Direct observation of long-range many-body coherence in quasi-one-dimensional attractive Bose gases

This study experimentally demonstrates that degenerate quasi-one-dimensional attractive Bose gases, when quenched from repulsive to unstable attractive interactions, exhibit long-range phase-coherent density waves distinct from soliton trains, which can spontaneously re-establish quasi-long-range coherence upon returning to the repulsive regime through the nucleation and annihilation of density defects.

The Gist

Imagine a crowded dance floor where everyone is moving in perfect unison. This is what physicists call a Bose-Einstein Condensate (BEC): a super-cold cloud of atoms acting as a single, giant "super-atom" where every particle marches to the same beat. This perfect synchronization is called coherence.

Usually, these dance floors are stable. But in this experiment, the scientists decided to shake things up. They took a calm, orderly crowd of atoms and suddenly changed the rules of the dance, turning the "repulsive" (pushing away) forces between atoms into "attractive" (pulling together) forces.

Here is the story of what happened, broken down into simple steps:

1. The "Dam Break" and the Chaos

Think of the atoms as water held back by a dam in a long, narrow channel. When the scientists switched the forces to "attractive," it was like suddenly removing the dam.

• The Instability: Instead of flowing smoothly, the water (atoms) started to slosh violently. Small ripples grew into massive waves. In physics, this is called Modulational Instability.
• The Shockwaves: Because the atoms were pulled together, they didn't just form random splashes. They created two giant waves crashing toward each other from the ends of the channel. When these waves met in the middle, they didn't just crash and stop; they created a chaotic, oscillating mess of density peaks and valleys.
• The "Soliton" Myth: In the past, scientists expected this chaos to turn into neat, separate packets of water called "solitons" (like distinct, self-contained waves that travel without losing shape). But here, the chaos was different. It was a continuous, breathing wave of density, not separate islands.

2. The "Amnesia" (Phase Scrambling)

As time went on, this chaotic dancing got worse. The atoms started to lose their memory of who they were dancing with.

• Imagine a choir singing in perfect harmony. Suddenly, everyone starts singing a different note at a different time. The music becomes noise.
• In the experiment, the "phase" (the timing of the atomic wave) got scrambled. The atoms developed "phase slips"—little glitches where the rhythm jumped by 180 degrees. The long-range harmony was broken, and the system looked like a disorganized mess.

3. The "Rewind" Button (Rephasing)

Here is the most magical part. The scientists didn't just let the chaos continue. They hit the "rewind" button.

• They slowly switched the forces back from "attractive" to "repulsive" (pushing away).
• The Surprise: If you had a messy pile of tangled headphones and you slowly pulled the wires apart, you might expect them to stay tangled. But here, as the atoms started pushing away from each other again, something incredible happened: The order returned.
• The chaotic, scrambled rhythm spontaneously fixed itself. The atoms found their beat again and started dancing in unison, restoring the long-range coherence.

4. How Did They Fix It? (The "Defect Cleanup")

How did the atoms fix themselves? The scientists discovered that the "glitches" (the phase slips) didn't just vanish; they annihilated each other.

• Think of the glitches as little knots in a rope. In a perfectly one-dimensional world (a single string), these knots would stay forever.
• But because the atoms were in a "quasi-one-dimensional" box (a very thin, but slightly wide channel), these knots could move sideways. They bumped into each other, collided, and cancelled out, turning into harmless sound waves that dissipated.
• It's like a room full of people tripping over each other. If they are stuck in a single-file line, they stay stuck. But if they have a little room to move side-to-side, they can step around each other, untangle the mess, and get back to walking in a straight line.

The Big Picture

This experiment is a breakthrough because it shows that order can spontaneously emerge from chaos, even in a quantum system that was previously thought to be too messy to recover.

• The Analogy: Imagine a room full of people running in panic (the attractive phase). They bump into each other, creating a chaotic crowd. Then, you tell them to stop and walk away from each other (the repulsive phase). Instead of leaving a mess, they somehow organize themselves back into a neat line, having "cleaned up" their own mistakes along the way.

This discovery helps us understand how complex quantum systems behave when they are out of balance, which is crucial for building future quantum computers and understanding the fundamental laws of the universe. It proves that even when a system looks broken, the right conditions can allow it to heal itself.

Technical Summary

1. Problem Statement

Macroscopic phase coherence is a hallmark of quantum many-body systems like Bose-Einstein condensates (BECs) and superfluids. While coherence in repulsive regimes is well-understood, studying it in attractive interaction regimes presents significant challenges due to modulational instability (MI).

• The Challenge: In attractive gases, MI typically amplifies density fluctuations, leading to wave collapse and the formation of localized solitons (bright solitons) or soliton trains. In dimensions $D > 1$, this leads to collapse; in 1D, it was theoretically predicted to form integrable structures like breathers or solitons.
• The Gap: It has been unclear whether long-range phase coherence survives the nonlinear stage of MI in attractive gases or if the system becomes completely phase-scrambled. Furthermore, the dynamics of rephasing (restoring coherence) after such a chaotic, unstable evolution had not been experimentally observed.

2. Methodology

The authors employed a combination of ultracold atom experiments and numerical simulations to investigate these dynamics.

