A DC SQUID is a superconducting loop interrupted by two Josephson junctions. Each junction carries a supercurrent I = Ic,i sin(δi), and single-valuedness of the pair wavefunction around the loop locks the two phase differences to the enclosed flux:
δ2 − δ1 = 2π Φ / Φ0 (fluxoid quantization, negligible self-inductance)
Φ0 = h / 2e ≈ 2.068 × 10⁻¹⁵ Wb (flux quantum, Cooper pairs → 2e)
Maximum (critical) current the loop can carry lossless:
Ic(Φ) = sqrt( Ic1² + Ic2² + 2·Ic1·Ic2·cos(2πΦ/Φ0) )
→ 2·Ic0·|cos(πΦ/Φ0)| for identical junctions (Ic1 = Ic2 = Ic0)
Overdamped (RSJ) time-averaged voltage once biased past Ic:
⟨V⟩ = Rn · sqrt( Ib² − Ic(Φ)² ) for Ib > Ic(Φ), else ⟨V⟩ = 0
Consistency check (done numerically before writing this engine): at Φ=0 the formula gives Ic = Ic1+Ic2 (junctions in phase, maximum constructive interference); at Φ=Φ0/2 it gives Ic = |Ic1−Ic2| (destructive interference, only the asymmetry survives). Both limits match the standard textbook DC-SQUID result, so the 3D source's math was carried over unchanged — no correction was needed here.
- Φ/Φ0 slider — applied flux threading the loop. Ic(Φ) oscillates with period exactly one flux quantum: this interference pattern is what makes a SQUID the most sensitive magnetometer known (~10⁻¹⁵ T).
- Asymmetry — unequal junctions (Ic2 ≠ Ic1) never fully cancel Ic at the destructive-interference points, filling in the oscillation minima.
- Bias current — drive current through the loop. Below Ic(Φ) the loop is a dissipationless supercurrent (V = 0, particles glow steady blue); above it, the junctions switch to a resistive branch and a real time-averaged voltage appears (particles flicker orange).
- Shunt resistance Rn — normal-state resistance of the junctions; it sets how steeply the voltage rises once the bias exceeds Ic, visible directly in the bottom-right I–V panel.
- The moving marker on the interference curve tracks your current Φ/Φ0 against the analytic Ic(Φ) trace, and the marker on the I–V panel tracks where your bias sits on the V(Ib) curve for that same Ic.
Real-world relevance: this exact Ic(Φ) interference pattern is read out in every commercial SQUID magnetometer and susceptometer, and the same two-junction phase-locking is the operating principle behind flux qubits used in superconducting quantum processors.