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.