Nonequilibrium plasmon liquid in a Josephson junction chain

This paper demonstrates the emergence of a strongly interacting, nonequilibrium liquid of plasmons in a Josephson junction chain by using multimode spectroscopy to reveal how strong driving transforms weak pairwise mode couplings into high-order cascaded interactions that redistribute energy non-locally across hundreds of modes.

The Gist

Imagine a long, crowded hallway filled with people (let's call them "energy dancers"). In a calm, quiet state, everyone is just standing in their own spot, maybe swaying slightly to their own rhythm. They don't really interact with the person next to them. This is how most quantum systems behave when they are at rest: a collection of independent, non-interacting waves.

But what happens if you start blasting loud music and shaking the floor? Suddenly, the dancers can't ignore each other. They bump, they push, they grab hands, and they start moving in complex, chaotic groups. The hallway stops being a collection of individuals and becomes a single, churning, liquid-like mass of movement.

This paper is about creating and studying exactly that kind of "churning mass" using a superconducting wire made of thousands of tiny electrical junctions (a Josephson junction chain). Here is the breakdown of their discovery:

1. The Playground: A Superconducting Highway

The scientists built a 5-millimeter-long chain of tiny electrical switches (Josephson junctions). Think of this chain as a super-highway for light waves (specifically, microwave photons).

• The Cars: The "cars" driving on this highway are called plasmons. They are ripples of energy moving through the wire.
• The Traffic Rules: In a normal, quiet state, these ripples act like independent cars. They drive their own lanes and rarely crash into each other.

2. The Experiment: Turning Up the Volume

The researchers wanted to see what happens when you force these waves to interact violently. They used a technique called multimode spectroscopy.

• The Analogy: Imagine you have a giant piano. Usually, if you press one key, only that note rings out. But in this experiment, they pressed two specific keys (pumps) very hard while listening to a third key (the read-out).
• The Weak Drive (The Gentle Tap): When they pressed the keys gently, the third key just shifted its pitch slightly. The waves were still mostly independent, just nudging each other.
• The Strong Drive (The Rock Concert): When they turned up the volume, things got crazy. The waves started crashing into each other so hard that they couldn't be distinguished as individual notes anymore. They formed a cascading chain reaction.

3. The Discovery: The "Plasmon Liquid"

The most exciting part of the paper is the discovery of the Nonequilibrium Plasmon Liquid.

• The Cascade Effect: When the energy is high, a wave doesn't just bump into one neighbor; it triggers a domino effect. Wave A hits Wave B, which hits Wave C, which hits Wave D, and so on. This happens so fast and so thoroughly that the energy spreads out across the entire chain instantly.
• The Liquid State: Instead of distinct, separate waves (like individual cars), the energy behaves like a liquid. If you poke the liquid in one spot, the ripple spreads everywhere immediately. The energy redistributes itself across hundreds of different modes (frequencies) in a way that looks like a continuous fluid rather than a collection of particles.

4. Why This Matters

This isn't just about fancy physics; it's about understanding how complex systems behave when they are pushed to their limits.

• The "Traffic Jam" vs. The "Flow": Usually, when things get crowded, they jam up (thermalize slowly). But here, the scientists found a regime where the interactions are so strong that the system flows like a super-fluid, even though it's being driven by external power.
• The "Phase Slip" Surprise: At the very highest energy levels, they saw something unexpected. It's as if the dancers in our hallway started tripping and falling over in a specific way (called "quantum phase slips"). This suggests that at extreme energies, the rules of the highway change completely, and new, chaotic behaviors emerge.

The Big Picture

Think of this research as learning how to control a tsunami of energy inside a tiny wire.

• Before: We knew how to make waves that didn't talk to each other.
• Now: We know how to make them talk, argue, and merge into a single, powerful, liquid-like entity.

This is a huge step forward for quantum engineering. If we want to build better quantum computers or super-sensitive sensors, we need to understand how to manage these "liquid" states of energy. The scientists have essentially found the "on/off" switch and the "volume knob" for turning a quiet quantum system into a roaring, interacting liquid, and they've mapped out exactly how the energy flows when you do it.

Technical Summary

1. Problem Statement

Equilibrium quantum systems are typically described as gases of weakly interacting normal modes. However, driving these systems far from equilibrium can drastically enhance mode-to-mode interactions, potentially leading to the emergence of a "liquid" state characterized by strong collective dynamics.

• The Gap: While multimode resonators are used in quantum computing and sensing, the frontier of high energy-density nonlinear dynamics remains poorly understood. Specifically, the transition from pairwise mode coupling to a strongly interacting, non-equilibrium "plasmon liquid" with near-continuum internal dynamics has not been experimentally characterized.
• The Question: What are the internal dynamical processes of this plasmon liquid, and how can they be observed and characterized in an engineered superconducting circuit?

2. Methodology

The authors utilized a 5 mm-long chain of Josephson junctions (JJ-chain) as a pristine realization of a one-dimensional multimode quantum system.

