Efficient algebraic solution for a time-dependent quantum harmonic oscillator

TL;DR

This paper introduces an efficient algebraic method using Lie algebra techniques to solve the dynamics of a time-dependent quantum harmonic oscillator, enabling accurate and numerically convenient analysis of quantum states.

Contribution

The authors develop an iterative analytical solution based on BCH-like relations and a time-splitting approach, offering a new, efficient way to analyze the oscillator's dynamics for any initial state.

Findings

Method reproduces known results accurately

Demonstrates efficiency in analyzing squeezing effects

Facilitates numerical calculations with simple recurrence relations

Abstract

Using operator ordering techniques based on BCH-like relations of the su(1,1) Lie algebra and a time-splitting approach,we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from literature in order to prove the usefulness of the method and, based on these references, we discuss efficiency in squeezing, when comparing the parametric resonance modulation and the Janszky-Adam scheme.