Harnessing Super-resolution: Continuous Wavefunction Dynamics

In recent years, technologies exploiting quantum optical effects have advanced rapidly, highlighting the need for accurate simulation of quantum state evolution across diverse processes. Yet, numerical simulation of quantum dynamics remains difficult due to the rapid growth of Hilbert space. Traditional Fock-basis methods become impractical in high-photon or multimode regimes, while continuous-variable (CV) representations, based on discretized position or momentum bases, still require scalable encoding strategies.
We address this challenge using tensor networks, specifically the Matrix Product State (MPS) formalism for representing both wave functions and operators in a continuous basis. This approach leverages the fact that MPS encodings are compact for smooth multivariable functions, enabling efficient propagation of the time-dependent wave function by solving the Schrödinger equation directly in MPS form.
As a benchmark, we simulate degenerate spontaneous parametric down-conversion (SPDC). For a pump initialized in a low-amplitude coherent state ($\alpha = 10$), MPS calculations exactly reproduce Fock-basis results. At higher amplitude ($\alpha = 100$), where Fock simulations become intractable, the MPS method remains efficient and stable.
These results demonstrate that tensor-network techniques enable super-resolution of quantum states and provide a scalable route to simulating multimode optical dynamics, including whispering-gallery-mode microresonators.