Corresponding author: ][email protected]

Magnetic phases of matter and spin transport within them are fundamental topics with broad implications, particularly in spintronics [1]. The spin-glass phase, a disordered system where spins are aligned randomly due to frustrated interactions, is especially intriguing [2, 3, 4, 5, 6], attracting interest across diverse scientific fields [2, 7, 3, 4, 5, 6, 8, 9], and earning Giorgio Parisi the 2021 Nobel Prize in Physics [7, 4, 6, 10]. However, despite extensive theoretical work, experimentally probing spin currents in disordered magnetization textures like in spin-glass phases, remains challenging [11, 12, 13, 14, 15, 16, 17, 18]. Precise control over spin currents and the study of phase transition dynamics in such complex systems, are still limited, underscoring the need for systems that offer greater control over spin transport in disordered magnetic phases.

In disordered systems, phenomena like weak localization [19, 20], and Anderson localization [21, 22] are predicted to occur. In Anderson localization, a disordered potential halts the diffusion of an electron wavefunction, leading to spatial localization with exponentially decaying tails. Lightwaves can also exhibit localization [23]. This has been observed in both completely random media [24, 19, 20, 25, 26, 27, 28, 29, 30, 31, 32], and disordered periodic media in the original sense of the Anderson model [33, 34, 35, 36, 37, 38, 39, 40], where localization arises from disordered scalar potentials. However, the study of localization in disordered vectorial potentials, such as those found in spin-glass magnetization textures—where disorder can be adjusted in both magnitude and orientation—has remained elusive. Additionally, the interplay between disorder and nonlinearity, although explored in optics, continues to pose open questions [41, 33, 34, 42, 23, 37, 43, 44].

Recently, spin transport in magnetic materials has been emulated using a sum-frequency generation (SFG) process within nonlinear photonic crystals (NLPCs) [45, 46, 47, 48, 49, 50, 51, 52]. In this approach, the amplitudes of the idler and signal fields define a pseudospin, whereas the nonlinear coupling—dependent on the pump field and the crystal’s second-order susceptibility—acts as a synthetic magnetization texture, M(r). The ability to manipulate the synthetic magnetization texture, either by spatially shaping the pump field [49, 50, 51], or by spatially modulating the NLPC [47, 49, 50, 53, 52], creates a versatile platform for studying spin transport phenomena in various 2D magnetization textures using light.

In this letter, we leverage this analogy to study spin transport in a spin-glass magnetic phase using an all-optical platform. By manipulating the nonlinear interaction parameters, we introduce a disordered transverse synthetic magnetization pattern with randomly oriented pseudospins, emulating a 2D disordered spin-glass. Our numerical and experimental results show that, unlike in the optical ordered ferromagnet analogue, where the idler-signal pseudospin exhibits ballistic broadening, disorder in the synthetic magnetization leads to transverse localization of both idler and signal fields. Additionally, we show that the localization strength is highly dependent on the nonlinear coupling and reveal a temporal decoherence effect manifesting as decaying Rabi oscillations. This novel pseudospin transverse localization phenomenon, driven by vectorial disorder, opens avenues for studying complex magnetic phases and their dynamical transitions using light.

We consider an SFG process with pump, idler, and signal fields at frequencies $\omega_{p},\omega_{i}$, and $\omega_{s}$, where $\omega_{s}=\omega_{p}+\omega_{i}$ due to energy conservation [54]. Assuming a quasi-phase-matched process [55, 54], slowly varying envelope, undepleted and long-wavelength pump approximations, the idler-signal dynamics within the NLPC can be expressed as an optical spin transport equation within a 2D magnetic material [45, 56, 48, 57, 58, 49, 50] (see Supplemental Material (SM) [59]):

