Spin transport emulation through nonlinear optics

Emulating 2D spin transport necessitates a pseudospin degree of freedom, an effective magnetization field acting on the pseudospin, and a space-time along which the dynamics is probed. We define the pseudospin degree of freedom by considering a nonlinear optical process involving two interacting frequencies, which can be geometrically represented on a Bloch sphere^25,32,44 (Fig. 1a–b). This degree of freedom is controlled by the inherent phase matching of the process and its complex coupling coefficient, which together define an effective magnetization field applied on the pseudospin^33,45 (Fig. 1c). Considering the propagation direction as a time axis, the transverse profile of a light beam defines the wavefunction of a massive particle in two spatial dimensions (see Figs. 1d and 2a), with its dynamics dictated by the variation of the effective magnetization field in space.

Fig. 1

Emulation of spin transport phenomena in nonlinear optics.

a Bloch sphere representation of a two-level system—here comprising the idler ($$ {\omega }_{\mathrm{i}}$$) and signal ($$ {\omega }_{\mathrm{s}}$$) frequencies—wherein the frequency degree of freedom acts as a pseudospin dimension. Consequently, an effective magnetization field $$ \mathbf{M}$$, driving the dynamics, can be defined. b Realization of two-frequency dynamics. In sum-frequency generation, the idler and pump photons generate a higher-frequency signal photon. For an undepleted pump, the interaction reduces to a two-level system of $$ {\omega }_{\mathrm{i}}$$ and $$ {\omega }_{\mathrm{s}}$$. A periodic spatial modulation of the nonlinearity compensates for the momentum mismatch $$ \mathrm{\Delta }k$$. c Parameter space defining the spatially-varying effective magnetization, $$ \mathbf{M}\left(\mathbf{r}\right)$$. The z-component is controlled by the momentum mismatch $$ \mathrm{\Delta }k\left(\mathbf{r}\right)$$—namely, the poling period (marked in green). The radial component is defined by the nonlinear coupling strength $$ ∣\kappa \left(\mathbf{r}\right)∣$$, proportional to the modulation duty cycle and to the pump field strength (marked in blue). The polar angle $$ \phi \left(\mathbf{r}\right)$$ is given by the relative phase between the poling phase and pump phase front (marked in purple). $$ \theta \left(\mathbf{r}\right)$$ denotes the elevation angle. All these parameters are tuneable and can be used to tailor arbitrary magnetization textures $$ \mathbf{M}\left(\mathbf{r}\right)$$. d Dynamics of the pseudospin and position of the light beam as it traverses a synthetic magnetization texture. In the photonic system, the propagation coordinate represents time, and the location of the light beam on the transverse plane is analogous to the position of the spin-1/2 particle (axis inset), while the spin degree of freedom is analogous to the color of the light (green/blue colors representing spin up/down). In a similar manner to the electron spin, adiabatically following the local magnetization direction, light undergoes adiabatic frequency conversion as it propagates.

Fig. 2

Topological Hall effect for light beams.

a A quadratic $$ \left({\chi }^{\left(2\right)}\right)$$ nonlinear photonic crystal with an imprinted synthetic skyrmion texture (gray and yellow for negative and positive nonlinearity poling) is illuminated by a broad pump beam (red). A tightly focused beam in the idler frequency (green) impinges the crystal at an angle $$ \theta $$ with respect to the optical axis (dashed black line). As the beam propagates, it experiences adiabatic frequency conversion to the signal frequency (blue) and back to the idler frequency, as the spectral pseudospin adiabatically follows the local direction of the effective skyrmionic magnetization (see also Fig. 1d). As a result, a real-space geometric phase is accumulated, and the synthetic gauge fields associated with it give rise to an emergent Lorentz force, causing the beam waist to deflect by an angle $$ \alpha $$ in the transverse plane (shifting the x-axis position of the beam at the output). b cross sections of a high-order skyrmionic nonlinear photonic crystal used in Fig. 3c,d, at crystal center $$ \left(Z=0\right)$$, $$ Z=2.5\mathrm{mm}$$, and $$ Z=5\mathrm{mm}$$. The nominal quasi-phase-matching period is $$ 10.24\mathrm{\mu }\mathrm{m}$$, and it changes between $$ 10.33\mathrm{\mu }\mathrm{m}$$ at the crystal center (the optical axis), to $$ 10.15\mathrm{\mu }\mathrm{m}$$ at the edges.

Full control over the dynamics is achieved by engineering the nonlinearity in the material, causing the detuning from phase matching and the complex coupling strength to change in space, for a fixed pump illumination. Alternately, the complex coupling can be tailored by shaping the pump field, together with a correspondingly engineered variation in the detuning. The nonlinear process and its tunable parameters are presented in Fig. 1b-c.

