Flux quantization

# Flux quantization

## When magnetism climbs stairs

The “huge” size of the quantum wave that describes superconductivity has a surprising consequence: the quantization of the magnetic field. Indeed, when an object becomes superconducting, the quantum wave describing the electron properties covers the entire object, however big the latter might be. This is what is original in these collective modes, where a single wave corresponds to all the electrons in the system. This superconducting wave has a “phase”. The notion of phase is very general: the phase of a wave for instance, consists in knowing whether we are in the trough or on the crest of the wave.

In classical physics, a wave corresponds to something that oscillates; to know the phase, you have to check at which place of the oscillation you are: the value of the phase enables to know whether we are at the beginning, in the middle or at the end of that oscillation. For a quantum wave, the definition of phase is more complicated, because no object is oscillating. Nevertheless, this phase has a determined value that can be constant or that can vary in the superconducting object.
When the superconducting object has a hole (in the case of a ring for instance), the phase variation must follow certain rules. Indeed, when you have reached a complete turn of the ring, the phase value needs to go back to the same value, because the phase cannot be discontinuous. Some variations are hence forbidden. Yet, when a magnetic flux is applied to a ring, this flux modifies the phase along the ring. In order for the phase to go back to its initial value after a complete turn, only certain values of magnetic flux are allowed to go through the ring. The flux is “quantized”, which means it only proceeds by stages and only takes integer multiple values of a basic value called flux quantum, f0, equal to 2,07 10-15 Tm2 (or Weber).

Schematic representation of a superconducting ring. The phase variation of the condensate along the ring is represented by a colour oscillation. The left image cannot exist, because the phase is not continuous (no joint after a turn of the ring). The right image corresponds to an acceptable situation.

To summarize, since the quantum wave is defined on the entire superconducting object, its phase must do an integer number of “oscillations” when we follow a full turn of the ring. Consequently, the magnetic flux in this ring is quantized, which means it must be a multiple of a minimum value, f0. This value is very small, but it was possible to measure it with experiments! Theoretically, the value of f0 depends on the electric charge of the bosons that form the superconducting condensate: the value we get for f0 corresponds to a charge twice as high as that of an isolated electron. Measuring f0 is hence a way to confirm that the quantum objects forming the superconducting condensate are electron pairs and not isolated electrons.

Flux quantization also enables to understand why, in a mixed state, one magnetic flux quantum goes through each vortex, whatever the chemical composition of the superconductor and whatever the precise value of the temperature and the magnetic field.

Another consequence of the existence of the phase of the condensate is the Josephson effect.