where $\Psi(\textbf{r})=\frac{1}{\sqrt{N}}\left(\sqrt{\frac{n_{i}}{\omega_{i}}}A_{i}(\textbf{r}),\sqrt{\frac{n_{s}}{\omega_{s}}}A_{s}(\textbf{r})\right)^{T}$ is the effective spinor in the rotating frame, with $N=\frac{n_{i}}{\omega_{i}}\left|A_{i}(\textbf{r})\right|^{2}+\frac{n_{s}}{\omega_{s}}\left|A_{s}(\textbf{r})\right|^{2}$ a constant of motion due to the Manley-Rowe relations [54], such that $||\Psi||=\sqrt{\Psi^{\dagger}\Psi}=1$. Here, $A_{j}(\textbf{r})$, $n_{j}$ with $j=i,s$ are the slowly varying envelopes and refractive indices of the idler and signal fields, respectively. $\bar{k}$ is the mean wave number of the idler and signal, $\mathbf{p}_{T}=-i\nabla_{T}=-i(\partial_{x},\partial_{y})$ is the transverse momentum operator in position-space representation, $\pmb{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the Pauli matrix-vector, and $\textbf{r}=(x,y,z)$ is a 3D position vector. $\textbf{M}_{T}(\textbf{r})$ is the transverse synthetic magnetization texture driving the dynamics between the interacting fields and is defined as $\textbf{M}_{T}(\textbf{r})=M_{0}(\textbf{r})\hat{\textbf{M}}_{T}(\phi(\textbf{r}))=M_{0}(\textbf{r})\left[\hat{\textbf{x}}\cos\phi(\textbf{r})+\hat{\textbf{y}}\sin\phi(\textbf{r})\right]$. Here, $M_{0}(\textbf{r})\propto|\chi^{(2)}(\textbf{r})A_{p}(\textbf{r})|$, where $\chi^{(2)}(\textbf{r})$ is the position-dependent second-order susceptibility, $A_{p}(\textbf{r})$ is the complex envelope of the pump field, and $\phi(\textbf{r})=\phi_{p}(\textbf{r})-\phi_{NLPC}(\textbf{r})$ is the relative phase between the pump envelope and the NLPC modulation.

In this framework, a 1D periodically-poled crystal with a wide Gaussian pump beam generates a uniform transverse magnetization texture, analogous to a ferromagnetic phase (Fig. 1(a)). To emulate a disordered spin-glass phase, illustrated in Fig. 1(b), we create a random synthetic magnetization pattern by shaping the pump field into a non-diffracting (ND-) speckle pattern [33, 60, 61, 62, 63], effectively producing propagation-invariant disorder, as required for transverse localization schemes [64, 33, 34, 23, 43, 39]. This is depicted on the left of Fig. 1(b). Alternatively, disorder can be introduced by sampling the crystal’s modulation phase, $\phi_{NLPC}(\textbf{r})$, from a uniform distribution within $\left[0,2\pi\right]_{x}\times\left[0,2\pi\right]_{y}$, while keeping it propagation-invariant by maintaining a constant pump envelope phase along the propagation direction using a wide Gaussian beam, as illustrated on the right of Fig. 1(b).

While pump shaping can be experimentally implemented with standard optical components and a 1D-modulated NLPC, it is limited by wave diffraction. In contrast, crystal shaping provides greater control over disorder strength and structure by independently tuning magnetization density and orientation, the two degrees of freedom defining vectorial disorder in spin glasses. However, it requires 3D-modulated NLPCs, currently restricted to short crystal lengths [65, 66, 67]. Consequently, we numerically study pseudospin localization using crystal shaping and experimentally verify our predictions through pump shaping. Additionally, Appendix A demonstrates how current 2D-modulated NLPC technology can induce 1D transverse localization.

We simulated the idler-signal pseudospin transport within a 25 mm-long synthetically disordered periodically-poled KTiOPO₄ (PPKTP) NLPC using the split-step Fourier method [68]. Given the statistical nature of the phenomenon, our results were obtained by averaging over 70 disorder realizations with identical strength and statistical properties. To quantify the disorder level in the synthetic transverse magnetization, we divided the grid into magnetic domains of varying sizes and random shapes [69], with $\phi_{NLPC}(\textbf{r})$ constant within each domain and uniformly sampled from $\left[0,2\pi\right]$ across domains. The disorder level, defined as $l=100\times\left(N_{d}-1\right)/\left(N_{p}-1\right)\left(\%\right)$, is determined by the number of domains, $N_{d}$, ranging from 1 (full order, resembling a ferromagnetic phase), to $N_{p}$ (full disorder, emulating a spin-glass phase), where $N_{p}$ is the total number of pixels in the transverse grid. Intermediate $N_{d}$ values represent a magnetic domain structure [70, 71] (see insets of Fig. 2).

To evaluate the localization strength, we measured the beam confinement during propagation by calculating the average effective beam width, $w_{\text{eff}}(z)=\langle P(z)\rangle^{-1/2}$, where $\langle\cdot\rangle$ denotes averaging over multiple disorder realizations, and $\mathcal{P}\equiv\frac{\iint I(x,y,z)^{2}\,dx\,dy}{\left(\iint I(x,y,z)\,dx\,dy\right)^{2}}$ is the inverse participation ratio [72, 64, 33, 73, 43, 74]. Here, $I(x,y,z)$ represents the intensity distribution after propagating a distance $z$ through the crystal, with the integration performed over the entire transverse grid.

