Single-fluid model for rotating annular supersolids and its experimental implications

This paper proposes a single-fluid model for rigidly rotating annular supersolids, demonstrating that their mixed classical-superfluid dynamics arise from a spatially varying global wavefunction phase and enabling experimental protocols to detect peculiar phenomena such as partially quantized supercurrents.

The Gist

Imagine a material that is two things at once: a rigid crystal, like a block of ice, and a superfluid, like a frictionless liquid that can flow forever without slowing down. Scientists call this a supersolid. It's a bit like a dance troupe where the dancers are locked in a rigid formation (the crystal) but can also slide around each other without any friction (the superfluid).

For a long time, physicists explained how these supersolids spin using a "two-fluid" model. They imagined the material as being made of two separate groups: a "solid" crowd that spins like a rigid wheel, and a "super" crowd that spins like a frictionless liquid.

The Big Idea: One Fluid, Two Personalities
This paper argues that the "two-fluid" idea is actually a bit of a trick. The authors propose a single-fluid model. They say there aren't two separate groups of atoms; there is just one giant group of atoms behaving in a complex, coordinated way.

Think of it like a conga line moving around a circular track.

• In a normal solid (like a spinning ice skater), everyone holds hands and moves at the exact same speed.
• In a normal superfluid, everyone moves at a speed determined by a strict rule (quantum mechanics), but they don't necessarily hold hands in a rigid line.
• In a supersolid, the dancers are holding hands in a rigid line (the crystal), but their speed varies depending on where they are in the line. Some parts of the line speed up, while others slow down, all to keep the whole formation moving smoothly.

The paper shows that this "speeding up and slowing down" is actually just the result of the quantum wave (the invisible rulebook guiding the atoms) changing its shape as it wraps around the circle.

The "Partially Quantized" Mystery
In normal superfluids, the amount of spin (angular momentum) an atom has is always a whole number multiple of a tiny quantum unit (like counting 1, 2, 3...). You can't have 1.5 spins.

However, in a supersolid, the authors show that the atoms can carry less than a full unit of spin. It's like if the dance troupe could spin at "1.5 steps" instead of just 1 or 2. This is called "partially quantized" current. The solid part of the crystal "steals" some of the spin, leaving the superfluid part with less than a full quantum unit.

How They Tested It (The "Phase Imprinting" Trick)
The researchers wanted to see if they could make these supersolids spin in specific ways. Usually, to make something spin, you just spin the container it's in (like spinning a bucket of water). But for supersolids, that's tricky because the "solid" part wants to spin with the bucket, while the "super" part wants to stay still or spin differently.

Instead, the authors used a clever trick called phase imprinting.

• The Analogy: Imagine you have a long, flexible ribbon lying on a table. If you want the ribbon to move, you could push the whole table (spinning the bucket). But instead, the authors used a "magic laser" to briefly touch the ribbon in a specific pattern. This "touched" the ribbon's quantum state, instantly forcing it to start moving in a specific way without physically pushing the container.
• The Result: They successfully created these "partially quantized" spinning states. They showed that they could make the supersolid spin with a specific amount of momentum that was between the usual whole numbers, proving their single-fluid theory right.

Measuring the Spin
How do you measure this weird spin? The authors proposed a new way to "read" the spin.

• The Analogy: Imagine the supersolid is a group of dancers holding hands. If you suddenly tell them to let go of each other (turning off the "super" part so they become just a normal crystal), the momentum they had while holding hands has to go somewhere.
• The Method: The researchers simulated a process where they slowly changed the material so the "super" part disappeared, leaving only the "solid" part. Because momentum is conserved, the "solid" part would suddenly start spinning faster to make up for the lost "super" spin. By measuring how fast the solid crystals spun at the end, they could calculate exactly how much spin the material had at the beginning, even if it was a weird, "partial" amount.

Why This Matters
This paper doesn't just fix a math problem; it gives scientists a new map for navigating these strange materials.

1. New Experiments: It tells experimentalists exactly how to use lasers to "imprint" specific spinning patterns onto these materials.
2. Better Understanding: It shows that the "solid" and "super" behaviors are actually two sides of the same coin, emerging from a single quantum wave, rather than two separate fluids fighting each other.
3. Broader Application: The authors note that this same logic applies to other systems where a fluid is forced into a pattern, like superfluids trapped in grids of light (optical lattices), not just supersolids.

In short, the paper replaces the idea of a "split personality" material with a "unified, shape-shifting" material, and provides the tools to make it dance in ways we've never seen before.

