Signatures of rigidity and second sound in dipolar supersolids

This paper proposes and simulates a dynamical protocol using quasi-one-dimensional double-well potentials to distinguish the rigidity and phase coherence of dipolar supersolids from other phases by observing damped crystal oscillations and second sound excitations triggered by merging fragments and phase-imprinted jumps.

The Gist

Imagine a material that is two things at once: a rigid crystal, like a diamond, and a superfluid, like a frictionless liquid that can flow forever without losing energy. This strange state of matter is called a supersolid. For decades, physicists have been trying to prove it exists and understand how it behaves.

This paper proposes a clever "experiment" (done via computer simulations) to test the stiffness and flow of these supersolids using a specific type of cold atom gas (Dysprosium). Here is the story of what they did, explained simply.

The Setup: A Frozen Dance Floor

Imagine you have a long, narrow hallway (a 1D trap) filled with millions of atoms.

• The Atoms: These aren't normal atoms; they are like tiny magnets (dipoles) that attract and repel each other in complex ways.
• The Barrier: In the middle of the hallway, there is an invisible wall splitting the atoms into two separate groups.
• The Goal: The researchers want to see what happens when they suddenly knock down that wall and let the two groups merge.

The Three Characters

Depending on how strong the magnetic attraction is, the atoms behave in three different ways:

1. The Superfluid: A smooth, wavy liquid with no structure.
2. The Isolated Droplets: The atoms clump into separate, solid balls (like marbles) that don't talk to each other. They are rigid but disconnected.
3. The Supersolid (The Star): The atoms form a crystal pattern (like a row of marbles), but they are all connected by an invisible, frictionless "superfluid" glue. They are rigid like a crystal but can flow like a liquid.

Experiment 1: The "Rigidity Test" (Knocking Down the Wall)

The researchers simulate removing the central wall.

• In the "Marble" phase (Isolated Droplets): When the wall drops, the marbles bounce back and forth like balls on a spring. They keep bouncing forever because there is no friction. They are perfectly rigid.
• In the "Supersolid" phase: The marbles also bounce, but they slowly stop.
  • The Analogy: Imagine the marbles are running on a track made of thick honey (the superfluid background). As the marbles bounce, they drag the honey with them, creating friction.
  • The Discovery: The rate at which they stop bouncing tells the scientists exactly how much "honey" (superfluid) is connecting the marbles. If they stop quickly, there is a lot of superfluid connection. If they keep bouncing, they are more like isolated marbles. This is a new way to measure the "stiffness" and "stickiness" of the supersolid.

Experiment 2: The "Second Sound" (The Phase Jump)

This is the most exciting part. The researchers don't just knock down the wall; they also give the two groups of atoms a "nudge" in opposite directions before merging them. Think of it as giving the left group a high-five and the right group a high-five, but in a way that creates a "phase jump" (a sudden shift in their rhythm).

• The Solitary Wave: When the wall drops, this nudge creates a "dark soliton." Imagine a wave moving through a crowd where the people part ways to let a hole pass through. In a normal liquid, this hole bounces back and forth.
• The Twist: In the supersolid, this "hole" (the soliton) doesn't bounce. Instead, it acts like a magnet. It grabs the rigid crystal structure and pulls it.
• The Second Sound: As the soliton pulls the crystal to the right, the invisible superfluid glue underneath is forced to flow to the left to keep the total momentum balanced.
  • The Analogy: Imagine a person (the crystal) walking on a treadmill (the superfluid). If the person walks forward, the treadmill belt moves backward. In a supersolid, you can see both the person walking and the belt moving in opposite directions at the same time.
  • Why it matters: This "opposite flow" is called Second Sound. It's a unique signature of supersolids that proves the material is truly a mix of solid and liquid. The researchers found they could control how fast the crystal moved just by changing the size of the initial "nudge."

The "Quantum Canary"

The paper uses a great metaphor: Dark solitons are like "quantum canaries."
In old coal mines, miners brought canaries to detect toxic gas. If the canary stopped singing, the miners knew something was wrong.
Here, the "canary" is the solitary wave. By watching how this wave behaves (does it bounce? does it drift? does it pull the crystal?), scientists can detect the hidden properties of the supersolid, like its rigidity and how well the liquid and solid parts are connected.

The Bottom Line

This paper proposes a recipe to:

1. Prove Rigidity: By watching how fast the crystal "bounces" and stops, we can measure how much superfluid is holding it together.
2. Detect Second Sound: By creating a specific wave, we can make the solid part and the liquid part flow in opposite directions, which is the "smoking gun" proof that a supersolid exists.

It's like learning to drive a car that is simultaneously a solid block of ice and a flowing river, and figuring out how to steer it by watching how the ice slides and the water rushes in opposite directions.

Technical Summary

1. Problem Statement

The paper addresses the challenge of dynamically probing the dual nature of dipolar supersolids—a phase of matter exhibiting both crystalline order (broken translational symmetry) and superfluidity (global phase coherence). While static properties like density modulation have been observed, dynamically characterizing the rigidity of the crystal lattice and the superfluid connectivity between droplets remains difficult. Specifically, the authors aim to:

• Distinguish between isolated droplet arrays (incoherent) and true supersolids (coherent).
• Quantify the superfluid fraction and its role in damping crystal oscillations.
• Investigate the excitation of second sound (out-of-phase motion of superfluid and crystal components) and the behavior of dark solitary waves within a supersolid background.