Figure 2(a) shows the average effective width of the idler-signal beam at the crystal output as a function of disorder level for a fixed pump intensity, revealing that it decreases with increasing disorder, stabilizing above $l=20\%$. Figure 2(b) presents the evolution of the average effective width, $w_{\text{eff}}(z)$, along the propagation axis for different disorder levels. Generally, $w_{\text{eff}}(z)$ follows a power-law behavior with propagation distance, $w_{\text{eff}}\propto z^{\nu}$ [72, 64, 30, 43, 33]. In the absence of disorder $\left(l=0\%\right)$, corresponding to an optical ferromagnet, ballistic transport is expected with $w_{\text{eff}}$ increasing linearly with $z$, resulting in $\nu\approx 1$ [72, 30, 43, 33]. This is confirmed by the linear fit for $l=0\%$ in Fig. 2(b) and the wide Gaussian output beam intensity shown in Fig. 2(c), arising from dominant diffraction effects. Here, the intensity distribution follows $I\propto\exp\left(-2\frac{\textbf{r}_{T}^{2}}{\sigma^{2}}\right)$, where $\textbf{r}_{T}=(x,y)$ is the transverse position vector, and $\sigma$ denotes the Gaussian beam width.

With weak disorder, the transport shifts towards diffusion, with $\nu\approx 0.5$ [72, 33]. In our system, at $l=0.75\%$ the best-fit curve yields $\nu\approx 0.53$, with the corresponding ensemble-averaged output intensity shown in Fig. 2(d). For higher disorder levels, the beam initially undergoes diffusive transport but then localizes after a certain propagation distance. In these cases, $\nu$ decreases to values within $0<\nu<\frac{1}{2}$, depending on the disorder level [64, 72, 33, 73, 30, 43]. As disorder increases, localization becomes stronger, and for disorder levels above $l=20\%$, $\nu$ stabilizes around 0.34. Figure 2(e) shows that the ensemble-averaged output intensity becomes transversely localized, exhibiting an exponential decay profile $\langle I\rangle\propto\exp\left(-2\frac{|\textbf{r}_{T}|}{\xi_{\text{loc}}}\right)$, with $|\textbf{r}_{T}|=\sqrt{x^{2}+y^{2}}$ representing the transverse distance from the beam’s center, and $\xi_{\text{loc}}$ is the localization length, defining the beam width where diffusion ceases and localization dominates [75].

The transition from ballistic transport in the 2D optical ferromagnet to localization in the 2D optical spin-glass is further evident when comparing the intensity profiles on a logarithmic scale: a parabolic curve for the optical ferromagnet versus a linear curve for the optical spin-glass (see insets of Figs. 2(c) and 2(e), respectively). This highlights that the idler-signal light beam, representing the spin current, undergoes transverse localization due to the induced disorder. Notably, while Fig. 2 demonstrates the localization of the two-frequency idler-signal beam under disorder, localization occurs separately for both the idler and signal, with each becoming transversely localized (see SM [59]).

Our numerical model allows us to isolate the effects of nonlinearity and disorder on this localization phenomenon, enabling us to examine the interplay between the two. Figure  3(a) shows that at a fixed disorder level, the average effective width at the crystal output decreases with increasing pump intensity, stabilizing for pump intensities above 100 MW/cm². This indicates that disorder alone cannot fully explain this localization, and a strong coupling between the idler and signal beams, facilitated by a high pump intensity, is also necessary. If the effect were purely linear and perturbative, localization would occur even with a weak signal field and nearly undepleted idler. However, our findings demonstrate that it is the combination of strong disorder and non-perturbative nonlinear coupling that drives localization. Figure  3(b) illustrates this: at $l=26\%$, the logarithmic plot of the ensemble-averaged output intensity for different pump intensities shows that, at low pump intensities, the SFG process occurs without achieving localization, with the intensity profile displaying diffusive transport. However, as the pump intensity increases, the idler-signal beam becomes transversely localized, forming a narrow, linear intensity profile.

The relationship between nonlinearity and localization is summarized in 3(c), showing that stronger nonlinearity (higher pump intensity) not only enhances localization by reducing the localization length but also induces the characteristic exponential decay of localization at lower disorder levels [33, 34, 23, 37]. These findings offer new insights into the interplay between nonlinearity and localization, a topic extensively studied in contexts such as $\chi^{(3)}$-nonlinearity [41, 33, 34, 37], and $\chi^{(2)}$-nonlinearity of second harmonic generation [24, 76]. From a spintronics perspective, our results suggest that in spin-glass phases, where the magnetization texture introduces a disordered vectorial potential, the magnetization density (analogous to nonlinearity strength in our optical system) plays a central role in spin current localization, alongside the randomness in magnetic moment alignment.

Another interesting aspect of this localization involves the energy exchange between the idler and signal fields, which can be modeled as a two-level system coupled by the nonlinear coupling [56, 45, 50]. Under phase-matching, this coupling leads to Rabi oscillations, where energy oscillates between the idler and signal [56, 50]. However, in the optical disordered spin-glass, where pseudospin localization occurs, our simulations reveal a pseudospin decoherence effect, manifesting as decaying Rabi oscillations. This effect is further discussed in Appendix B.

