⚡ Josephson Junction — AC & DC Josephson Effects

DC mode current-biases the junction: you set I/I_c and the simulation integrates the phase equation to find whether a static phase (zero voltage) or a rotating phase (finite voltage) results. AC mode voltage-biases it instead: you set V directly and the phase is forced to advance at the fixed rate dφ/dt = 2eV/ħ, always producing an oscillating supercurrent.

The damping control sets β_c^−½ , the inverse square root of the McCumber parameter that appears in the RCSJ equation of motion. Low values correspond to an underdamped, high-Q junction whose I-V curve is hysteretic, while high values correspond to an overdamped junction that switches smoothly between the superconducting and resistive branches with no hysteresis.

A closed loop means the phase φ oscillates around a fixed point without ever completing a full rotation — the junction is superconducting and carries a static (or bounded) supercurrent. A trajectory that drifts continuously across the panel means φ is winding indefinitely, which corresponds to a running, time-averaged voltage: the junction has switched to its resistive branch.

It sweeps the DC drive current from zero up to 2.5·I_c, settles the phase dynamics at each value, and measures the resulting time-averaged voltage ⟨dφ/dt⟩. Below I_c the average voltage stays at zero because the phase can sit still; once the drive exceeds I_c no static phase can balance the current, so the phase starts rotating and the average voltage rises sharply — the knee marks the critical current.

Once I/I_c passes 1, the junction can no longer sustain a constant phase difference, so φ begins rotating continuously instead of settling at a fixed value. Each full rotation of φ by 2π corresponds to one flux quantum passing through the junction, and the time-averaged rate of rotation shows up as a genuine voltage across it, moving the state indicator from "SC" to "Resistive".