Localized quasiparticles in a fluxonium with quasi-two-dimensional amorphous kinetic inductors

This paper investigates tungsten silicide fluxonium qubits and resonators fabricated from quasi-two-dimensional amorphous films, revealing that energy loss is primarily driven by localized quasiparticles trapped in spatial variations of the superconducting gap, with loss increasing alongside the level of disorder.

The Gist

Imagine you are trying to build a super-fast, super-precise digital clock that runs on electricity. To make this clock work, you need a special kind of "spring" (an inductor) that stores energy in a magnetic field. In the world of quantum computers, these springs are crucial for holding information.

Usually, these springs are made of standard metals. But scientists want springs that are tiny yet strong. To get this, they use a special, messy material called Tungsten Silicide (WSi). Think of WSi not as a smooth highway, but as a bumpy, rocky path. This "messiness" is actually a feature, not a bug; it makes the material act like a very strong spring in a very small space.

However, there's a catch. When you make these materials too thin (thinner than a human hair, of course), they start to behave strangely. The paper you read investigates exactly what goes wrong when we use these ultra-thin, rocky springs in quantum computers.

Here is the story of what they found, broken down into simple concepts:

1. The "Rocky Road" Problem

Imagine the electrons (the tiny particles carrying electricity) are like cars driving on a road.

• In a normal metal: The road is smooth asphalt. The cars drive fast and easily.
• In WSi: The road is full of potholes and rocks. The cars have to slow down and struggle. This struggle creates a lot of "friction," which scientists call Kinetic Inductance. This is good because it lets us make tiny, powerful springs.

But, because the road is so bumpy, some cars get stuck in the potholes. In physics terms, these stuck cars are called Quasiparticles. They are like electrons that have lost their partners (Cooper pairs) and are now wandering around, causing trouble.

2. The Experiment: Testing the Springs

The researchers built two types of test tracks to see how well these WSi springs work:

• The Resonator: Think of this as a swing set. They pushed the swing and watched how long it kept swinging before stopping. The longer it swings, the better the material is.
• The Fluxonium Qubit: This is the actual "clock" or memory unit of a quantum computer. They tried to keep a specific state of energy alive in this clock and measured how long it lasted before fading away.

They tested two versions of the WSi road:

• Thick Road (10 nm): A bit smoother, with fewer potholes.
• Thin Road (3 nm): Very bumpy, with lots of potholes.

3. The Big Discovery: The "Stuck Cars" are the Culprit

The scientists found that the thinner the road, the worse the performance. The swings stopped sooner, and the quantum clock lost its memory faster.

Why? Because in the thin, bumpy roads, the "stuck cars" (quasiparticles) are everywhere.

• The Analogy: Imagine trying to run a race on a track where random people are standing in your way. The more people standing there (more disorder/thinner film), the harder it is to finish the race.
• The Twist: When they turned up the power (pushed the swing harder), something interesting happened. The "stuck cars" got shaken loose from the potholes and started running again. For a moment, the track became smoother, and the performance got better. But if they pushed too hard, they broke the cars apart entirely, and performance crashed again.

4. Why This Matters

This paper is a detective story. Before this, scientists knew WSi was a great material for making tiny quantum springs, but they didn't know why the quantum computers using it were sometimes losing information so quickly.

They proved that the main villain isn't the material itself, but the trapped quasiparticles caused by the material's disorder.

• The Good News: Now that we know the problem, we can fix it. We can design better quantum computers by managing these "stuck cars" or choosing materials that don't trap them as easily.
• The Future: This research helps us build better "super-inductors." These are the building blocks for the next generation of quantum computers, which promise to solve problems that are impossible for today's computers.

Summary in a Nutshell

The researchers took a messy, bumpy material (WSi) and tried to use it to build tiny springs for quantum computers. They discovered that while the material is great for making small springs, its "bumpiness" traps tiny energy particles (quasiparticles) that act like potholes, slowing down the computer and causing errors. By understanding exactly how these potholes work, they can now figure out how to pave a smoother road for the future of quantum technology.

Technical Summary

1. Problem Statement

Disordered superconducting materials with high kinetic inductance are essential for creating non-linear circuit elements and high-impedance environments in quantum circuits (e.g., fluxonium qubits, parametric amplifiers). However, their utility is often limited by high intrinsic dissipation (loss).

• The Challenge: While materials like granular aluminum, NbTiN, and TiN have been studied, the specific loss mechanisms in amorphous tungsten silicide (WSi)—a material widely used for single-photon detectors but less explored in coherent quantum circuits—remain unclear.
• The Question: Is the loss in WSi-based devices dominated by dielectric losses (two-level systems), geometric factors, or quasiparticle dynamics? Specifically, how do spatial variations in the superconducting gap in quasi-two-dimensional (2D) amorphous films affect device performance?

2. Methodology

The authors fabricated and characterized microwave resonators and fluxonium qubits using quasi-2D amorphous WSi films.

