Hydrodynamic attractor in periodically driven ultracold quantum gases

This paper demonstrates that ultracold quantum gases with externally modulated scattering lengths exhibit a novel cyclic hydrodynamic attractor during oscillating isotropic expansion, extending the concept of hydrodynamic attractors beyond the previously studied monotonic expansion scenarios.

The Gist

Imagine you are watching a crowd of people in a large room. If you suddenly push them all to one side, they will scramble, bump into each other, and eventually settle into a calm, uniform distribution. Physicists call this "equilibrium."

For decades, scientists have studied how systems like this settle down. They found that even before the crowd is perfectly calm, they start moving in a predictable, fluid-like way. This early, predictable movement is called a Hydrodynamic Attractor. Think of it like a "magnetic track" that the system falls onto, regardless of whether you pushed the crowd from the left, the right, or the center. Once on the track, the crowd behaves the same way.

However, until now, scientists only studied this "track" in systems that were expanding forever (like a balloon inflating and never stopping). This paper asks a new question: What happens if you shake the system back and forth?

Here is the story of what the authors discovered, explained simply:

1. The Experiment: A Quantum Trampoline

The authors are studying ultracold quantum gases. Imagine a cloud of atoms cooled down to almost absolute zero, where they behave like a single, super-fluid wave rather than individual particles.

Usually, to study how these gases flow, scientists let them expand. But here, the researchers propose a new trick: periodic driving.

• The Analogy: Imagine the gas is a trampoline. Instead of just letting the trampoline sag and stay down (expansion), you jump on it rhythmically, making it bounce up and down (expansion and contraction) over and over again.
• The Tool: They do this by magically changing how the atoms "bump" into each other (the scattering length) using magnetic fields. It's like changing the stiffness of the trampoline springs in rhythm with your jumps.

2. The Discovery: A New Kind of Track

When the gas expands and contracts rhythmically, the authors found that it doesn't just settle into the old, predictable "track" (which they call the Navier-Stokes limit). Instead, it finds a new, cyclic track.

• The Old Way (Monotonic Expansion): Imagine a ball rolling down a hill. It eventually stops at the bottom. The "attractor" is the bottom of the hill.
• The New Way (Periodic Driving): Imagine a ball rolling on a washboard road that goes up and down. The ball doesn't stop at the bottom; it settles into a rhythm, bouncing up and down in a specific, repeating loop.
• The Result: No matter how you start the ball (pushing it hard or gently), it quickly forgets your initial push and settles into this specific "bouncing loop." This loop is the Cyclic Attractor.

3. Why the Old Rules Don't Work

In the past, scientists used a simple rule (Navier-Stokes hydrodynamics) to predict how fluids behave. It's like using a simple map for a flat city.

• The Problem: When the gas is being shaken back and forth, the simple map fails. The gas moves in a complex, elliptical loop that the simple map can't predict.
• The Insight: The authors used a more advanced theory (Müller-Israel-Stewart) to predict the shape of this new loop. They found that the gas creates a "lag" or a "phase shift." It's like trying to dance to a fast song; your body can't keep up perfectly with the beat, so you develop a specific, slightly delayed rhythm. This delay creates the unique shape of the new attractor.

4. The Nonlinear Twist: The Drifting Loop

When the shaking gets very strong (large amplitude), things get even more interesting.

• The Analogy: Imagine you are jumping on that trampoline very hard. You aren't just bouncing; you are getting hotter and hotter because of the friction and energy you are putting in.
• The Result: The "loop" the gas follows doesn't close perfectly. It slowly drifts or spirals because the gas is heating up and changing its properties as it moves. Even though the loop is drifting, it is still an "attractor" because all different starting points still converge onto this same drifting path.

5. Why This Matters

This is a big deal for two reasons:

1. It's Testable: Unlike high-energy nuclear collisions (where particles smash together at the speed of light and disappear in a fraction of a second), ultracold atoms in a lab move slowly. Scientists can watch this "bouncing loop" happen in real-time. They can actually see the attractor form.
2. It Connects Worlds: This research bridges the gap between two very different fields: High-Energy Physics (studying the Big Bang and black holes) and Condensed Matter Physics (studying cold atoms). It shows that the same mathematical "tracks" that govern the universe's birth also govern a jar of cold atoms in a lab.

Summary

The paper says: "If you shake a quantum fluid rhythmically, it doesn't just settle down; it finds a new, repeating dance pattern that it sticks to, no matter how you start the dance. This pattern is a new type of 'hydrodynamic attractor' that we can finally watch and measure in a lab."

It's like discovering that if you swing a pendulum back and forth, it doesn't just stop; it finds a perfect, repeating swing that ignores how hard you initially pushed it. And now, we have the tools to watch that swing happen.

Technical Summary

1. Problem Statement

Hydrodynamic attractors describe the phenomenon where strongly interacting systems rapidly lose memory of their initial conditions and evolve along a universal trajectory toward hydrodynamic behavior, even before reaching local thermal equilibrium. Historically, these attractors have been studied primarily in the context of monotonically expanding systems, such as the Bjorken flow in high-energy heavy-ion collisions (Quark-Gluon Plasma).

In these monotonic scenarios, the system eventually converges to the Navier-Stokes (NS) hydrodynamic limit at late times. However, the authors identify a gap in understanding: how do hydrodynamic attractors behave in periodically driven systems? Specifically, they investigate whether a system undergoing cyclic expansion and contraction relaxes to a standard Navier-Stokes limit or exhibits a novel dynamical behavior. This is particularly relevant for ultracold quantum gases, where experimental techniques allow for the periodic modulation of scattering lengths to simulate fluid expansion.

