1. Introduction
In superconductors and superfluids, vortices are not continuous classical structures but discrete quantum objects. Their circulation is quantized: it can only take values that are multiples of an elementary quantum. These quantum vortices have remarkable topological properties and play a crucial role in the macroscopic properties of quantum materials.
2. Superconductivity and the Meissner Effect
A type-I superconductor completely expels the magnetic field from its interior (Meissner effect) below its critical temperature. A type-II superconductor allows the field to penetrate as quantized flux tubes — Abrikosov vortices — above a first critical field H_c1. These vortices organize into a hexagonal crystal lattice (Abrikosov lattice).
3. Abrikosov Vortices
Each Abrikosov vortex carries exactly one quantum of magnetic flux Φ_0 = h/(2e) ≈ 2.07 × 10^-15 Wb. At the vortex core, superconductivity is destroyed over a coherence length ξ. Around the core, supercurrents circulate in a spiral over a penetration depth λ. The vortex topology is characterized by an integer winding number.
4. Superfluids and Quantized Vortices
In superfluid helium-4, vortices have quantized circulation κ = h/m. When a superfluid rotates, it cannot rotate like a rigid body: it develops a lattice of quantized vortices whose density is proportional to the angular velocity. This phenomenon has been observed experimentally and is direct evidence of the quantum nature of the superfluid.
5. Conclusion
Quantum vortices illustrate how quantum mechanics imposes topological constraints on rotational structures. The quantization of circulation is a direct consequence of the wave nature of the wave function, and the resulting vortices are topologically stable objects with properties having no classical analogue.