• Material Fabrication:
  • Films were co-sputtered on sapphire wafers with a fixed stoichiometry ($W_{0.85}Si_{0.15}$).
  • Two distinct film thicknesses were used to tune kinetic inductance ($L_K$):
    • ~3 nm thickness: $L_K \approx 300$ pH/□ (quasi-2D, thickness < coherence length $\xi \approx 7$ nm).
    • ~10 nm thickness: $L_K \approx 100$ pH/□.
  • Films were capped with a 2 nm Si layer to prevent aging.
• Device Architecture:
  • Resonators: Both distributed (coplanar waveguide) and lumped-element resonators were fabricated. Crucially, the geometry was designed to alter the electric field participation ratio of the WSi surface by two orders of magnitude between the two types, allowing the authors to distinguish between surface dielectric loss and bulk/inductive loss.
  • Fluxonium Qubits: WSi wires served as the high-inductance shunt (linear inductor) in fluxonium circuits, with standard aluminum Josephson junctions and capacitors.
• Measurement Techniques:
  • Resonator Spectroscopy: Measured internal quality factors ($Q_{int}$) and resonance frequency shifts as a function of drive power (photon number) and frequency.
  • Qubit Spectroscopy: Measured relaxation times ($T_1$) of the fluxonium excited states across different flux biases (tuning qubit frequency).
  • Modeling: Used finite-element simulations (Sonnet/HFSS) to calculate field participation and compared experimental data against theoretical models for quasiparticle-induced loss and dielectric loss.

3. Key Contributions

• First Integration of WSi in Fluxonium: Demonstrated the feasibility of using amorphous WSi as a linear inductor in fluxonium qubits, achieving performance comparable to other disordered superconductors.
• Identification of Dominant Loss Mechanism: Provided strong evidence that localized quasiparticles trapped in spatial fluctuations of the superconducting gap are the primary source of loss, rather than dielectric two-level systems (TLS).
• Dimensionality Effects: Showed that reducing the film thickness below the coherence length (enhancing phase fluctuations) significantly increases the density of localized quasiparticles and degrades performance.
• Nonlinear Dynamics: Observed non-monotonic power dependence in resonator quality factors, attributed to the recombination of localized quasiparticles at low power and pair-breaking at high power.

4. Key Results

A. Resonator Measurements

• Quality Factors: $Q_{int}$ ranged from $10^4$ to $10^5$. Thinner films ($300$ pH/□) exhibited lower $Q$ than thicker films ($100$ pH/□), indicating higher loss in the more disordered/quasi-2D regime.
• Loss Mechanism Differentiation:
  • Despite a 100x difference in electric field participation between distributed and lumped resonators, the quality factors were similar for the same film. This rules out surface dielectric loss (TLS) as the dominant factor.
  • $Q_{int}$ increased with resonance frequency, a signature characteristic of quasiparticle loss (scaling as $1/\sqrt{f}$).
• Power Dependence:
  • At low photon numbers, $Q_{int}$ increased with power. This is explained by the excitation of localized quasiparticles out of shallow traps, allowing them to recombine faster.
  • At high power, $Q_{int}$ dropped due to Cooper pair breaking.
  • Resonance frequency shifts showed a non-monotonic behavior (initial increase due to reduced kinetic inductance from recombination, followed by a drop due to Kerr shift).

B. Fluxonium Qubit Measurements

• Relaxation Times ($T_1$): Measured $T_1$ increased with qubit frequency, consistent with the quasiparticle loss model.
• Quasiparticle Density: The extracted quasiparticle density ratios ($x_{qp}$) from the qubit data ($\sim 10^{-5}$) matched those derived from resonator measurements.
• Model Validation:
  • The authors modeled the WSi inductor as an array of Josephson junctions (representing superconducting grains separated by insulating barriers).
  • Theoretical predictions for relaxation rates due to quasiparticle tunneling across these "junctions" matched the experimental $T_1$ data perfectly.
  • In contrast, a dielectric loss model predicted that $T_1$ should decrease with frequency, which contradicted the experimental data.

5. Significance and Conclusion

• Material Optimization: The study establishes that while WSi is a viable material for high-kinetic-inductance circuits, its performance is fundamentally limited by localized quasiparticles arising from disorder-induced spatial variations in the superconducting gap.
• Design Implications: To improve coherence, future designs must focus on minimizing these spatial fluctuations or engineering the material to reduce the density of localized states, rather than just optimizing surface treatments for TLS.
• Scalability: The successful integration of WSi into fluxonium qubits opens the door for using WSi nanowires as nonlinear elements (e.g., weak-link Josephson elements or quantum phase slip junctions) in advanced quantum circuits.
• Fundamental Physics: The work provides a clear experimental validation of the "granular" model of disorder in amorphous superconductors, where the material behaves effectively as a network of Josephson junctions, with quasiparticle tunneling being the primary dissipation channel.

In summary, this paper resolves the loss mechanism in amorphous WSi devices, identifying localized quasiparticles as the bottleneck, and provides a roadmap for utilizing high-kinetic-inductance materials in next-generation superconducting quantum hardware.