The Josephson Junction: where quantum phase becomes an engineering variable

Level 0 · The picture

Two pools of water sit side by side at exactly the same level, separated by a thin wall. Common sense says nothing should flow between them: water only moves when one side is higher than the other. Now make the wall thin enough and porous enough, and a steady trickle flows through it anyway — with no height difference at all.

That is a Josephson junction. The pools are two pieces of superconducting metal, cooled until electricity flows through them with no resistance. The wall is an insulating layer a few atoms thick. And the "trickle with no height difference" is a real electric current that flows across the gap with no voltage driving it. Brian Josephson predicted it in 1962, at the age of twenty-two, and Philip Anderson and John Rowell saw it the next year.

Why does it happen? Because a superconductor is not a collection of separate electrons any more. Below its transition temperature the electrons pair up, and every pair joins one enormous shared quantum wave that fills the whole piece of metal. When two such waves come within a few atoms of each other, they overlap through the barrier and lock together — the way two pendulum clocks hung on the same wall fall into step. The lock is what carries the current.

There is a second surprise. If you do apply a voltage across the gap, the current does not simply grow. It oscillates, at a frequency set by the voltage and two constants of nature — about 484 billion cycles per second for every volt. Nothing about the metal, the barrier, or the temperature enters that number. It is so exact that since 1990 the world's standard of the volt has been defined by it.

Ways to think about it

• A superconductor is one wave. A junction is two waves touching. The physics of the junction is the physics of two waves trying to agree on where their crests are.
• Zero voltage, steady current: the two waves are locked with a fixed offset.
• Constant voltage, oscillating current: the offset is winding round and round like a clock hand.

Level 1 · Foundations

Every wave has a phase — where its crest is right now. Two waves of the same frequency can be in step, out of step, or anywhere between, and the difference in their phases is a single number, an angle between 0 and 360 degrees. For a Josephson junction, that one angle — the phase difference δ across the barrier — is the whole story.

Rule one (the DC Josephson effect). The current across the gap depends only on the phase difference. Set the phase offset, and a fixed current flows — with no voltage. The largest current the junction can carry this way is called its critical current.

Rule two (the AC Josephson effect). A voltage across the gap makes the phase difference advance at a steady rate — the clock hand turns — and because the current depends on the phase, the current oscillates. The rate is exactly proportional to the voltage.

That is all a junction does, and it is enough to build an astonishing range of instruments:

• Voltage standards. Shine microwaves of a known frequency on a junction and its current–voltage curve becomes a staircase of steps whose heights are set by the frequency and fundamental constants alone. Laboratories around the world calibrate the volt this way.
• SQUIDs. Put two junctions in a ring and the ring's current depends on the magnetic flux through it with exquisite sensitivity — enough to map the magnetic fields of a beating heart or a thinking brain from outside the body.
• Qubits. A junction in a small circuit behaves like a single artificial atom with two lowest energy levels. Those are the qubits inside most of today's superconducting quantum computers.
• Emitters. Because a biased junction oscillates at a frequency you can tune with voltage, an array of junctions is a tunable source — of microwaves today, and in the proposal this site follows, of something more exotic.

Level 2 · The two rules

Here are Josephson's two relations written down. Everything else in this course is commentary on them.

First Josephson relation (DC)(1)

$$I=I_c\sin \delta$$

Second Josephson relation (AC)(2)

$$\frac{d\delta }{dt}=\frac{2e}{\hslash }V$$

Why the factor of two. The carriers are pairs of electrons, so the charge that appears in the rule is 2e. That factor was itself a confirmation that the current is carried by Cooper pairs.

The Josephson constant. Rule two says a voltage V produces oscillation at f = (2e/h) V. The ratio 2e/h is now a defined constant: K_J = 483 597.848… GHz per volt. Turn the rule around — shine microwaves of frequency f on a junction and its voltage locks to exact multiples of hf/2e — and you have the voltage standard. Those exact multiples are the Shapiro steps.

A worked example. Apply one millivolt across a junction. Rule two gives an oscillation frequency of 483.6 GHz per volt × 0.001 V ≈ 484 MHz — a UHF radio frequency. Apply one microvolt and you get about 484 kHz. To reach the 24 GHz of the gaser design in Level 5 you need a bias of roughly 50 microvolts. The junction is a voltage-to-frequency converter of perfect linearity, which is exactly the property an emitter designer wants.

