The circuit that goes through a wall

How long the crossing takes closes by naming two directions the subject of tunnelling goes next: dissipative tunnelling, “where a coupling to an environment supplies the clock and suppresses the rate”, and macroscopic quantum tunnelling, “where the coordinate under the barrier is a current in a circuit rather than a particle”. They turn out to be one subject, and it was settled by one experiment, carried out at Berkeley in 1984 and 1985 by Michel Devoret, John Martinis and John Clarke. Forty years later it earned them the 2025 Nobel Prize in Physics.

The question they answered had been sharpened by Anthony Leggett around 1980. The wall that is not quite a wall describes a quantum particle leaking through a barrier it cannot climb, and every example there is microscopic: an electron, an alpha particle, a proton in a hydrogen bond. Quantum mechanics says nothing about size. But there was no evidence that a variable describing a large object — something made of many billions of particles, measured with ordinary electronics — obeys it as a single coordinate, with superposition and tunnelling and discrete levels, rather than behaving classically because its many parts are coupled to so much else. Leggett proposed that a superconducting circuit was the place to look, and with Amir Caldeira he worked out what an experiment would have to see.

A phase in a tilted washboard #

A Josephson junction is two superconductors separated by an insulating layer a nanometre thick. Each superconductor is described by a single quantum phase, shared by all its electron pairs, and the voltage that is a frequency shows that the difference between the two phases, $\varphi$, is the variable that matters: the supercurrent through the barrier goes as $\sin\varphi$, and the voltage across it is the rate at which $\varphi$ turns.

Drive a steady current through the junction and $\varphi$ behaves like a particle in a potential. The junction’s capacitance plays the part of the particle’s mass, and the potential is a washboard — a cosine from the junction’s coupling, tilted by the bias current. With no bias the board is level and the phase sits in any of its wells. As the current rises the board tilts and the barriers holding the phase in its well shrink, vanishing entirely at the critical current. While the phase sits in a well the junction carries its current with zero voltage. If the phase escapes, it runs away down the board, turning ever faster, and a voltage appears across the junction almost instantly. An escape is therefore a single, sharp, electrical event, and counting them — ramping the current up again and again and recording the current at which the voltage appears — measures the escape rate.

A few quantum levels in a circuit’s well #

If the phase is a quantum coordinate, the well has discrete levels, spaced by the frequency of small oscillations of the phase times Planck’s constant. For the junction in the drawing that frequency is a few gigahertz, and the level spacing corresponds to a temperature of a couple of hundred millikelvin — large enough for a refrigerator at a few tens of millikelvin to leave the phase almost always in the lowest level. Near the critical current the well is shallow and holds only a handful of levels; the drawing’s holds five, crowded together towards the top because the well is not a parabola, the same crowding that lets a molecule’s spectrum weigh its bond.

Martinis, Devoret and Clarke observed these levels directly in 1985. They shone microwaves on the junction and found that the escape rate jumped at particular frequencies — when the microwaves matched the spacing between levels and lifted the phase to a higher one, from which it tunnelled out far faster — and the resonant frequencies moved with the bias current exactly as the level spacing should. A coordinate of a circuit had a spectrum, like an atom. That observation, as much as the tunnelling, is what the 2025 prize cited: “the discovery of macroscopic quantum mechanical tunnelling and energy quantisation in an electric circuit”.

How an escape is counted #

An individual escape is easy to see and hard to time, so the measurement is statistical. The bias current is ramped up from zero, again and again, thousands of times a second. On each ramp the phase sits in its well until, at some current, it escapes and the voltage jumps; the current at that moment is recorded and the junction reset. Because escape is random, the switching current differs from ramp to ramp, and after tens of thousands of ramps the recorded currents form a histogram a few tenths of a per cent wide. Theodore Fulton and L. N. Dunkleberger showed in 1974 how to turn such a histogram into the escape rate at each current: the fraction of ramps that survive to a given current, and how quickly they are then lost, give the rate directly.

The method measures escape rates over many orders of magnitude without timing any single event, and it measures them at every bias along the way. Its weakness is that anything else that makes the phase escape — a burst of noise from the room-temperature electronics, a stray photon down a wire — looks exactly like a thermal or quantum escape. The experiment’s filters, cooled to the refrigerator’s temperature and built to absorb microwaves from outside, were as important to the result as the junction.

