The voltage that is a frequency

In 1962 a twenty-two-year-old research student at Cambridge, Brian Josephson, worked out what should happen when two superconductors are separated by an insulating layer thin enough for electrons to tunnel through it. The answer he got was so strange that one of the architects of the theory of superconductivity published an argument that it could not be right. It was right. It was seen the following year, it earned Josephson a Nobel Prize at thirty-three, and it now sits at the foundation of how electrical units are defined.

The strangeness is in two short equations, and the figures on this page are those equations integrated.

A current set by a phase difference #

A superconductor is described by a single complex number spread over the whole piece of metal — an order parameter with an amplitude and a phase — and the two in the flux quantum follows from requiring that phase to come back to itself around a ring. In one isolated piece the absolute value of the phase means nothing. Between two pieces the difference of phases is physical, and what it controls is the current.

Separate the two superconductors by a barrier a nanometre or two thick. Pairs of electrons can tunnel across it, as a particle tunnels through a wall that is not quite a wall, and the tunnelling couples the two condensates with an energy that depends on their phase difference $\varphi$ as $-E_J\cos\varphi$. A current is the rate of flow of charge, and in this system it is fixed by how that energy changes with the phase:

$$I = I_c \sin\varphi.$$

That is the first Josephson relation. It says a current up to $I_c$ can flow through the barrier with no voltage across it — a supercurrent through an insulator — and that the current’s size is set by a quantum phase difference between two pieces of metal. Nothing like it exists in a normal conductor, where current is pushed by a voltage against a resistance.

The zero-voltage branch of the figure is that supercurrent. Drive more than $I_c$ and the sine cannot supply it, and the junction has to do something else.

A voltage makes the phase run #

The second relation is what happens then. A voltage $V$ across the junction gives a pair of electrons on one side an energy $2eV$ more than a pair on the other, and in quantum mechanics a difference of energy is a difference in the rate at which phases rotate. So the phase difference advances steadily:

$$\frac{d\varphi}{dt} = \frac{2eV}{\hbar}.$$

Put the two together and a steady voltage produces a supercurrent that oscillates, $I_c\sin(2eVt/\hbar)$, at a frequency $V/\Phi_0$ where $\Phi_0 = h/2e$ is the flux quantum. The conversion factor is enormous. One microvolt gives 483.6 megahertz; one millivolt, nearly half a terahertz. A voltage across a Josephson junction is a frequency, and the constant that converts one into the other contains only the charge of an electron and Planck’s constant.

A real junction is not driven by a perfect voltage source, and the figure uses the model that describes most practical ones: the junction in parallel with a resistance, fed by a current. The current divides between the supercurrent and the resistor, the voltage across the resistor sets the rate at which the phase advances, and the result is a single equation for the phase, $\dot\varphi \propto I - I_c\sin\varphi$. Below the critical current it has a steady solution and the phase stands still. Above it the phase has to advance, and it does so unevenly: slowly where $\sin\varphi$ nearly carries the bias, quickly where it does not.

The voltage is therefore a train of pulses rather than a steady value, and each pulse is one complete turn of the phase — one flux quantum crossing the junction. The mean voltage is the number of turns per second times $\Phi_0$, which is the figure’s check: three slips in the window, and a mean of 0.831, the value $\sqrt{i^2-1}$ that the equation predicts exactly.

Why the prediction was doubted #

The objection to Josephson’s result was reasonable, and seeing why it fails says something about what a condensate is. Tunnelling through an insulator is improbable: an electron crosses a barrier a nanometre thick with a small amplitude, and a pair crossing together should, by the ordinary rules, need that small amplitude twice — the square of something small, and far too weak to carry a current anyone could measure. John Bardeen, who with Leon Cooper and Robert Schrieffer had built the theory of superconductivity, argued exactly that in print.

The flaw is in treating the pairs as independent. In a superconductor every pair shares one phase, and what tunnels is the condensate’s amplitude as a whole: the contributions of all the pairs add in step rather than at random, so the current is first order in the coupling and not second. It is the difference between two lamps, whose light never interferes and adds in intensity, and one laser, whose light adds in amplitude. Philip Anderson and John Rowell saw the zero-voltage supercurrent in 1963, and the way it responded to a magnetic field left no doubt about what it was.

That response is itself a surprise. A magnetic field threading the barrier makes the phase difference vary across the junction’s width — the way a magnet leaves a phase on a path it never touches — so different parts of the junction carry current in different directions and partly cancel. The largest supercurrent the junction can carry, against the flux through its barrier, follows $|\sin(\pi\Phi/\Phi_0)/(\pi\Phi/\Phi_0)|$: the same function as the amplitude of light diffracted by a single slit, with the flux quantum in the place of the wavelength, and it falls to zero whenever a whole number of flux quanta threads the barrier. A junction in a field is a diffraction experiment performed by a condensate on itself.