To experimentally verify our predictions, we employed the pump shaping approach and created an all-optical disordered transverse synthetic magnetization pattern using an ND-speckled pump. The experimental setup, detailed in SM [59], involved an SFG process within a 1D-modulated PPKTP crystal, between a pulsed pump laser at 1064.5 nm and a continuous-wave idler laser at 1550 nm. The ND-speckled pump beam was generated using an axicon lens and rotating diffuser, producing multiple disorder realizations with identical properties [33, 63]. To corroborate our results, we simulated the interaction using the pump shaping approach with the same experimental parameters.

When employing a Gaussian pump beam, effectively creating an optical ferromagnet, we observed a broad Gaussian signal beam at the crystal output, as shown in Figs. 4(a) and  4(d) for the simulation and experiment, respectively. This reflects dominant diffraction effects, corresponding to ballistic transport of spin current in a ferromagnetic phase. Conversely, when employing an ND-speckled pump beam, effectively creating an optical disordered spin-glass, the signal diffraction is suppressed. As predicted, the ensemble-averaged signal output intensity becomes transversely localized with exponentially decaying tails, as shown in Figs. 4(b) and  4(e) for the simulation and experiment, respectively. Importantly, the localization signature is evident from the linear profiles on a logarithmic scale for the optical spin-glass (insets of Figs. 4(b) and  4(e)), compared to the parabolic profiles in the optical ferromagnet (insets of Figs. 4(a) and  4(d)) [77].

Figure  4(g) shows the normalized effective width of the signal at the crystal output, $w_{\text{eff}}(L)$, at increasing pump intensities. As the intensity approaches the maximum intensity available in our experimental setup (51.4 MW/cm²), $w_{\text{eff}}(L)$ decreases, with our simulation predicting further narrowing at higher intensities. Furthermore, examining the logarithmic ensemble-averaged signal output intensity profiles for various pump intensities, as depicted in Fig. 4(h), reveals that at low intensities, localization is not yet achieved, and the profile is non-exponential. As the pump intensity increases, the signal beam becomes transversely localized, with the logarithmic profile transitioning to a clear linear shape, and the localization length decreases. These findings confirm our numerical predictions, demonstrating that both high pump intensity and disorder are necessary for idler-signal pseudospin localization.

To summarize, we have numerically and experimentally demonstrated a new, controllable transverse localization of light phenomenon in a nonlinear optical system analogous to a disordered spin-glass phase, enhancing our understanding of localization in disordered vectorial potentials. This may shed light on phenomena such as ferromagnetic-metal to spin-glass-insulator phase transitions [78, 12, 13, 79], similar to metal-insulator transitions in disordered scalar potentials [21]. Our controllable system opens avenues for investigating other complex magnetization textures, including spin-glass phases with structured and correlated disorder [11, 80, 81, 82], and spin-ice phases [83, 84, 16]. Additionally, expanding synthetic disorder to all three dimensions of the NLPC could emulate temporal modulation in spintronic systems [49], allowing exploration beyond equilibrium dynamics.

Extending our framework to the depleted pump regime, where idler/signal back-action perturbs the synthetic magnetization [49, 50], could reveal further details about the interplay between nonlinearity and localization. This may also provide insights into static and dynamic spin-transfer torque in spin-glass [85]. Additionally, our approach can be applied to other nonlinear processes, such as difference frequency generation [86], $\chi^{(3)}$-nonlinearity [87, 88, 89], and other two-level systems, including polarization states of light [90, 53, 91], and coupled optical waveguides [92], potentially uncovering additional light localization phenomena.

Finally, our framework can be expanded into the quantum domain, where spin currents could be implemented using quantum light. This opens the possibility of exploring quantum spin phenomena in vectorial potentials with two-frequency idler-signal entangled photons [93, 48, 94, 95, 96], including studies of quantum correlations in quantum random walks with disorder [97, 98, 99, 100].

Acknowledgments—S.I. acknowledges support from the Colton Scholarship Foundation and the Weinstein Institute for signal processing. A.K. acknowledges support from the Urbanek-Chodorow postdoctoral fellowship of Stanford University and the Zuckerman STEM leadership postdoctoral program. S.T. acknowledges support from the Yad Hanadiv Foundation through the Rothschild fellowship and the Helen Diller Quantum Center postdoctoral fellowship. Both A.K. and S.T. acknowledge support from the VATAT-Quantum fellowship and the Viterbi fellowship of the Technion—Israel Institute of Technology. This work was funded by the Israel Science Foundation, grants 969/22 and 3117/23.