One-at-a-time accidents turn into a coherent macroscopic flow ── and then into the definition of the volt
So far tunneling has been a rare accident, one particle at a time: an alpha particle once in 4.5 billion years, a proton once in nine billion. That changes here. Sandwich two superconductors around an insulating film about 1 nm thick, attach wires, and ── current flows with no voltage applied. What sets the amount is neither the voltage nor the temperature nor the resistance of the film, but only the difference between the "phases" carried by the two superconductors. Brian Josephson, a graduate student in Cambridge, predicted this in 1962 at the age of 22; Anderson and Rowell confirmed it the following year. The subject of the tunneling shifts from "particle" to "phase" ── and that single step now underpins the definition of the volt and the hardware of most quantum computers.
Why superconductivity happens is left to Episode 5. All we need here is the result.
Inside a superconductor, electrons pair up (Cooper pairs) and all of them behave as one and the same wave. So a superconductor, whether 1 cm or 1 m across, is described by a single wavefunction for the whole thing.
$$\Psi=\sqrt{n}\;e^{i\theta}$$\(n\) is the density of Cooper pairs, and \(\theta\) is the phase. In an ordinary material \(10^{23}\) electrons carry wildly different phases which average into meaninglessness (Episode 3 of the sister series "Renormalization That Clicks," on decoherence). In a superconductor it collapses to a single number.
A natural question follows: can you observe a phase? Not on its own. The absolute value of the phase has no physical meaning (it is a gauge freedom); what is observable is only the difference between the phases of two superconductors. So you need two of them, joined.
Sandwich a thin insulator (1–2 nm). Cooper pairs tunnel through it exactly as in Episode 1. Then:
DC Josephson effect ── current flowing at zero voltage:
$$I=I_c\sin\Delta\theta$$AC Josephson effect ── apply a voltage and the phase difference starts winding:
$$\frac{d(\Delta\theta)}{dt}=\frac{2eV}{\hbar}$$The second is especially potent. If the phase difference winds at a constant rate then \(\sin\Delta\theta\) oscillates ── i.e. a DC voltage produces an AC current. The frequency is
$$f=\frac{2eV}{h}=483.6\ \mathrm{THz}\ \text{per volt}$$One microvolt gives 483.6 MHz. Voltage turns directly into frequency.
The behaviour of a Josephson junction becomes transparent if you think of the phase difference \(\delta\) as the position of a particle. Writing the energy:
The first term is the cosine corrugation (the washboard), the second is a tilt set by the current.
· With no current (\(I=0\)) the board is level. The phase difference sits still at the bottom of a valley ── zero voltage.
· Increase the current and the board tilts. As long as a valley remains, the phase difference is still trapped ── still zero voltage. That is what "current flows with no voltage" really is.
· Once \(I\) exceeds \(I_c\) the valleys vanish and the phase difference rolls away. Since \(d\delta/dt\ne0\), the AC Josephson equation gives \(V\ne0\). The moment at which superconductivity appears to "break."
And here is where Episode 3 comes back. Even while a valley remains, the phase difference can escape by tunneling through the wall. Two ways out ── climb over thermally, or slip through. Exactly the contest from Episode 3, reappearing verbatim.
The figure is that washboard. Tilt it with the current slider. The valleys grow shallower and vanish as \(I/I_c\to1\).
At the same time, Episode 3's \(T_0\) is displayed on the right. As the valley shallows, the summit frequency \(\omega_p\) falls, and so does \(T_0=\hbar\omega_p/2\pi k_B\). "How cold you must go before escape switches from thermal to tunneling" is computed live. For a typical junction it is a few tens of millikelvin ── which is why macroscopic quantum tunneling could only be observed in the 1980s, with dilution refrigerators.
