🔬 mesoscale physics

Controlled Parity of Cooper Pair Tunneling in a Hybrid Superconducting Qubit

The authors demonstrate a gate-tunable "harmonic parity qubit" based on a hybrid InAs/Al nanowire SQUID that successfully suppresses odd Josephson harmonics to achieve a controllable even-parity tunneling regime, offering a simple building block for Fourier engineering in superconducting circuits.

Original authors: David Feldstein-Bofill, Leo Uhre Jacobsen, Ksenia Shagalov, Zhenhai Sun, Casper Wied, Shikhar Singh, Anders Kringhøj, Jacob Hastrup, András Gyenis, Karsten Flensberg, Svend Krøjer, Morten Kjaergaard

Published 2026-09-01

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CC BY 4.0

Original authors: David Feldstein-Bofill, Leo Uhre Jacobsen, Ksenia Shagalov, Zhenhai Sun, Casper Wied, Shikhar Singh, Anders Kringhøj, Jacob Hastrup, András Gyenis, Karsten Flensberg, Svend Krøjer, Morten Kjaergaard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of quantum computing, the most powerful machines rely on circuits made of superconducting materials, which conduct electricity with zero resistance. The heart of these circuits is a tiny component called a Josephson junction, a bridge where electrons can tunnel through a barrier. What makes these circuits special is not just that they conduct electricity, but that they do so in a way that creates a unique kind of friction or nonlinearity. This nonlinearity allows scientists to create artificial atoms, known as qubits, which can exist in multiple states at once. For decades, the standard design for these bridges has been remarkably simple, behaving like a single, smooth wave where electrons cross one by one in pairs. This predictable behavior has been the workhorse of the field, enabling the creation of processors that are now being used to solve complex problems. However, this simplicity also limits what these circuits can do. Scientists have long suspected that if they could engineer these bridges to allow electrons to cross in groups of two, three, or more at the same time, they could create qubits that are naturally protected from the noise and errors that plague current machines. The challenge has been finding a way to control exactly how these electrons group together, shifting the balance from single pairs to larger groups without building impossibly complex structures.

A team of researchers at the University of Copenhagen has now demonstrated a way to do exactly this, creating a new type of circuit element that can be tuned to change how electrons tunnel. They call their invention the "harmonic parity qubit." In their experiment, they built a device that combines two different types of bridges into a single loop. On one side of the loop, they placed two standard aluminum-oxide junctions, which act as a stable, unchanging backbone. On the other side, they placed a junction made from a tiny semiconductor nanowire coated in aluminum. This nanowire junction is special because its properties can be adjusted by applying a voltage to a nearby gate, much like turning a knob to change the flow of traffic. By carefully balancing these two sides and applying a specific magnetic field, the researchers were able to make the electrons tunnel in pairs of pairs, effectively canceling out the usual single-pair tunneling.

The researchers tested their device by measuring how it responded to microwave signals while they adjusted the gate voltage across a wide range of settings. They found that by simply turning the voltage knob, they could shift the device from behaving like a standard quantum bit to behaving like a completely different kind of system. At one extreme, the device acted normally, with electrons tunneling one pair at a time. As they adjusted the voltage, the behavior became a mix of both single and double tunneling. Finally, at a specific setting, they reached a point where the single-pair tunneling was suppressed by a factor of one hundred compared to the tunneling of two pairs at once. In this state, the energy landscape of the device changed shape, forming a double-well potential where the lowest energy states were separated by a barrier that only allowed electrons to cross in groups of two. This is a significant achievement because it proves that the "parity" of the tunneling—whether an odd or even number of pairs cross—can be controlled with a single external knob.

This control is crucial for the future of quantum computing because it offers a path toward building qubits that are inherently protected from errors. In many current designs, the state of a qubit can be easily flipped by random noise, leading to calculation errors. However, if a qubit is designed so that its lowest energy states only allow electrons to move in even-numbered groups, the laws of physics prevent certain types of noise from disturbing the system. The new device achieves this by creating a situation where the odd-numbered tunneling processes cancel each other out, leaving only the even-numbered ones. The researchers showed that they could tune the ratio of even to odd tunneling across a range of nearly two orders of magnitude, effectively turning the device into a programmable element for quantum circuits. They did not just simulate this effect; they measured it directly using spectroscopy, observing the specific energy transitions that confirmed the dominance of the even-numbered tunneling.

The device they built is compact and relatively simple, consisting of a superconducting loop with a few junctions and a single gate electrode. This simplicity is a major advantage over previous attempts to achieve similar results, which often required complex arrays of many junctions or difficult-to-control hybrid structures. By using a single gate to tune the balance between the two sides of the loop, the researchers created a reliable way to switch between different quantum behaviors. Their work demonstrates that the fundamental properties of a superconducting circuit can be engineered on the fly, allowing scientists to design the energy landscape of a qubit to suit specific needs. This opens the door to a new generation of quantum devices where the protection against errors is built into the very fabric of the circuit, rather than relying on complex software corrections. The ability to control whether electrons tunnel in singles or doubles provides a new tool for building more robust and powerful quantum computers, moving the field closer to the goal of creating machines that can operate reliably in the real world.

