Linked and knotted synthetic magnetic fields

We show that the realisation of synthetic magnetic fields via light-matter coupling in the Lambda-scheme implements a natural geometrical construction of magnetic fields, namely as the pullback of the area element of the sphere to Euclidean space via certain maps. For suitable maps, this construction generates linked and knotted magnetic fields, and the synthetic realisation amounts to the identification of the map with the ratio of two Rabi frequencies which represent the coupling of the internal energy levels of an ultracold atom. We consider examples of maps which can be physically realised in terms of Rabi frequencies and which lead to linked and knotted synthetic magnetic fields acting on the neutral atomic gas. We also show that the ground state of the Bose-Einstein condensate may inherit topological properties of the synthetic gauge field, with linked and knotted vortex lines appearing in some cases.