Spin Hamiltonian in the modulated momenta of light

Juan Feng; Zengya Li; Luqi Yuan; Erez Hasman; Bo Wang; Xianfeng Chen

doi:10.1117/1.AP.7.4.046001

Advanced Photonics, Volume. 7, Issue 4, 046001(2025)

Spin Hamiltonian in the modulated momenta of light

Juan Feng¹, Zengya Li¹, Luqi Yuan¹, Erez Hasman², Bo Wang^1、*, and Xianfeng Chen^1,3,4

¹Shanghai Jiao Tong University, School of Physics and Astronomy, State Key Laboratory of Photonics and Communications, Shanghai, China

²Technion – Israel Institute of Technology, Russell Berrie Nanotechnology Institute, and Helen Diller Quantum Center, Atomic-Scale Photonics Laboratory, Haifa, Israel

³Shanghai Research Center for Quantum Sciences, Shanghai, China

⁴Shandong Normal University, Collaborative Innovation Center of Light Manipulations and Applications, Jinan, China

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Fig. 1. Schematic of the optical spin model simulator. (a) The spin array is represented as the phases of light. The spin-lattice constant is $Λ$ , and the size of the array is $L \times L$ . (b) An example of the spin interaction function with only the NN term. The black arrows represent the XY spin vectors mapped to the phases of light. (c) The Fourier transform $V (k)$ of NN interaction function. (d) Simplified experimental setup. A plane wave laser beam (wavelength of 532 nm) is phase-modulated via a reflective SLM, collected by a lens, and detected by a CMOS camera in the momentum space (Fourier plane). Feedback is implemented by a computer for data processing, calculating the Hamiltonian, and updating the phase distributions. (e) An example of light momentum-space intensity distribution captured by the camera.

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Fig. 2. Phase diagram of a $J_{1} - J_{2} - J_{3}$ model at $T = 0$ . (a) The predicted phase diagram at $T = 0$ as a function of $R_{1}$ and $R_{2}$ . The red dots denote the locations where we conduct experimental demonstrations. (b) The ground states correspond to the four regions in panel (a). AF, antiferromagnetic state; SAF, super-antiferromagnetic state; the notation ( $2 \times 2$ ) means that the unit cell of the ground state is composed of a $2 \times 2$ spin array. The state ( $4 \times 2$ ) is also known as the staggered-stripe pattern, ( $2 \times 1$ ) is the single-stripe pattern, and ( $4 \times 4$ )s are the double-stripe pattern.

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Fig. 3. Optically solving a typical ground state of the $J_{1} - J_{2} - J_{3}$ model. (a) The spin-interaction function $J (r_{i j})$ for $(R_{1}, R_{2}) = (0.5, 0.9)$ . The dots array stands for the locations of spins. The colormap for $J (r_{i j})$ is represented by taking one of the spins at $r_{i} = 0$ . The $V (k)$ is the Fourier transform of $J (r_{i j})$ , with $V (k) \sim \sum_{i j} J (r_{i j}) e^{i k \cdot r_{i j}}$ . Panels (b) and (c) are the phase distributions (upper panel) and corresponding momentum-space intensity distributions (lower panel) for an initial spin distribution (b) and the solved spin configuration (c). (d) The recorded $H_{r} - H_{k}$ during the optical annealing (dots). All experimentally measured $H_{k}$ are divided by the minimal ( $- H_{k}$ ) to obtain the normalized values. The black line is a fitted result, and the black arrow indicates the evolution direction. (e) The observed $H_{k}$ during optical annealing. (f) $H_{r}$ as a function of $T$ . The dots are experimental results, the curve is the averaged simulation result, and the shaded area is the simulation variance from statistics. Note that $H_{r}$ is normalized by the number of spin-spin interactions, and we have set the Boltzmann constant as $k_{B} = 1$ .

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$Experimental demonstrations for the J1-J2-J3 model phase diagram with optical momentum modulations. From the top to the bottom panels: the spin-interaction function J(rij), the Fourier transform V(k), the solved ground state φ(r)sol., and the corresponding diffraction image I(k)sol..$

Fig. 4. Experimental demonstrations for the $J_{1} - J_{2} - J_{3}$ model phase diagram with optical momentum modulations. From the top to the bottom panels: the spin-interaction function $J (r_{i j})$ , the Fourier transform $V (k)$ , the solved ground state $φ (r)_{sol .}$ , and the corresponding diffraction image $I (k)_{sol .}$ .

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Fig. 5. Observation of BKT dynamics. (a) Observed spin distributions at different temperatures. (b) The number of vortices as a function of $T$ . The dots are experimentally extracted from panel (a), and the curve is an averaged simulation result, with a statistical variance denoted as the shaded area. (c) Observed $H_{r}$ as a function of $T$ . (d) The observed vortex-pair state from quenching. The curved black arrows indicate the opposite signs of topological charges. (e) The experimentally recorded $H_{k}$ during quenching. The inset shows the observed $H_{k} - H_{r}$ correspondence.

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Juan Feng, Zengya Li, Luqi Yuan, Erez Hasman, Bo Wang, Xianfeng Chen, "Spin Hamiltonian in the modulated momenta of light," Adv. Photon. 7, 046001 (2025)

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