Technical Summary

Technical Summary: Single-fluid model for rotating annular supersolids and its experimental implications

Problem Statement
The supersolid phase of matter, characterized by the simultaneous breaking of global gauge symmetry and translational symmetry, presents a theoretical challenge regarding its rotational dynamics. While the standard two-fluid model (analogous to finite-temperature superfluids) has been applied to describe supersolids as a mixture of a superfluid component and a "crystalline" normal component, this framework is conceptually inconsistent at zero temperature ($T=0$). In supersolids, the normal component arises from the crystalline lattice structure within the same mean-field macroscopic wavefunction, rather than from a thermal population. Consequently, separating the system into two distinct fluids with opposite behaviors contradicts the quantum nature of the ground state. Furthermore, experimental protocols for rotating supersolids in annular geometries remain unexplored, particularly regarding the detection of "partially quantized" supercurrents where the angular momentum per particle is reduced ($L/N = w f_s \hbar < w \hbar$) due to the superfluid fraction $f_s < 1$.

Methodology
The authors propose a unified single-fluid model based on the continuity equation to describe rigidly rotating annular supersolids.

1. Theoretical Derivation: Starting from a single-component wavefunction $\psi(x,t)$ with a spatially periodic density $\rho(x)$, the authors derive a microscopic velocity field $v_w(x)$ for a system rotating at velocity $V = \Omega R$. By integrating the continuity equation under the irrotational condition (fixed winding number $w$), they obtain an analytical expression for the velocity field that varies spatially within the unit cell.
2. Phase Profile Calculation: The model yields a specific phase profile $\phi_w(x, \Omega)$ that dictates the current properties. This allows for the derivation of an analytical expression for the total angular momentum $L$ and kinetic energy $E$, which decompose into classical and superfluid contributions without invoking separate fluid components.
3. Numerical Simulations: The authors validate the model using numerical simulations of the Gross-Pitaevskii equation (GPE) for $N = 5 \times 10^4$ atoms of $^{162}\text{Dy}$ in a ring trap ($R = 5\,\mu\text{m}$).
   • Phase Imprinting: They simulate a protocol where a spatially dependent optical potential $V_{PI}(x)$ is applied to imprint the calculated phase profile onto the ground state, effectively setting the desired angular momentum.
   • Quench Dynamics: They simulate a ramp-down of the scattering length to transition the system from a supersolid phase ($f_s > 0$) to a droplet crystal phase ($f_s \to 0$) to measure angular momentum transfer.
   • Optical Lattice Extension: The model is tested on superfluids in rotating optical lattices to verify its applicability to density-modulated superfluids beyond intrinsic supersolids.

Key Contributions and Results

• Single-Fluid Description: The paper demonstrates that the seemingly classical and superfluid contributions to rotation emerge naturally from the spatially varying phase of a single global wavefunction. The derived angular momentum $L = I_c(1-f_s)\Omega + N w f_s \hbar$ matches the form of the two-fluid model but is derived from a fundamental single-fluid perspective.
• Velocity Field and Phase Structure: The model predicts that even in the $w=0$ manifold (zero circulation), a non-zero velocity field exists where atoms in density maxima move opposite to the superfluid background to satisfy irrotationality. The phase profile is explicitly calculated, showing how it accommodates both rigid rotation and quantized circulation.
• Experimental Protocols:
  • Phase Imprinting: The authors propose a method to excite specific rotational states (including metastable persistent currents with $w=1$ and $\Omega=0$) by imprinting the calculated phase profile. Simulations confirm this method creates rigid body rotation of the density and accurately populates the predicted energy manifolds.
  • Angular Momentum Measurement: An innovative measurement protocol is introduced. By adiabatically ramping the system to a droplet crystal phase ($f_s \to 0$), the initial superfluid angular momentum is transferred entirely to the classical rotational motion of the droplets. Measuring the final velocity of the droplets allows for the direct determination of the total initial angular momentum $L$, sensitive to the partial quantization ($f_s$) rather than just the winding number $w$.
• Energy Barriers and Stability: The analysis of the energy landscape reveals that persistent current states ($w=1$) are local energy minima protected by an energy barrier $E_b$. This barrier vanishes when $f_s \leq 0.5$, indicating a transition where the classical state becomes energetically favorable over the persistent current for the same angular momentum.
• Generalizability: The model is shown to apply to superfluids in optical lattices where translational symmetry is explicitly broken by an external potential, confirming that the physics of partially quantized currents is a general feature of density-modulated superfluids.

Significance
The paper resolves the conceptual contradiction of applying a two-fluid model to a zero-temperature supersolid by providing a rigorous single-fluid derivation based on the global wavefunction. This framework provides the necessary theoretical tools to design experimental protocols for exciting and detecting the unique rotational dynamics of supersolids. Specifically, the proposed phase imprinting and angular momentum measurement techniques offer a pathway to experimentally verify the existence of partially quantized supercurrents, a phenomenon that has remained elusive. The results suggest that these findings extend beyond dipolar supersolids to a broader class of spatially modulated superfluids, including annular Josephson junctions and vortex necklaces, thereby opening new avenues for investigating superflow decay and atomtronic configurations.