🔬 mesoscale physics

Higher Josephson harmonics in a tunable double-junction transmon qubit

This paper demonstrates a tunable double-junction transmon qubit featuring a Josephson potential with magnetic-flux-controlled harmonics, where spectroscopy reveals a second harmonic amplitude up to ~10% and identifies a flux sweet spot that eliminates dispersive shifts by balancing couplings to internal and qubit modes.

Original authors: Ksenia Shagalov, David Feldstein-Bofill, Leo Uhre Jakobsen, Zhenhai Sun, Casper Wied, Amalie T. J. Paulsen, Johann Bock Severin, Malthe A. Marciniak, Clinton A. Potts, Anders Kringhøj, Jacob Hastrup

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ksenia Shagalov, David Feldstein-Bofill, Leo Uhre Jakobsen, Zhenhai Sun, Casper Wied, Amalie T. J. Paulsen, Johann Bock Severin, Malthe A. Marciniak, Clinton A. Potts, Anders Kringhøj, Jacob Hastrup, 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

Imagine the world of quantum computing as a high-stakes game of musical chairs, but instead of chairs, the players are tiny packets of energy called "qubits." To make these qubits work, scientists build them out of superconducting circuits—loops of wire that conduct electricity with zero resistance when cooled to near absolute zero. The secret sauce that makes these circuits behave like quantum bits rather than just regular wires is a special component called a Josephson junction. Think of a Josephson junction as a magical, ultra-thin door between two superconducting rooms. It lets pairs of electrons (called Cooper pairs) sneak through, but only if they do it in a very specific, rhythmic way. This rhythm creates a "potential energy landscape," which you can picture as a wavy hill or a valley where the electrons like to sit.

For decades, scientists have treated these hills as simple, smooth waves, like a perfect sine wave you might see in a basic physics textbook. This simple shape is what gives qubits their special "anharmonicity"—a fancy word for the fact that the energy steps between levels are uneven, which is crucial for telling the computer's "0" and "1" states apart. However, recent experiments have hinted that these hills might be more complex than we thought. Just like a musical note can have hidden overtones (higher harmonics) that change its color or tone, these quantum junctions might have hidden "higher harmonics" in their energy waves. If we can learn to control these hidden harmonics, we could build qubits that are much tougher against noise and errors, potentially leading to a future where quantum computers are stable enough to solve problems we can't even imagine today.

In this new study, a team of researchers from the University of Copenhagen and collaborators have built a clever new gadget to test this idea. They created a "tunable double-junction transmon," which is essentially a circuit made of two Josephson junctions linked together in a row: one is a single junction, and the other is a loop with two junctions inside it (called a SQUID). Imagine this setup as a musical instrument where you can tighten or loosen a string (by applying a magnetic field) to change the shape of the sound it makes. By threading a magnetic flux through the SQUID loop, the team could dynamically reshape the energy landscape of their qubit, effectively tuning the "musical notes" of the circuit.

What they found is quite exciting. By measuring the energy transitions of their qubit, they discovered that they could generate a "second harmonic"—a higher-frequency ripple in the energy wave—that was surprisingly strong. In their most symmetric setup, this second harmonic reached about 10% of the strength of the main wave. To put that in perspective, previous experiments with single junctions had only seen tiny, almost invisible ripples. This is like going from hearing a faint whisper of a second note to hearing a clear, distinct harmony. The researchers confirmed this by building a detailed computer model that included an "internal mode"—a hidden vibration of the tiny island of superconducting material sitting between the two junctions. This internal mode acts like a hidden gear in a machine that slightly reshapes the main motion, and their model showed that this hidden gear was responsible for the extra harmonics they observed.

The team also stumbled upon a very special "sweet spot" in their experiment. As they adjusted the magnetic flux, they found a specific point where the usual interaction between the qubit and their readout device (a resonator) completely canceled out. Normally, the qubit would push the resonator's frequency up or down, but at this magic point, the push from the main qubit mode was perfectly balanced by a pull from the internal mode, resulting in zero net shift. This is a bit like two people pushing a swing in opposite directions with equal force; the swing stays perfectly still. This discovery is significant because it suggests a way to make qubits that are less sensitive to certain types of noise, potentially leading to more robust quantum computers.

The paper doesn't claim to have solved all the problems of quantum computing, nor does it say these new qubits are ready for your living room. Instead, it provides a solid, measured demonstration that we can engineer these complex energy landscapes in a fully superconducting system. The researchers showed that by playing with the asymmetry between their two junctions, they could access a rich variety of quantum behaviors that were previously hard to reach. They ruled out the idea that these effects were just random noise or simple artifacts, proving instead that they were a direct result of the circuit's design and the interplay between the main qubit and its internal "twin." This work opens a new door for designing "protected qubits"—quantum bits that are naturally shielded from errors—and suggests that the future of quantum hardware might involve tuning these hidden harmonics to create custom-built quantum devices.

