Highly-linear flux-to-voltage transducer based on superconducting quantum interference proximity transistors

This paper presents the experimental demonstration of a highly linear bi-SQUIPT flux transducer that utilizes a dual-loop architecture to cancel non-linearities, achieving a 60 dB spurious-free dynamic range and femtowatt power dissipation while operating stably up to 600 mK for scalable cryogenic quantum electronics.

The Gist

Imagine you are trying to listen to a whisper in a room full of shouting people. In the world of quantum computing and ultra-sensitive sensors, scientists need to measure tiny magnetic fields (the "whispers") with incredible precision. For decades, the gold standard for this job has been a device called a SQUID (Superconducting Quantum Interference Device).

However, the traditional SQUID has a major flaw: it's like a clock face. If you turn the magnetic field knob, the needle goes up and down in a perfect circle. It repeats itself every time it hits the top. This means if the signal gets too strong, the device gets confused, doesn't know which "lap" of the clock it's on, and the data becomes distorted. To fix this, engineers usually have to build a complex "feedback loop" (like a thermostat) to keep the needle locked in one spot. But this adds bulk, heat, and complexity, which is bad for delicate quantum computers that need to stay super cold and quiet.

The New Solution: The "Bi-SQUIPT"

The researchers in this paper have built a new device called a Bi-SQUIPT. Think of it as a clever upgrade that solves the "clock face" problem without needing the bulky thermostat.

Here is how it works, using some everyday analogies:

1. The Old Way vs. The New Way

• The Old SQUID: Imagine a single person trying to push a heavy swing. They push, the swing goes up, comes down, and goes back. If you push too hard, the swing goes over the top and the rhythm breaks.
• The Bi-SQUIPT: Now, imagine two people pushing two swings side-by-side. But here's the trick: they are pushing in opposite directions relative to each other. When one person's swing goes up (creating a voltage spike), the other person's swing goes down (creating a voltage dip).

2. The "Cancelling Act"

The genius of this device is differential readout. The scientists connect the two swings so they measure the difference between them.

• Because the two swings are slightly different (one might be a bit stiffer than the other), their individual "wobbles" (non-linearities) cancel each other out.
• The result? Instead of a wobbly, curved line, you get a straight, flat road. The device can handle much larger signals without getting confused or distorting the data.

3. The "Proximity" Magic

The name "SQUIPT" stands for Superconducting Quantum Interference Proximity Transistor.

• The Analogy: Imagine a superconductor (a material that conducts electricity with zero resistance) as a super-fast highway. Usually, electrons zip along smoothly.
• In a SQUIPT, the scientists put a "speed bump" (a weak link made of normal metal) in the middle of the highway.
• When a magnetic field passes over the loop, it changes the "texture" of the road at that speed bump. It's like the road suddenly becoming bumpy or smooth depending on the magnetic field.
• The device measures how hard it is for electrons to cross this bump. Because the road texture changes so smoothly with the magnetic field, the output is incredibly linear (straight).

Why is this a Big Deal?

The paper highlights three massive advantages:

1. Super Straight Line (High Linearity):
   The device is so linear that it can handle signals with a "Spurious-Free Dynamic Range" of 60 dB.

   • Analogy: Imagine trying to hear a pin drop while a jet engine is roaring. A normal device would get overwhelmed by the roar. This new device can hear the pin drop clearly, even with the jet engine running, without the sound getting garbled. It rivals much larger, more expensive arrays of sensors.

2. Ice Cold Efficiency (Low Power):
   Quantum computers must be kept near absolute zero. Any heat generated by the sensors can melt the delicate quantum states.

   • Analogy: Traditional sensors are like old incandescent lightbulbs—they get hot and waste energy. The Bi-SQUIPT is like a tiny LED that uses almost no power (femtowatts). It generates so little heat it's practically invisible to the cooling system.

3. Stability in the Cold:
   The device works perfectly up to 600 millikelvin (which is still incredibly cold, but "warm" for quantum standards).

   • Analogy: It's like a winter coat that keeps you warm even when the wind picks up, whereas older devices would freeze and stop working if the temperature rose just a tiny bit.

The Bottom Line

The scientists have built a compact, ultra-efficient, and incredibly accurate magnetic sensor. By using a "two-person team" approach (dual loops) to cancel out errors, they created a device that is as accurate as a giant array of sensors but fits in a tiny space and uses almost no power.

This is a game-changer for quantum computing. It means we can read the status of quantum bits (qubits) much more clearly and with less heat, paving the way for larger, more powerful quantum computers that don't require massive, complex cooling systems for every single sensor.

Technical Summary

1. Problem Statement

Superconducting Quantum Interference Devices (SQUIDs) are the gold standard for ultra-sensitive magnetometry but suffer from a fundamental limitation: their flux-to-voltage transfer function is inherently periodic and quasi-sinusoidal. This non-linearity severely restricts their dynamic range and introduces harmonic distortion when processing large signals.

• Current Mitigation: Conventional SQUIDs use Flux-Locked Loop (FLL) electronics to linearize the response. However, FLLs introduce electronic complexity, bandwidth constraints, and additional noise, hindering scalability for large-scale quantum circuits.
• Existing Alternatives:
  • Bi-SQUIDs: Attempt to linearize the response by adding a third Josephson junction. While theoretically promising (>120 dB linearity), experimental realizations struggle to match predictions (saturating below ~50 dB) due to stringent fabrication symmetry requirements. Furthermore, they often operate in the nano-to-microwatt power dissipation regime, imposing a heavy thermal load on dilution refrigerators.
  • Superconducting Quantum Arrays (SQAs): Offer high linearity but require large physical footprints and high power consumption, making them unsuitable for dense integration at the millikelvin stage.
• The Gap: There is a critical need for a flux-to-voltage transducer that combines intrinsic linearity, ultra-low power dissipation (femtowatt range), and compactness suitable for cryogenic quantum electronics.