In what follows, we consider the three-wave mixing process of sum-frequency generation in a quadratic nonlinear photonic crystal between an idler (E_i), signal (E_s) and pump (E_p) electric fields (Fig. 1b). Assuming the pump field is strong and nondepleted, the interplay is effectively only between the idler and signal fields. In this setting, the transverse propagation of the light beam and its frequency (idler or signal) emulate the motion of a spin-1/2 particle at a certain spin state, traversing a two-dimensional magnetization texture (see Fig. 1d).

Under the long pump wavelength approximation⁴⁶, the paraxial coupled wave equations for the signal and idler slowly varying envelopes are:

A local gauge transformation to the rotating frame can be applied to Eq. (1) by defining $$ U\left(\mathbf{r}\right)=\mathrm{diag}\left(e^{-i\mathrm{\Phi }\left(\mathbf{r}\right)/2},e^{i\mathrm{\Phi }\left(\mathbf{r}\right)/2}\right)$$, while rewriting it for the transformed two-component field vector $$ \mathrm{\Psi }=U^{†}{\left(E_{\mathrm{i}},E_{\mathrm{s}}\right)}^T$$:

The topological Hall effect for light beams

We demonstrate the capabilities of our approach by emulating the topological Hall effect (THE), in which polarized spin currents are deflected by a topologically nontrivial magnetization texture (such as magnetic skyrmions^11–13). In the adiabatic regime of the THE, the orientation of electron spin follows the local normalized magnetation direction $$ \widehat{\mathbf{M}}=\mathbf{M}/∣\mathbf{M}∣$$, causing the electron wavefunction to deflect due to an acquired geometric phase (illustrated in Fig. 1d).

We implement the local gauge transformation $$ U^{\prime }\left(\mathbf{r}\right)=\exp \left[-i\theta \left(\mathbf{r}\right)\mathit{\sigma }\cdot \widehat{\mathit{\phi }}\left(\mathbf{r}\right)/2\right]$$ on the spinor wavefunction $$ \mathrm{\Psi }$$ of Eq. (2), aligning the synthetic optical spin with the local magnetization direction^11,47. Here, $$ \theta \left(\mathbf{r}\right)=\arccos \widehat{\mathbf{M}}\cdot \widehat{\mathbf{z}}$$ and $$ \widehat{\mathit{\phi }}\left(\mathbf{r}\right)=\widehat{\mathbf{z}}\times \widehat{\mathbf{M}}/∣\widehat{\mathbf{z}}\times \widehat{\mathbf{M}}∣$$, are the elevation angle and polar unit vector, as defined in Fig. 1c. The transformed state $$ {\mathrm{\Psi }}^{\prime }=U^{\prime †}\mathrm{\Psi }$$ then satisfies the equation of motion:

The synthetic magnetic field in Eqs. (4) and (5) is directly related to the geometric phase acquired during propagation of light through the crystal, and it’s magnitude is solely dependent on the real-space Berry curvature of the magnetization⁴⁷. As such, only topologically nontrivial magnetization textures can induce a deflection of the beam trajectory associated with the THE (see Supplementary Material), with the so-called skyrmion number $$ S=\int d^2\mathbf{r}\widehat{\mathbf{M}}\cdot \left({\partial }_x\widehat{\mathbf{M}}\times {\partial }_y\widehat{\mathbf{M}}\right)/4\pi \ne 0$$ as the relevant topological invariant. On the other hand, the synthetic electric field term in Eqs. (4) and (5) depends on the coupling strength explicitly through $$ M_{\mathrm{T}}\propto \kappa $$, and is thus related to the dynamic phase accumulated during the interaction (leading, in the fully phase-matched case, to a photonic Stern-Gerlach effect^45,46 wherein $$ \mathcal{F}=q_{\mathrm{s}}{\mathbf{\nabla }}_{\mathrm{T}}{M}_{\mathrm{T}}$$).

Simulating the topological Hall effect from single skyrmions with light

Using our formalism, it is possible to explore the topological Hall effect occurring in otherwise thermodynamically unstable magnetization textures, such as single high-order skyrmions¹⁴. To this end, let us define a circularly-symmetric magnetization texture with a single topological defect as follows:

For the given magnetization in Eq. (6), the synthetic electric and magnetic fields are:

Though quite a specific solution, Eqs. 6,7 are still general enough such that the simulation of several meaningful magnetization textures may be achieved, through the simple task of guessing the distribution function $$ m\left(\rho \right)$$, winding number $$ n$$ and the constant phase $$ \eta $$. For example, a Néel-type skyrmion⁴⁸ (or Néel-type antiskyrmion) can be described by $$ m\left(\rho \right)=\mp \cos \left(\pi \rho /R\right)$$, winding number $$ n=\pm 1$$, and phase factor $$ \eta =0$$. Similarly, changes can be made to accommodate Bloch-type skyrmions⁴⁸ ($$ \eta =\pi /2$$), or high-order skyrmions ($$ ∣n∣>1$$).

