Higher Josephson harmonics in a tunable double-junction transmon qubit

TL;DR

The authors demonstrate an all-SIS tunable double-junction transmon by placing a SIS junction in series with a SQUID, enabling flux-driven control of higher Josephson harmonics. A two-mode circuit model, augmented by a Born–Oppenheimer correction to account for an internal island mode, reveals substantial higher-harmonic content (notably a second harmonic up to ~0.1 of the first) and a flux-dependent anharmonicity, with a flux sweet spot where dispersive shifts cancel. These findings are supported by full circuit analysis and dispersion calculations, and are contrasted with a fixed-frequency SIS transmon that shows only small higher harmonics. The work suggests a path toward protected qubits and customizable nonlinear microwave devices using an all-SIS platform, without the coherence penalties of hybrid architectures.

Abstract

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.

Figures (9)

• Figure 1: Qubit circuit and device design. (a) Circuit schematic of the tunable double-junction transmon with a single JJ (blue) and a SQUID loop (red) in series, shunted by a large capacitor (green). (b) SEM of the tunable double-junction transmon showing the large island which forms the capacitor (green). A quarter-wave resonator is capacitively coupled to the qubit for readout. A flux line threads magnetic flux through the SQUID loop, and the drive line facilitates the XY control. (c) SEM image of the single JJ (blue) and the SQUID loop (red) in series. (d) The potentials $U$ plotted for each SIS junction (blue, red), which in series effectively correspond to the potential from Eq. \ref{['eq_pot_Akhmerov']}, shown for $\lambda = 1$ (purple). (e) Harmonic content of the potential from Eq. \ref{['eq_pot_Akhmerov']} as a function of $\lambda$. (f) $\lambda$ as a function of the ratio between the Josephson energies of the two junctions.
• Figure 2: Flux-dependent two-tone spectroscopy of the tunable double-junction transmon (second cooldown) and fit using the two-mode model from Eq. \ref{['eq: full_ham']}. (a) Two-tone spectroscopy of the $f_{01}$ (light blue), $2 \cdot f_{02} / 2$ (yellow), $3 \cdot f_{03} / 3$ (pink) and $4 \cdot f_{04} / 4$ (brown) transitions as a function of $\Phi_\mathrm{e}$. The $f_{01}$ and $f_{02}$ transitions are fitted to the two-mode model of Eq. \ref{['eq: full_ham']} (solid gray), and the resulting parameters are used to predict $f_{03}$ and $f_{04}$ (dotted gray). (b) Extracted anharmonicity (green) obtained from the $f_{01}$ and $f_{02}/2$ transitions together with the corresponding fit (solid gray). The analytical expression from Eq. \ref{['eq_anh']}, based on the potential in Eq. \ref{['eq_pot_Akhmerov']}, is shown for comparison (dashed teal). Inset: simulated anharmonicity versus flux for a tunable transmon with comparable device parameters.
• Figure 3: Measured anharmonicity as a function of flux and extracted harmonic content of the tunable double-junction transmon (second cooldown). (a) Measured anharmonicity (green) compared with the Born–Oppenheimer (BO) prediction (blue). Vertical dashed lines mark $\Phi_\mathrm{e}=0$ and $\Phi_\mathrm{e}=0.25\Phi_0$. (b) Extracted second, third, and fourth harmonics normalized to the first harmonic from the BO model; at $\Phi_\mathrm{e}=0$ we obtain $U_k = [1,0.107,0.023,0.006]$. (c) Four observed transitions at $\Phi_\mathrm{e}=0$ with Lorentzian fits (solid) and BO-predicted frequencies (dashed), showing good agreement. (d) Four observed transitions at $\Phi_\mathrm{e}=0.25\Phi_0$; here the BO model captures the relative level spacings (anharmonicity) but not the absolute transition frequencies.
• Figure 4: Flux dependence of the total dispersive shift in the tunable double-junction transmon (first cooldown). (a) Measured data points of the qubit $f_{01}$ (light blue) and of the resonator $f_\textrm{res}$ (dark red) with the bare frequency of the resonator $f^\textrm{bare}_\textrm{res}$ indicated by the red dashed line. Extending the two-mode model from Eq. \ref{['eq: full_ham']} (see Supplementary Material SM), we fit $f_\textrm{res}$ (gray) and extract the simulated internal-mode frequency $f^\mathrm{sim}_\mathrm{int}$ (purple) and the qubit’s $f^\mathrm{sim}_{03}$ transition, revealing an avoided crossing near $\Phi_\mathrm{e}=\Phi_0/2$. (b) Resonator spectroscopy versus readout amplitude at three representative flux points. At $\Phi_\mathrm{e}=0$ (left), the qubit contribution dominates, shifting the resonator upward. At the balance point (middle), the qubit and internal-mode contributions cancel, giving zero net dispersive shift. Near $\Phi_\mathrm{e}=\Phi_0/2$ (right), the internal mode dominates, shifting the resonator downward.
• Figure 5: Extraction of the transitions $f_{01}$, $f_{02} / 2$, $f_{03} / 3$ and $f_{04} / 4$ at $\Phi_\textrm{e} = 0$ , at $\Phi_\textrm{e} = 0.31\Phi_0$ which corresponds to $\lambda \approx 0.98$ and at $\Phi_\textrm{e} = 0.5\Phi_0$ (CD2). (a) Two-tone spectroscopy versus amplitude. (b) Raw data averaged over the amplitude to facilitate the fitting of the transitions to a Lorentzian function.