The figure computes the pattern the slow way rather than quoting it. The field makes the phase difference advance by one turn across the barrier for every flux quantum threading it, each strip of the barrier carries the sine of its own phase, and the critical current is the largest total the strips can reach together. At exactly one flux quantum the phase runs through a whole turn across the width and every strip carrying current one way has a partner carrying it the other: the total is zero whatever the phase at the edge. Between whole numbers the cancellation is incomplete, and the side lobes are what is left over — 21.7 per cent of the zero-field current at 1.43 flux quanta, the same fraction as the first bright fringe beside a slit’s central maximum. Measuring that pattern in 1963 was how a small, fragile supercurrent was shown to be the phase and not a short circuit through a pinhole in the barrier: a pinhole carries current that a field does not switch off at whole flux quanta.

An overdamped pendulum on a slope #

The junction equation is one that turns up across physics, and recognising it makes the dynamics obvious. $\dot\varphi = i - \sin\varphi$ is the equation of a pendulum so heavily damped that its inertia is negligible — pushed by a steady torque $i$, pulled back by gravity through $\sin\varphi$, and moving at a speed proportional to the net torque, which is the overdamped way of coming to rest with a drive added.

A small torque tilts the pendulum and it stays there: that is the supercurrent, at a fixed phase. A torque larger than the pendulum’s weight can resist sends it over the top, and it goes round and round — slowly as it climbs past horizontal, quickly as it falls through the bottom. The rate of rotation is the voltage. The picture explains the pulses, the square-root threshold of the mean voltage and the way the junction behaves under a periodic drive, and the same equation describes a charge-density wave sliding through a crystal and a clock being entrained by a stronger one.

Steps a microwave draws #

Add a microwave current to the steady bias, and the pendulum is driven by a torque that oscillates as well as one that pushes. A driven rotor can lock to its drive, going round exactly once — or exactly twice, or $n$ times — for every cycle of the oscillating torque, and while it is locked a change in the steady torque does not change its rate of rotation.

For the junction, a rotation rate is a voltage, so a locked rotation is a voltage that holds still as the bias current changes: a vertical step on the current–voltage curve, at

$$V_n = n\,\frac{h f}{2e}.$$

Sidney Shapiro saw these steps in 1963. The figure’s steps sit where the formula puts them to within half a per cent, over ranges of bias current that grow with the microwave power. The size of each step in current depends on the junction; its voltage does not. It contains the drive frequency, Planck’s constant and the charge of the electron, and nothing about the metal, the barrier, the temperature or the resistance.

A volt counted in cycles #

That property is extraordinary for a measurement standard. Frequencies are the quantities physics can measure best — the best atomic clocks count cycles to parts in $10^{18}$ — and a Josephson step converts a frequency into a voltage through a constant whose only ingredients are fundamental.

A single step is small — 144.7 microvolts at 70 gigahertz — so a practical standard puts many junctions in series on one chip, all driven by the same microwave source, and counts their steps. Ten volts takes 69,085 junction-steps at that frequency, made from about twenty thousand junctions each locked to a step of a few. The voltage is then known as well as the frequency and the count, and its uncertainty is limited by everything except the constant.

The international electrical community adopted a conventional value for $2e/h$ in 1990 and began realising the volt this way. Whether the constant really is the same for every junction was tested by driving junctions made of different superconductors from one source and comparing their steps directly, and they agree to parts in $10^{16}$ and beyond: the relation belongs to the phase, not to the material. In 2019 the redefinition of the SI fixed the values of $e$ and $h$ exactly, which made $2e/h$ exact too, and the Josephson volt became a definition rather than a very good measurement.

The same step is part of how mass is now measured. A Kibble balance weighs a kilogram by balancing its weight against the force on a current-carrying coil in a magnetic field, and then measures the voltage the coil generates when moved; the product of a voltage and a current, each realised with a quantum standard — a Josephson junction for the voltage and the quantum Hall effect for the resistance — gives the mass in terms of Planck’s constant. A chip of superconducting junctions is, indirectly, one of the ways a kilogram is weighed.

The same arrays now make voltages that change. Divide the junctions of a chip into segments of 1, 2, 4, 8 and onward, each biased onto its first step or held at zero independently, and the chip becomes a digital-to-analogue converter whose every output level is a count of flux quanta per microwave cycle: a stepped sine wave with each step exact. Drive a junction with short current pulses instead of a sinusoid, and each pulse transfers exactly one flux quantum, so the time-averaged voltage is the pulse rate times $h/2e$ and a sequence of pulses chosen by a computer synthesises an alternating voltage whose amplitude is defined as well as the direct one. Both are built on the relation the second figure shows — a voltage pulse is one turn of the phase — used as a unit of charge-flow that cannot be divided.

