Bosonic quantum mixtures with competing interactions: quantum liquid droplets and supersolids

This paper reviews the formation and properties of ultradilute quantum liquid droplets stabilized by quantum fluctuations in bosonic mixtures and dipolar gases, while also exploring the emergence and characteristics of supersolidity in both dipolar systems and spin-orbit-coupled mixtures.

The Gist

The Big Picture: A Tale of Two Exotic States

Imagine you are a physicist playing with a cloud of ultra-cold atoms. Usually, these atoms act like a gas: they bounce around, spread out, and fill whatever container you put them in. But this paper describes how scientists can trick these atoms into doing two very strange things:

1. Turning into a self-bound liquid droplet (like a drop of water that doesn't need a cup to hold it).
2. Turning into a "Supersolid" (a material that is rigid like a crystal but flows like a super-fluid).

The secret ingredient to making these magic states happen is competition. The scientists set up a tug-of-war between different forces acting on the atoms.

---

Part 1: The Quantum Liquid Droplet (The "Self-Bound Drop")

The Problem: Gas vs. Liquid

In the real world, liquids (like water) exist because molecules attract each other to stick together, but they also repel each other slightly so they don't crush into a single point. If you have a gas of atoms that only attract each other, they collapse and disappear. If they only repel, they fly apart.

The Solution: The "Tug-of-War"

In these experiments, scientists use two types of atoms (or two different "personalities" of the same atom). They tune the interactions so that:

• Force A (Mean-Field): The atoms want to attract each other and collapse.
• Force B (Quantum Fluctuations): A weird, invisible "jitter" from quantum mechanics pushes them apart.

Usually, the attraction is so strong that the "jitter" doesn't matter. But the scientists fine-tune the system so the attraction is almost canceled out. Suddenly, the tiny "jitter" (quantum fluctuations) becomes the hero. It's strong enough to stop the collapse but weak enough to let the atoms stick together.

The Analogy: Imagine a group of people holding hands in a circle.

• Attraction: They are all leaning in, trying to hug each other so tightly they collapse into a pile.
• Repulsion (The Jitter): They are all shivering and vibrating slightly.
• The Result: If they lean in just a tiny bit too much, the shivering becomes the only thing keeping them from collapsing into a single point. They form a tight, stable circle that holds its shape without anyone pushing them from the outside. This is a Quantum Liquid Droplet.

Key Features:

• Self-Bound: If you take away the magnetic "bowl" holding the atoms, they don't fly apart. They stay together as a drop.
• Ultra-Dilute: Even though it's a liquid, it's incredibly empty compared to water. It's like a cloud that has the structure of a solid.

---

Part 2: The Supersolid (The "Flowing Crystal")

What is a Supersolid?

This sounds like a contradiction.

• A Solid is rigid; atoms are locked in a grid (like a marching band standing still).
• A Superfluid flows without friction; atoms move freely (like a river).
• A Supersolid is both. It has a crystal pattern, but you can pour it through a pipe without it getting stuck.

How They Make It Happen

The paper discusses two ways to make this, comparing them like two different recipes:

Recipe A: The Dipolar Gas (The "Magnetic Magnet")

• The Setup: Use atoms that act like tiny magnets (Dipolar gases).
• The Mechanism: The magnets attract each other in some directions and repel in others. This competition creates a chain of droplets (like a string of pearls).
• The Magic: If the droplets are close enough, atoms can "tunnel" (teleport) from one droplet to the next. This connects them all into one giant, flowing crystal.
• The Catch: This relies heavily on the "jitter" (quantum fluctuations) we talked about earlier to keep the droplets from collapsing.

Recipe B: The Spin-Orbit Coupled Mixture (The "Dancing Partners")

• The Setup: Use a mixture of atoms and shine lasers on them to create "Spin-Orbit Coupling."
• The Mechanism: This is like forcing the atoms to dance. The lasers link an atom's internal "spin" (like a compass needle) to its movement.
• The Result: The atoms naturally want to form two different groups moving in opposite directions. These two groups interfere with each other, creating a striped pattern (like a zebra).
• The Magic: Even though they are striped (crystal), the two groups are still connected and flowing together (superfluid).
• The Difference: Unlike the magnetic recipe, this one works even without the "jitter" help. It's a "mean-field" effect, meaning it's a more straightforward dance choreography.

