New method for analyzing the evolution of coherent state in a generalized harmonic oscillator

In this work, we introduce a generalized harmonic oscillator (GHO) model, which is constructed based on normal harmonic oscillators (HO) and inverted harmonic oscillators (IHO). Employing theoretical frameworks, we meticulously derive the normal ordering form of the time evolution operator and its corresponding quantum transformation for the GHO. Our proposed approach enables a comprehensive comparative analysis of the time evolution of the input coherent states (CSs) in the HO and the IHO. By adjusting the frequency value within the GHO, we conduct a detailed study through the evaluation of key parameters such as fidelity, squeezing effect, and photon distribution characteristics. This novel method provides a fresh perspective on understanding the quantum state evolution under different quantum systems, opening up new research directions in the field of quantum dynamics.