Technical Summary: Controlled Parity of Cooper Pair Tunneling in a Hybrid Superconducting Qubit

Problem Statement
Superconducting quantum circuits rely on the nonlinearity provided by the Josephson energy–phase relation. While standard Josephson junctions are dominated by the fundamental $\cos(\phi)$ term (representing single Cooper pair tunneling), higher-order Fourier harmonics $\cos(k\phi)$ corresponding to correlated tunneling of $k$ Cooper pairs can qualitatively alter device properties, including selection rules, wavefunction parity, and noise sensitivity. Controlling the parity of the dominant tunneling process—specifically, suppressing odd harmonics to create an even-harmonic-dominated regime (e.g., $\cos(2\phi)$ )—is a critical goal for realizing intrinsically protected qubits and advanced Hamiltonian engineering. However, achieving this control has historically required complex multi-junction or all-hybrid architectures that are difficult to tune with a single external parameter. Existing hybrid superconductor-semiconductor-superconductor (S-Sm-S) junctions offer tunable harmonics but lack a compact, reliable mechanism to switch between odd- and even-dominated regimes within a single circuit element.

Methodology
The authors introduce a "Harmonic Parity Qubit" (HPQ), a hybrid circuit element designed to programmatically control the parity of Josephson harmonics. The device is a Superconducting Quantum Interference Device (SQUID) with two distinct arms:

Fixed Arm: Two Aluminum-oxide (Al/AlOx/Al) tunnel junctions (SIS-SIS) in series. This arm provides a stable, fabrication-determined harmonic backbone with fixed higher-order harmonic content.
Tunable Arm: A gate-tunable Indium Arsenide/Aluminum (InAs/Al) nanowire junction (S-Sm-S). This arm provides a non-sinusoidal energy-phase relation where the transmission channels, and thus the Fourier coefficients of the potential, can be continuously tuned via an electrostatic gate voltage ( $V_g$ ).

The total potential of the HPQ is the sum of the potentials from both arms. By operating at half-flux quantum frustration ( $\Phi_e = \Phi_0/2$ ), the authors exploit the interference between the two arms. At this flux point, the sine components of the potential vanish, and the cosine coefficients become the sum or difference of the coefficients from the two arms depending on their parity ( $k$ ). Specifically, odd harmonics ( $k=1, 3, \dots$ ) subtract, while even harmonics ( $k=2, 4, \dots$ ) add.

The experimental methodology involved:

Fabricating a chip containing three nominally identical qubit islands: a reference transmon, a double-junction SIS-SIS transmon, and the HPQ.
Performing two-tone spectroscopy across 85 distinct gate voltage settings ( $V_g$ from $-7$ V to $9.8$ V) and varying external flux.
Fitting the observed multi-transition spectra to a full circuit Hamiltonian to reconstruct the energy-phase relation and extract the Fourier coefficients ( $c_k$ ) of the Josephson potential.

Key Contributions and Results

Demonstration of Harmonic Parity Control: The study successfully demonstrates that a single gate voltage can tune the relative weight of even and odd harmonics in a compact superconducting element. By sweeping $V_g$ , the device transitions through three distinct regimes:
- Odd-dominated ( $2\pi$ -periodic): At low gate voltages, the potential is dominated by the fundamental $\cos(\phi)$ term, resembling a standard transmon or "quarton" qubit.
- Mixed-harmonic: An intermediate regime where $\cos(\phi)$ and $\cos(2\phi)$ are comparable, resulting in a shallow double-well potential.
- Even-dominated ( $\pi$ -periodic): At high gate voltages, the fundamental $\cos(\phi)$ term is suppressed by more than two orders of magnitude relative to the second harmonic. The potential becomes dominated by $\cos(2\phi)$ and higher even terms, creating a deep double-well potential with minima near $\pm \pi/2$ .
Quantitative Suppression of Odd Harmonics: At half-flux quantum and optimal gate settings ( $V_g \approx 7.2$ V), the first harmonic coefficient ( $c_1$ ) crosses zero and changes sign, effectively canceling the odd components. The ratio of even-to-odd harmonic weights ( $|c_{even}/c_{odd}|$ ) was tuned over nearly two orders of magnitude, reaching values up to $\approx 34.5$ . In the even-dominated regime, the ground and first excited states exhibit distinct Cooper-pair parity, with the ground state supported on even charge states and the excited state on odd charge states.
Reconstruction of the Harmonic Landscape: Through spectroscopy at 85 gate points, the authors reconstructed the full harmonic landscape. They confirmed that the total even-harmonic contribution remains robust (15–20 GHz scale) while the total odd-harmonic contribution can be reduced to near-zero (0.6 GHz) at the cancellation point.

Significance
The paper claims that this work provides a "manageable control knob" for the Josephson potential, enabling the deliberate design of energy-phase relations within a single superconducting circuit. By combining a fixed SIS-SIS backbone with a tunable S-Sm-S junction, the HPQ achieves reliable, single-gate control over harmonic parity without the instability associated with coordinating multiple gates on separate junctions.

The authors state that this ability to engineer the parity structure of the Josephson potential in situ opens a new pathway for:

Intrinsically protected qubits: Realizing $\pi$ -periodic elements that enforce Cooper-pair parity conservation, a key resource for protected qubit architectures.
Fourier engineering: Tailoring energy barriers and nonlinearity on demand.
Circuit design: Providing a new building block for complex superconducting quantum circuits that exploit multi-Cooper-pair tunneling processes.

The work establishes that the harmonic content of Josephson elements is not merely a fixed byproduct of fabrication but can be a programmable resource, offering a simplified route to the even-harmonic-dominated regimes previously accessible only through complex architectures.

Technical Summary: Controlled Parity of Cooper Pair Tunneling in a Hybrid Superconducting Qubit

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