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 λ, which controls the transparency of the junction, and a phase jump δ, 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 ⟨p̂0⟩ = Ic(δ,λ)sinδ. For opaque junctions Ic becomes independent of δ and the first Josephson relation I=Icsinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the sinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure sinδ law: Ic acquires a dependence on δ already at order λ−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 δ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to δ̇, 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.
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BibTeX
@misc{blanco2026modelling,
title = {{Modelling Josephson Junctions with Boundary Conditions: Towards High-Fidelity Models for Superconducting Qubits}},
author = {Luis Bueno Blanco and Juan Manuel Pérez-Pardo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13417},
eprint = {2610.13417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13417},
}
RIS
TY - UNPB
TI - Modelling Josephson Junctions with Boundary Conditions: Towards High-Fidelity Models for Superconducting Qubits
AU - Luis Bueno Blanco
AU - Juan Manuel Pérez-Pardo
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13417
UR - https://arxiv.org/abs/2610.13417
AB - 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 λ, which controls the transparency of the junction, and a phase jump δ, 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 ⟨p̂_0⟩ = I_c(δ,λ)sinδ. For opaque junctions I_c becomes independent of δ and the first Josephson relation I=I_csinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the sinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure sinδ law: I_c acquires a dependence on δ already at order λ^−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 δ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to δ̇, 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.
ER -