Flux quantization

Flux quantization follows from the requirement that the superconducting wavefunction return to the same value after going once around a closed path. The condensate can be represented by a complex order parameter whose phase changes continuously along the superconductor; around a loop, its total phase change must be an integer multiple of 2π.¹ Combining this condition with the relation between phase gradient and electromagnetic vector potential yields a quantized gauge-invariant fluxoid.

For a sufficiently thick superconducting loop, where the current-density contribution is small, the result is approximately Φ = nΦ₀, with integer n. The constant Φ₀ = h/2e is based on the charge of a Cooper pair rather than an individual electron. This factor of two was a major clue that superconductivity involves paired carriers.

Superconducting rings can preserve a trapped flux state because magnetic fields are expelled during the transition into the superconducting state and circulating currents oppose changes in the enclosed flux. Early experiments by B. S. Deaver and W. M. Fairbank, and independently by Robert Doll and Martin Näbauer, observed flux in discrete units and established the value associated with paired charge.²

The exact conserved quantity is generally the fluxoid, not simply the magnetic flux. A fluxoid includes both the magnetic-flux term and a term proportional to the supercurrent and penetration depth. In large, thick-walled rings the distinction is often negligible; in thin films, narrow wires, and mesoscopic devices it can be experimentally significant.³

Josephson junctions convert flux quantization into a controllable circuit effect. A junction weakly connects two superconductors, allowing a supercurrent whose value depends on the difference between their quantum phases. In a superconducting quantum interference device, or SQUID, magnetic flux changes the relative phase around a loop and produces periodic oscillations in the critical current or voltage.⁴

Because Φ₀ is fixed by fundamental constants, Josephson circuits are used to realize highly precise voltage standards, while SQUIDs detect extremely small magnetic fields in geophysics, medicine, materials research, and biomagnetism. Flux qubits also encode quantum states using distinct circulating-current or flux configurations, although practical devices must suppress environmental noise and unwanted transitions.⁵

Flux quantization does not require every superconducting loop to contain exactly one quantum: the integer n can be large, and transitions between states occur through phase slips or vortex motion. In type-II superconductors, magnetic flux penetrates as quantized vortices, each carrying Φ₀; their arrangement and motion strongly influence electrical resistance and the performance of superconducting magnets.⁶

Some unconventional superconductors raise subtler possibilities. If the condensate has multiple components or unusual pairing symmetry, defects can support fractional flux quanta or altered fluxoid structures under particular boundary conditions. Such effects are not the ordinary rule for a conventional single-component superconductor, and their interpretation depends on the material, geometry, and junction design. Flux quantization therefore probes both the charge of the condensate and the topology of its order-parameter phase.

Flux quantum

The fundamental unit of superconducting flux, Φ0 = h/2e.

Fluxoid

A gauge-invariant combination of magnetic flux and the supercurrent contribution in a superconducting loop.

Cooper pair

A correlated pair of electrons that forms the charge-carrying condensate in conventional superconductors.

Phase slip

A process in which the superconducting phase changes by approximately 2π, allowing a loop to move between fluxoid states.

SQUID

A superconducting quantum interference device used to measure very small magnetic flux changes.