Generation of dipolar supersolids through a barrier sweep in droplet lattices

This paper proposes and analyzes a dynamical protocol using a sweeping repulsive barrier to transform an incoherent array of dipolar quantum droplets into a coherent supersolid state, characterized by a persistent superfluid background and synchronized crystal oscillations.

The Gist

Imagine you have a long row of tiny, isolated islands made of frozen water droplets floating in a calm, frictionless lake. In the world of quantum physics, these are dipolar droplets—clumps of atoms that naturally want to stay apart from each other, like magnets with the same pole facing each other.

Normally, these islands are "incoherent." Think of them as a group of strangers standing on separate islands, all looking in different directions, not talking to each other, and not moving in sync. They are solid (like a crystal) but they aren't "superfluid" (they can't flow without friction).

The scientists in this paper asked a simple question: Can we force these isolated islands to start talking to each other and move as one perfect team, creating a "supersolid"?

A supersolid is a magical state of matter that sounds impossible: it is rigid like a crystal (you can hold it in your hand) but flows like a superfluid (it can slip through cracks without losing energy).

The Experiment: The "Gaussian Barrier" as a Pusher

To make this happen, the researchers didn't just wait for the atoms to figure it out. They used a "pusher."

Imagine a giant, invisible, repulsive wall (a laser beam shaped like a hill) moving slowly down the river of islands.

1. The Setup: They start with a line of 80,000 Dysprosium atoms arranged in a neat row of droplets.
2. The Action: They sweep this repulsive wall through the line of droplets.
3. The Collision: As the wall hits the first island, it pushes it. That island bumps into the next one, which bumps into the next. It's like a game of billiards, but with quantum clouds.

What Happens Next? The "Avalanche"

When the wall pushes the first droplet, it doesn't just stop. It creates a chain reaction:

• The Crash: The droplets crash into each other. In this quantum world, crashing isn't just a bump; it causes atoms to "tunnel" (teleport) from one droplet to another.
• The Spillover: Some atoms get knocked loose from the islands and fall into the water between them.
• The Lake Forms: These loose atoms don't just sit there; they form a continuous, flowing "lake" (a superfluid background) that connects all the islands.

The Result: A Supersolid is Born

Here is the magic part. Even though the islands are now connected by this flowing lake, they don't dissolve. They stay as distinct, solid islands. But now, because they are connected by the superfluid lake, they start doing something amazing:

• They Dance in Sync: Instead of wobbling randomly, all the islands start oscillating (wiggling) back and forth at the exact same time, like a choir singing in perfect harmony.
• The Flow: The "lake" between them flows without friction.

This is the supersolid: The islands provide the solid structure (the crystal), and the connecting lake provides the frictionless flow (the superfluid).

Why Does Speed Matter?

The researchers found that the speed of the "pusher wall" is critical:

• Too Fast: If the wall zooms through too quickly, it's like a car speeding past a puddle. It barely touches the droplets, and they don't have time to interact. Nothing happens.
• Too Slow: If it moves too slowly, it might just push the whole line without causing the necessary "crashes" to release the atoms.
• Just Right: At a specific, moderate speed, the wall hits the droplets hard enough to knock atoms loose and create the connecting lake, but not so hard that it melts the islands entirely.

The Takeaway

This paper shows that you don't need to change the ingredients (the atoms) to create this exotic state of matter. You just need to push them in the right way.

By sweeping a laser barrier through a line of quantum droplets, the scientists successfully engineered a state where matter is both solid and fluid at the same time. It's like turning a row of disconnected, frozen statues into a synchronized dance troupe that can also glide across the floor without tripping. This opens the door to creating these strange, beautiful states of matter in the lab using simple, controllable tools.

Technical Summary

1. Problem Statement

While supersolidity (a state exhibiting both crystalline order and frictionless superfluid flow) has been experimentally realized in dipolar quantum gases, the mechanisms for its dynamical generation from an incoherent state remain an open question. Previous methods often rely on interaction quenches or specific initial conditions. The authors address the challenge of catalyzing the transition from a fully incoherent, elongated droplet lattice (where droplets are isolated and lack phase coherence) to a supersolid state using an external, experimentally accessible perturbation. Specifically, they investigate whether sweeping a repulsive potential barrier through a droplet array can induce the necessary inelastic collisions and energy redistribution to establish global phase coherence and a superfluid background.

2. Methodology

The study employs a theoretical framework based on the extended Gross-Pitaevskii equation (eGPE) incorporating quantum corrections.

