# Time evolution of two harmonic oscillators with cross-Kerr interactions

## TL;DR

This paper provides an exact analytical solution for the time evolution of two coupled quantum harmonic oscillators with cross-Kerr interactions, enabling detailed analysis of their nonlinear dynamics without approximations.

## Contribution

It introduces a method to decouple the evolution operator and derive explicit solutions for a nonlinear two-oscillator system, advancing understanding of full nonlinearity effects.

## Key findings

- Exact analytical expressions for system evolution
- Ability to analyze nonlinear effects without approximations
- Potential applications in optomechanics

## Abstract

We study the time evolution of two coupled quantum harmonic oscillators interacting through nonlinear optomechanical-like Hamiltonians that include cross-Kerr interactions. We employ techniques developed to decouple the time-evolution operator and obtain the analytical solution for the time evolution of the system. We apply these results to obtain explicit expressions of a few quantities of interest. Our results do not require approximations and therefore allow to study the nature and implications of the full nonlinearity of the system. Potential applications and extensions to optomechanics are also discussed.