• Experimental Setup:

  • System: Cesium ($^{133}\text{Cs}$) Bose-Einstein condensates loaded into two parallel quasi-1D box traps (length $l=40\,\mu\text{m}$).
  • Geometry: High aspect ratio traps ($\omega_y \approx 68\,\text{Hz}$, $\omega_z \approx 2.2\,\text{kHz}$) ensure quasi-1D behavior, though finite transverse width allows for 2D effects.
  • Protocol:
    1. Initial State: A repulsive superfluid ($a \approx 105a_0$) with pre-thermalized density fluctuations.
    2. Quench Down: The scattering length is rapidly quenched to a weakly attractive value ($a \approx -5a_0$) via a magnetic Feshbach resonance.
    3. Hold Time ($t$): The system evolves for variable times (up to $50\,\text{ms}$) in the attractive regime.
    4. Quench Up (Rephasing): The interaction is ramped back to the initial repulsive value over a variable ramp time ($t_r$).
  • Diagnostics:
    • In-situ Imaging: To measure density profiles and density-density correlations ($C_n$).
    • Matter-Wave Interferometry: Two parallel gases are released and allowed to expand and interfere (Time-of-Flight). This extracts the relative phase $\phi(x)$, allowing the calculation of the phase correlation function ($C_\phi$) and the counting of phase slips (defects).

• Numerical Simulations:

  • 1D Gross-Pitaevskii Equation (GPE): Used to model ideal dam-break scenarios and dispersive shock waves (DSWs).
  • 2D GPE: Used to incorporate realistic experimental conditions, including finite transverse confinement, initial noise, and three-body atom loss. This was crucial for explaining defect annihilation mechanisms.

3. Key Contributions and Results

A. Dynamics in the Attractive Regime (Quench Down)

Upon quenching to attraction, the system exhibits a complex interplay between two mechanisms:

1. Noise-Amplified Modulations: Random fluctuations seed MI, creating density modulations with a periodicity matching the healing length ($\sim 4.4\,\mu\text{m}$).
2. Dispersive Shock Waves (DSWs): The sharp edges of the box trap generate DSWs that propagate inward, a phenomenon analogous to the "dam-break" problem in nonlinear hydrodynamics.

Key Observations:

• Non-Solitonic Behavior: Unlike the iconic soliton trains seen in harmonically trapped attractive gases, the box-trapped gas develops phase-coherent density waves.
• Phase Coherence Evolution:
  • Initially ($t \sim 7\,\text{ms}$), the system retains finite-range phase coherence despite density modulations.
  • At intermediate times ($t \sim 13\,\text{ms}$), the competition between DSWs and noise-induced modulations leads to a transient increase in phase coherence range.
  • At long times ($t \gtrsim 31\,\text{ms}$), the system becomes phase-scrambled. The correlation length shrinks to the healing length ($\xi$), and the number of phase slips ($N_j$) increases, indicating a loss of global order.

B. Rephasing Dynamics (Quench Up)

The most significant finding is the spontaneous re-establishment of quasi-long-range coherence when the interaction is ramped back to repulsion.

• Defect Annihilation: As the system transitions to repulsion, the density modulations convert into dark solitons (defects).
• Role of Dimensionality: In a pure 1D integrable system, these solitons would be stable. However, the quasi-1D (2D) nature of the experiment allows these defects to evolve into vortex dipoles.
• Mechanism: These vortex dipoles undergo inelastic collisions and interact with the trap boundaries, leading to their annihilation and decay into sound waves.
• Outcome: As defects are annihilated, the phase jumps disappear, and global phase coherence is restored.
  • Ramp Time Dependence: Slower ramp times ($t_r \gtrsim 132\,\text{ms}$) allow sufficient time for defect annihilation, resulting in a recovery of coherence comparable to the initial state. Faster ramps leave more defects, preserving phase scrambling.
• Energy Conservation: Analysis of kinetic, interaction, and potential energies confirms that this rephasing occurs without significant heating or cooling, ruling out evaporative cooling as the cause of coherence recovery.

4. Significance

• Beyond Mean-Field Physics: The study explores a regime where mean-field theories (like standard 1D GPE) fail to predict the full dynamics due to the importance of dimensionality and defect annihilation.
• Non-Equilibrium Quantum Dynamics: It provides a direct experimental realization of rephasing in a system that has undergone a chaotic, unstable evolution. This challenges the notion that attractive interactions inevitably lead to irreversible loss of coherence.
• Universal Physics: The interplay of DSWs, noise amplification, and defect dynamics connects ultracold atoms to broader nonlinear physics phenomena in plasmas, optics, and granular systems.
• Defect Dynamics: The work highlights the critical role of extended dimensionality (even in quasi-1D systems) in enabling defect annihilation mechanisms that are forbidden in strictly 1D integrable models.

Conclusion

This paper demonstrates that quasi-1D attractive Bose gases, despite undergoing modulational instability and phase scrambling, can spontaneously recover long-range phase coherence upon returning to a repulsive regime. This recovery is driven by the annihilation of density defects (vortex dipoles) facilitated by the system's finite transverse dimensionality, offering new insights into non-equilibrium quantum many-body physics and the robustness of quantum coherence.