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 trillion 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.

Turn the voltage, watch the frequency follow

A bias voltage across the gap makes the phase difference wind round, and the current oscillates at a rate the voltage alone sets — about 484 megahertz for every microvolt, with nothing about the metal, the barrier or the temperature in it. The marks along the axis are the eleven operating points Stephenson tabulates for his junction array, and the line through them is the Josephson constant itself, two fixed numbers of the SI divided by each other. Move the dial and the frequency, the wavelength and the emitter that would radiate it all follow.

Bias voltage across the junction

Bias voltage across the junction

49.73

Return to the band the paper selects

Jump to a published operating point

1.296 GHz (2.68 µV)2.32 GHz (4.80 µV)3.4 GHz (7.03 µV)5.76 GHz (11.91 µV)10.368 GHz (21.44 µV)24.048 GHz — the band the paper selects (49.73 µV)47.088 GHz (97.37 µV)76.233 GHz (157.64 µV)120 GHz (248.14 µV)

• The Josephson constant, 2e/h
• the band the paper selects

Josephson frequency:

24.049 GHz

The Josephson constant, 2e/h:

483.598 MHz/µV

This row's own ratio, from the table's rounded cells:

483.571 MHz/µV (-0.005%)

Free-space wavelength:

1.247 cm

Quarter-wave emitter pad:

0.312 cm

Full-wavelength spacings across a 150 mm wafer:

12.0

What the junction is doing: The dial is standing on a published row of the table, and that row's two rounded cells divide to within a thousandth of the constant the line draws.

Where the line comes from. It is the Josephson constant itself, K_J = 2e/h. Since 2019 both the charge of the electron and Planck's constant are fixed numbers of the SI, so this slope is exact arithmetic on two defined values rather than a measurement, and it has defined the volt since 1990. The published rows were never fitted to it, and all eleven of them land on it to better than a tenth of a percent. That agreement is the whole point: a relation built only from the charge of a pair and Planck's constant has no room in it for the metal, the barrier or the temperature. Dividing one published row's rounded cells by each other comes close but does not reproduce the constant, so the read-out shows that quotient beside the constant and labelled as a row's ratio, never in the constant's place.

Why this band. The paper spaces its emitter pads by full wavelength increments to keep them in phase, so the wavelength decides how many can share a wafer. Below about four gigahertz fewer than two spacings fit across a 150 millimetre wafer; at the selected band twelve do. Run the dial down and watch the count fall — the author's own reason for the choice, reached from his own figures.

Scope: the second Josephson relation drawn as a line, with a published table of operating points laid on it. Critical current, junction materials, linewidth and emitted power are not modelled, and nothing about gravitational radiation is asserted or measured here.

Voltage standards have used the Josephson voltage-to-frequency relation since 1990. The junction array proposed as a gravitational-wave emitter is a separate paper design with an open problem its own author names, and the course takes that up at Level 5.

The source sheet: Stephenson 2026, Stimulated Emission of Gravitational Waves via Dissimilar Superconducting Josephson Junctions — Section 5.1, Table 1

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)

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

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 THz per volt × 0.001 V ≈ 484 GHz — a submillimetre-wave frequency. Apply one microvolt and you get about 484 MHz, a UHF radio frequency. 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)

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

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)

$$\delta = \theta_2 - \theta_1 - \frac{2e}{\hbar}\int_1^2 \mathbf{A}\cdot d\mathbf{l}$$

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)

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

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)

$$I_{\max}(\Phi) = 2I_c\left|\cos\!\left(\frac{\pi\Phi}{\Phi_0}\right)\right|, \qquad \Phi_0 = \frac{h}{2e}$$

The interferometer with the field switched off

Split an electron beam, send the two halves either side of a magnetic flux that is completely confined and completely shielded, and bring them back together. On the paths themselves the magnetic field is exactly zero — the shaded corridor below is field-free the whole way. The vector potential there is not zero, and the fringes move. Dial the enclosed flux and watch the pattern slide sideways by exactly one fringe for every flux quantum. Then try the second control. It changes the gauge — the potential itself, at every point on both paths, as the arrows show — and the fringes do not move at all. Both controls are live; only one of them changes anything a detector can see.

Two controls: one moves the fringes, one cannot

Enclosed flux, in flux quanta h/e

0.00

Advance by one whole flux quantumReturn to zero enclosed flux

Gauge choice λ — the freedom in the potential itself

0.00

Return to the textbook gauge

• Upper path
• Lower path
• Confined flux
• Vector potential on the paths

What is happening: The enclosed flux is a whole number of quanta, so the phase difference is a whole number of turns and the pattern sits exactly where it started. This is the periodicity that makes the effect a measurement rather than a drift.

