Emulating spin transport with nonlinear optics, from high-order skyrmions to the topological Hall effect

Exploring material magnetization led to countless fundamental discoveries and applications, culminating in the field of spintronics. Recently, research effort in this field focused on magnetic skyrmions – topologically robust chiral magnetization textures, capable of storing information and routing spin currents via the topological Hall effect. In this article, we propose an optical system emulating any 2D spin transport phenomena with unprecedented controllability, by employing three-wave mixing in 3D nonlinear photonic crystals. Precise photonic crystal engineering, as well as active all-optical control, enable the realization of effective magnetization textures beyond the limits of thermodynamic stability in current materials. As a proof-of-concept, we theoretically design skyrmionic nonlinear photonic crystals with arbitrary topologies and propose an optical system exhibiting the topological Hall effect. Our work paves the way towards quantum spintronics simulations and novel optoelectronic applications inspired by spintronics, for both classical and quantum optical information processing.


Supplementary Note 1: Derivation of the topological Hall effect dynamics
The coupled wave equations in a sum frequency generation process with undepleted pump are 1 : where is the accumulated phase mismatch.
Working in the long wavelength pump approximation ( p ≫ i , s ), we can assume i~s~̅ and define the coupling strength = 2 eff 2 ̅ 2 p , we then rewrite the coupled wave equations as a two-level dynamics Under this analogy the momentum operator is given by the transverse Laplacian operator T = − T , and time is analogous to the propagation coordinate, . Note that since Φ depends on , the dynamics is "time-dependent". To eliminate this, we move to a rotating frame by making the local transformation where Now we multiply by † on the left and insert ( i We have that † [ (̃ĩ Explicitly for our choice of : So, we have that: Renaming the terms: as the equivalent magnetization vector, and as an emergent gauge field, and as the two-component spinor, we have We now follow what is commonly done in the literature for the derivation of the magnetic topological Hall effect in the adiabatic regime, for example see 2 . It is beneficial to move to a local frame of reference, where the spin z-direction points along the magnetization. This is done by yet another local transformation: where where and are the polar and azimuthal angles and This transforms our equation yet again: and are the new emergent gauge field and potential.

Supplementary Note 2: Derivation of the synthetic gauge fields
The first term in ′ is the same as in Ref. 2 , giving rise to an emergent magnetic field which equals the skyrmion density. It is given by: The second term in ′ arises because of the original gauge field we started with. Since ′ rotated the magnetization direction ̂ to ̂, the second term in ′ is intuitively calculated by understanding it as a rotation of the ̂ direction in the opposite orientation (̂ reflected in → − ). Writing ̂= coŝ+ sin coŝ+ sin sin̂ in Cartesian coordinates, we have that So that in total ′ = 1 2 ( ⋅̂) T − 1 2 sin ( ⋅̂) T + sin 2 2 T + (cos − sin ( ⋅̂))( T Φ/2), The potential is calculated in a similar manner to the first term in ′ In the adiabatic approximation, the "spin" of our light beam (its two-component color) follows the "magnetization" direction (the nonlinear coupling). Under this approximation the spin operator is locally equal to in the magnetization frame. Equivalently there is no crossing between the two spin eigenstates. Therefore, in the gauge potentials we may keep only the terms proportional to , and, we also assume no dependence of on ( = 0, the equivalent of time invariance), finally giving Since the dynamics is decoupled, we may readily omit the operator while defining an effective "charge" s = ±1 corresponding to each eigenvalue of , which we associate with the different spin eigenstates depending whether they point in parallel or antiparallel to the local magnetization. The synthetic magnetic field can be obtained from the vector potential Note that for constant (which is commonly the case considered in the literature), the term Finally, this yields the synthetic magnetic field and the synthetic electric field The Lorentz force is expressed as where = T / ̅ , the beam angle with respect to the optical axis, serves as an effective velocity of the light beam in the transverse plane.
For a phase-matched interaction, = 0 and = ± T T , which is the Stern-Gerlach force exerted on particles according to their spin 3,4 .

Supplementary Note 3: Single skyrmion magnetization texture
Let us consider the following magnetization vector ̂= √1 − 2 ( ) cos( + )̂+ √1 − 2 ( ) sin( + )̂+ ( )̂, We have that cos = ( ) and = giving the following magnetic and electric fields For the general skyrmion described in Supplementary Equation 33, the skyrmion number is For skyrmions, the magnetization in the direction changes its sign across the domain wall, such that = ± is always an integer.

Neel-and Bloch-type skyrmions
Example: linear domain wall In Fig. 4 of the main text, we analyze the dynamics of THE in the presence of different domain wall distributions: the variation of the out-of-plane magnetization from one direction to the opposite. Below, we give an example of such domain wall variation, which differs from the conventional Neel-or Bloch-type skyrmion domain wall.
For a different type of domain wall, say a linear variation ( ) = 1 − 2 / as in Fig. 4b in the main text: The electric and magnetic fields are

Supplementary Note 4: Simulation results for the signal field
Below are simulation results for the THE starting with the signal frequency, instead of the idler frequency, shown in Fig. 3 of the paper. Simulation conditions are the same as described in the main text. We compare deflection from four different skyrmion numbers: = ±1, ±4.
The deflection is opposite to the one of the idler case, since the signal eigenstate acquires a geometric phase with the opposite sign. This Fourier series ensures that C( ) = ±1, as required by poling techniques. Practically, for the interaction only the Fourier components with = ±1 contribute, as they are closest to phase-matching. The interaction is therefore encoded in these Fourier components such that with Φ = Δ , as expected.