2. Methodology

The authors employ a combination of numerical simulations and phenomenological modeling:

• System Setup: They simulate $N = 8 \times 10^4$ atoms of $^{164}$Dy (highly magnetic dipolar atoms) confined in a quasi-one-dimensional (quasi-1D) double-well potential. The dipolar interaction strength is tuned via the ratio of dipolar to contact scattering lengths ($\epsilon_{dd}$).
• Governing Equation: The dynamics are modeled using the 3D extended Gross-Pitaevskii equation (eGPE). This equation includes:
  • Kinetic energy and external trapping potential.
  • Short-range contact interactions.
  • Long-range anisotropic dipolar interactions.
  • The Lee-Huang-Yang (LHY) correction term, which provides repulsive quantum fluctuations necessary to stabilize the supersolid phase against collapse.
• Dynamical Protocol:
  1. Barrier Removal: The gas is split into two fragments by a central barrier. The barrier is suddenly removed (quench), allowing the fragments to merge.
  2. Phase Imprinting: In specific scenarios, a phase jump ($\Delta\phi = \pi$ or $\pi/2$) is imprinted across the two fragments before barrier removal to generate solitonic excitations.
• Analytical Modeling: To interpret the complex eGPE results, the authors develop a damped coupled-oscillator model. They treat droplet peaks as massive particles ($M_I$ for inner, $M_O$ for outer) connected by springs (representing elastic forces) and subject to damping coefficients ($\gamma$) representing coupling to the superfluid background.

3. Key Contributions and Results

A. Rigidity and Damping as a Probe of Superfluidity

• Isolated Droplets ($\epsilon_{dd} = 1.48$): Upon barrier removal, the droplet lattice exhibits undamped, rigid-body oscillations. The droplets oscillate out-of-phase but retain their shape, behaving like a classical crystal with high stiffness.
• Supersolids ($\epsilon_{dd} = 1.36$): The merged system exhibits damped oscillations. The damping rate ($\Gamma$) is directly correlated with the superfluid fraction.
  • Mechanism: The superfluid background acts as a "viscous" medium. As the crystal oscillates, it couples to the background, dissipating energy.
  • Quantification: The damping rate increases as the system moves deeper into the supersolid regime (lower $\epsilon_{dd}$), providing a direct dynamical measure of phase coherence and superfluid connectivity between droplets.
• Elastic Properties: By fitting the oscillator model, the authors extract the Young's modulus ($E$). They find that $E$ increases with $\epsilon_{dd}$, indicating that the system becomes stiffer (more solid-like) as it transitions toward the isolated droplet regime.

B. Excitation of Second Sound via Solitary Waves

• Phase Imprinting: When a $\pi$-phase jump is imprinted across the barrier in the supersolid phase, a dark solitary wave (density dip) forms at the trap center.
• Soliton Dynamics: Unlike in pure superfluids where solitons oscillate, the dipolar soliton in the supersolid is initially stationary but eventually drifts due to interactions with the crystal lattice and sound waves.
• Second Sound Signature: The drifting soliton transfers momentum to the crystal lattice, causing the entire crystal to move in one direction. Crucially, the superfluid background flows in the opposite direction to conserve the center of mass.
  • This counter-flow is the definitive signature of second sound.
  • The drift velocity ($v_d$) of the crystal is tunable by adjusting the phase jump amplitude ($\Delta\phi$).
  • The authors derive a momentum conservation relation ($n_s v_s = n_d v_d$) that accurately predicts the drift velocity based on the superfluid velocity.

C. Validation of Models

• The damped coupled-oscillator model (with nearest-neighbor springs) successfully reproduces the eGPE dynamics for both isolated droplets and supersolids.
• The inclusion of next-to-nearest neighbor (NNN) couplings did not significantly improve the fit, suggesting that nearest-neighbor interactions dominate the crystal dynamics in this quasi-1D geometry.
• The model confirms that the system behaves as a viscoelastic material, exhibiting both elastic (crystal) and dissipative (superfluid) characteristics.

4. Significance

• New Diagnostic Tool: The paper proposes a robust, dynamical protocol to experimentally distinguish between incoherent droplet arrays and true supersolids. The damping rate of crystal oscillations serves as a direct proxy for the superfluid fraction.
• Observation of Second Sound: It provides a clear pathway to observe and control second sound in dipolar supersolids, a phenomenon previously difficult to isolate dynamically. The ability to tune the drift velocity via phase imprinting offers a new method for manipulating supersolid flow.
• Soliton as a Probe: The study reveals that dark solitary waves in supersolids act as unique "quantum canaries." Instead of oscillating, they transfer momentum to the lattice, acting as a controlled mechanism to excite second sound.
• Theoretical Framework: The successful application of a classical damped oscillator model to a quantum many-body system offers a simplified yet accurate phenomenological tool for understanding the elastic and dissipative properties of supersolids.

Conclusion

Bougas et al. demonstrate that merging dipolar quantum gases in a double-well potential reveals distinct dynamical signatures of the supersolid phase. The transition from undamped rigid oscillations (droplets) to damped oscillations (supersolids) quantifies superfluid connectivity, while phase-imprinted solitons provide a controllable route to excite and observe second sound. These findings bridge the gap between theoretical predictions of supersolid elasticity and experimental observables in ultracold atomic gases.