2. Methodology

The authors present the first experimental demonstration of the bi-SQUIPT (Bi-SQUID based on Superconducting Quantum Interference Proximity Transistors).

• Device Architecture:
  • The device consists of two SQUIPTs connected in parallel, sharing a common ground loop.
  • SQUIPT Principle: Unlike SQUIDs that rely on phase-dependent supercurrents, SQUIPTs exploit the flux-controlled modulation of the quasiparticle density of states (DOS) in a proximitized weak link (a normal-metal Cu nanowire coupled to superconducting Al leads).
  • Differential Readout: The two SQUIPTs are biased with constant currents ($I_{b,a}$ and $I_{b,b}$). The output voltage is measured differentially ($V = V_a - V_b$).
  • Flux Control: The system utilizes two types of magnetic flux:
    • Common Mode Flux (CMF): A uniform field threading both loops (controlled by a superconducting coil).
    • Differential Mode Flux (DMF): An independent flux applied to one loop (controlled by a dedicated flux line).
• Fabrication:
  • Devices were fabricated using shadow mask evaporation in an electron beam evaporator.
  • Structure: Al superconducting loops closed on a Cu weak link, tunnel-coupled to an Al/Al-oxide probe.
  • The specific device tested used two SQUIPTs with significantly different tunnel resistances ($1.18\,\text{M}\Omega$ and $35.71\,\text{M}\Omega$).
• Characterization Strategy:
  • The team measured $I-V$ curves as a function of magnetic flux to characterize individual SQUIPTs.
  • They optimized bias currents to maximize voltage swing despite resistance mismatches.
  • They mapped the transfer function ($V$ vs. $\Phi$) by varying CMF and DMF.
  • Linearity Analysis: Since direct high-frequency SFDR measurement was limited by cryogenic line bandwidth (~kHz), they performed a numerical estimation. They fitted the experimental transfer function to spline curves, simulated the response to monochromatic inputs, and calculated the Spurious-Free Dynamic Range (SFDR) via Fourier transform.

3. Key Contributions

1. First Experimental Demonstration of bi-SQUIPT: The paper validates the theoretical proposal that a differential dual-loop architecture using proximity transistors can achieve high linearity.
2. Robustness to Fabrication Variations: A major contribution is demonstrating that the device remains highly linear even with significant mismatches in tunnel resistances (an order of magnitude difference). This is achieved by independently tuning the bias currents for each SQUIPT to balance the voltage outputs, eliminating the need for perfect fabrication symmetry required by Bi-SQUIDs.
3. Ultra-Low Power Dissipation: The device operates in the femtowatt power dissipation regime due to the high impedance of the tunnel probes, a significant improvement over Bi-SQUIDs (nano/microwatts) and SQAs.
4. High Linearity Performance: The device achieves a Spurious-Free Dynamic Range (SFDR) of up to 60 dB, comparable to large SQUID arrays but in a compact, dual-cell form factor.

4. Key Results

• Voltage Swing: The bi-SQUIPT achieved a peak-to-peak voltage swing of approximately 120 µV, which is roughly twice that of a single SQUIPT.
• Linearity (SFDR):
  • At optimal operating points (specific DMF values), the calculated SFDR reached 60 dB.
  • This matches theoretical predictions and rivals the linearity of much larger Superconducting Quantum Arrays (SQAs).
  • The linearity is highly sensitive to the operating point; moving away from the "sweet spot" causes a dramatic drop in SFDR.
• Sensitivity vs. Dynamic Range Trade-off:
  • Lower DMF values yield higher sensitivity (up to $0.6\,\text{mV}/\Phi_0$) but a smaller dynamic range (~$0.1\Phi_0$).
  • Higher DMF values reduce sensitivity but expand the linear dynamic range.
• Thermal Stability:
  • The device maintains stable operation up to 600 mK.
  • At 600 mK, the SFDR remains at 36 dB.
  • Linearity degrades significantly above 600 mK (approaching 800 mK) as the superconducting order parameter in the Aluminum loop decays.
• Fabrication Tolerance: The device successfully compensated for a ~30x difference in tunnel resistance between the two SQUIPTs simply by adjusting bias currents ($55.0\,\text{pA}$ vs. $4.9\,\text{pA}$).

5. Significance

The bi-SQUIPT represents a paradigm shift for cryogenic quantum electronics:

• Scalability: By eliminating the need for complex FLL electronics and offering a compact footprint, it enables the integration of high-linearity sensors directly into quantum circuits.
• Thermal Management: The femtowatt power dissipation is crucial for dilution refrigerators operating below 100 mK, where heat load is a primary constraint.
• Versatility: The ability to tune the trade-off between sensitivity and dynamic range via CMF and DMF control makes it adaptable for various applications, from qubit readout to high-density cryogenic diagnostics.
• Technological Maturity: The demonstration that the device is robust against fabrication imperfections suggests it is a viable candidate for mass production and integration into next-generation quantum computing architectures.

In summary, the bi-SQUIPT successfully bridges the gap between the low dissipation of SQUIPTs and the high linearity of Bi-SQUIDs/SQAs, offering a compact, ultra-low-power, and highly linear solution for the future of quantum sensing.