A Josephson junction consists of two superconductors separated by a thin insulating barrier. Cooper pairs tunnel through the barrier coherently, producing macroscopic quantum interference phenomena: the DC and AC Josephson effects, foundational to superconducting quantum devices.
I = Ic · sin(phi) (supercurrent)
dphi/dt = 2eV / hbar (AC phase evolution)
f_J = 2eV / h (Josephson frequency)
~ 483.6 MHz / microVolt
beta_c = 2e Ic R^2 C / hbar (McCumber parameter)Brian Josephson predicted these effects in 1962 as a 22-year-old PhD student, winning the Nobel Prize in Physics in 1973. The Josephson constant KJ = 2e/h = 483,597.848… GHz/V is one of the most precisely measured constants in physics, forming the basis of the international volt standard since 1990.
A Josephson junction is a quantum device consisting of two superconductors separated by a thin insulating barrier (or weak link) through which Cooper pairs can tunnel coherently without any voltage drop, producing a dissipationless supercurrent whose magnitude is I = Ic sin(φ).
The DC Josephson effect is the flow of a supercurrent I = Ic sin(φ) across the junction when no voltage is applied. The current depends only on the quantum phase difference φ between the two superconductors. Any current up to Ic flows without resistance.
When a constant voltage V is applied across the junction, the phase evolves as dφ/dt = 2eV/ℏ, producing an oscillating supercurrent at the Josephson frequency fJ = 2eV/h ≈ 483.6 MHz/µV. This links voltage directly to frequency, enabling precision voltage metrology.
The RSJ model represents the junction as an ideal Josephson element in parallel with a shunt resistance R (and optionally capacitance C). The total current I = Ic sin(φ) + V/R + C dV/dt governs all dynamics. It captures both the DC phase-locking regime and the AC voltage-biased oscillatory regime.
The McCumber parameter βc = 2eIcR²C/ℏ characterises junction damping. For βc < 1 the junction is overdamped with no hysteresis in the I-V characteristic. For βc > 1 the junction is underdamped and shows hysteretic switching between the superconducting and resistive branches.
Because fJ = (2e/h) × V, measuring the microwave frequency of Josephson oscillations gives an absolute determination of DC voltage. The ratio 2e/h (the Josephson constant KJ = 483,597.848 GHz/V) is known to better than 10 significant figures, making Josephson arrays the primary voltage standard at national metrology institutes worldwide.
A Superconducting QUantum Interference Device (SQUID) contains one or two Josephson junctions in a superconducting loop. Magnetic flux threading the loop shifts the interference pattern of Cooper-pair wavefunctions, modulating the effective critical current. SQUIDs can detect magnetic fields as small as a few femtotesla, making them the most sensitive magnetometers available.
Each superconductor is described by a macroscopic quantum wavefunction Ψ = |Ψ|eiθ. The phase difference φ = θ2 − θ1 across the junction determines both the supercurrent magnitude (sin φ) and, through the second Josephson relation, the voltage (dφ/dt = 2eV/ℏ). It is therefore the canonical conjugate variable for the Cooper-pair number difference.
When the bias current exceeds Ic, no static phase difference can sustain the required supercurrent. The phase begins to rotate continuously (phase slip at rate dφ/dt = 2eV/ℏ), generating a time-averaged voltage across the junction. The system switches from the zero-voltage superconducting branch to the resistive (voltage-carrying) branch of the I-V characteristic.
Josephson junctions provide the nonlinear inductance needed to create an anharmonic quantum oscillator — a superconducting qubit. Designs such as the transmon, flux qubit, and fluxonium all use junctions to make individual energy-level transitions addressable by microwave pulses. They operate at millikelvin temperatures to suppress thermal decoherence.
The phase dynamics of an RSJ junction are mathematically identical to those of a damped particle sliding on a tilted washboard potential U(φ) = −Ic cos φ − (Iℏ/2e)φ. When the bias current I < Ic the particle sits in a potential well (DC effect). When I > Ic it rolls downhill, and the average velocity corresponds to the Josephson voltage.
This simulation integrates the RCSJ (resistively and capacitively shunted junction) model of a Josephson junction, in which the supercurrent I = Ic·sin(φ) depends only on the quantum phase difference φ between two weakly coupled superconductors. In DC mode a fixed drive current is applied and the phase equation of motion is solved with 4th-order Runge–Kutta, revealing whether the junction stays dissipationless or switches to a resistive state. In AC mode a fixed voltage forces the phase to wind at a constant rate, reproducing the Josephson voltage-frequency relation dφ/dt = 2eV/ħ. Three linked panels — phase portrait, time-domain supercurrent and I-V characteristic — let you watch these two regimes and their crossover in real time.
The phase portrait plots φ mod 2π against dφ/dt, revealing closed orbits when the junction is superconducting and drifting trajectories once it switches to the resistive branch. The time-domain panel traces the supercurrent Is(t) against the ±Ic bounds, while the I-V panel sweeps drive current to build the average-voltage characteristic, showing the sharp knee at the critical current.
Switch the Mode selector between DC Josephson (current-biased) and AC Josephson (voltage-biased). Set the critical current Ic (0.1–2.0), the drive current ratio I/Ic (0–2.0) for DC mode, and the voltage bias V (0.05–3.0) for AC mode. The damping slider sets βc−½ (0.05–2.0), the inverse square root of the McCumber parameter, tuning the junction from overdamped to underdamped. Pause, Reset or open the Info panel with the on-screen buttons.
Because the Josephson frequency fJ = 2eV/h links voltage directly to frequency through fundamental constants, arrays of junctions driven by microwaves produce quantised voltage steps (Shapiro steps) precise enough to define the international volt standard, while loops of junctions called SQUIDs measure magnetic fields down to the femtotesla scale.
DC mode current-biases the junction: you set I/Ic 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 βc−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·Ic, settles the phase dynamics at each value, and measures the resulting time-averaged voltage ⟨dφ/dt⟩. Below Ic the average voltage stays at zero because the phase can sit still; once the drive exceeds Ic 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/Ic 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".