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An Abrikosov vortex has a core where the superconducting order parameter is suppressed, surrounding circulating electric currents, and a winding of its quantum phase. A singly wound vortex carries one superconducting flux quantum in the appropriate bulk flux-quantization limit. The exact contour statement is fluxoid and flux quantization in a superconductor; the enclosed magnetic flux alone need not equal a flux quantum on a contour with appreciable electric current.

Assume a triangular array of straight, singly quantized Abrikosov vortices with nearest-neighbor distance $a$. Here $\overline{B}>0$ is the magnitude of the spatially averaged internal magnetic field, and ${\Phi }_0=h/(2e)$ is the superconducting flux quantum; $e>0$ is the elementary charge. Two lattice vectors of length $a$ meeting at $60^{\circ }$ span area $A_c=\sqrt{3}{a}^2/2$. One flux quantum per cell givesThus multiplying $\overline{B}$ by five decreases $a$ by $\sqrt{5}$; it does not multiply the flux of each vortex by five. Using the exact SI Planck constant and elementary charge, ${\Phi }_0\simeq 2.06783\times 10^{-15}\mathrm{Wb}$.^[ref]

For chosen internal fields $0.20\mathrm{T}$ and $1.0\mathrm{T}$, the spacings are approximately $109\mathrm{nm}$ and $48.9\mathrm{nm}$. These are geometric model calculations, not measurements from a particular sample. The local magnetic field varies within a cell; replacing $\overline{B}$ by the field at a core would not give this counting relation.

A spatially uniform Josephson junction in a uniform transverse in-plane magnetic field has a linear gauge-invariant phase profile across its width. Integrating its sinusoidal local current-phase relation of a superconducting weak link gives a signed sinc coefficient multiplying the sine of the central phase. Maximizing over that phase gives the displayed nonnegative critical-current envelope. Zeros occur at nonzero integer multiples of the superconducting flux quantum. The relevant magnetic thickness includes electrode penetration when appreciable, and self-fields or nonuniform local critical current density alter the simple pattern.

Rearrange the source's condensate-current expression and integrate around a closed path lying in the superconducting material:Single-valuedness of the superconducting wavefunction requires $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$, and Stokes theorem gives $\oint \mathbf{A}\cdot d\mathbf{l}=\Phi$. Thus the exact result is fluxoid and flux quantization in a superconductor:Reversing the winding orientation absorbs the sign of $q$. Since a Cooper pair has $|q|=2e$, the positive superconducting flux quantum isFor a thick ring, choose the path deep enough that the supercurrent is negligible; then the enclosed magnetic flux is $n{\Phi }_0$. This is the flux quantization usually intended in the question.

Qualification of the unqualified flux claim: a thin ring can have a nonzero kinetic contribution around every available path, so it is the fluxoid that is exactly quantized. For example, with zero applied flux and geometric and kinetic inductances $L_g=L_k$, a metastable winding-$1$ state has $(L_g+L_k)I={\Phi }_0$ but $\Phi =L_{g}I={\Phi }_0/2$. This concrete counterexample satisfies the given current law while its magnetic flux is not an integer multiple of ${\Phi }_0$.

A Josephson junction converts a relative condensate phase into a measurable current. For superconductors the relevant coherent object is a Cooper pair, with electric charge $q=-2e$. Its Gauge-invariant Josephson phase isWith a consistent directed-current convention, the weak-link relation is $I_s=I_c\sin \delta$. A voltage $V=U_2-U_1$ gives $\dot{\delta }=2eV/\hslash$, so a constant voltage produces the AC Josephson effect at frequency $2e|V|/h$. For neutral atom reservoirs the phase evolution instead follows their chemical-potential difference, $\hslash \dot{\delta }={\mu }_1-{\mu }_2$ with the displayed phase labeling. The current transfers atoms rather than electric charge, but still probes coherent matter-wave amplitudes.

A DC SQUID provides two coherent paths around a superconducting loop. Neglect loop self-inductance, take equal junction critical currents $I_0$, and let $\Phi$ be the enclosed flux. The relative junction phase differs by $2\pi \Phi /{\Phi }_0$, with superconducting flux quantum ${\Phi }_0=h/(2e)$. Adding the two sinusoidal currents and maximizing over their common phase givesFor unequal junctions the corresponding result is $\sqrt{I_1^2+I_2^2+2I_1{I}_2\cos (2\pi \Phi /{\Phi }_0)}$. Finite inductance and junction asymmetry alter the ideal modulation, so a measured minimum need not be zero. Flux changes a phase difference between the paths; the detector reads the resulting critical-current or voltage modulation. The Jaklevic Josephson interference experiment established superconducting interference experimentally.^[ref]

A bosonic Josephson junction uses two weakly linked Bose-Einstein condensates. In the Albiez double-well Josephson experiment, the population distribution was measured in situ and the relative phase inferred from interference fringes. The apparatus therefore tests both number transfer and phase coherence, rather than interpreting a density oscillation alone as a phase measurement. Interactions make the dynamics nonlinear: large imbalance can produce self-trapping instead of simple tunneling oscillations.^[ref]

The usefulness is interferometric sensitivity: a small phase perturbation can change a measurable current. It requires a coherent weak link, controlled phase noise and the appropriate particle's phase evolution; a generic insulating barrier between incoherent reservoirs is not already a matter-wave interferometer.

