Skyrmions are stable topological field configurations which have been observed only in a very small number of physical systems - most recently, in optics [1]. Their most famous implementation is in magnetic materials [2], where they are currently considered a highly promising route for magnetic information storage and transfer [3]. They also possess a unique influence on spin-polarized electrons – dubbed as the topological Hall effect [4] – due to the gauge field they apply. In photonics, artificial gauge fields were used to implement magnetic fields for photons [5] and were recently utilized in a synthetic photonic dimension to demonstrate topological edge states [6]. Among many examples, one of particular interest is the synthetic spin-1/2 dimension created by sum frequency generation (SFG) with a strong, undepleted pump beam, recently shown to produce the geometric phase required for artificial gauge fields [7,8].

Here, we theoretically show how a skyrmion magnetization can be created in a synthetic spin-1/2 dimension by utilizing the spectral degree of freedom of light and by using an engineered 3D periodically-poled nonlinear photonic crystal [9]. We then explore the synthetic gauge fields experienced by a tightly focused beam in the transverse plane of the crystal, showing that its dynamics are governed by two artificial fields: Magnetic, equivalent to the one creating the topological Hall effect; and an electric field which is unique to our system.

We find that an effective Lorentz force with a spin-dependent charge is exerted on the two frequency eigenstates, thereby inducing a transverse deflection to the beam. The origin of the emergent magnetic field is purely topological (the accumulated Berry phase), while the electric field results from a dynamical phase instead. Simulations of idler and signal beams in an initial oblique direction with respect to the propagation axis show a nonreciprocal deflection when the direction is reversed - a clear signature of the topological Hall effect.

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[8] A. Karnieli et al., "Experimental observation of the geometric phase in nonlinear frequency conversion," Optica 6, 1401-1405 (2019).
[9] T. Xu et al., "Three-dimensional nonlinear photonic crystal in ferroelectric barium calcium titanate," Nat. Photonics 12, 591-595 (2018).