Level 3 · Undergraduate

Where the rules come from. Feynman's derivation, in Volume III of his lectures, takes two pages. Describe each superconductor by a single macroscopic wavefunction ψ = √n · e^(iθ), where n is the pair density and θ the phase. Let the two wavefunctions couple weakly through the barrier with a coupling energy K. Write the Schrödinger equation for the two coupled amplitudes and take real and imaginary parts. The rate of change of pair density on each side is the current, and it comes out proportional to sin(θ₂ − θ₁); the rate of change of the phase difference comes out proportional to the energy difference between the sides, which for a pair of charge 2e in a voltage V is 2eV. Both Josephson relations fall out at once. The whole effect is the quantum mechanics of two coupled states — the same mathematics as the ammonia molecule or a two-level atom — applied to a macroscopic object.

The Josephson energy(3)

$$U(\delta )=-E_J\cos \delta ,E_J=\frac{\hslash I_c}{2e}$$

Gauge invariance: the vector potential returns. When a magnetic field threads the junction, the phase difference that appears in rule one is not the bare θ₂ − θ₁ but the gauge-invariant phase difference, which subtracts the line integral of the vector potential across the barrier:

Gauge-invariant phase difference(4)

$$\delta ={\theta }_2-{\theta }_1-\frac{2e}{\hslash }{\int }_1^2\mathbf{A}\cdot d\mathbf{l}$$

Definitive The Josephson relations, Shapiro steps, flux quantisation and the SQUID are measured to parts in a billion and are the basis of international metrology.

Ways to think about it

• A junction is a two-level quantum system you can wire to a battery.
• Rule two is a conversion factor between voltage and frequency with no material constants in it — the purest such relation in physics.
• The phase difference is only meaningful once the vector potential is included; the junction is the Aharonov–Bohm effect made into a circuit element.

Level 4 · Graduate

The tilted washboard. Add a real circuit around the junction — a capacitance C in parallel, a resistance R for the normal electrons — and drive it with a current I. The phase difference δ then obeys the equation of a particle of "mass" proportional to C moving in a tilted washboard potential: the cosine of the Josephson energy, tipped by the drive current.

The RCSJ model(5)

$$\frac{\hslash C}{2e}\ddot{\delta }+\frac{\hslash }{2eR}\dot{\delta }+I_c\sin \delta =I$$

The SQUID. Put two junctions in a superconducting ring. The phases around the ring must return to themselves, and by equation (4) the vector potential's loop integral — the flux — enters the sum. The result is that the ring's maximum supercurrent oscillates with the applied flux, with a period of exactly one flux quantum h/2e:

The DC SQUID(6)

$$I_{\max }(\Phi )=2I_c\left|\cos \left(\frac{\pi \Phi }{{\Phi }_0}\right)\right|,{\Phi }_0=\frac{h}{2e}$$

The qubit. Shunt a junction with a large capacitor so that the Josephson energy dominates the charging energy, and the two lowest levels of the cosine well form a nearly ideal two-level system whose transition frequency sits in the microwave band and whose properties are insensitive to stray charge. That is the transmon (Koch et al., 2007), the workhorse of superconducting quantum processors. Its state is manipulated by microwave pulses through exactly the locking physics of Level 2, and read out through a resonator. A processor with a hundred qubits is a hundred Josephson junctions, each a macroscopic quantum phase held in superposition.

Definitive The RCSJ model, SQUID interferometry and transmon qubits are established engineering.

Level 5 · Research frontier

The junction as an emitter. A biased junction oscillates at a frequency set by voltage alone, so an array of junctions driven in step is a tunable, coherent source. Arrays of thousands of junctions already produce useful terahertz radiation. Gary Stephenson's 2026 proposal, discussed at length on the channel this site follows, pushes the idea one step further: a phased array of Josephson junctions on a silicon wafer, tuned to about 24 GHz by the AC Josephson relation, with emitters spaced one wavelength apart and the beam steered by time delay, intended to emit not microwaves but high-frequency gravitational waves — a "gaser", a gravitational-wave laser, for communication through rock and seawater where radio cannot go.

