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

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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.
 
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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