• System Physics: The chain acts as a low-speed-of-light microwave resonator. The plasma modes are described by an effective bosonic Hubbard Hamiltonian containing a linear term ($H_0$) and a quartic interaction term ($H_{int}$) arising from the Josephson potential nonlinearity.
• Experimental Platform:
  • Fabricated using Al/AlOx/Al junctions on a silicon chip.
  • Operated at a base temperature of 12 mK in a dilution refrigerator.
  • Connected to microwave lines for transmission ($S_{21}$) and reflection measurements.
• Spectroscopy Techniques:
  1. Multimode Microwave Spectroscopy: Used to probe the system under weak and strong driving conditions.
  2. Three-Tone Spectroscopy: Two strong pump tones drive modes $p$ and $q$, while a weak read-out tone probes mode $k$. This allows the study of energy and momentum-conserving scattering processes ($m, n \to k, l$).
  3. Noise Power Measurements: Used to directly detect scattered photons and measure mode occupations in the non-equilibrium steady state (NESS).
  4. Incoherent Broadband Drives: A noise source was used to drive a subset of modes (1–13) to induce a NESS and study intrinsic linewidth broadening.

3. Key Contributions

• Microscopic Origin of Nonlinearity: The authors determined that the dominant interaction mechanism in the weak-drive regime is the gradient nonlinearity of the Josephson potential (scaling with wavenumber $k$), rather than quantum phase slips.
• Observation of Cascaded Couplings: They demonstrated that strong driving leads to cascaded photon scattering, where a mode couples not just to nearest neighbors, but to modes separated by integer multiples of the pump detuning ($k \pm i\delta$), creating a complex web of hybridized modes.
• Realization of a Plasmon Liquid: By scaling to many-mode drives, they stimulated interactions between hundreds of modes, resulting in a non-thermal steady state with near-continuum internal dynamics, explicitly verifying the emergence of a strongly interacting, non-equilibrium plasmon liquid.
• Kinetic Framework Validation: They extended the kinetic framework for thermal transport in 1D quantum liquids to describe the system, successfully matching experimental linewidth broadening and mode occupation distributions with theoretical kinetic equations.

4. Key Results

A. Weak-Drive Regime: Pairwise Coupling

• Avoided Crossings: When two modes ($p, q$) are pumped, the read-out mode $k$ couples to "nearest-neighbor" modes ($k \pm \delta$) via energy-conserving four-wave mixing.
• Coupling Strength: The effective coupling strength $g$ was measured to scale as $g \propto \sqrt{n_p n_q}$ (where $n$ is photon occupation), confirming the quartic nature of the interaction.
• Scaling: The coupling strength $g$ was found to increase linearly with the mode number $k$ ($g \propto k$). This confirmed that the nonlinearity arises from the gradient expansion of the Josephson potential, as opposed to quantum phase slips (which would scale inversely with $k$).

B. Strong-Drive Regime: Cascaded Photons

• High-Order Processes: As drive power increased, the system exhibited cascaded couplings. A photon in mode $k$ could scatter to $k \pm \delta$, which then scatters to $k \pm 2\delta$, and so on.
• Spectral Signatures: Transmission spectra showed non-Lorentzian lineshapes with multiple peaks and avoided crossings corresponding to $i$-th order scattering processes ($i = \pm 1, \pm 2, \dots$).
• Direct Detection: Noise measurements confirmed the emission of scattered photons at frequencies corresponding to these higher-order processes, demonstrating near-complete, cascaded down-conversion efficiency.

C. Many-Mode Drives: The Plasmon Liquid

• Linewidth Broadening: Under incoherent broadband driving of the first 13 modes, the linewidths of unpumped modes ($k > 13$) increased significantly ($\delta\kappa$).
  • At intermediate drives, $\delta\kappa \propto k^2$, consistent with intrinsic scattering from gradient nonlinearities.
  • At the strongest drives, a non-monotonic dependence emerged with a low-frequency scaling of $\delta\kappa \propto k^{-2}$, suggesting the onset of quantum phase slips in the highly non-equilibrium regime.
• Non-Local Redistribution: The system reached a Non-Equilibrium Steady State (NESS) where photon occupation was redistributed globally. Modes far from the driven set ($k \gg 13$) showed occupation numbers orders of magnitude higher than thermal equilibrium, proving that intrinsic scattering efficiently redistributes energy across the entire spectrum.
• Hydrodynamic Regime: The interaction-induced broadening exceeded all other loss mechanisms, indicating the system had entered a hydrodynamic regime for plasmons.

5. Significance

• Fundamental Physics: This work provides the first experimental verification of a strongly interacting, non-equilibrium plasmon liquid in a superconducting circuit. It bridges the gap between classical wave turbulence theories and quantum many-body physics.
• Quantum Simulation: The JJ-chain serves as a versatile platform for simulating 1D quantum liquids and studying phenomena like quasiparticle lifetimes and thermalization in regimes previously inaccessible.
• Device Engineering: The findings have implications for the design of driven quantum devices, highlighting the limits imposed by intrinsic multimode interactions and the potential for engineering non-Hermitian lattices or wave turbulence.
• Theoretical Validation: The study validates the use of kinetic equations (Boltzmann-like) to describe quantum systems far from equilibrium, even when the system is dominated by classical-like nonlinear wave dynamics.

In summary, Bubis et al. successfully mapped the transition from weakly interacting normal modes to a strongly coupled, liquid-like state of plasmons, establishing Josephson junction chains as a premier platform for exploring non-equilibrium quantum matter.