• System: A quasi-one-dimensional (quasi-1D) array of $N = 8 \times 10^4$ Dysprosium ($^{164}\text{Dy}$) atoms. The system is confined in a harmonic trap with frequencies $(\omega_x, \omega_y, \omega_z) = 2\pi \times (19, 53, 81)$ Hz, creating an elongated "cigar" geometry.
• Interaction Regime: The scattering length is tuned to $a = 86 a_0$ (where $a_0$ is the Bohr radius), placing the system in the droplet regime (dipolar interaction strength $\epsilon_{dd} > 1$). In this regime, the ground state consists of isolated droplets, not a supersolid.
• Theoretical Model: The dynamics are governed by the eGPE:
  $$i\hbar\partial_t\Psi = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{trap}} + g|\Psi|^2 + V_{\text{dd}} * |\Psi|^2 + \gamma |\Psi|^3 \right] \Psi$$
  Crucially, the equation includes the Lee-Huang-Yang (LHY) correction term ($\gamma |\Psi|^3$), which stabilizes the high-density droplets against collapse.
• Driving Protocol: A repulsive Gaussian barrier (height $V_0$, width $w$) is swept through the droplet array at a variable velocity $v_0$. The barrier parameters are chosen to be comparable to the chemical potential of the ground state ($V_0 \approx 10 \hbar\omega_x$) to induce significant perturbation.
• Numerical Simulation: The authors solve the eGPE using a split-step Crank-Nicolson method in 3D. They also simulate the effects of three-body recombination losses (Appendix A) to ensure robustness against experimental decay mechanisms.

3. Key Contributions

• Novel Protocol: Proposes a dynamical route to supersolidity starting from a fully incoherent droplet lattice (where no superfluid background exists initially) by sweeping a moving barrier.
• Mechanism Elucidation: Identifies the specific physical processes driving the transition: inelastic collisions between droplets, particle tunneling, and self-evaporation, which collectively build up a superfluid background.
• Parametric Mapping: Systematically maps the "supersolidity" phase space as a function of barrier velocity and height, defining the critical thresholds required for the transition.
• Robustness Analysis: Demonstrates that the supersolid state persists even in the presence of significant three-body losses, a critical factor for experimental viability.

4. Key Results

A. Dynamical Stages of the Transition

The evolution of the system is divided into three distinct regimes:

1. Stage I (Pre-collision): The barrier approaches the array. The droplets remain stationary (in the absence of losses) or drift slightly (with losses). No superfluid background exists.
2. Stage II (Collision & Avalanche): The barrier collides with the first droplet, triggering a cascade of inelastic collisions.
   • Droplets emit particles (self-evaporation) and undergo tunneling.
   • This creates an "avalanche" effect where subsequent droplets are excited by emitted particles and collisions, leading to significant atom redistribution.
3. Stage III (Supersolid Nucleation): A persistent superfluid background forms beneath the droplets. The individual droplets begin to oscillate in-sync (collective motion) atop this background, indicating the establishment of global phase coherence.

B. Diagnostic Observables

The authors confirm supersolidity through multiple complementary measures:

• Momentum Distribution: A gradual transfer of population from higher momentum modes to the zero-momentum peak ($k=0$) is observed. The emergence of a sharp $k=0$ peak is a hallmark of superfluidity.
• Center-of-Mass (CoM) Motion: Post-collision, the droplet array exhibits synchronized oscillations with a frequency ($\approx 3$ Hz) significantly lower than the trap frequency ($19$ Hz). This low-frequency mode is characteristic of dipolar supersolids (e.g., out-of-phase Goldstone modes) and distinct from normal superfluids.
• Superfluid Fraction ($f_s$): Using an order parameter based on the density profile, the authors show that $f_s$ increases from 0 (pure droplets) to finite values ($0 < f_s < 0.4$) after the barrier sweep.
  • Velocity Dependence: Supersolidity forms for moderate velocities ($v_0 \lesssim v_g$, where $v_g$ is the characteristic velocity). At high velocities ($v_0 \gg v_g$), the barrier passes too quickly (diabatic regime), and the array remains unperturbed.
  • Height Dependence: A minimum barrier height ($V_0 \gtrsim 7 \hbar\omega_x$) is required to impart enough energy to melt the lattice into a supersolid. If $V_0$ is too high, the lattice melts completely into a disordered fluid.
• Energy Redistribution: The chemical potential $\mu(t)$ shows a sharp increase during collisions followed by damped oscillations, indicating energy transfer from the barrier to the system and subsequent redistribution between kinetic, dipolar, and LHY energy terms.

C. Impact of System Parameters

• Three-Body Losses: Simulations including loss terms show that while the number of atoms and the amplitude of the superfluid background decrease, the supersolid phase still nucleates. The droplets continue to oscillate in-sync, though with reduced amplitude.
• Array Size: The phenomenon is robust across different droplet numbers (2, 4, and 6 crystals), though the specific resonant velocities for optimal supersolid generation shift depending on the system size.

5. Significance

• Experimental Feasibility: The proposed protocol relies on standard experimental tools (focused laser beams for barriers) and does not require tuning interaction strengths (Feshbach resonances) during the process, making it highly accessible for current ultracold atom laboratories.
• Fundamental Physics: The work provides a clear mechanism for how phase coherence can emerge from an incoherent state through driven dynamics, bridging the gap between isolated droplets and supersolids.
• Future Directions: The authors suggest this method can be extended to:
  • Study defect formation (e.g., dark solitons) in dipolar superfluids.
  • Explore supersolid generation in higher dimensions (2D/3D lattices).
  • Investigate vortex lattice nucleation using rotating barriers.

In conclusion, the paper establishes that sweeping a repulsive barrier through a dipolar droplet lattice is a powerful, controllable method to dynamically engineer supersolidity, characterized by the coexistence of rigid crystalline order and a superfluid background.