Spin Hamiltonian in the modulated momenta of light

Spatial photonic Ising machines (SPIMs) are promising computation devices that can be used to find the ground states of different spin Hamiltonians and solve large-scale optimization problems. The photonic architecture leverages the matrix multiplexing ability of light to accelerate the computing of spin Hamiltonian via free space light transform. However, the intrinsic long-range nature of spatial light only allows for uncontrolled all-to-all spin interaction. We explore the ability to establish arbitrary spin Hamiltonian by modulating the momentum of light. Arbitrary displacement-dependent spin interactions can be computed from different momenta of light, formulating as a generalized Plancherel theorem, which allows us to implement a SPIM with a minimal optical operation (that is, a single Fourier transform) to obtain the Hamiltonian of customized spin interaction. Experimentally, we unveil the exotic magnetic phase diagram of the generalized J1-J2-J3 model, shedding light on the ab initio magnetic states of iron chalcogenides. Moreover, we observe Berezinskii-Kosterlitz-Thouless dynamics by implementing an XY model. We open an avenue to controlling arbitrary spin interaction from the momentum space of light, offering a promising method for on-demand spin model simulation with a simple spatial light platform.