How does one time-evolve a quantum state with its kernel function?

[Sciencemaster]

TL;DR

I'd like to model the evolution of a squeezed state by representing it as a kernel function and applying a unitary transformation, but I'm having trouble doing this.

I'd like to model the evolution of a squeezed state and its properties (such as phase at different spatiotemporal coordinates). I know one can represent them using kernel functions (and I have found a paper that gives a kernel function for a squeezed state: https://arxiv.org/pdf/2105.05990.pdf). I've been told one can diagonalize the kernel function in terms of some eigenbasis and then represent the state in terms of a matrix representation with a truncated set of these eigenbasis functions, or alternatively just represent the kernel function in terms of a higher dimensional grid. Apparently you also need to represent the squeezing parameter in terms of a kernel function ($ξa^2→a\hat(k1)ξ(k1,k2)a\hat(k2)$) Once this is done, one can use unitary transformations on the matrix to actually simulate the system.
However, I'm having trouble doing this. Specifically, I'm stuck on trying to make a kernel function matrix from the paper and/or setting up an initial system (for example, let's say we have a Gaussian beam of squeezed light being emitted, and this simulation aims to time evolve this beam). Does anyone have any insights on how I can do this? Any help would be appreciated.