V(\mathbf{r})=0$ ) using a complex order parameter $\Psi_C$ .
Quench Protocol: A linear thermal quench is applied by ramping the chemical potential $\mu(t)$ from an initial value $\mu_i$ to a final value $\mu_f$ over a duration $\tau_Q$ . This drives the system through the critical point $\mu_c$ at a finite rate.
Dynamics: The evolution includes damping ( $\gamma$ ) modeling energy dissipation and complex Gaussian noise ( $d\eta$ ) accounting for thermal fluctuations.
Analysis: The authors analyze the system at the equilibration time ( $t_{eq}$ $t_{e q}$ ), defined by the transition from exponential to linear growth of the condensate population. Key diagnostics include:
- Vortex counting and topological charge identification.
- Decomposition of the density-weighted velocity field $\mathbf{u} = |\Psi_C|\mathbf{v}$ into compressible (irrotational) and incompressible (solenoidal) components.
- Calculation of kinetic energy spectra ( $E_i(k)$ and $E_c(k)$ ) and velocity structure functions to test for Kolmogorov scaling.

Key Contributions and Results

Spontaneous Generation of SQT: The simulations demonstrate that a thermal quench induces the spontaneous formation of a vortex-proliferated BEC. The vortex density ( $n_v$ ) and the equilibration time ( $t_{eq}$ ) follow the universal scaling laws predicted by the KZM:
- $t_{eq} \propto \tau_Q^{0.54 \pm 0.07}$ (consistent with the theoretical prediction $\tau_Q^{z\nu/(1+z\nu)}$ for mean-field exponents $\nu=1/2, z=2$ ).
- $n_v \propto \tau_Q^{-0.53 \pm 0.05}$ (consistent with the predicted $\tau_Q^{-2\nu/(1+z\nu)}$ ).
- Only vortices with topological charge $w = \pm 1$ are observed; higher winding numbers are energetically unstable.
Kolmogorov Scaling in the Incompressible Spectrum: The incompressible kinetic energy spectrum $E_i(k)$ exhibits a power-law scaling $E_i(k) \propto k^{-\alpha}$ within an inertial range.
- The fitted exponent $\alpha \approx 1.65\text{--}1.71$ is in agreement with the Kolmogorov $k^{-5/3}$ scaling ( $5/3 \approx 1.67$ ).
- The inertial range is bounded by the mean intervortex distance (energy injection scale) and the vortex core size (dissipation scale).
- The compressible spectrum $E_c(k)$ follows expected behaviors: $E_c(k) \propto k$ at low $k$ (dominated by thermal fluctuations at the simulation temperature) and $E_c(k) \propto k^{-3}$ at high $k$ (reflecting vortex core structure).
Universality and Data Collapse: The authors establish the nonequilibrium universality of SQT. By rescaling the momentum $k$ and the energy spectrum $E_i(k)$ using the KZ correlation length $\hat{\xi} \propto \tau_Q^{\nu/(1+z\nu)}$ and the associated energy scale, the spectra for different quench times ( $\tau_Q$ ) collapse onto a single universal curve. This confirms that the turbulent state is governed by the universal critical dynamics of the phase transition.
Extended Self-Similarity (ESS): Due to the limited separation of length scales in 2D BECs (resulting in a narrow inertial range), the authors utilize velocity structure functions $S_p(r)$ . The analysis reveals extended self-similarity, where $S_p(r) \propto [S_3(r)]^{\zeta(p)}$ . The measured exponents $\zeta(p)$ deviate from the standard K41 prediction ( $p/3$ ) but align well with the refined K62 model, indicating intermittency corrections similar to those found in classical and other quantum turbulence systems.

Significance
The paper claims to establish the existence of Spontaneous Quantum Turbulence generated purely by the Kibble-Zurek mechanism during the formation of a BEC. Its primary significance lies in bridging two distinct notions of universality:

The universal scaling of energy spectra in turbulent flows (Kolmogorov scaling).
The universal defect dynamics across continuous phase transitions (KZM).

By demonstrating that the Kolmogorov scaling of the incompressible kinetic energy admits a universal collapse when rescaled by KZM parameters, the work provides a unified framework for understanding how turbulence emerges spontaneously from symmetry breaking. The study highlights that while the KZM accurately describes the initial defect formation and the onset of turbulence, subsequent non-universal processes (such as vortex-antivortex annihilation and coarsening) eventually suppress the scaling, particularly for slow quenches where the system evolves nearly adiabatically. The findings offer a controlled theoretical platform for exploring the interplay between critical dynamics and hydrodynamic turbulence in quantum fluids.