Numerical simulations of nonlinear beam propagation inside a skyrmionic NLPC, based on the split-step Fourier method (see Methods), are presented in Fig. 3. We note that in the numerical calculation we only apply standard assumptions, such as reflection-less, paraxial propagation, rather than the approximations employed for the analytical solution. Figure 3a,b shows the idler beam’s transverse dynamics as it propagates through a crystal with a skyrmion magnetization of $$ S=-1$$: for opposite incident angles, the beam deflects to opposite transverse directions, in a clear manifestation of the THE^11,47. A reversed angle is analogous to a reversed velocity, and the opposite deflection is intimately connected to the nonreciprocity induced by the synthetic magnetic field.

Fig. 3

Simulating the topological Hall effect in skyrmionic nonlinear photonic crystals.

a, b Simulated beam shape and position in the transverse plane of the crystal, for selected depths of propagation inside the NLPC (the successive points are for $$ z$$ locations of $$ -10.0,-7.5,-5.0,-2.5,0,2.5,5.0,7.5$$ and $$ 10.0\mathrm{mm}$$). The theoretical trajectory of the beam’s center-of-mass motion is depicted by solid red lines. In a, an idler beam enters the NLPC at an entrance angle of 1.5 degrees, traversing a Néel-type antiskyrmion ($$ S=-1$$, $$ m\left(\rho \right)=\cos \left(\pi \rho /R\right),n=1,\eta =0,R=200\mathrm{\mu }\mathrm{m}$$) from left to right. Also inset is the transverse component of the synthetic magnetization in the NLPC, color-coded by its corresponding $$ z$$-component. In b, the dynamics are flipped (from right to left), and as a result, the beam is deflected in the opposite direction—a clear signature of the THE. c, d Same as in a, b, but with an engineered high-order antiskyrmion $$ \left(n=4\right)$$, causing a more pronounced difference in the deflection. e Deflection angle of the beam for varying skyrmion numbers. By considering the rotation of the velocity vector $$ \mathit{v}$$ by the magnetic field, the deflection angle of the beam’s center-of-mass should be $$ \alpha \propto S/R∣\mathit{v}∣$$ (see Supplementary Material). Simulation results show a clear linear dependence on the skyrmion number, for two different cases (red circles—$$ R=200\mathrm{\mu }\mathrm{m}$$, $$ ∣\mathit{v}∣=\sin \left(1.5^{\circ }\right)$$; green circles - $$ R=100\mathrm{\mu }\mathrm{m}$$ and $$ ∣\mathit{v}∣=\sin \left(1.2^{\circ }\right)$$). f Photon number, normalized by its initial value, in the idler (green continuous line) and signal (blue dashed line) frequencies as a function of the propagation inside the NLPC for case a, demonstrating adiabatic frequency conversion. Full simulation parameters are given in the Methods.

The effect becomes more pronounced for higher skyrmion numbers or smaller skyrmion radii, as illustrated in Fig. 3c–e, and in Supplementary Movies 1 and 2. Our theoretical model seems to describe the nonlinear optical system well, as evident by the predicted trajectories (red lines in Fig. 3a–d) and by the adiabatic frequency conversion during propagation (Fig. 3f), in complete analogy to the electron spin in the THE. The signal beam behaves similarly, although it is deflected to the opposite direction, as expected (see the Supplementary Material).

Domain walls engineering

Interestingly, the dynamics in the THE regime can be significantly altered by the exact variation of the magnetization from one out-of-plane direction to its opposite—called the domain wall. The ability to accurately engineer 3D features of NLPCs allows us to precisely design domain walls, including the adjustment of the skyrmion winding, size and radial profile $$ m\left(\rho \right)$$. Figure 4a–c shows the simulated transverse dynamics of light beams propagating through $$ S=-2$$ crystals with cubic (Fig. 4a), linear (Fig. 4b), and exponential (Fig. 4c) domain walls (see also Supplementary movies 3 and 4).

Fig. 4

Tailoring domain walls to engineer the topological Hall effect for light.

