The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. In experimentally relevant parameter regimes, we numerically find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.
Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental
cosφ term, this relation can also contain higher Fourier harmonics
cos(kφ) corresponding to correlated tunneling of
k Cooper pairs. The parity of the dominant tunneling process, i.e.~whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit" (HPQ); an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near
±π/2. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.
Leveraging the higher harmonics content of the Josephson potential in a superconducting circuit offers a promising route in the search for new qubits with increased protection against decoherence. In this work, we demonstrate how the flux tunability of a hybrid semiconductor-superconductor Josephson junction based on a single full-shell nanowire enables this possibility. Near one flux quantum,
Φ≈Φ0=h/2e, we find that the qubit system can be tuned from a gatemon regime to a parity-protected regime with qubit eigenstates localized in phase space in the
0 and
π minima of the Josephson potential (
cos2φ0). Estimates of qubit coherence and relaxation times due to different noise sources are presented.
We study a recently demonstrated design for a high-performance tunable coupler suitable for superconducting Xmon and planar transmon qubits [Y. Chen et al., Phys. Rev. Lett. 113, 220502 (2014)]. The coupler circuit uses a single flux-biased Josephson junction and acts as a tunable current divider. We calculate the effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of the qubit and coupler junctions perturbatively. We find that the qubit nonlinearity has two principal effects: The first is to suppress the magnitude of the transverse ${\ensuremath{\sigma}}^{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{x}$ coupling from that obtained in the harmonic approximation by about 15%, assuming typical qubit parameters. The second is to induce a small diagonal ${\ensuremath{\sigma}}^{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{z}$ coupling. The effects of the coupler junction nonlinearity are negligible in the parameter regime considered. The approach used here can be applied to other complex nonlinear circuits arising in the design of superconducting hardware for quantum information processing.
Abstract Hybrid superconductor-semiconductor Josephson field-effect transistors (JoFETs) function as Josephson junctions with a gate-tunable critical current. Additionally, they can feature a non- sinusoidal current-phase relation (CPR) containing multiple harmonics of the superconducting phase difference, a so-far underutilized property. In this work, we exploit this multi-harmonicity to create a Josephson circuit element with an almost perfectly π-periodic CPR, indicative of a largely dominant charge-4e supercurrent transport. Such a Josephson element was recently proposed as the basic building block of a protected superconducting qubit. Here, it is realized using a superconducting quantum interference device (SQUID) with low-inductance aluminum arms and two nominally identical JoFETs. The latter are fabricated from a SiGe/Ge/SiGe quantum-well heterostructure embedding a high-mobility two-dimensional hole gas. By carefully adjusting the JoFET gate voltages and finely tuning the magnetic flux through the SQUID close to half a flux quantum, we achieve a regime where the sin(2φ) component accounts for more than 95 % of the total supercurrent. This result demonstrates a new promising route for the realization of superconducting qubits with enhanced coherence properties.
Incorporating van der Waals (vdW) superconductors into Josephson elements extends circuit-QED beyond conventional Al/AlO
x/Al tunnel junctions and enables microwave probes of unconventional condensates and subgap excitations. In this work, we realize a flux-tunable transmon whose nonlinear inductive element is an Al/AlO
x/4Hb-TaS
2 Josephson junction. The tunnel barrier is formed by sequential deposition and full in-situ oxidation of ultrathin Al layers on an exfoliated 4Hb-TaS
2 flake, followed by deposition of a top Al electrode, yielding a robust, repeatable hybrid junction process compatible with standard transmon fabrication. Embedding the device in a three-dimensional copper cavity, we observe a SQUID-like flux-dependent spectrum that is quantitatively reproduced by a standard dressed transmon--cavity Hamiltonian, from which we extract parameters in the transmon regime. Across measured devices we obtain sub-microsecond energy relaxation (
T1 from
0.08 to
0.69 μs), while Ramsey measurements indicate dephasing faster than our
16 ns time resolution. We also find a pronounced discrepancy between the Josephson energy inferred from spectroscopy and that expected from the Ambegaokar--Baratoff relation using room-temperature junction resistances, pointing to nontrivial junction physics in the hybrid Al/AlO
x/4Hb-TaS
2 system. Although we do not resolve material-specific subgap modes in the present geometry, this work establishes a practical route to integrating 4Hb-TaS
2 into coherent quantum circuits and provides a baseline for future edge-sensitive designs aimed at enhancing coupling to boundary and subgap degrees of freedom in vdW superconductors.