Josephson Effect

The Josephson effect is a phenomenon in which a supercurrent flows between two weakly coupled superconductors, with its magnitude controlled by the difference between their quantum phases. The coupling structure is called a Josephson junction; a common implementation consists of two superconductors separated by a thin insulating barrier. A steady supercurrent can flow without an applied voltage, whereas a constant nonzero voltage causes the supercurrent to oscillate. Predicted by Brian D. Josephson in 1962, the effect connects macroscopic electrical behavior with quantum mechanics. (nobelprize.org)

Physical basis and discovery

In conventional superconductors, electrons form correlated Cooper pairs. Their collective state has a macroscopic quantum phase, described through a complex superconducting order parameter. When two superconductors are sufficiently weakly coupled, their relative phase influences the transfer of pairs between them. Across an insulating barrier, this transfer occurs through quantum tunneling rather than ordinary conduction through the insulator. The resulting supercurrent is distinct from the dissipative current carried by individual quasiparticle excitations. (nobelprize.org)

Josephson developed his prediction while a research student at Cambridge. His work extended the understanding of tunneling between superconductors by showing that a barrier could transmit a phase-dependent supercurrent even at zero voltage. He received half of the 1973 Nobel Prize in Physics for these theoretical predictions. In 1963, Sidney Shapiro experimentally observed microwave-induced voltage steps, providing an important verification of the predicted voltage–frequency relationship. (nobelprize.org)

The effect is not confined to insulating tunnel barriers. Superconducting weak links can also be formed using narrow constrictions or conducting regions between superconducting electrodes. Their detailed current–phase characteristics may differ from those of an ideal tunnel junction. (arxiv.org)

Josephson relations

For a conventional ideal tunnel junction, the basic relations are

Here $I_s$ is the supercurrent, $I_c$ the critical current, $\varphi$ the gauge-invariant phase difference, and $V$ the voltage across the junction. The positive quantity $e$ is the elementary charge, while $\hbar=h/(2\pi)$, with $h$ the Planck constant. The critical current depends on the junction’s physical properties; the voltage–phase coefficient is determined by fundamental constants. (arxiv.org)

The DC Josephson effect follows when $V=0$: the phase can remain constant, supporting a stationary supercurrent whose magnitude does not exceed $I_c$. Zero voltage does not require zero current. Conversely, in the AC Josephson effect, a constant voltage produces a phase that advances uniformly in time and an oscillating supercurrent with frequency

The Josephson constant is $K_J=2e/h$, approximately $483\,597.8484\ \mathrm{GHz/V}$. Thus, a voltage of one microvolt corresponds to an oscillation frequency of approximately 483.6 megahertz. These oscillations arise from a DC bias rather than requiring an externally applied alternating voltage. (ptb.de)

Microwave response and magnetic interference

When a junction is driven by microwaves of frequency $f$, its phase evolution can synchronize with the external drive. Its current–voltage characteristic then develops plateaus called Shapiro steps, at average voltages

where $n$ is an integer. Changing the bias current within a step does not change its quantized voltage. The width of each step depends on the drive amplitude and the junction’s electrical properties. This response is often called the inverse AC Josephson effect. (tsapps.nist.gov)

Magnetic fields also affect superconducting phase differences. A superconducting loop containing weak links therefore converts changes in magnetic flux into changes in its electrical response. This principle underlies the superconducting quantum interference device, or SQUID. Its characteristic flux scale is the superconducting flux quantum, $\Phi_0=h/(2e)$, the same combination of constants appearing in the voltage-step relation. SQUIDs detect extremely weak magnetic signals, including those generated by biological electrical activity. (nist.gov)

Junction dynamics and circuit behavior

Real junctions combine the Josephson supercurrent with capacitance and dissipative conduction. In the resistively and capacitively shunted junction model, the total current is represented by

Together with the voltage–phase relation, this gives a nonlinear differential equation governing the phase. The resistance $R$ represents damping, while the capacitance $C$ permits dynamical charge storage. Depending on damping and bias conditions, a junction can show oscillations and hysteresis between zero-voltage and finite-voltage states. (arxiv.org)

The associated coupling energy is

This nonlinear energy dependence makes a junction fundamentally different from an ordinary linear inductor. In sufficiently isolated circuits, quantum dynamics of the collective phase can produce discrete energy levels and macroscopic quantum tunneling. These circuit-level phenomena should be distinguished from Cooper-pair tunneling through the junction barrier itself. (arxiv.org)

Applications

Josephson voltage standards use synchronized junction arrays to generate reproducible voltages from a known frequency. Connecting many junctions in series raises the output to useful calibration levels. Since the revised International System of Units took effect on May 20, 2019, both $h$ and $e$ have fixed exact values, making $K_J$ exact within the SI. Practical instruments nevertheless retain measurement uncertainties from their operation and implementation. (tsapps.nist.gov)

In superconducting qubits, Josephson nonlinearity enables the unequal energy-level spacings needed to control selected transitions. These devices are fabricated as electrical circuits rather than individual microscopic particles, and their coherence depends on controlling unwanted environmental coupling and noise. (arxiv.org)

Josephson junctions also support single-flux-quantum logic. A $2\pi$ phase advance produces a voltage pulse whose time integral is $\Phi_0$. Such pulses can carry digital information through superconducting circuits and provide precisely quantized building blocks for waveform synthesis. (nist.gov)