Everything about the electromagnetic side of that design is established: the tunability is rule two, the coherence is the locking of Level 2, and phased-array steering is standard radar engineering. The open question, which Stephenson names himself, is the quantum efficiency — how much of the array's oscillation can be converted into gravitational radiation. The general-relativistic coupling that governs that conversion (the same G/c⁴ factor that appears in the Gertsenshtein effect) is very small, which is why the proposal specifies a first experiment rather than a finished device. A related theoretical route — a pair of junctions arranged so that their oscillating charge has a time-dependent quadrupole moment, the configuration that radiates gravitationally — was worked out by Victor Atanasov in 2019.

Speculative Junction arrays as gravitational-wave emitters: a fully specified proposal with a named open question and a first experiment; no signal yet.

What Charles Chase adds. In his August 2026 interview Chase describes the goal of putting matter waves in phase using the vector potential, and moves directly to the physics of superconductors and junctions, because they are the systems in which a macroscopic quantum phase already exists and can be wired to a battery. The junction is, in that sense, the proof that "phase as an engineering variable" is not a metaphor: the volt is defined by it.

What to watch

1. A first measurement from a phase-locked junction array built to Stephenson's specification, with the detector and the expected signal level stated in advance.
2. Terahertz junction-array sources continuing to improve in coherence and power — the same engineering the gaser needs, published in the mainstream literature.
3. Superconducting quantum processors crossing the fault-tolerance threshold: the strongest ongoing demonstration that Josephson phase can be controlled at scale.

Teaching aids

Three demonstrations you can run

1. Two pools, one wall. Two clear containers at the same water level joined by a strip of paper towel or a porous ceramic: water creeps across with no height difference. Ask what "drives" it. (Capillary action is not the Josephson mechanism, but the lesson — flow without a pressure difference is possible when the two sides are coupled — is the right first picture.)
2. Metronomes on a board. Put three or four metronomes on a board resting on two soda cans and start them out of step. Within a minute they synchronise. That is phase locking through a shared coupling — the same behaviour as two superconductors coupled through a barrier, and the same mathematics (the Kuramoto model) Chase's slides cite for the matter-wave beam.
3. The washboard. A marble on a tilted corrugated surface (a piece of corrugated cardboard on a book) shows the RCSJ picture in one glance: below a certain tilt the marble rests (zero voltage); past it, the marble rolls dip to dip (oscillation).

Self-check (answers below)

1. What flows across a Josephson junction at zero voltage, and what sets how much?
2. Apply exactly one microvolt across a junction. At what frequency does the current oscillate?
3. Why does the charge 2e appear in the Josephson relations rather than e?
4. A SQUID's critical current has just completed one full oscillation. By how much did the flux through the ring change?
5. In one sentence: why can a junction be a qubit when an ordinary capacitor-and-inductor circuit cannot?

Answers. (1) Cooper pairs, as a supercurrent; the amount is I_c sin δ, set by the phase difference. (2) About 484 kHz, from f = (2e/h)V — 483.6 GHz per volt times one millionth of a volt. (3) The carriers are electron pairs. (4) One flux quantum, h/2e ≈ 2.07 × 10⁻¹⁵ Wb. (5) The junction's cosine energy is nonlinear, so its energy levels are unevenly spaced and the lowest two can be addressed on their own; a linear circuit has equally spaced levels and cannot.

One-page summary for the wall

• A superconductor is one macroscopic quantum wave; a junction is two waves coupled through a thin barrier.
• Rule one: I = I_c sin δ — current with no voltage. Rule two: dδ/dt = 2eV/ħ — voltage makes the phase turn, so the current oscillates at 483.6 GHz per volt.
• The phase difference that matters includes the vector potential; that is why flux quantises and why two junctions in a ring make a magnetometer.
• Washboard picture: rest in a dip (supercurrent) or roll (oscillation). Add a capacitor and the well's lowest two levels are a qubit.
• The frontier: arrays of phase-locked junctions as coherent emitters — terahertz today; gravitational waves in the gaser proposal, with the quantum efficiency as the open question.