An Abrikosov vortex combines a core of suppressed superconducting order parameter, circulating electric current and a $2\pi$ winding of the quantum phase. In the bulk limit a singly wound vortex carries the superconducting flux quantum ${\Phi }_0=h/(2e)$. More precisely it is the fluxoid that is quantized; appreciable current on the integration contour prevents identifying enclosed flux alone with the integer.

For a type-II superconductor, the Ginzburg-Landau parameter is $\kappa =\lambda /\xi >1/\sqrt{2}$. The superconducting mixed state allows vortices between the lower critical magnetic field and upper critical magnetic field. A short superconducting coherence length permits a high orbital upper field,Thus surviving a large applied field generally requires accommodating many vortices, not excluding flux everywhere. The Abrikosov vortex lattice provides their equilibrium arrangement in the simplest isotropic model. Material anisotropy, disorder and thermal fluctuations can change that arrangement and its stability. Spin-polarization limits can also constrain an actual upper field, beyond this orbital estimate.

Carrying a large transport current density is a separate problem. For a straight vortex with tangent $\hat{\mathbf{t}}$, the driving Lorentz force per unit length is ${\mathbf{f}}_L=\mathbf{J}\times {\Phi }_0\hat{\mathbf{t}}$. Vortex motion produces a mean electric field $\mathbf{E}=\mathbf{B}\times {\mathbf{v}}_L$ and hence dissipative power density $\mathbf{J}\cdot \mathbf{E}$. A large $B_{c2}$ alone therefore does not imply a large usable superconducting critical current.

Flux pinning holds vortices at favorable defects: suppressing the order parameter costs less where superconductivity is already weakened. Precipitates, dislocations, grain boundaries and deliberately introduced damage can provide pinning, subject to material-specific trade-offs. A defect scale comparable with a core size can be effective; a defect that destroys connected superconducting paths is harmful. In a coarse force balance, a maximum pinning force density $F_p$ gives $J_{c}B\simeq F_p$ for transverse transport. Vortex depinning and thermally activated flux creep can produce a voltage below an ideal static threshold.

High-field magnet conductors consequently need both a large upper field and strong, persistent pinning while retaining mechanical integrity and electrical connectivity. Niobium-titanium and niobium-tin conductors illustrate why superconducting composition and microstructural processing are developed together. A critical-current measurement should specify field, temperature and voltage criterion; it is not simply a measurement of the temperature at which resistance first vanishes.

In 1986, Akira Tonomura and collaborators coated a ferromagnet shaped as a torus with superconducting niobium and copper. A cooled stage held it in a 150 kV field-emission electron microscope. An electron biprism overlapped object and reference waves; film holograms were reconstructed optically. Room-temperature leakage screening selected samples; varying copper thickness tested electron penetration.^[ref]

Below the superconducting transition, through-hole and outside electron waves differed in wave phase by zero or $\pi$ modulo $2\pi$. Warming through it changed the interference reversibly. Leakage and coherent-penetration bounds do not prove zero magnetic field.^[ref]

Why these two phases? The shield's superconducting flux quantum is $h/(2e)$, whereas the probing electron has electric charge $-e$. Substituting $\Phi =Nh/(2e)$ givesAn odd $N$ therefore moves a carrier fringe by half its spacing; an even $N$ leaves it unchanged modulo a full spacing. The phase quantization provides an internal check on the superconducting enclosure. It does not mean that zero wave phase proves zero trapped magnetic flux.^[ref]

The readout is electron holography in the off-axis holography geometry: a known tilted reference converts an unknown relative wave phase into a spatial carrier displacement. Reconstructing a magnetic wave phase assumes stable reference phase, adequate coherence and a calibrated carrier; electrostatic charging and material traversal can also shift wave phase. Optical wave phase amplification rescales the displayed fringe displacement. The experiment's shielding and comparison tests support the magnetic inference; an interferogram by itself measures the total relative wave phase, rather than separately certifying every contribution. The ideal field-free model describes the limiting geometry, while finite leakage and penetration remain questions of experimental bounds.

The superconducting mixed state is the equilibrium vortex regime of a type-II superconductor, between its lower critical magnetic field and upper critical magnetic field in the ideal bulk description. Magnetic flux enters through Abrikosov vortices, while the surrounding material retains a nonzero superconducting order parameter.

Each vortex has a suppressed-order core and circulating electric currents; in the appropriate bulk limit, a singly wound vortex carries one superconducting flux quantum. Their mutual interactions can produce an Abrikosov vortex lattice. Increasing the field increases the vortex density until superconductivity is destroyed at the upper critical field. Defects can immobilize vortices through flux pinning, so a prepared sample's magnetization need not coincide with the equilibrium curve. Spatial coexistence here means vortex cores within superconducting matter, rather than arbitrary macroscopic patches of two phases.^[ref]