Josephson Junction & SQUID Simulator: Supercurrent Tunneling

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Is = 5 μA — zero-voltage DC Josephson supercurrent

With Ic = 10 μA and bias current of 5 μA, the junction carries a lossless supercurrent at zero voltage. The phase difference is φ = 30°.

Formula

Is = Ic × sin(φ) (DC Josephson relation)
dφ/dt = 2eV/ℏ (AC Josephson relation)
Ic_squid = 2Ic × |cos(π × Φ/Φ₀)| (DC SQUID)

Tunneling of Cooper Pairs

In 1962, a 22-year-old Cambridge graduate student named Brian Josephson predicted that Cooper pairs could tunnel through a thin insulating barrier between two superconductors, maintaining phase coherence. This was remarkable — individual electrons tunnel easily, but bound pairs tunneling coherently was unexpected. The prediction was confirmed within a year and earned Josephson the 1973 Nobel Prize.

DC and AC Effects

The DC Josephson effect states that a supercurrent Is = Ic sin(φ) flows at zero voltage, where φ is the phase difference across the junction and Ic is the maximum (critical) current. When I exceeds Ic, the phase begins to evolve in time according to dφ/dt = 2eV/ℏ — the AC Josephson effect. This oscillating phase produces an alternating supercurrent at a frequency proportional to voltage, precisely 483.6 GHz per millivolt.

The SQUID

A DC SQUID places two Josephson junctions in a superconducting loop. Quantum interference between the two paths makes the critical current oscillate as a function of the magnetic flux threading the loop, with period Φ₀ = h/2e. By biasing near a steep part of this pattern, the SQUID converts tiny flux changes into large voltage signals. SQUIDs can detect magnetic fields of 10⁻¹⁵ T — sensitive enough to measure brain activity (magnetoencephalography).

Quantum Computing and Metrology

Josephson junctions are the building blocks of superconducting qubits — the leading quantum computing platform. The transmon qubit, used by IBM, Google, and others, is essentially a Josephson junction shunted by a capacitor, creating a nonlinear quantum oscillator. In metrology, Josephson junction arrays define the international voltage standard with parts-per-billion accuracy via the exact frequency-voltage relation.

FAQ

What is the Josephson effect?

The Josephson effect is the flow of supercurrent across a thin insulating barrier (weak link) between two superconductors. Predicted by Brian Josephson in 1962 at age 22, it arises from quantum tunneling of Cooper pairs. The DC effect produces current without voltage; the AC effect generates microwave radiation at a voltage-dependent frequency.

What is a SQUID?

A Superconducting Quantum Interference Device (SQUID) consists of a superconducting loop interrupted by one or two Josephson junctions. It converts magnetic flux into a measurable voltage, achieving sensitivity of ~10⁻¹⁵ T — the most sensitive magnetometer known. SQUIDs are used in brain imaging (MEG), geological surveys, and fundamental physics experiments.

What is the Josephson frequency-voltage relation?

The AC Josephson relation states f = 2eV/h, giving exactly 483.5979 GHz per millivolt. This precise relationship is used as the international voltage standard — a Josephson junction array driven at a known microwave frequency defines the volt to parts-per-billion accuracy.

How are Josephson junctions fabricated?

The most common type is a superconductor-insulator-superconductor (SIS) trilayer, typically Nb/AlOx/Nb, with an oxide barrier about 1 nm thick. They are also made with constrictions (Dayem bridges) or normal metals (SNS junctions). Modern superconducting quantum computers use hundreds of thousands of Josephson junctions as qubits.

Sources

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