# Time-evolution of nonlinear optomechanical systems: Interplay of mechanical squeezing and non-Gaussianity

## TL;DR

This paper presents an analytical method for solving the time evolution of nonlinear optomechanical systems with arbitrary parameters, and analyzes how mechanical squeezing influences the system's non-Gaussianity over time.

## Contribution

The authors develop a Lie algebra-based approach to solve the dynamics of nonlinear optomechanical systems with time-dependent parameters, enabling analysis of non-Gaussianity evolution.

## Key findings

- Adding mechanical squeezing generally reduces non-Gaussianity.
- Sinusoidal squeezing leads to Mathieu equations for system dynamics.
- Non-Gaussianity increases with time and squeezing amplitude in certain regimes.

## Abstract

We solve the time evolution of a nonlinear optomechanical Hamiltonian with arbitrary time-dependent mechanical displacement, mechanical single-mode squeezing and a time-dependent optomechanical coupling up to the solution of two second-order differential equations. The solution is based on identifying a minimal and finite Lie algebra that generates the time-evolution of the system. This reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. To demonstrate the applicability of our method, we compute the degree of non-Gaussianity of the time-evolved state of the system by means of a measure based on the relative entropy of the non-Gaussian state and its closest Gaussian reference state. We find that the addition of a constant mechanical squeezing term to the standard optomechanical Hamiltonian generally decreases the overall non-Gaussian character of the state. For sinusoidally modulated squeezing, the two second-order differential equations mentioned above take the form of the Mathieu equation. We derive perturbative solutions for a small squeezing amplitude at parametric resonance and show that they correspond to the rotating-wave approximation at times larger than the scale set by the mechanical frequency. We find that the non-Gaussianity of the state increases with both time and the squeezing parameter in this specific regime.