Escape by heat, and escape without it #

There are two ways out of a well. The classical way is over the top: thermal noise occasionally kicks the phase hard enough to clear the barrier, at a rate proportional to $e^{-\Delta U/kT}$ — the exponential that decides everything, and the same law by which chemical reactions go faster when warm. The quantum way is through: the phase’s wavefunction extends into the barrier and leaks out the other side, at a rate that depends on the barrier’s height and width and on the mass of the coordinate, and not on the temperature at all.

The drawing puts them together. At a hundred millikelvin the thermal rate is enormous and the tunnelling rate irrelevant. As the junction cools, the thermal rate falls exponentially and eventually passes below the tunnelling rate, which stays fixed. Below that crossover — about $\hbar\omega_p/2\pi k$, a few tens of millikelvin here — escapes continue at a steady rate, and cooling the junction further changes nothing. A classical coordinate would stop escaping altogether as its temperature approached zero. A quantum one keeps going.

A temperature the circuit refuses to go below #

The measured quantity in the 1985 experiment was exactly this curve. For each temperature of the refrigerator, the escape rate was measured and converted into an escape temperature: the temperature a purely thermal process would need to give that rate. At high temperatures the escape temperature equalled the refrigerator’s, confirming that escapes were thermal. Below a few tens of millikelvin the escape temperature stopped falling and levelled off, while the refrigerator went on cooling.

The levelling-off alone was not new: earlier experiments had seen flattening of this kind and it could have been caused by anything that kept the junction warmer than the thermometer said — electrical noise leaking down the wires, or heating by the measurement itself. What made the Berkeley result convincing was that everything that could fake it was measured. The same junction was deliberately biased into a regime where its escape was classical, and its measured escape temperature there tracked the refrigerator’s, showing that the junction was as cold as claimed. Its capacitance, critical current and the damping from the circuit around it were measured independently, by the microwave resonances, and put into the tunnelling theory with no free parameter. The predicted plateau and the measured one agreed. The phase difference across a piece of metal was tunnelling.

Friction slows tunnelling #

The part of the problem that made it hard, and that makes it matter far beyond Josephson junctions, is dissipation. A junction does not sit in a vacuum. It is wired to a current source and a voltmeter, and the wiring has resistance, so the phase’s motion is damped: energy it gains leaks away into the electrons of the circuit. A classical coordinate with friction is still classical. What does friction do to a quantum one?

Caldeira and Leggett’s answer, in 1981 and 1983, is in the drawing. Dissipation suppresses tunnelling, multiplying its exponent by a factor $1 + 0.87/Q$ for a lightly damped junction with quality factor $Q$. With little damping the effect is small; with heavy damping it is large, and a junction wired carelessly might not tunnel measurably at all. The physical reading is the one where the interference goes develops for any superposition: an environment that absorbs energy from the phase’s motion also acquires a record of that motion, and a coordinate that is being recorded loses the ability to be in the superposition of positions that tunnelling requires. Friction is decoherence, and decoherence slows the passage through the wall.

That is why the 1985 experiment needed its circuit characterised as carefully as its junction: the tunnelling rate depends on the environment, and a prediction without the environment would have been a prediction of nothing. And it is why the result mattered. It showed that the effect of the environment on a macroscopic quantum variable could be calculated from the circuit’s measured impedance — which is exactly the knowledge needed to build a circuit whose quantum behaviour survives for long enough to be used.

Why a circuit, of all things #

Leggett’s choice was not obvious. A circuit variable seems the least quantum thing there is — it is what an electrical engineer measures with a voltmeter. But the phase across a superconducting junction has three properties that made it the best candidate available. Its mass, set by the capacitance, is tiny compared with any mechanical object’s, so its quantum levels are widely spaced. Its only coupling to the rest of the world is electrical, through wires whose impedance can be measured and designed, so its dissipation can be made small and, crucially, known. And its value can be read out by a single, unambiguous event — the appearance of a voltage — without the readout disturbing it before that moment.

A pendulum has none of these. Its angle is a coordinate of a massive body, its quantum levels are spaced by far less than any achievable temperature, and it is coupled to air, to its pivot and to the vibration of the building by channels nobody can list. The junction is a pendulum — its equation of motion is the pendulum’s, with the bias current as a constant torque — built so that its friction is small, measurable and electrical.