Magnetic field on the electron paths

zero, at every setting

Vector potential on the upper path, in this gauge

0.00 nWb/m

Vector potential on the lower path, in this gauge

0.00 nWb/m

Phase the gauge adds to each half

0.00 radians — the same for both halves, so it cancels

Phase difference between the two halves

0.00 radians

Fringe shift

0.00 fringe widths

What no choice of gauge can change

the field on the paths, the phase difference, the fringe shift

The precision rule this instrument exists to protect: the vector potential controls phase. No energy is stored in it, and none needs to be — the phase difference is set by the flux the two paths enclose between them, which is exactly the quantity you are dialling. And the folklore reading — that the effect proves the potential is physically real at a point — is exactly what the second control is here to foreclose: the honest observable is the loop integral, never the potential at a point.

Nothing happening is the result here, not a broken control. The potential is not unique: for any smooth function χ you may replace A by A + ∇χ, which changes the potential at every point and changes the magnetic field nowhere, because the curl of a gradient is zero. This instrument performs that substitution for real and then integrates the full potential, gauge term included, around the closed loop the two halves of the beam make between them. Each half does pick up a phase from the gauge term — exactly 2λ radians in this family — and both halves pick up the same one, so it cancels out of their difference. What survives is the loop integral, which is the enclosed flux, and that is the observable.

Tonomura's 1986 experiment answered the obvious objection by construction: the flux sat inside a toroidal magnet wrapped in a superconducting shield, so stray field was not merely small on the electron wave but excluded from it. The shift was still there, and it was still one fringe per flux quantum.

Scope: a drawn two-path interferometer with a perfectly confined flux, which is the geometry of the published experiment. Beam energy, path length and magnet size are not modelled, and no absolute intensity is asserted. Φ₀ = h/e ≈ 4.136e-15 Wb.

The source sheet: Tonomura et al. 1986, Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave

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.

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.

The gaser, measured against the instruments

Stephenson's wafer is specific enough to be measured, so measure it. Here is the megahertz-to-gigahertz band with the published detectors laid across it, and 24.048 gigahertz — the frequency the gaser is designed to emit at — drawn as a line through them. Then the distance the author names himself: roughly twenty orders of magnitude of coupling still to find. A zoomable scale would hide that, because above about a million readers stop reading a logarithmic axis logarithmically. So the orders are counted instead, one rung at a time, next to two distances of known size.

What to look at

The band: who is listening at 24 gigahertzThe distance, counted in orders of magnitude

• Strain noise per root hertz
• An upper limit already set on a background
• The gaser

The published instruments

The 100-megahertz recycling interferometer (1E-16)The Fermilab Holometer (1E-21)Superconducting transducers above a gigahertz (1E-21)Quartz bulk acoustic-wave resonators (2E-22)A microwave cavity on ADMX parameters (3E-22)Graviton–magnon resonance (7.6E-22)ARCADE 2, recast (1E-24)The Li–Baker detector (1E-32)

Back to the gaser's own frequency

The gaser's emission frequency:

24.0 GHz

The published ideal case:

1.2E-1 W / 6E-10 W

The shortfall the author states:

20

Where that leaves the device today:

-11.7

Reading: A microwave cavity on ADMX parameters. 3E-22 — strain per root hertz. Its published band stops short of 24.048 gigahertz.

Berlin and colleagues show that an axion haloscope is already most of a gravitational-wave detector: an eight-tesla field, a tenth of a cubic metre of cavity, a quality factor of 100,000, one kelvin and a minute of integration reaches about 3 × 10⁻²² at a gigahertz, and a phase-sensitive matched filter takes it to about 10⁻²³. The hardware exists; what changes is what it is asked.

The published ideal case assumes nearly perfect conversion, and the author is explicit that the conversion is the open problem. Subtract the shortfall he states and the ledger closes where the drawing shows it.

This is what taking a proposal seriously looks like. A programme that names no frequency and no output cannot appear on these axes at all; the gaser can, because it names a wafer, a bias voltage, an emitter count, a predicted power, a detector and its own missing number. The distance is real, and so is the fact that it is stated rather than discovered by a critic.

Three things would move the line. The frequency scaling Stephenson gives on the same broadcast: strain rises with the square of the frequency and radiated power with the fourth, so the whole ledger moves with the band a future wafer is designed for. The coherence law the array exists for: radiated power grows with the square of the number of emitters kept in step, so 81 pads on a 150 mm wafer is the smallest version of the device, not the largest. And the conversion efficiency itself — the doping sweep across d-wave and s-wave regimes that the paper recommends as the first experiment is the measurement that would put a real number where twenty orders of magnitude currently stand.

Read the source sheets. The 100-megahertz recycling interferometerA microwave cavity on ADMX parametersARCADE 2, recastThe Li–Baker detectorThe gaser

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, who also writes the objection to his own device into the paper: under the standard description of a junction held at constant potential the charge densities stay constant, so nothing oscillates and nothing radiates, and the proposal rests on the junction's non-linear dynamics departing from that solution.

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 MHz, from f = (2e/h)V — 483.6 THz 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 THz 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.