An accurate understanding of the Josephson effect is the keystone of quantum information processing with superconducting hardware. Here we show that the celebrated sinφ current-phaserelation (CφR) of Josephson junctions (JJs) fails to fully describe the energy spectra of transmon artificial atoms across various samples and laboratories. While the microscopic theory of JJs contains higher harmonics in the CφR, these have generally been assumed to give insignificant corrections for tunnel JJs, due to the low transparency of the conduction channels. However, this assumption might not be justified given the disordered nature of the commonly used AlOx tunnel barriers. Indeed, a mesoscopic model of tunneling through an inhomogeneous AlOx barrier predicts contributions from higher Josephson harmonics of several %. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The measurement of Josephson harmonics in the CφR of standard tunnel junctions prompts a reevaluation of current models for superconducting hardware and it offers a highly sensitive probe towards optimizing tunnel barrier uniformity.
Two-dimensional (2D) transition-metal dichalcogenide (TMD) superconductors have unique and desirable properties for integration with conventional superconducting circuits. These includethe ability to form atomically-flat and clean interfaces with stable tunnel barriers, increased kinetic inductance due to the atomically-thin geometry, and resilience to very high in-plane magnetic fields. However, integration of 2D TMD superconductors in conventional superconducting circuits, particularly those employing microwave drive and readout of qubits, requires that a fully superconducting contact be made between the 2D material and a three-dimensional (3D) superconductor. Here, we present an edge contact method for creating zero-resistance contacts between 2D \nbse and 3D aluminum. These hybrid Al-NbSe_2 Josephson junctions (JJs) display a Fraunhofer response to magnetic field with micron2-scale effective areas as the thin NbSe_2 allows field to uniformly penetrate the flake. We present a model for the supercurrent flow in a 2D-3D superconducting structure by numerical solution of the Ginzburg-Landau equations and find good agreement with experiment. The devices formed from 2D TMD superconductors are strongly influenced by the geometry of the flakes themselves as well as the placement of the contacts to bulk 3D superconducting leads. These results demonstrate our ability to graft 2D TMD superconductors and nano-devices onto conventional 3D superconducting materials, opening the way to a new generation of hybrid superconducting quantum circuits.