How Do We Know It's Real?

The scientists proved these states exist by:

1. Taking Pictures: They zoomed in and saw the striped pattern (the crystal part).
2. Listening to the Music: They shook the system and listened to how it vibrated. A normal solid has one type of vibration; a superfluid has another. A supersolid has both types of vibrations at the same time.
3. Spinning It: In the magnetic experiments, they spun the gas and saw vortices (tiny tornadoes). Only a superfluid can do this.

---

Why Does This Matter?

Think of these experiments as a playground for the future of physics.

• Universality: These "Quantum Droplets" are universal. Their behavior doesn't depend on the specific type of atom, just on the rules of quantum mechanics.
• New Materials: Understanding how to make matter flow like a liquid but hold shape like a solid could lead to new technologies, perhaps in super-efficient energy transport or quantum computing.
• The Interface: The paper concludes by suggesting that if we mix these two ideas (Spin-Orbit Coupling + Quantum Droplets), we might discover even stranger states of matter where the "jitter" creates the crystal structure itself.

Summary in One Sentence

By carefully balancing the forces that pull atoms together and the quantum "jitter" that pushes them apart, scientists have created a new kind of matter that is a self-contained liquid drop and a flowing crystal at the same time.

Technical Summary

Based on the lecture notes by Sarah Hirthe and Leticia Tarruell, here is a detailed technical summary of the paper "Bosonic quantum mixtures with competing interactions: quantum liquid droplets and supersolids."

1. Problem Statement

The paper addresses two fundamental challenges in ultracold quantum gas physics:

1. Stabilizing Ultralight Liquids: In standard weakly interacting Bose gases, attractive mean-field interactions lead to catastrophic collapse. The problem is how to stabilize a self-bound liquid phase in the absence of external confinement without relying on strong interactions or dense packing (Van der Waals paradigm).
2. Realizing Supersolidity: Supersolidity—the simultaneous breaking of translational symmetry (crystalline order) and $U(1)$ gauge symmetry (superfluidity)—is a counterintuitive state of matter. While theoretically predicted decades ago, its experimental realization in equilibrium systems has been elusive. The paper investigates how competing interactions and quantum fluctuations can engineer this state in two distinct platforms: dipolar gases and spin-orbit-coupled mixtures.

2. Methodology

The authors employ a combination of theoretical frameworks and experimental analysis:

• Theoretical Frameworks:
  • Beyond-Mean-Field Theory: Utilizing the Lee-Huang-Yang (LHY) correction to the equation of state. This involves calculating the zero-point energy of Bogoliubov quasiparticles, which provides a repulsive term scaling as $n^{5/2}$, contrasting with the attractive mean-field term scaling as $n^2$.
  • Extended Gross-Pitaevskii Equation (eGPE): A modified mean-field equation incorporating the LHY term as a non-local or local density-dependent potential to describe the dynamics of quantum droplets.
  • Effective Mixture Models: Describing spin-orbit-coupled (SOC) systems as effective mixtures of dressed states with renormalized scattering lengths and masses.
  • Bogoliubov Analysis: Analyzing excitation spectra to identify soft modes (rotons) and Goldstone modes associated with symmetry breaking.
• Experimental Platforms:
  • Bose-Bose Mixtures: Using mixtures of atomic species (e.g., $^{39}$K, $^{41}$K-$^{87}$Rb) or internal spin states where interspin interactions ($g_{\uparrow\downarrow}$) are tuned via Feshbach resonances to compete with intraspin interactions.
  • Dipolar Gases: Utilizing highly magnetic atoms (Dysprosium, Erbium) where the anisotropic dipole-dipole interaction competes with contact interactions.
  • Spin-Orbit Coupling: Implementing Raman coupling between internal atomic states to create synthetic gauge fields and modify the single-particle dispersion relation.

