Sound propagation in striped supersolid cold gases at zero temperature

This paper presents a unified hydrodynamic framework describing the anisotropic two-sound spectrum in stripe-phase supersolids formed by ultracold dipolar gases and spin-orbit-coupled BECs, highlighting how the lack of Galilean invariance in the latter case uniquely alters the identification of the normal density component.

The Gist

Imagine a substance that is both a solid and a liquid at the same time. It's rigid enough to hold a shape like a crystal, yet it can flow without any friction like a super-liquid. Scientists call this a supersolid.

This paper is about studying how sound travels through these strange supersolids. Specifically, the researchers looked at two different ways to create these supersolids in a lab using super-cold gases:

1. Dipolar Gases: Atoms that act like tiny bar magnets.
2. Spin-Orbit Coupled (SOC) Gases: Atoms that are "jammed" together by lasers in a way that makes their internal spin and their movement inseparable.

Here is the breakdown of their findings, translated into everyday language.

1. The "Striped" Pattern

In these supersolids, the atoms don't just sit randomly. They arrange themselves into stripes, like the stripes on a zebra or the lines on a notebook.

• The Analogy: Imagine a crowd of people at a concert. In a normal gas, they are a messy, jumbled crowd. In a supersolid, they spontaneously line up in neat rows and columns, but they can still slide past each other without bumping into anyone (superfluidity).

2. The Two Types of Sound

In a normal fluid (like water), there is only one way sound travels: a compression wave (squeezing the water). In a supersolid, because the atoms are arranged in stripes, there are two distinct types of sound that can travel:

• Sound Type A (The "Squeeze"): This is like normal sound. You squeeze the stripes together and let them pop back. It involves the density of the atoms changing.
• Sound Type B (The "Slide"): This is the weird one. Imagine the stripes themselves sliding back and forth, like a deck of cards being pushed sideways. The atoms don't necessarily get squeezed; the pattern just shifts. This is the sound of the "crystal" part of the supersolid.

3. The Direction Matters (Anisotropy)

The most interesting finding is that the speed of these sounds depends entirely on which way you are listening.

• The Analogy: Think of a wooden floor. It's easy to slide a rug along the grain of the wood, but hard to slide it across the grain.
• The Result: If you send a sound wave parallel to the stripes, it behaves one way. If you send it perpendicular (across) the stripes, it behaves completely differently. The "Slide" sound (Type B) is very fast in some directions and disappears in others.

4. The Two Different Worlds

The paper compares the two platforms (Magnets vs. Lasers) and finds they are similar but have a crucial difference:

• The Magnet Platform (Dipolar): This behaves somewhat like a standard crystal. The atoms follow the usual rules of physics where if you push the whole system, everything moves together.
• The Laser Platform (SOC): This is the tricky one. Because of the lasers, the atoms have lost a fundamental rule of physics called Galilean Invariance.
  • The Analogy: Imagine you are on a train.
    • In the Magnet case, if the train moves, you move with it. You and the train are one unit.
    • In the Laser case, the "train" (the laser field) is moving, but you (the atoms) are partially stuck to the tracks and partially moving with the train. You are "locked" to the laser in a way that makes the math of how you move very different.
  • The Consequence: In the Laser platform, the "normal" part of the fluid (the part that doesn't flow perfectly) is actually negative in a mathematical sense! It's like having a debt of mass that cancels out some of the real mass. This creates a unique "three-fluid" behavior where the sound waves are a mix of the atoms, the stripes, and the laser field itself.

5. Why Does This Matter?

The researchers created a unified theory (a single set of rules) that can describe sound in both of these very different systems.

• The Big Picture: Even though the microscopic details (the magnets vs. the lasers) are totally different, the "macroscopic" behavior (how the sound waves move) looks very similar.
• The Connection to Liquid Crystals: They compared these supersolids to smectic liquid crystals (the stuff in LCD screens). They found that the "Slide" sound in the supersolid is very similar to the way layers in a liquid crystal wiggle. This helps scientists understand both quantum gases and everyday materials better.

Summary

The paper tells us that in these magical "striped" supersolids, sound doesn't just travel in a straight line. It splits into two modes: a "squeeze" and a "slide." While the two ways of making these solids (magnets vs. lasers) are different, they both create a world where sound is anisotropic (direction-dependent) and behaves like a mix of a solid crystal and a frictionless liquid. The laser-based version is particularly weird because the lasers themselves act like a third ingredient in the fluid, changing the rules of how sound moves.

Technical Summary

1. Problem Statement

The paper addresses the theoretical description of sound propagation in striped supersolid phases realized in two distinct ultracold quantum gas platforms:

1. Dipolar Gases: Where supersolidity arises from the interplay of short-range contact interactions and long-range dipolar interactions.
2. Spin-Orbit-Coupled (SOC) Bose-Einstein Condensates (BECs): Where supersolidity is induced by Raman lasers coupling the atomic spin and momentum.

While both systems exhibit spontaneous breaking of $U(1)$ gauge symmetry (superfluidity) and translational symmetry (crystalline order), they differ fundamentally at the microscopic level. Specifically, SOC systems lack Galilean invariance due to the momentum-dependent coupling, whereas dipolar gases preserve it. The authors aim to develop a unified hydrodynamic framework to describe the anisotropic sound modes in both platforms, clarifying how the lack of Galilean invariance in SOC systems alters the identification of normal density and the resulting sound spectrum.

