Modelling Josephson Junctions with Boundary Conditions: Towards High-Fidelity Models for Superconducting Qubits

Abstract:Josephson junctions are the central nonlinear element of superconducting qubits, yet their standard description is reached through a chain of approximations. We propose a purely quantum-mechanical model of the junction, formulated directly in Hilbert space: a charged particle moves on a one-dimensional loop, and the junction is a point at which the wave function obeys self-adjoint boundary conditions, labelled by a coupling strength $\lambda$, which controls the transparency of the junction, and a phase jump $\delta$, which plays the role of the superconducting phase difference. For a loop with a single junction we solve the spectral problem and show that the ground-state momentum expectation, identified with the supercurrent, factorises exactly as $\langle\hat{p}_0\rangle = I_c(\delta,\lambda)\sin\delta$. For opaque junctions $I_c$ becomes independent of $\delta$ and the first Josephson relation $I=I_c\sin\delta$ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the $\sin\delta$ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure $\sin\delta$ law: $I_c$ acquires a dependence on $\delta$ already at order $\lambda^{-2}$, compatible with the higher Josephson harmonics recently found necessary to describe superconducting qubit spectra, and a fully transparent contact yields a current linear in $\delta$. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to $\dot\delta$, so that the second Josephson relation emerges without being imposed. The boundary-condition approach thus reproduces both Josephson relations from first principles and predicts corrections to them, offering a route towards higher-fidelity models of superconducting qubits.

Quantum Physics (quant-ph); Superconductivity (cond-mat.supr-con)

arXiv:2610.13417 [quant-ph]

(or arXiv:2610.13417v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.13417 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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