The RCSJ (resistively-and-capacitively-shunted junction) model treats a current-biased Josephson junction exactly like a damped pendulum being pushed by a constant torque. In normalized units (time in units of the inverse plasma frequency ωp0⁻¹):
φ̈ + (1/Q)·φ̇ + sin φ = i
φ = superconducting phase difference (the "pendulum angle")
φ̇ ∝ instantaneous junction voltage V (Josephson voltage-phase relation)
i = I_bias / I_c (normalized bias current — the constant "torque")
Q = √β_c = √(2e I_c C R² / ħ) (Stewart–McCumber damping parameter)
This is a genuinely different regime from a closed, undriven washboard: here the phase particle is both driven (tilted washboard, slope ∝ i) and damped (shunt resistance R dissipates energy ∝ φ̇/Q). Two stable behaviours compete:
- Zero-voltage state — the particle sits trapped in a local minimum of the tilted washboard U(φ) = −cos φ − iφ; time-averaged voltage is zero (superconducting).
- Running state — once i exceeds the switching current ic = 1, no minimum exists at all and the particle rolls forever, φ̇ oscillating around a nonzero mean — a finite DC voltage appears (resistive branch).
For an underdamped junction (large Q) these two branches overlap: once running, the particle keeps enough kinetic energy to coast back over the barriers even after i is lowered below 1, so it only re-traps into the zero-voltage state once i drops below a smaller retrapping current ir ≈ 4/(πQ) (Stewart–McCumber's classic asymptotic result). Sweeping i up then down therefore traces a hysteresis loop in the current–voltage curve — the middle strip below plots exactly that loop as you move the bias slider.
- Bias current i — the tilt of the washboard. Cross i = 1 to force a switch into the running state.
- Quality factor Q — junction damping. Low Q (overdamped, small shunt R) makes ir ≈ ic (no hysteresis); high Q (underdamped) opens a wide hysteresis loop.
- Kick / Reset — force the phase particle directly into the running or trapped state to explore both branches at the same i without waiting for a spontaneous switch.
This is the mechanism SQUID and RF-SQUID readout electronics are built around, and it is mechanically the same equation as an underdamped pendulum on a slowly tilting table — completely independent physics from the closed, undriven anharmonic-oscillator picture used to derive a transmon's qubit frequency.