The same relation wherever a phase is shared #

Nothing in the second Josephson relation is specific to electrons. It says that a difference of energy per particle between two coupled condensates makes their phase difference run, and it holds wherever a macroscopic phase exists. Two reservoirs of superfluid helium joined through an array of apertures a few nanometres across, held at slightly different pressures, carry a flow that oscillates at the frequency the difference in chemical potential sets. It was observed from 1997 onwards, and in helium-4 the oscillating flow, turned into sound, is a pure tone. The whirlpool that comes in one size is the same single-valued phase quantising a circulation instead of a current.

A long junction adds space to the dynamics. When the barrier is longer than a characteristic length the phase difference varies along it and obeys a wave equation with a sine in it, and one flux quantum trapped in the barrier travels along it as a solitary wave that keeps its shape — the same kind of object as the pulse two failures keep alive in a fibre, with a nonlinearity and a dispersion balancing exactly. And a two-dimensional grid of small junctions, each coupling the phases of its neighbours, is a model system in which a transition with nothing to order can be watched as vortices of phase bind into pairs and unbind.

The phase also makes the junction the most sensitive detector of magnetic field there is. Two junctions in a superconducting loop form an interferometer whose critical current swings through a full cycle for every flux quantum threading the loop — about twenty picotesla over a square centimetre, under a millionth of the Earth’s field — and reading a small fraction of that cycle is how the faint fields of the brain and the heart are measured. The flux quantum that the two in the flux quantum derives from a ring becomes a ruler once a junction is put in the ring.

Where the model stops #

The junction has no capacitance. The shunted-junction model drawn here is the overdamped limit, in which the phase has no inertia. A real tunnel junction is also a capacitor, and adding capacitance gives the pendulum a mass: its current–voltage curve becomes hysteretic, jumping to a finite voltage at the critical current and staying there as the current is reduced. The junctions in voltage standards are of this kind, and their steps cross zero current, which is what lets a whole array be biased at once.

Thermal noise has been left out. At a finite temperature the phase is kicked by fluctuations, and a junction whose coupling energy is not large compared with $k_BT$ slips its phase spontaneously even below the critical current, rounding the corner of the current–voltage curve and blurring the edges of the steps.

The current–phase relation is taken as a pure sine. That is right for a tunnel barrier. Other weak links — a narrow constriction, a normal-metal bridge — have current–phase relations of other shapes, and some have components at twice the phase. The voltage–frequency relation is unaffected, because it comes from the energy of a pair rather than from the shape of the barrier.

And the drive is uniform. A voltage standard needs every one of twenty thousand junctions to see the same microwave amplitude closely enough to sit on its intended step, which makes the design of the chip as a microwave circuit most of the engineering.

What the pictures cannot show #

None of the figures draws the phase itself, and the phase is the whole of the physics. It is a quantum variable of a piece of metal containing $10^{22}$ electrons, defined across the entire superconductor, and the junction is a device for measuring the difference between two such variables with a voltmeter. That a quantity like that can be read with an ordinary instrument is what makes superconductivity a macroscopic quantum phenomenon, and a current–voltage curve gives no sense of how unusual it is.

Nor do the averaged curves show the dynamics under drive. For some combinations of amplitude and frequency, and especially with capacitance included, the driven junction does not lock or rotate regularly at all but moves chaotically, and the mean voltage between steps is an average over motion with no simple pattern. The flat steps are the islands of order in a much richer map.

Still open: whether a circuit this coherent can be in two states at once #

The Josephson junction treats the phase difference as a classical variable that happens to be quantum in origin: it has a value, it rotates, it locks. But the phase is conjugate to the number of pairs that have crossed the junction, and in a small enough junction with a small enough capacitance the two cannot both be definite. The circuit then has quantised energy levels of its own, and its phase can tunnel out of a well of the washboard potential rather than being thermally kicked over the top.

Seeing that — a circuit of macroscopic wires behaving as a single quantum object, with discrete energy levels and tunnelling — was the work recognised by the 2025 Nobel Prize in Physics, and it is the basis of superconducting quantum bits. How long such a circuit can hold a superposition of two of its states, and what in its surroundings destroys the superposition, is the question the junction opens once its phase is treated as the quantum variable it always was.

The habit worth carrying away is the one Shapiro’s steps demonstrate. When a measured quantity depends on nothing about the object that produces it, the object has become a way of reading a constant. The critical current, the resistance and the material of the junction all drop out of the step voltage, and what is left is a frequency and the ratio of two fundamental constants — which is exactly what a standard should be made of.