The AC Josephson relation \(f=2eV/h\) is a device that converts voltage into frequency. Frequency is the quantity humanity measures most accurately (atomic clocks), so this becomes a voltage standard.
| Use | What it is |
|---|---|
| voltage standard | Since 1990 the Josephson effect has been the world's voltage standard. The 2019 SI revision made \(e\) and \(h\) defined constants, so \(K_J=2e/h=483597.8484\) GHz/V is now exact. The markings on the voltmeter you are using come, ultimately, from this equation |
| SQUID | Put two junctions in a ring and interference occurs in units of the flux quantum \(\Phi_0=h/2e\). Fields at the femtotesla (10⁻¹⁵ T) level become measurable, which is used in magnetoencephalography and in mineral prospecting. That is ten billionths of the Earth's field |
| superconducting qubits | The valleys of the washboard are not harmonic (it is a cosine). That nonlinearity makes the energy levels unequally spaced, so the lowest two can be isolated as a qubit. The heart of every superconducting quantum computer running today is a Josephson junction |
| Episodes 1–3 | This episode | |
|---|---|---|
| What tunnels | one particle | the phase difference (one macroscopic variable) |
| How often | extremely rarely | a steady current |
| What sets it | height, width, mass | the phase difference |
| Observation | only visible after collecting statistics | read directly on an ammeter |
When phases line up, tunneling turns from statistics into a phenomenon. "Almost never happens, one at a time" becomes "happens all the time" once \(10^{10}\) of them line up ── that is this episode in one line.
Established: the DC and AC Josephson effects (predicted by Josephson 1962, observed by Anderson–Rowell 1963, Nobel Prize 1973); \(I=I_c\sin\Delta\theta\) and \(d(\Delta\theta)/dt=2eV/\hbar\); \(K_J=2e/h=483597.8484\) GHz/V (exact since the 2019 SI) and its use as the voltage standard; the flux quantum \(\Phi_0=h/2e\) and the observation of flux quantisation (Deaver–Fairbank, Doll–Näbauer, 1961); the washboard potential \(U(\delta)=-E_J(\cos\delta+s\delta)\) with barrier \(\Delta U=2E_J[\sqrt{1-s^2}-s\arccos s]\) and plasma frequency \(\omega_p=\omega_{p0}(1-s^2)^{1/4}\); the observation of macroscopic quantum tunneling in Josephson junctions (Voss–Webb 1981, Devoret–Martinis–Clarke 1985); SQUID sensitivities and the application to superconducting qubits. All established physics and technology.
Points to note: (1) "A superconductor is described by a single wavefunction" is a mean-field picture (Ginzburg–Landau / BCS). Strictly, phase and particle number are conjugate, so a sharper phase means a fuzzier number (that very uncertainty is how a charge qubit works). (2) The absolute phase has no physical meaning (it is gauge-dependent); only the difference does. The body says this, but the properly gauge-invariant form includes a line integral of the vector potential. (3) The figure is an ideal, dissipationless junction (the zero-resistance limit of the RCSJ model). Real junctions are damped by quasiparticles, which lowers \(T_0\) and suppresses escape. (4) The figure's \(\omega_{p0}=\sqrt{2eI_c/\hbar C}\) assumes a junction capacitance \(C=1\) pF. (5) The Josephson effect itself is the tunneling of Cooper pairs, and as the next episode argues, it is not the mechanism of superconductivity.
A superconductor is described as a whole by one wavefunction \(\Psi=\sqrt n\,e^{i\theta}\). Join two through a thin insulator and Cooper pairs tunnel, giving \(I=I_c\sin\Delta\theta\) ── current with no voltage. Apply a voltage and the phase difference winds, \(d(\Delta\theta)/dt=2eV/\hbar\), turning a DC voltage into an AC at 483.6 THz per volt.
Treat the phase difference as the position of a particle and the whole thing becomes a tilted washboard. Current tilts the board; where the valleys disappear is the critical current. Even with a valley left, the phase difference can escape, and whether it escapes thermally or by tunneling is decided by Episode 3's \(T_0=\hbar\omega_p/2\pi k_B\), applied verbatim ── a few tens of millikelvin for a typical junction.
The big change is that the subject of the tunneling moved from "one particle" to "the phase, a macroscopic variable." A quantity decided collectively by \(10^{10}\) Cooper pairs crosses the wall as a unit (macroscopic quantum tunneling). A rare accident became a steady phenomenon readable on an ammeter, and is now the definition of the volt and the heart of quantum computers.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the current slider tilts the washboard and shallows the valley. At the same time it displays the temperature T₀ at which escape switches from thermal to tunneling. "See the answer" opens each solution.