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Use a complex pair order parameter $\Psi =|\Psi |e^{i\theta }$ with signed charge $q$, $|q|=2e$. A local isotropic Ginzburg-Landau theory of superconductivity has free energyFor $\alpha <0$, $\beta >0$, the uniform amplitude is $|{\Psi }_0{|}^2=-\alpha /\beta$. Variation gives the static field equation and supercurrentThe vector potential obeys Maxwell's equations, including $\nabla \times \mathbf{B}={\mu }_0({\mathbf{j}}_s+{\mathbf{j}}_n)+{\mu }_0{ϵ}_0{\partial }_t\mathbf{E}$ and $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$, with $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$. The static approximation drops the displacement current. A common dissipative near-transition dynamical extension is the time-dependent Ginzburg-Landau equationIt is a phenomenological relaxation law in its applicable regime, not a universal microscopic Schrödinger equation for every superconductor. Its covariant time derivative and the covariant spatial derivative ensure consistent phase and electromagnetic gauge transformations.

The amplitude and screening lengths areFor $\kappa >1/\sqrt{2}$ the ideal model is type II. An Abrikosov vortex has a $2\pi n$ phase winding, a core with suppressed $|\Psi |$ of size $\xi$, and circulating screening currents over $\lambda$. Single-valuedness implies fluxoid and flux quantization in a superconductor:The sign of $n$ depends on the chosen circulation and charge convention. Far from the core, where the contour current term is negligible, each singly wound vortex carries one flux quantum in magnitude.

The equilibrium lower critical magnetic field marks favorable entry of the first vortex; for $\kappa \gg 1$, ${\mu }_0{H}_{c1}\simeq [{\Phi }_0/(4\pi {\lambda }^2)](\ln \kappa +C)$ with an order-one core correction $C$. The upper critical magnetic field follows by linearizing the order-parameter equation in the field:Between these fields lies the superconducting mixed state, often with an Abrikosov vortex lattice. The ideal equilibrium entry and exit fields need not yield large irreversible loops. Flux pinning and surface barriers impede vortex motion; field sweeps then retain history-dependent flux and produce magnetic hysteresis and dissipative losses.

A superconducting weak link, such as a tunnel junction or narrow constriction, couples the phases of two superconductors. With consistent choices of phase and voltage orientation, the Josephson effect givesThe Gauge-invariant Josephson phase includes the line integral of $\mathbf{A}$ across the link. A zero voltage can sustain a phase-dependent steady current; a constant nonzero voltage makes that contribution oscillate. Microwave irradiation can lock this oscillation to an applied frequency, giving voltage steps at $V_n=nhf/(2e)$, an experimental test of coherent pair transport rather than an ordinary resistive current law.

A DC SQUID places two weak links in a loop. For identical junction critical currents $I_0$ and negligible self-inductance, flux quantization fixes their relative phase difference to $2\pi {\Phi }_{\mathrm{ext}}/{\Phi }_0$. Adding the two sinusoidal currents and maximizing over their common phase givesThe flux-periodic response provides sensitive magnetic measurements. Junction asymmetry or appreciable loop inductance changes the ideal contrast and can introduce additional hysteresis; the phase-coherent interference principle remains. The Jaklevic Josephson interference experiment is a motivating experimental realization of this link between a superconducting loop, weak links and magnetic flux.

A SQUID converts magnetic flux into a periodic interference response of superconducting weak links. A DC SQUID has two Josephson junctions in a loop. For equal junction critical currents $I_0$ and negligible loop self-inductance, their Gauge-invariant Josephson phases differ by $2\pi \Phi /{\Phi }_0$, with ${\Phi }_0=h/(2e)$. Adding $I_0\sin {\delta }_1+I_0\sin {\delta }_2$ and maximizing over the common phase givesThus the critical current repeats for every added flux quantum. A suitable current bias and dissipative junction dynamics convert the changing critical current into an observable voltage modulation. Junction asymmetry prevents exact zero minima, and appreciable self-inductance makes the flux include the circulating current's contribution; these require a modified response model. The Jaklevic Josephson interference experiment supplies an early experimental realization of the interference principle.

An RF SQUID instead uses one junction in a loop inductively coupled to a radio-frequency resonator. In either design, a pickup coil and flux transformer can couple a small external field into the sensor, and flux feedback can keep it near a sensitive operating point over a larger measurement range. Applications include weak-field magnetometry, measurements of magnetic susceptibility and persistent currents, geophysical surveys and precision low-current detection. Gradiometers suppress common external fields, while thermal, junction and environmental noise set practical sensitivity. The instrument measures a flux-dependent electrical response; sensitivity is not established solely by the small size 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.