Higher Josephson harmonics in a tunable double-junction transmon qubit

Tunable Josephson harmonics open new avenues for qubit design. We demonstrate a superconducting circuit element consisting of a tunnel junction in series with a SQUID loop, yielding a Josephson potential whose harmonic content is strongly tunable by magnetic flux. Through spectroscopy of the first four qubit transitions, together with an effective single-mode model renormalized by the internal mode, we resolve a second harmonic with an amplitude up to $\sim10%$ of the fundamental. We identify a flux sweet spot where the dispersive shift vanishes, achieved by balancing the dispersive couplings to the internal and qubit modes. This highly tunable element provides a route toward protected qubits and customizable nonlinear microwave devices.

💡 Research Summary

This paper presents the realization and comprehensive characterization of a novel superconducting circuit element: the “tunable double-junction transmon qubit.” The device consists of a single Superconductor-Insulator-Superconductor (SIS) Josephson junction placed in series with a superconducting quantum interference device (SQUID) loop, the whole structure shunted by a large capacitor to form a transmon-like qubit.

The core innovation lies in using an external magnetic flux through the SQUID to control the asymmetry between the two Josephson junctions in situ. This asymmetry parameter (λ) directly tunes the shape of the effective Josephson potential experienced by the qubit mode. Theoretically, when the junctions are symmetric (λ=1), the potential becomes strongly non-sinusoidal, containing significant higher harmonic components (proportional to cos(kφ) with k>1) compared to the conventional cos(φ) potential.

Through detailed two-tone spectroscopy across multiple cooldowns, the authors mapped the first four qubit transition frequencies (f01, f02, f03, f04) as a function of magnetic flux. The data was accurately modeled using a full two-mode Hamiltonian that accounts for both the qubit phase and the phase across the internal island between the junctions. A key experimental finding is the non-monotonic dependence of the qubit anharmonicity on flux, which starkly contrasts with the behavior of a standard flux-tunable transmon and is a signature of the modified potential.

To quantify the higher harmonic content, the authors developed an effective single-mode model incorporating a Born-Oppenheimer correction for the internal mode. This model excellently reproduces the measured anharmonicity and allows for the extraction of Fourier coefficients Uk. At zero applied flux, they resolve a second harmonic U2 with an amplitude of approximately 10.7% of the fundamental U1, a value substantially larger than previously reported for individual SIS junctions.

Furthermore, the paper investigates the impact of the internal mode on qubit readout. By performing resonator spectroscopy alongside qubit spectroscopy, they demonstrate that the internal mode couples dispersively to the readout resonator. A remarkable phenomenon is observed: at a specific flux bias (Φe ≈ 0.44Φ0), the dispersive shift contributed by the qubit (χ_q) and that contributed by the internal mode (χ_int) cancel each other out, resulting in a net zero dispersive shift of the resonator frequency despite the persistent coupling. This “flux sweet spot” emerges from balancing the couplings to the two modes.

In summary, this work demonstrates a highly tunable, all-superconducting platform for engineering Josephson harmonics in situ. By leveraging a simple series combination of two SIS junctions, it achieves significant higher harmonic content—a key ingredient for designing noise-protected qubits—while maintaining the fabrication and coherence advantages of standard SIS technology. The additional control knob over the dispersive shift via the internal mode also opens new avenues for developing advanced qubit readout techniques and customizable nonlinear microwave devices.