Technical Summary: Higher Josephson Harmonics in a Tunable Double-Junction Transmon Qubit

Problem Statement
Superconducting circuits, particularly Josephson junctions (JJs), are conventionally modeled by a single-harmonic Josephson potential, $U(\phi) = -E_J \cos \phi$ . This single-harmonic approximation underpins standard circuit properties like anharmonicity and charge-noise sensitivity. However, recent experiments indicate that Superconductor–Insulator–Superconductor (SIS) junctions can exhibit higher harmonics (terms proportional to $\cos(k\phi)$ for $k>1$ ) due to junction pinholes or lead inductance. While higher harmonic engineering is a promising route for designing protected qubits and superconducting diodes, existing methods to generate them—such as Superconductor–Normal-metal–Superconductor (SNS) junctions or hybrid superconductor-semiconductor weak links—often suffer from frequency instability and reduced coherence due to losses in the semiconducting regions. Furthermore, previous SIS-based approaches to tune harmonics have been limited in their tunability and the magnitude of the higher harmonic content achieved.

Methodology
The authors introduce and characterize a tunable circuit element consisting of a single SIS junction in series with a SQUID loop, shunted by a large capacitor to operate in the transmon regime. This "tunable double-junction transmon" allows for in situ control of the effective energy-phase relation via an external magnetic flux ( $\Phi_e$ ) threading the SQUID loop.

The study employs the following methodological approaches:

Device Fabrication and Design: A device was fabricated featuring a single JJ and a SQUID loop in series. The middle superconducting island between the junctions forms a capacitive node.
Spectroscopy: The authors performed two-tone spectroscopy to measure the first four qubit transitions ( $f_{01}, f_{02}, f_{03}, f_{04}$ ) as a function of external flux.
Theoretical Modeling:
- Reduced Model: A simplified model was derived where the charging energy of the middle island is initially omitted, treating the two junctions as a single effective element with a potential $U_{red.}(\phi_q) = -E_{J\Sigma}\sqrt{1-\lambda \sin^2(\phi_q/2)}$ , where $\lambda$ is a junction asymmetry parameter dependent on flux.
- Two-Mode Hamiltonian: A full Hamiltonian was constructed including both phase degrees of freedom ( $\phi_1, \phi_2$ ) across the individual junctions, accounting for charging energies ( $E_{C1}, E_{C2}$ ) and mutual coupling ( $g_{12}$ ). This model was used to fit the experimental spectra and extract circuit parameters.
- Born-Oppenheimer (BO) Approximation: To quantify the higher harmonic content, the authors derived an effective single-mode model. By treating the internal mode (associated with the middle island) as high-frequency relative to the qubit, they applied a BO approximation to derive an effective potential $U_{BO}(\phi_q)$ that includes corrections from the internal mode. This allowed for the extraction of harmonic coefficients $U_k$ .
- Dispersive Shift Analysis: The model was extended to include dispersive coupling to a readout resonator to analyze the ground-state dispersive shift ( $\chi_0$ ) arising from both the qubit and internal modes.

Key Results

Tunable Higher Harmonics: The device demonstrates a Josephson potential with strongly tunable harmonic content. At the symmetric point ( $E_{J1} \approx E_{J2}$ ), the second harmonic amplitude reaches approximately 10% of the fundamental ( $|U_2/U_1| \approx 0.107$ ), significantly exceeding values previously reported for single SIS junctions.
Anharmonicity Behavior: Unlike conventional transmons, where anharmonicity varies monotonically with flux, this device exhibits a pronounced non-monotonic dependence. At zero flux, the anharmonicity reaches $\hbar\alpha \approx -E_C/3$ , deviating from the standard transmon value of $\approx -E_C$ .
Model Validation: The two-mode Hamiltonian accurately reproduces the measured transition frequencies across all flux values. The Born-Oppenheimer single-mode model successfully captures the anharmonicity and harmonic coefficients, showing excellent agreement with experimental data at $\Phi_e = 0$ .
Internal Mode Dynamics: The internal mode associated with the middle island renormalizes the qubit spectrum. Spectroscopy reveals an avoided crossing between the qubit and internal modes near $\Phi_e = 0.5\Phi_0$ .
Flux Sweet Spot for Dispersive Shift: The authors identified a specific flux point ( $\Phi_e \approx 0.44\Phi_0$ ) where the dispersive couplings of the qubit mode and the internal mode to the readout resonator cancel each other out. At this point, the total ground-state dispersive shift ( $\chi_0$ ) vanishes.

Significance and Claims
The paper claims that the tunable SIS-SIS element provides a fully superconducting route to controlling Josephson harmonics without the coherence penalties associated with hybrid SNS or semiconductor platforms. By balancing the dispersive couplings of the qubit and internal modes, the device achieves a "sweet spot" where the ground-state dispersive shift vanishes. This feature, combined with the ability to tune the potential landscape, offers a pathway toward:

Protected Qubits: The ability to engineer specific harmonic ratios (e.g., suppressing the fundamental harmonic) is a prerequisite for designs like the $\cos(2\phi)$ qubit.
Customizable Nonlinear Devices: The element serves as a building block for superconducting circuits requiring tailored nonlinearities.
Improved Readout: The vanishing dispersive shift at a specific flux point suggests potential applications in next-generation qubit readout schemes where minimizing state-dependent shifts is advantageous.

The authors emphasize that the internal mode, residing in the transmon regime and separated by a large energy gap, does not introduce significant additional decoherence, and leakage is suppressed due to the absence of direct drive matrix elements coupling the modes.

Technical Summary: Higher Josephson Harmonics in a Tunable Double-Junction Transmon Qubit

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