WSi weak link element with a non-sinusoidal current-phase relation

This paper demonstrates that a tungsten silicide constriction embedded in a 3D RF-SQUID exhibits strong nonlinearity consistent with a sawtooth-like current-phase relation or quantum phase slip behavior, enabling the measurement of relaxation times for metastable persistent-current states.

The Gist

The Big Picture: Building a Better "Quantum Switch"

Imagine you are trying to build a computer that uses the laws of quantum physics (the weird rules that govern atoms) to solve problems. To make this work, you need a special kind of switch that can be in two states at once (a "qubit").

Most of these switches are made using a specific type of barrier (like a thin layer of aluminum oxide) that acts like a tunnel. However, these tunnels can be messy. They sometimes have tiny, unwanted "glitches" (fluctuators) that make the computer unstable, or they have extra "parasitic" parts that make them hard to control.

The Goal of this Paper:
The researchers wanted to see if they could make a cleaner, simpler switch by removing the "tunnel" barrier entirely. Instead, they used a tiny, narrow bridge made of a special material called Tungsten Silicide (WSi). They wanted to see if this bridge could act as a "weak link" that behaves like a quantum switch, but without the messy tunnel.

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The Experiment: The "Magnetic Roller Coaster"

To test this, the team built a device called an RF-SQUID. Think of this as a superconducting loop (a ring of wire with no resistance) that has a tiny gap in it. This gap is the "weak link" made of the WSi material.

They placed this ring inside a copper box (a cavity) and shined microwave signals at it, like tuning a radio. They also had a way to push magnetic fields through the ring, acting like a remote control to change the shape of the energy landscape.

The Analogy: The Ball in the Valley

Imagine the energy inside this ring is like a landscape with hills and valleys.

• The Ball: A tiny particle (representing the quantum state) sits in one of these valleys.
• The Shape of the Valley: This depends on the material.
  • Normal Switches (Sinusoidal): Usually, these valleys look like smooth, round bowls (like a standard sine wave).
  • This New Switch (Sawtooth): The researchers found that their WSi bridge created valleys that looked like sawteeth or sharp, jagged peaks.

When they changed the magnetic field, they watched how the "ball" moved. They measured the frequency at which the device "sang" (resonated).

• The Result: The way the frequency changed matched the "sawtooth" pattern perfectly. It didn't look like a smooth curve; it looked like a series of flat steps that suddenly dropped. This proved that the WSi bridge wasn't acting like a standard tunnel, but like a unique, sharp-edged quantum element.

They also tested a second theory: that the bridge might act like a Quantum Phase Slip.

• The Analogy: Imagine a rope tied in a knot. Sometimes, the knot can suddenly "slip" and untie itself, changing the state of the rope. In their material, the "knot" (the quantum phase) slips through the narrow bridge.
• The Result: This theory also fit the data perfectly. The device behaved as if it was a "sawtooth" switch OR a "knot-slipping" switch. Both models described the data equally well.

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The "Sleeping Giant": Long-Lasting States

One of the most exciting findings was about how long these states last.

In many quantum computers, the "ball" in the valley is unstable. It rolls out of the valley quickly (in nanoseconds or microseconds) because the walls are too thin or the energy is too high. This is like trying to balance a pencil on its tip; it falls over immediately.

What they found:
Because the WSi bridge creates such deep, sharp "sawtooth" valleys, the ball gets stuck very securely.

• The Analogy: Imagine the ball is in a deep, narrow canyon with very high, steep walls. It takes a massive amount of energy for the ball to climb out.
• The Measurement: They prepared the device in a specific state and then just waited. They watched to see how long it took for the state to "decay" (fall out of the valley).
• The Result: The state lasted for over an hour. In the world of quantum computing, where things usually disappear in a blink of an eye, an hour is an eternity. It's like the difference between a house of cards collapsing instantly and a stone fortress standing for a century.

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Summary of Claims

1. New Material: They successfully used a disordered, amorphous material (Tungsten Silicide) as a "weak link" in a superconducting circuit.
2. Non-Sinusoidal Behavior: Unlike standard switches that have smooth, round energy curves, this material creates a "sawtooth" shape. This is a desirable trait for making better quantum computers because it offers more protection against errors.
3. Two Models Fit: The data fits two different mathematical descriptions:
   • A Josephson Junction with a sawtooth shape.
   • A Quantum Phase Slip element (where the quantum "knot" slips through).
   • Note: The paper states that based on this specific experiment, they cannot tell which of the two models is the exact truth, but both work to describe the behavior.
4. Extreme Stability: They demonstrated that the quantum states trapped in this material are incredibly stable, with relaxation times (how long they last) reaching over an hour.

What the Paper Does Not Claim

• They do not claim to have built a working quantum computer yet.
• They do not claim this material is the "best" for every application, only that it is a viable new option for creating nonlinear elements.
• They do not discuss medical uses or commercial products; this is purely fundamental physics research.