Simulated beam shape and position in the transverse plane of the crystal, for selected depths of propagation inside the NLPC and different domain wall distributions ($$ R=200\mathrm{\mu }\mathrm{m},n=2,\eta =0$$, and beam angle $$ 1.5^{\circ }$$ in all panels; insets show the domain wall cross sections). a cubic polynomial dependence $$ m\left(\rho \right)=1-2{\left(\rho /R\right)}^3$$ (with $$ \mathcal{B}\propto \rho $$). b linear dependence $$ m\left(\rho \right)=1-2\rho /R$$ (with $$ \mathcal{B}\propto 1/\rho $$). c exponential dependence $$ m\left(\rho \right)=\exp \left(-\rho /R\right)\left[1-\left(\rho /R\right)\left(1+e\right)\right]$$ (with $$ \mathcal{B}\propto \exp \left(-\rho /R\right)/\rho $$). Inset is the transverse component of the synthetic magnetization in the NLPC, color-coded by its corresponding $$ z$$-component. It appears that the deflection becomes more dominant as the domain wall transition effectively occurs in a smaller area. d Simulated center-of-mass trajectories of the light beams traversing the different domain walls of a–c, as compared to a conventional Néel-type skyrmion ($$ m\left(\rho \right)=\cos \left(\pi \rho /R\right)$$, with $$ \mathcal{B}\propto \mathrm{sinc}\left(\pi \rho /R\right)$$).

While all three configurations possess the same radius and topological invariant, they exert a different force on the light beams, resulting in different center-of-mass trajectories. The trajectories are compared in Fig. 4d, along with that found in conventional Néel-type skyrmions with cosine domain walls. Evidently, topologically equivalent magnetic textures yield different THE signatures, with the underlying mechanism being the strong singularity emerging in the synthetic magnetic field distribution (Fig. 4b, c). Interestingly, a larger and more localized Berry curvature enhances the Lorentz force, similarly to how a larger local optical angular-momentum density increases the torque applied on particles by optical vortices⁴⁹, even though both effects initially stem from global topological charges.

Active all-optical control over the topological Hall effect with light

Effective magnetization textures in nonlinear optical media can also be induced directly by light, enabling diverse opportunities for all-optically-controlled devices. In the context of generating effective skyrmion magnetizations, pump fields carrying orbital angular momentum⁴⁹ (OAM) can provide the required chirality²⁵ and coupling strength variation, through their phase and intensity profiles (Fig. 1c). Thus, the NLPC design is greatly simplified, requiring only a radially varying periodicity (see Fig. 5a).

Fig. 5

All-optical angular-momentum control of the topological Hall effect for light.

a Cross section of the three-dimensional nonlinear photonic crystal along the optical axis, showing the radial variation in the modulation period, $$ \mathrm{\Lambda }\left(\rho \right)$$ ($$ \rho $$ denotes the radial coordinate with respect to the optical axis). The nominal quasi-phase-matching period $$ \mathrm{\Lambda }\left(\rho \right)$$ is $$ 10.24\mathrm{\mu }\mathrm{m}$$, and it changes between $$ 10.33\mathrm{\mu }\mathrm{m}$$ at the crystal center, to $$ 10.15\mathrm{\mu }\mathrm{m}$$ at the edges. Note that in this case the required $$ {\chi }^{\left(2\right)}$$ pattern is rotationally symmetric, given by $$ {\chi }^{\left(2\right)}=\mathrm{sign}\left\{\cos \left[2\pi Z/\mathrm{\Lambda }\left(\rho \right)\right]\right\}$$. b, c a quadratic ($$ {\chi }^{\left(2\right)}$$) nonlinear photonic crystal with a radially symmetric variation in the periodicity is illuminated by a broad pump beam carrying orbital angular momentum of $$ \pm ħl$$ (red). The pump beam profile and phase front, together with the 3D NLPC structure, induce a topologically nontrivial magnetization texture, with opposite skyrmion numbers $$ S=\pm l$$. As a result, an incident idler beam is deflected according to the topological Hall effect. d, e Simulation results for the optically-controlled topological Hall effect, for a pump beam carrying OAM of d $$ +2ħ$$ and e $$ -2ħ$$, with a waist of $$ w_0=100\mathrm{\mu }\mathrm{m}$$ ($$ S=\pm 2$$, $$ n=l=\pm 2,\eta =0,R=200\mathrm{\mu }\mathrm{m}$$ and beam angle of $$ 1.1^{\circ }$$). The idler beam is deflected in opposite directions according to the skyrmion number.

In this manner, the pump and crystal together induce an effective skyrmion of order $$ S=l$$, where $$ ħl$$ is the pump OAM, whereas active control over the THE is enabled by changing the OAM. Examples are given in Fig. 5, where we simulate the THE from high-order skyrmions with Gauss-Laguerre pump beams carrying OAM of $$ \pm 2ħ$$. As expected, it appears that for opposite OAM values, the deflection is reversed, in accordance with the skyrmion number changing its sign.

Aside from spatially modulating the pump, temporal modulation can also bring about new degrees of control. Since optical nonlinear effects relax in ultrafast time scales, modulation frequency should only be limited by the propagation time through the NLPC, allowing working rates on the order of tens of GHz for the crystal lengths considered in this work. The simplest modulation is, of course, mere intensity modulation, which turns the THE on or off. However, it is also possible to modulate the angular momentum of the pump in time, thus changing between the deflection properties of the THE.