2. Methodology

The authors employ a multi-theoretical approach to model the system, ranging from linear response to nonlinear kinetic theory:

• System Model: They consider a three-dimensional Fermi gas in an isotropic trap where the inverse scattering length $a^{-1}(t)$ is modulated periodically: $a^{-1}(t) = a^{-1}_0 (1 + A \sin \omega t)$. This modulation is mathematically equivalent to a homogeneous expansion/contraction of the gas with a rate $\theta = \partial_\mu u^\mu$.
• Linear Regime (Müller-Israel-Stewart Theory):
  • For small drive amplitudes ($A \ll 1$), the system is described by the Müller-Israel-Stewart (MIS) hydrodynamic equations.
  • The evolution of the bulk pressure $\Pi(t)$ is governed by a relaxation equation: $\tau_\zeta \dot{\Pi} = -\Pi + \Pi_{NS}$, where $\tau_\zeta$ is the relaxation time and $\Pi_{NS}$ is the Navier-Stokes expectation.
  • The authors analyze the phase space trajectory of the actual bulk pressure $\Pi(t)$ versus the Navier-Stokes prediction $\Pi_{NS}(t)$.
• Nonlinear Regime (Relativistic Kinetic Theory):
  • To study large drive amplitudes where linear response fails, the authors utilize the relativistic Boltzmann equation in the Relaxation Time Approximation (RTA) for massive particles.
  • They solve the Boltzmann equation in a time-dependent metric (FLRW-like) $ds^2 = dt^2 - b(t)^2(dx^2+dy^2+dz^2)$, where $b(t) = 1 + A \sin \omega t$.
  • Crucially, they self-consistently determine the instantaneous effective temperature $T(t)$ by equating the instantaneous energy density of the non-equilibrium distribution to the equilibrium energy density at $T(t)$. This accounts for entropy production and heating.

3. Key Contributions

• Discovery of Cyclic Attractors: The paper demonstrates that periodically driven systems do not converge to the Navier-Stokes limit, even at late times. Instead, they settle into a cyclic hydrodynamic attractor.
• Novel Diagnostic Tool: The authors propose plotting the actual anisotropy (bulk pressure) against the Navier-Stokes expectation. In monotonic expansion, trajectories converge to the diagonal ($y=x$) at late times. In periodic driving, the trajectory forms a closed (or slowly drifting) loop (an ellipse in the linear regime) that remains distinct from the diagonal, providing a clear signature of non-Navier-Stokes behavior.
• Experimental Feasibility: The work bridges high-energy nuclear physics and ultracold atom physics, proposing that existing techniques for modulating scattering lengths (via Feshbach resonances) can be used to observe these attractors in real-time.

4. Key Results

A. Linear Response (MIS Theory)

• Elliptic Attractor: For small amplitudes, different initial conditions rapidly converge to a universal elliptical trajectory in the $(\Pi_{NS}, \Pi)$ phase space.
• Frequency Dependence: The shape of the ellipse depends on the dimensionless drive frequency $\omega \tau_\zeta$.
  • In the slow drive limit ($\omega \tau_\zeta \ll 1$), the ellipse is narrow and close to the diagonal (approaching NS behavior).
  • As frequency increases, the ellipse widens, and the phase shift $\phi$ between the drive and the response increases ($\tan \phi = \omega \tau_\zeta$).
• No Convergence to NS: Unlike monotonic expansion where viscous corrections vanish as $\tau \to \infty$, the periodic drive maintains a persistent deviation from Navier-Stokes dynamics due to the continuous competition between driving and relaxation.

B. Nonlinear Regime (Kinetic Theory)

• Non-Periodic Drift: For large drive amplitudes ($A \sim 0.5 - 0.8$), the attractor is no longer a strictly closed loop. The trajectory drifts over time.
• Cause of Drift: This drift is attributed to entropy production (dissipative heating). The system heats up over cycles, causing the equation of state and transport coefficients (specifically the ratio $\zeta/\tau_R P_{eq}$) to evolve.
• Universal Convergence: Despite the drift and the changing equation of state, different initial conditions still converge rapidly (within $t \approx 3\tau_R$) to the same universal trajectory. This confirms the existence of a nonlinear attractor.
• Asymmetry: The attractor shape becomes asymmetric due to the time-evolving temperature and equation of state, deviating significantly from the linear MIS prediction.

5. Significance and Implications

• Experimental Validation: This work provides a concrete proposal for the first experimental observation of hydrodynamic attractors in a controlled setting. Ultracold atoms offer the advantage of independent control over drive frequency and amplitude, unlike the intrinsically coupled parameters in heavy-ion collisions.
• Beyond Navier-Stokes: It establishes that hydrodynamic behavior is not synonymous with Navier-Stokes dynamics. Systems can exhibit robust, universal hydrodynamic-like evolution (attractors) that fundamentally differ from standard viscous hydrodynamics, especially in driven, non-equilibrium settings.
• Cross-Disciplinary Bridge: The findings strengthen the phenomenological connection between high-energy nuclear physics (QGP) and condensed matter physics (ultracold gases), suggesting that attractor physics is a universal feature of strongly interacting quantum fluids.
• New Observables: The proposed phase-space plot ($\Pi$ vs. $\Pi_{NS}$) serves as a robust observable to distinguish between hydrodynamic and non-hydrodynamic regimes, offering a new tool for characterizing transport coefficients like bulk viscosity and relaxation times in quantum gases.

In summary, Mazeliauskas and Enss extend the concept of hydrodynamic attractors from monotonic expansion to periodic driving, revealing a new class of cyclic attractors that persist indefinitely and offer a promising pathway for experimental verification in ultracold quantum gases.