In short, the researchers found a new way to build a "quantum switch" that is sharper, cleaner, and holds its state for a very long time, opening the door to potentially more robust quantum devices.

Technical Summary

Technical Summary: WSi Weak Link Element with a Non-Sinusoidal Current-Phase Relation

Problem Statement
Nonlinearity is a fundamental requirement for encoding quantum states with non-uniform energy spacing, a prerequisite for implementing qubits, coherent gates, and signal processing in superconducting quantum circuits. While traditional superconducting qubits rely on Josephson junctions (typically aluminum-oxide tunnel barriers), these elements suffer from specific limitations, including the presence of unwanted two-level fluctuators, unpredictable circuit parameters, parasitic capacitances, and restrictions on operating temperatures. Alternative approaches utilizing superconducting weak links—nanoscale constrictions with dimensions comparable to the coherence length—offer a route to nonlinearity without introducing distinct barrier materials. However, the specific behavior of weak links made from amorphous, high-kinetic inductance materials like tungsten silicide (WSi) within a superconducting loop requires further characterization to determine if they function as Josephson junctions with non-sinusoidal current-phase relations (CPR) or as quantum phase slip (QPS) elements.

Methodology
The authors fabricated a radio-frequency superconducting quantum interference device (RF-SQUID) incorporating a WSi nanowire as the nonlinear weak link element. The device consists of a superconducting loop containing a 40 × 40 nm WSi nanowire (3 nm thick, with a kinetic sheet inductance of $\approx 300$ pH/$\square$), shunted by a large on-chip aluminum capacitor ($C_B = 646$ fF). This circuit is embedded in a three-dimensional copper cavity to facilitate strong coupling and readout.

The experimental methodology involved:

1. Microwave Spectroscopy: Measuring the transmission magnitude ($|S_{21}|$) across the cavity while ramping the external magnetic flux ($\Phi_{ext}$) to observe flux-dependent resonance frequencies.
2. Theoretical Modeling: Two distinct lumped-element circuit models were developed to interpret the data:
   • Josephson Junction (JJ) Model: Treating the constriction as a weak-link junction with a tunable energy-phase relation $E(\phi)$. The authors tested both sinusoidal and sawtooth-like CPRs.
   • Quantum Phase Slip (QPS) Model: Treating the nanowire as a dual element where coherent tunneling of fluxons occurs, described by an energy-charge relation. This required a symplectic geometry-based approach to circuit quantization to derive the Hamiltonian.
3. Time-Domain Measurements: Using single-shot readout to monitor the decay of metastable, persistent-current states trapped in local potential minima. This involved initializing the system, ramping the flux to a metastable state, and tracking the transmission amplitude over time to measure relaxation lifetimes.

Key Results

• Non-Sinusoidal CPR: The experimental resonance frequency data exhibited a flux-dependent behavior characterized by a relatively flat response over multiple flux quanta, followed by sharp, hysteretic jumps at critical flux values ($\Phi_c^{ext} \approx 1.56 \Phi_0$). This behavior deviated significantly from the predictions of a standard sinusoidal CPR.
• Model Consistency: Both theoretical models provided excellent fits to the experimental data.
  • The JJ model required a strongly skewed, sawtooth-like CPR (skewness parameter $\chi \approx 1$) to reproduce the flat resonance curves and the specific shape of the downturn near the flux jumps.
  • The QPS model also successfully described the data, attributing the frequency downturn to quantum mechanical coupling between neighboring potential wells rather than the shape of the CPR.
  • The authors note that the low-energy spectra of both models are indistinguishable in the realized parameter regime, meaning the resonance measurements alone cannot definitively distinguish between the two physical interpretations.
• Long-Lived Metastable States: Time-domain measurements revealed exceptionally long relaxation times for excited fluxon states. Depending on the barrier height (controlled by the external flux detuning), relaxation times ranged from tens of seconds to over one hour. The longest observed lifetime exceeded 3600 seconds. This longevity is attributed to the large characteristic junction energy ($E_0^{WSi} \approx 10^55$ GHz) and small charging energy, which strongly localize states in different potential valleys, thereby reducing transition matrix elements between eigenstates.

Significance and Claims
The paper demonstrates the feasibility of using amorphous tungsten silicide (WSi) nanowires as nonlinear elements in superconducting circuits. The primary contribution is the experimental observation of a weak link behaving consistently with either a Josephson junction possessing a sawtooth-like CPR or a quantum phase slip element.

The authors claim that this work opens possibilities for realizing weak-link quantum devices. Specifically, the observation of hour-long relaxation times in metastable states highlights the potential of WSi weak links for creating protected quantum states with reduced sensitivity to charge noise and offset charges. The work establishes WSi not only as a material for linear inductors (as previously used in fluxonium qubits) and single-photon detectors, but also as a viable candidate for nonlinear circuit elements capable of supporting complex quantum dynamics. The paper concludes modestly, noting that while the specific physical mechanism (JJ vs. QPS) remains ambiguous based on the current data, the material's nonlinear properties are robust and controllable via external flux.