2. Methodology

The authors employ a macroscopic hydrodynamic (HD) approach based on conservation laws and broken symmetries.

• Hydrodynamic Variables: The theory tracks the evolution of:
  • Total density ($\rho$).
  • Current density ($\mathbf{j}$).
  • Superfluid velocity ($\mathbf{v}_s$).
  • Gradient of the stripe displacement ($\nabla u$).
• Linearization: The HD equations are linearized around the ground state to derive the dispersion relations for small-amplitude excitations (sound waves).
• Parameter Extraction:
  • Dipolar Case: Parameters (compressibility, elastic constants, superfluid fraction) are extracted numerically using the Extended Gross-Pitaevskii Equation (eGPE), which includes the Lee-Huang-Yang (LHY) quantum fluctuation term necessary for supersolid stability. Real-time simulations validate the HD predictions.
  • SOC Case: Parameters are derived semianalytically. A key step involves decomposing the normal density into components related to the motion of the density fringes and the "locked" density associated with the Raman lasers.
• Symmetry Analysis: The authors explicitly account for the breaking of Galilean invariance in the SOC case, which necessitates a redefinition of the conserved momentum and the normal fluid component.

3. Key Contributions

A. Unified Hydrodynamic Formalism

The paper establishes that despite microscopic differences, both systems can be described by similar HD equations. However, the current equation differs significantly:

• Dipolar Gases: The normal density is defined by the difference between total density and superfluid density ($\rho_n = \rho_0 - \rho_s$).
• SOC Gases: Due to the lack of Galilean invariance, the normal density is split into three components:
  $$\rho_0 = \rho_s + \rho_{fr} + \rho_{laser}$$
  Where $\rho_s$ is the superfluid density, $\rho_{fr}$ is the density participating in fringe motion, and $\rho_{laser}$ is the density "locked" to the Raman lasers. Notably, in the supersolid phase, $\rho_{laser}$ is negative, a unique feature of SOC supersolids that acts as a zero-temperature analog to the three-fluid model.

B. Identification of Sound Modes

The theory predicts two distinct sound modes in the stripe phase, corresponding to the two broken symmetries:

1. First Sound (Longitudinal): Primarily a density wave, analogous to the first sound in standard superfluids but modified by the crystalline order.
2. Second Sound (Transverse/Crystalline): A mode associated with the oscillation of the stripe displacement (phonon of the crystal). In the SOC case, this mode exhibits significant spin oscillations.

C. Anisotropy and Directional Dependence

The sound speeds are highly anisotropic, depending on the angle $\phi$ between the propagation direction and the stripe orientation:

• Propagation Parallel to Stripes ($\phi = 0$): Both modes are longitudinal mixtures of superfluid and crystalline oscillations.
• Propagation Perpendicular to Stripes ($\phi = \pi/2$):
  • The modes decouple.
  • One mode is a standard superfluid sound ($c_1 = \sqrt{\kappa^{-1}}$).
  • The other is a purely transverse dispersive mode ($c_2 = \sqrt{\lambda/\rho_n}$) associated with stripe rotation. In SOC systems, this transverse mode remains robust due to a large rotational restoring force.

4. Key Results

• Sound Speeds: The authors derived explicit formulas for the two sound speeds $c_{1,2}(\phi)$.
  • For dipolar gases, the second sound speed vanishes when the magnetic field is aligned vertically ($\alpha=0$) because the rotational restoring force ($\lambda_y$) disappears.
  • For SOC gases, the second sound is less anisotropic because the Raman lasers provide a rotational restoring force ($\Lambda_y$) comparable to the compressional constant ($\Lambda_x$).
• Mode Characterization:
  • Dipolar: The second sound is a transverse mode involving stripe displacement.
  • SOC: The second sound involves not only stripe displacement but also spin polarization oscillations ($\delta s_z$), a direct consequence of the spin-orbit coupling.
• Validation: The hydrodynamic predictions for dipolar gases were benchmarked against real-time eGPE simulations, showing excellent agreement for both longitudinal and transverse modes.
• Comparison with Liquid Crystals: The authors draw parallels with the smectic-A phase of liquid crystals. They note that while the mathematical structure is similar, the physical origin differs: the "first sound" in these supersolids is a collisionless sound (unlike the collisional sound in liquid crystals), and the "second sound" is modified by superfluidity.

5. Significance

• Theoretical Unification: The work provides a comprehensive framework to compare two experimentally distinct routes to supersolidity, highlighting how fundamental symmetries (Galilean invariance) dictate macroscopic hydrodynamic behavior.
• Experimental Guidance: The paper predicts specific anisotropic behaviors and spin-oscillation signatures in SOC supersolids. These predictions offer clear targets for experimental verification, particularly regarding the transverse nature of the second sound and the role of the negative "laser density" component.
• New Physics in SOC Systems: The identification of a negative normal density component ($\rho_{laser}$) in the SOC supersolid phase challenges standard intuition about fluid components and suggests a unique "three-fluid" behavior even at zero temperature.
• Bridge to Condensed Matter: By linking ultracold gas physics to the hydrodynamics of liquid crystals and standard solids, the paper enriches the understanding of Goldstone modes in systems with broken translational symmetry.

In conclusion, the paper successfully demonstrates that while the microscopic origins of supersolidity differ, the macroscopic sound propagation is governed by a unified set of hydrodynamic principles, with the specific nature of the sound modes serving as a fingerprint of the underlying symmetry breaking and invariance properties of the system.