Tensor-network approach to quantum optical state evolution beyond the Fock basis

Understanding the quantum evolution of light in nonlinear media is central to the development of next-generation quantum technologies. Yet, modeling these processes remains computationally demanding, as the required resources grow rapidly with photon number and phase-space resolution. Here, we introduce a tensor-network approach that efficiently captures the dynamics of nonlinear optical systems in a continuous-variable representation. Using the matrix product state (MPS) formalism, both quantum states and operators are encoded in a highly compressed form, enabling direct numerical integration of the Schrödinger equation. We demonstrate the method by simulating degenerate spontaneous parametric down-conversion (SPDC) and show that it accurately reproduces established theoretical benchmarks—energy conservation, pump depletion, and quadrature squeezing—even in regimes where conventional Fock-basis simulations become infeasible. For high-intensity pump fields (\alpha=100) ( α = 100 ) , the MPS representation achieves compression ratios below 3×10^{-4} 3 × 10 − 4 while preserving physical fidelity. This framework opens a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond the reach of traditional methods.