3. Key Contributions

A. Quantum Liquid Droplets

• Mechanism of Stabilization: The paper elucidates the mechanism proposed by D. S. Petrov where fine-tuning competing interactions in a Bose-Bose mixture (or dipolar gas) cancels the attractive mean-field energy ($\delta g \to 0$). In this regime, the repulsive LHY quantum fluctuation term dominates, stabilizing a self-bound liquid droplet.
• Universality: Unlike classical liquids, these droplets are "ultradilute" (densities $\sim 10^8$ times lower than liquid helium) and universal; their properties depend only on scattering lengths, not the range of the interaction potential.
• Self-Bound Nature: The droplets exist without external traps. They exhibit a critical atom number ($N_c$) below which quantum pressure dissociates them into a gas.
• Dipolar vs. Mixture: The authors compare droplets in dipolar gases (anisotropic, stabilized by dipole-dipole vs. contact competition) with those in mixtures (isotropic, stabilized by interspin vs. intraspin competition).

B. Supersolidity

The paper reviews two distinct routes to supersolidity:

1. Dipolar Supersolids (Beyond Mean-Field):
   • Formed by arrays of quantum droplets stabilized by fluctuations.
   • Global phase coherence is established via particle exchange (tunneling) between droplets.
   • Requires a delicate balance where the system is in the "droplet array" regime but not fully phase-separated.
2. Spin-Orbit-Coupled (SOC) Supersolids (Mean-Field):
   • Emerges in repulsive Bose mixtures subjected to Raman-induced SOC.
   • The SOC creates a dispersion relation with two minima. When both minima are occupied, matter-wave interference creates a spontaneous density modulation (stripes).
   • Crucially, this phase can be described within a mean-field framework (unlike dipolar supersolids), as the "effective mixture" of dressed states becomes miscible.

4. Key Results

• Experimental Verification of Droplets:

  • Direct observation of self-bound droplets in $^{164}$Dy (dipolar), $^{39}$K mixtures, and $^{23}$Na-$^{133}$Cs polar molecules.
  • Measurement of the liquid-to-gas transition: As atom number decreases due to three-body losses, droplets dissociate at a critical $N_c$, confirmed by a sudden increase in cloud size.
  • Collision Dynamics: Observation of droplet merging vs. bouncing, providing evidence for surface tension and the crossover from compressible to incompressible regimes.
  • Soliton-Droplet Transition: In 1D waveguides, a bistable region was observed where solitons and droplets coexist, depending on interaction strength and atom number.

• Characterization of Supersolids:

  • Symmetry Breaking: Direct imaging of periodic density modulation (stripes) in SOC systems (using matter-wave magnification) and dipolar systems.
  • Phase Coherence: Verification of global phase coherence via interference fringes (time-of-flight) and Talbot interferometry.
  • Excitation Spectra:
    • Identification of two distinct Goldstone modes: a superfluid mode (breathing) and a crystal mode (compression/phonon).
    • Observation of mode softening: The crystal compression mode frequency drops to zero at the transition from the supersolid stripe phase to the plane-wave phase.
    • Vortices: Detection of quantized vortices in dipolar supersolids, providing a "smoking gun" for superfluidity.

• Comparison of Platforms:

  • Dipolar: Supersolidity arises from beyond-mean-field fluctuations stabilizing droplet arrays.
  • SOC: Supersolidity arises from mean-field interference of dressed states.
  • Excitations: Both platforms host compressible crystal structures (phonons), distinguishing them from cavity-based supersolids which often have rigid lattices.

5. Significance and Future Directions

• Fundamental Physics: The work provides the first robust experimental realization of supersolidity in equilibrium quantum gases, confirming that quantum fluctuations can stabilize exotic phases of matter previously thought impossible.
• Universality vs. Specificity: It highlights how different microscopic mechanisms (fluctuation-stabilized droplets vs. mean-field SOC interference) can lead to the same macroscopic phenomenology (supersolidity).
• Open Questions:
  • Superfluidity in Droplets: Proving superfluidity in isolated quantum droplets (via vortex nucleation) remains an open challenge due to short lifetimes and high densities.
  • Finite Temperature: The role of temperature on droplet stability and the critical atom number is not yet fully understood.
  • 2D Supersolids: Extending SOC supersolids to 2D crystals and investigating the Higgs mode (amplitude oscillations of the order parameter).
  • Hybrid Systems: The potential for creating dipolar mixtures to combine the benefits of both platforms (e.g., catalyzing supersolidity in unmodulated BECs).

In conclusion, these lecture notes synthesize a decade of research demonstrating that competing interactions in bosonic mixtures and dipolar gases serve as a powerful quantum simulator for engineering states of matter where quantum fluctuations are not just corrections, but the primary stabilizing force.