The same physics in a magnet #

Once dissipative tunnelling had a theory, it was sought in other collective variables, and the next clear case was magnetic. Nothing keeps a magnetisation for ever describes a small magnetic particle whose magnetisation sits in one of two wells and switches by thermal activation over the barrier between them. A molecular magnet — a cluster of a dozen manganese ions whose spins lock together into one large spin, crystallised by the billion into identical copies — is such a particle made small enough to show levels. In 1996 Jonathan Friedman, Myriam Sarachik and colleagues found that the hysteresis loop of the crystal $\mathrm{Mn_{12}}$-acetate had steps: at particular values of the applied field, where levels on the two sides of the barrier lined up, the magnetisation suddenly relaxed much faster. The magnetisation was tunnelling through the barrier, resonantly, and the steps were the levels.

The magnet and the junction obey the same mathematics: a single collective coordinate, a double or tilted well, levels, a crossover temperature below which escape becomes temperature-independent, and a rate controlled by the coordinate’s coupling to its environment — the lattice vibrations and nuclear spins, in the magnet’s case. What both taught is that “macroscopic” is not the variable that decides whether quantum mechanics shows. Isolation is.

From one junction to a quantum computer #

The junction in the drawings is the ancestor of the superconducting qubit. Its two lowest levels, separated by a few gigahertz and addressable with microwaves, are a two-level quantum system; the phase can be put into superpositions of them with a microwave pulse and read out by whether it tunnels. Over the following decades the design was changed to protect those two levels from noise — the transmon, the design used in most superconducting quantum processors, is a junction shunted by a large capacitor, which makes its levels insensitive to stray charges — but the physics is the physics of the washboard. Martinis went on to lead the team at Google that in 2019 demonstrated a fifty-three-qubit processor; Devoret’s group at Yale developed much of the modern theory of circuit quantum electrodynamics. A coordinate of a circuit, shown in 1985 to obey quantum mechanics, is now engineered by the thousand.

It is also the circuit that forgets its charge seen from the other side. A small junction has a well-defined charge and an uncertain phase; a large one, like the junction here, a well-defined phase and an uncertain charge. The two regimes are the two ends of one uncertainty relation between the number of pairs that have crossed the junction and the phase difference across it, and the transmon lives deliberately near the phase end.

Where the washboard picture stops #

A single coordinate. The whole analysis treats the phase as one degree of freedom moving in a potential, with everything else — the quasiparticles, the phonons, the electromagnetic modes of the wiring — lumped into a damping. That is the Caldeira–Leggett model, and its success is the evidence that it is right for these circuits; it would not be for a junction with many internal modes of comparable frequency.

The formulas near the crossover. The drawn rates add a thermal rate and a quantum rate, each correct far from the crossover. Near it the true rate interpolates smoothly between them, with quantum corrections to the thermal rate and thermal corrections to the tunnelling, and the drawn curves are only approximate there.

A cubic well. The tunnelling exponent, $7.2\,\Delta U/\hbar\omega_p$, is for a well shaped like a cubic polynomial, which the washboard becomes near the critical current. Further from it, where the well is deeper and more like a cosine, the numerical factor changes and the tunnelling rate becomes immeasurably small anyway.

What the pictures cannot show #

The drawings show a coordinate and its potential, and they make the phase look like a marble in a groove. Nothing moves in the junction the way the marble does. The phase difference is a property of the whole quantum state of two superconductors, and when it tunnels, no particle crosses any barrier: the state of all the electron pairs in both electrodes changes, collectively, from one describing a stationary phase to one describing a phase running down the washboard, with a voltage appearing across the metal. The marble is a faithful mathematical picture and a misleading physical one, and the surprise the experiment carried — that a variable describing billions of particles at once has a single quantum wavefunction — is exactly what the picture hides.

Still open: how large a superposition can be #

The junction shows that a collective variable can tunnel and have discrete levels. Whether it can be put into a superposition of two states that differ macroscopically — two different directions of a current flowing round a loop, each involving billions of electrons — was the next question, and superconducting loops were shown around 2000 to support superpositions of opposite currents. How “large” those superpositions are is disputed, because different measures of macroscopic distinctness disagree about whether the states differ by billions of electrons or by a handful. The broader question — whether quantum mechanics holds without modification for arbitrarily large, massive or complex systems, or whether something, perhaps gravity, destroys superpositions beyond some scale — remains open, and is being pursued with circuits, with vibrating crystals and with interfering molecules.

The habit worth carrying away is to ask what a coordinate is coupled to before asking whether it is quantum. Size does not decide whether a variable obeys quantum mechanics; its coupling to its surroundings does — and a variable describing billions of electrons tunnels through a wall like a single particle, and shows energy levels like an atom, once the circuit around it is quiet enough to keep no record of which way it went.