Modelling Josephson Junctions with Boundary Conditions: Towards High-Fidelity Models for Superconducting Qubits
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 $\langle\hat{p}_0\rangle = I_c(δ,λ)\sinδ$. For opaque junctions $I_c$ becomes independent of $δ$ and the first Josephson relation $I=I_c\sinδ$ 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 $\dotδ$, 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.